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Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions.
In this Blog post we will learn:
In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it’s biased. By rolling it a few times and analyzing the outcomes, you’d be engaging in the essence of hypothesis testing.
Think of hypothesis testing as the scientific method of the statistics world. Suppose you hear claims like “This new drug works wonders!” or “Our new website design boosts sales.” How do you know if these statements hold water? Enter hypothesis testing.
Before diving into testing, we must formulate hypotheses. The null hypothesis (H0) represents the default assumption, while the alternative hypothesis (H1) challenges it.
For instance, in drug testing, H0 : “The new drug is no better than the existing one,” H1 : “The new drug is superior .”
When You collect and analyze data to test H0 and H1 hypotheses. Based on your analysis, you decide whether to reject the null hypothesis in favor of the alternative, or fail to reject / Accept the null hypothesis.
The significance level, often denoted by $α$, represents the probability of rejecting the null hypothesis when it is actually true.
In other words, it’s the risk you’re willing to take of making a Type I error (false positive).
Type I Error (False Positive) :
Example : If a drug is not effective (truth), but a clinical trial incorrectly concludes that it is effective (based on the sample data), then a Type I error has occurred.
Type II Error (False Negative) :
Example : If a drug is effective (truth), but a clinical trial incorrectly concludes that it is not effective (based on the sample data), then a Type II error has occurred.
Balancing the Errors :
In practice, there’s a trade-off between Type I and Type II errors. Reducing the risk of one typically increases the risk of the other. For example, if you want to decrease the probability of a Type I error (by setting a lower significance level), you might increase the probability of a Type II error unless you compensate by collecting more data or making other adjustments.
It’s essential to understand the consequences of both types of errors in any given context. In some situations, a Type I error might be more severe, while in others, a Type II error might be of greater concern. This understanding guides researchers in designing their experiments and choosing appropriate significance levels.
Test statistic : A test statistic is a single number that helps us understand how far our sample data is from what we’d expect under a null hypothesis (a basic assumption we’re trying to test against). Generally, the larger the test statistic, the more evidence we have against our null hypothesis. It helps us decide whether the differences we observe in our data are due to random chance or if there’s an actual effect.
P-value : The P-value tells us how likely we would get our observed results (or something more extreme) if the null hypothesis were true. It’s a value between 0 and 1. – A smaller P-value (typically below 0.05) means that the observation is rare under the null hypothesis, so we might reject the null hypothesis. – A larger P-value suggests that what we observed could easily happen by random chance, so we might not reject the null hypothesis.
Relationship between $α$ and P-Value
When conducting a hypothesis test:
We then calculate the p-value from our sample data and the test statistic.
Finally, we compare the p-value to our chosen $α$:
Imagine we are investigating whether a new drug is effective at treating headaches faster than drug B.
Setting Up the Experiment : You gather 100 people who suffer from headaches. Half of them (50 people) are given the new drug (let’s call this the ‘Drug Group’), and the other half are given a sugar pill, which doesn’t contain any medication.
Calculate Test statistic and P-Value : After the experiment, you analyze the data. The “test statistic” is a number that helps you understand the difference between the two groups in terms of standard units.
For instance, let’s say:
The test statistic helps you understand how significant this 1-hour difference is. If the groups are large and the spread of healing times in each group is small, then this difference might be significant. But if there’s a huge variation in healing times, the 1-hour difference might not be so special.
Imagine the P-value as answering this question: “If the new drug had NO real effect, what’s the probability that I’d see a difference as extreme (or more extreme) as the one I found, just by random chance?”
For instance:
For simplicity, let’s say we’re using a t-test (common for comparing means). Let’s dive into Python:
Making a Decision : “The results are statistically significant! p-value < 0.05 , The drug seems to have an effect!” If not, we’d say, “Looks like the drug isn’t as miraculous as we thought.”
Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.
F statistic formula – explained, correlation – connecting the dots, the role of correlation in data analysis, sampling and sampling distributions – a comprehensive guide on sampling and sampling distributions, law of large numbers – a deep dive into the world of statistics, central limit theorem – a deep dive into central limit theorem and its significance in statistics, similar articles, complete introduction to linear regression in r, how to implement common statistical significance tests and find the p value, logistic regression – a complete tutorial with examples in r.
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Nilimesh Halder, PhD
Analyst’s corner
1. Introduction to Hypothesis Testing - Definition and significance in research and data analysis. - Brief historical background.
2. Fundamentals of Hypothesis Testing - Null and Alternative Hypothesis: Definitions and examples. - Types of Errors: Type I and Type II errors with examples.
3. The Process of Hypothesis Testing - Step-by-step guide: From defining hypotheses to decision making. - Examples to illustrate each step.
4. Statistical Tests in Hypothesis Testing - Overview of different statistical tests (t-test, chi-square test, ANOVA, etc.). - Criteria for selecting the appropriate test.
5. P-Values and Significance Levels - Understanding P-values: Definition and interpretation. - Significance Levels: Explaining alpha values and their implications.
6. Common Misconceptions and Mistakes in Hypothesis Testing - Addressing misconceptions about p-values and…
Principal Analytics Specialist - AI, Analytics & Data Science ( https://nilimesh.substack.com/ ). Find my PDF articles at https://nilimesh.gumroad.com/l/bkmdgt
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S.3 hypothesis testing.
In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail.
The general idea of hypothesis testing involves:
Every hypothesis test — regardless of the population parameter involved — requires the above three steps.
Is normal body temperature really 98.6 degrees f section .
Consider the population of many, many adults. A researcher hypothesized that the average adult body temperature is lower than the often-advertised 98.6 degrees F. That is, the researcher wants an answer to the question: "Is the average adult body temperature 98.6 degrees? Or is it lower?" To answer his research question, the researcher starts by assuming that the average adult body temperature was 98.6 degrees F.
Then, the researcher went out and tried to find evidence that refutes his initial assumption. In doing so, he selects a random sample of 130 adults. The average body temperature of the 130 sampled adults is 98.25 degrees.
Then, the researcher uses the data he collected to make a decision about his initial assumption. It is either likely or unlikely that the researcher would collect the evidence he did given his initial assumption that the average adult body temperature is 98.6 degrees:
In statistics, we generally don't make claims that require us to believe that a very unusual event happened. That is, in the practice of statistics, if the evidence (data) we collected is unlikely in light of the initial assumption, then we reject our initial assumption.
Criminal trial analogy section .
One place where you can consistently see the general idea of hypothesis testing in action is in criminal trials held in the United States. Our criminal justice system assumes "the defendant is innocent until proven guilty." That is, our initial assumption is that the defendant is innocent.
In the practice of statistics, we make our initial assumption when we state our two competing hypotheses -- the null hypothesis ( H 0 ) and the alternative hypothesis ( H A ). Here, our hypotheses are:
In statistics, we always assume the null hypothesis is true . That is, the null hypothesis is always our initial assumption.
The prosecution team then collects evidence — such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, and handwriting samples — with the hopes of finding "sufficient evidence" to make the assumption of innocence refutable.
In statistics, the data are the evidence.
The jury then makes a decision based on the available evidence:
In statistics, we always make one of two decisions. We either "reject the null hypothesis" or we "fail to reject the null hypothesis."
Did you notice the use of the phrase "behave as if" in the previous discussion? We "behave as if" the defendant is guilty; we do not "prove" that the defendant is guilty. And, we "behave as if" the defendant is innocent; we do not "prove" that the defendant is innocent.
This is a very important distinction! We make our decision based on evidence not on 100% guaranteed proof. Again:
We merely state that there is enough evidence to behave one way or the other. This is always true in statistics! Because of this, whatever the decision, there is always a chance that we made an error .
Let's review the two types of errors that can be made in criminal trials:
Jury Decision | Truth | ||
---|---|---|---|
Not Guilty | Guilty | ||
Not Guilty | OK | ERROR | |
Guilty | ERROR | OK |
Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing.
Decision | |||
---|---|---|---|
Null Hypothesis | Alternative Hypothesis | ||
Do not Reject Null | OK | Type II Error | |
Reject Null | Type I Error | OK |
Note that, in statistics, we call the two types of errors by two different names -- one is called a "Type I error," and the other is called a "Type II error." Here are the formal definitions of the two types of errors:
There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!
Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption. If it is likely , we do not reject the null hypothesis. If it is unlikely , then we reject the null hypothesis in favor of the alternative hypothesis. Effectively, then, making the decision reduces to determining "likely" or "unlikely."
In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:
In the next two sections, we review the procedures behind each of these two approaches. To make our review concrete, let's imagine that μ is the average grade point average of all American students who major in mathematics. We first review the critical value approach for conducting each of the following three hypothesis tests about the population mean $\mu$:
: = 3 | : > 3 | |
: = 3 | : < 3 | |
: = 3 | : ≠ 3 |
Upon completing the review of the critical value approach, we review the P -value approach for conducting each of the above three hypothesis tests about the population mean \(\mu\). The procedures that we review here for both approaches easily extend to hypothesis tests about any other population parameter.
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From controlling for testing errors to selecting the right test.
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Hypothesis testing is a method of statistical inference that considers the null hypothesis H ₀ vs. the alternative hypothesis H a , where we are typically looking to assess evidence against H ₀ . Such a test is used to compare data sets against one another, or compare a data set against some external standard. The former being a two sample test (independent or matched pairs), and the latter being a one sample test. For example, “does group A have a higher pain tolerance than group B?” or “is the mean age of the control group 21?”, respectively. A hypothesis test ends with a decision based on a pre-specified level of significance α–either to reject the null hypothesis when we have strong enough evidence against it, or fail to reject the null.
Now the question is, how do we know if we have strong enough evidence against this null hypothesis? To answer this, we must first understand the different types of errors when it comes to testing.
Either errors is considered bad, and so when concluding a decision for the hypothesis test, we wish to minimize the probability of such errors. Typically, we denote α = P(Type I Error) and β = P(Type II Error).
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You will learn the essentials of hypothesis tests, from fundamental concepts to practical applications in statistics.
Hypothesis testing is a statistical tool used to make decisions based on data.
It involves making assumptions about a population parameter and testing its validity using a population sample.
Hypothesis tests help us draw conclusions and make informed decisions in various fields like business, research, and science.
The null hypothesis (H0) is an initial claim about a population parameter, typically representing no effect or no difference.
The alternative hypothesis (H1) opposes the null hypothesis, suggesting an effect or difference.
Hypothesis tests aim to determine if there is evidence for the null hypothesis rejection in favor of the alternative hypothesis.
The significance level (α), often set at 0.05 or 5%, serves as a threshold for determining if we should reject the null hypothesis.
A p-value, calculated during hypothesis testing, represents the probability of observing the test statistic if the null hypothesis is true.
Suppose the p-value is less than the significance level. We reject the null hypothesis, in that case, indicating that the alternative hypothesis is more likely.
Parametric tests assume the data follows a specific probability distribution, usually the normal distribution. Examples include the Student’s t-test.
Non-parametric tests do not require such assumptions and are helpful when dealing with data that do not meet the assumptions of parametric tests. Examples include the Mann-Whitney U test.
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Independent samples t-test: This analysis compares the means of two independent groups.
Paired samples t-test: Compares the means of two related groups (e.g., before and after treatment).
Chi-squared test: Determines if there is a significant association, in a contingency table, between two categorical variables.
Analysis of Variance (ANOVA): Compares the means of three or more independent groups to determine whether significant differences exist.
Pearson’s Correlation Coefficient (Pearson’s r): Quantifies the strength and direction of a linear association between two continuous variables.
Simple Linear Regression: Evaluate whether a significant linear relationship exists between a predictor variable (X) and a continuous outcome variable (y).
Logistic Regression: Determines the relationship between one or more predictor variables (continuous or categorical) and a binary outcome variable (e.g., success or failure).
Levene’s Test: Tests the equality of variances between two or more groups, often used as an assumption checks for ANOVA.
Shapiro-Wilk Test: Assesses the null hypothesis that a data sample is drawn from a population with a normal distribution.
Hypothesis Test | Description | Application |
---|---|---|
Compares means of two independent groups | Comparing scores of two groups of students | |
Compares means of two related groups (e.g., before and after treatment) | Comparing weight loss before and after a diet program | |
Determines significant associations between two categorical variables in a contingency table | Analyzing the relationship between education and income | |
Compares means of three or more independent groups | Evaluating the impact of different teaching methods on test scores | |
Measures the strength and direction of a linear relationship between two continuous variables | Studying the correlation between height and weight | |
Determines a significant linear relationship between a predictor variable and an outcome variable | Predicting sales based on advertising budget | |
Determines the relationship between predictor variables and a binary outcome variable | Predicting the probability of loan default based on credit score | |
Tests the equality of variances between two or more groups | Checking the assumption of equal variances for ANOVA | |
Tests if a data sample is from a normally distributed population | Assessing normality assumption for parametric tests |
To interpret the hypothesis test results, compare the p-value to the chosen significance level.
If the p-value falls below the significance level, reject the null hypothesis and infer that a notable effect or difference exists.
Otherwise, fail to reject the null hypothesis, meaning there is insufficient evidence to support the alternative hypothesis.
In addition to understanding the basics of hypothesis tests, it’s crucial to consider other relevant information when interpreting the results.
For example, factors such as effect size, statistical power, and confidence intervals can provide valuable insights and help you make more informed decisions.
Effect size
The effect size represents a quantitative measurement of the strength or magnitude of the observed relationship or effect between variables. It aids in evaluating the practical significance of the results. A statistically significant outcome may not necessarily imply practical relevance. At the same time, a substantial effect size can suggest meaningful findings, even when statistical significance appears marginal.
Statistical power
The power of a test represents the likelihood of accurately rejecting the null hypothesis when it is incorrect. In other words, it’s the likelihood that the test will detect an effect when it exists. Factors affecting the power of a test include the sample size, effect size, and significance level. Enhanced power reduces the likelihood of making an error of Type II — failing to reject the null hypothesis when it ought to be rejected.
Confidence intervals
A confidence interval represents a range where the true population parameter is expected to be found with a specified confidence level (e.g., 95%). Confidence intervals provide additional context to hypothesis testing, helping to assess the estimate’s precision and offering a better understanding of the uncertainty surrounding the results.
By considering these additional aspects when interpreting the results of hypothesis tests, you can gain a more comprehensive understanding of the data and make more informed conclusions.
Hypothesis testing is an indispensable statistical tool for drawing meaningful inferences and making informed data-based decisions.
By comprehending the essential concepts such as null and alternative hypotheses, significance levels, p-values, and the distinction between parametric and non-parametric tests, you can proficiently apply hypothesis testing to a wide range of real-world situations.
Additionally, understanding the importance of effect sizes, statistical power, and confidence intervals will enhance your ability to interpret the results and make better decisions.
With many applications across various fields, including medicine, psychology, business, and environmental sciences, hypothesis testing is a versatile and valuable method for research and data analysis.
A comprehensive grasp of hypothesis testing techniques will enable professionals and researchers to strengthen their decision-making processes, optimize strategies, and deepen their understanding of the relationships between variables, leading to more impactful results and discoveries.
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Hypothesis testing is the act of testing a hypothesis or a supposition in relation to a statistical parameter. Analysts implement hypothesis testing in order to test if a hypothesis is plausible or not.
In data science and statistics , hypothesis testing is an important step as it involves the verification of an assumption that could help develop a statistical parameter. For instance, a researcher establishes a hypothesis assuming that the average of all odd numbers is an even number.
In order to find the plausibility of this hypothesis, the researcher will have to test the hypothesis using hypothesis testing methods. Unlike a hypothesis that is ‘supposed’ to stand true on the basis of little or no evidence, hypothesis testing is required to have plausible evidence in order to establish that a statistical hypothesis is true.
Perhaps this is where statistics play an important role. A number of components are involved in this process. But before understanding the process involved in hypothesis testing in research methodology, we shall first understand the types of hypotheses that are involved in the process. Let us get started!
In data sampling, different types of hypothesis are involved in finding whether the tested samples test positive for a hypothesis or not. In this segment, we shall discover the different types of hypotheses and understand the role they play in hypothesis testing.
Alternative Hypothesis (H1) or the research hypothesis states that there is a relationship between two variables (where one variable affects the other). The alternative hypothesis is the main driving force for hypothesis testing.
It implies that the two variables are related to each other and the relationship that exists between them is not due to chance or coincidence.
When the process of hypothesis testing is carried out, the alternative hypothesis is the main subject of the testing process. The analyst intends to test the alternative hypothesis and verifies its plausibility.
The Null Hypothesis (H0) aims to nullify the alternative hypothesis by implying that there exists no relation between two variables in statistics. It states that the effect of one variable on the other is solely due to chance and no empirical cause lies behind it.
The null hypothesis is established alongside the alternative hypothesis and is recognized as important as the latter. In hypothesis testing, the null hypothesis has a major role to play as it influences the testing against the alternative hypothesis.
(Must read: What is ANOVA test? )
The Non-directional hypothesis states that the relation between two variables has no direction.
Simply put, it asserts that there exists a relation between two variables, but does not recognize the direction of effect, whether variable A affects variable B or vice versa.
The Directional hypothesis, on the other hand, asserts the direction of effect of the relationship that exists between two variables.
Herein, the hypothesis clearly states that variable A affects variable B, or vice versa.
A statistical hypothesis is a hypothesis that can be verified to be plausible on the basis of statistics.
By using data sampling and statistical knowledge, one can determine the plausibility of a statistical hypothesis and find out if it stands true or not.
(Related blog: z-test vs t-test )
Now that we have understood the types of hypotheses and the role they play in hypothesis testing, let us now move on to understand the process in a better manner.
In hypothesis testing, a researcher is first required to establish two hypotheses - alternative hypothesis and null hypothesis in order to begin with the procedure.
To establish these two hypotheses, one is required to study data samples, find a plausible pattern among the samples, and pen down a statistical hypothesis that they wish to test.
A random population of samples can be drawn, to begin with hypothesis testing. Among the two hypotheses, alternative and null, only one can be verified to be true. Perhaps the presence of both hypotheses is required to make the process successful.
At the end of the hypothesis testing procedure, either of the hypotheses will be rejected and the other one will be supported. Even though one of the two hypotheses turns out to be true, no hypothesis can ever be verified 100%.
(Read also: Types of data sampling techniques )
Therefore, a hypothesis can only be supported based on the statistical samples and verified data. Here is a step-by-step guide for hypothesis testing.
First things first, one is required to establish two hypotheses - alternative and null, that will set the foundation for hypothesis testing.
These hypotheses initiate the testing process that involves the researcher working on data samples in order to either support the alternative hypothesis or the null hypothesis.
Once the hypotheses have been formulated, it is now time to generate a testing plan. A testing plan or an analysis plan involves the accumulation of data samples, determining which statistic is to be considered and laying out the sample size.
All these factors are very important while one is working on hypothesis testing.
As soon as a testing plan is ready, it is time to move on to the analysis part. Analysis of data samples involves configuring statistical values of samples, drawing them together, and deriving a pattern out of these samples.
While analyzing the data samples, a researcher needs to determine a set of things -
Significance Level - The level of significance in hypothesis testing indicates if a statistical result could have significance if the null hypothesis stands to be true.
Testing Method - The testing method involves a type of sampling-distribution and a test statistic that leads to hypothesis testing. There are a number of testing methods that can assist in the analysis of data samples.
Test statistic - Test statistic is a numerical summary of a data set that can be used to perform hypothesis testing.
P-value - The P-value interpretation is the probability of finding a sample statistic to be as extreme as the test statistic, indicating the plausibility of the null hypothesis.
The analysis of data samples leads to the inference of results that establishes whether the alternative hypothesis stands true or not. When the P-value is less than the significance level, the null hypothesis is rejected and the alternative hypothesis turns out to be plausible.
As we have already looked into different aspects of hypothesis testing, we shall now look into the different methods of hypothesis testing. All in all, there are 2 most common types of hypothesis testing methods. They are as follows -
The frequentist hypothesis or the traditional approach to hypothesis testing is a hypothesis testing method that aims on making assumptions by considering current data.
The supposed truths and assumptions are based on the current data and a set of 2 hypotheses are formulated. A very popular subtype of the frequentist approach is the Null Hypothesis Significance Testing (NHST).
The NHST approach (involving the null and alternative hypothesis) has been one of the most sought-after methods of hypothesis testing in the field of statistics ever since its inception in the mid-1950s.
A much unconventional and modern method of hypothesis testing, the Bayesian Hypothesis Testing claims to test a particular hypothesis in accordance with the past data samples, known as prior probability, and current data that lead to the plausibility of a hypothesis.
The result obtained indicates the posterior probability of the hypothesis. In this method, the researcher relies on ‘prior probability and posterior probability’ to conduct hypothesis testing on hand.
On the basis of this prior probability, the Bayesian approach tests a hypothesis to be true or false. The Bayes factor, a major component of this method, indicates the likelihood ratio among the null hypothesis and the alternative hypothesis.
The Bayes factor is the indicator of the plausibility of either of the two hypotheses that are established for hypothesis testing.
(Also read - Introduction to Bayesian Statistics )
To conclude, hypothesis testing, a way to verify the plausibility of a supposed assumption can be done through different methods - the Bayesian approach or the Frequentist approach.
Although the Bayesian approach relies on the prior probability of data samples, the frequentist approach assumes without a probability. A number of elements involved in hypothesis testing are - significance level, p-level, test statistic, and method of hypothesis testing.
(Also read: Introduction to probability distributions )
A significant way to determine whether a hypothesis stands true or not is to verify the data samples and identify the plausible hypothesis among the null hypothesis and alternative hypothesis.
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Formal hypothesis testing is perhaps the most prominent and widely-employed form of statistical analysis. It is sometimes seen as the most rigorous and definitive part of a statistical analysis, but it is also the source of many statistical controversies. The currently-prevalent approach to hypothesis testing dates to developments that took place between 1925 and 1940, especially the work of Ronald Fisher , Jerzy Neyman , and Egon Pearson .
In recent years, many prominent statisticians have argued that less emphasis should be placed on the formal hypothesis testing approaches developed in the early twentieth century, with a correspondingly greater emphasis on other forms of uncertainty analysis. Our goal here is to give an overview of some of the well-established and widely-used approaches for hypothesis testing. We will also provide some perspectives on how these tools can be effectively used, and discuss their limitations. We will also discuss some new approaches to hypothesis testing that may eventually come to be as prominent as these classical approaches.
A falsifiable hypothesis is a statement, or hypothesis, that can be contradicted with evidence. In empirical (data-driven) research, this evidence will always be obtained through the data. In statistical hypothesis testing, the hypothesis that we formally test is called the null hypothesis . The alternative hypothesis is a second hypothesis that is our proposed explanation for what happens if the null hypothesis is wrong.
The key element of a statistical hypothesis test is the test statistic , which (like any statistic) is a function of the data. A test statistic takes our entire dataset, and reduces it to one number. This one number ideally should contain all the information in the data that is relevant for assessing the two hypotheses of interest, and exclude any aspects of the data that are irrelevant for assessing the two hypotheses. The test statistic measures evidence against the null hypothesis. Most test statistics are constructed so that a value of zero represents the lowest possible level of evidence against the null hypothesis. Test statistic values that deviate from zero represent greater levels of evidence against the null hypothesis. The larger the magnitude of the test statistic, the stronger the evidence against the null hypothesis.
A major theme of statistical research is to devise effective ways to construct test statistics. Many useful ways to do this have been devised, and there is no single approach that is always the best. In this introductory course, we will focus on tests that starting with an estimate of a quantity that is relevant for assessing the hypotheses, then proceed by standardizing this estimate by dividing it by its standard error. This approach is sometimes referred to as “Wald testing”, after Abraham Wald .
As a basic example, let’s consider risk perception related to COVID-19. As you will see below, hypothesis testing can appear at first to be a fairly elaborate exercise. Using this example, we describe each aspect of this exercise in detail below.
The data shown below are simulated but are designed to reflect actual surveys conducted in the United States in March of 2020. Partipants were asked whether they perceive that they have a substantial risk of dying if they are infected with the novel coronavirus. The number of people stating each response, stratified on age, are shown below (only two age groups are shown):
High risk | Not high risk | |
---|---|---|
Age < 30 | 25 | 202 |
Age 60-69 | 30 | 124 |
Each subject’s response is binary – they either perceive themselves to be high risk, or not to be at high risk. When working with this type of data, we are usually interested in the proportion of people who provide each response within each stratum (age group). These are conditional proportions, conditioning on the age group. The numerical values of the conditional proportions are given below:
High risk | Not high risk | |
---|---|---|
Age < 30 | 0.110 | 0.890 |
Age 60-69 | 0.195 | 0.805 |
There are four conditional proportions in the table above – the proportion of younger people who perceive themselves to be at higher risk, 0.110=25/(25+202); the proportion of younger people who do not perceive themselves to be at high risk, 0.890=202/(25+202); the proportion of older people who perceive themselves to be at high risk 0.195=30/(30+124); and the proportion of older people who do not perceive themselves to be at high risk, 0.805=124/(30+124).
The trend in the data is that younger people perceive themselves to be at lower risk of dying than older people, by a difference of 0.195-0.110=0.085 (in terms of proportions). But is this trend only present in this sample, or is it generalizable to a broader population (say the entire US population)? That is the goal of conducting a statistical hypothesis test in this setting.
Corresponding to our data above is the unobserved population structure, which we can denote as follows
High risk | Not high risk | |
---|---|---|
Age < 30 | \(p\) | \(1-p\) |
Age 60-69 | \(q\) | \(1-q\) |
The symbols \(p\) and \(q\) in the table above are population parameters . These are quantitites that we do not know, and wish to assess using the data. In this case, our null hypothesis can be expressed as the statement \(p = q\) . We can estimate \(p\) using the sample proportion \(\hat{p} = 0.110\) , and similarly estimate \(q\) using \(\hat{q} = 0.195\) . However these estimates do not immediately provide us with a way of expressing the evidence relating to the hypothesis that \(p=q\) . This is provided by the test statistic.
As noted above, a test statistic is a reduction of the data to one number that captures all of the relevant information for assessing the hypotheses. A natural first choice for a test statistic here would be the difference in sample proportions between the two age groups, which is 0.195 - 0.110 = 0.085. There is a difference of 0.085 between the perceived risks of death in the younger and older age groups.
The difference in rates (0.085) does not on its own make a good test statistic, although it is a good start toward obtaining one. The reason for this is that the evidence underlying this difference in rates depends also on the absolute rates (0.110 and 0.195), and on the sample sizes (227 and 154). If we only know that the difference in rates is 0.085, this is not sufficient to evaluate the hypothesis in a statistical manner. A given difference in rates is much stronger evidence if it is obtained from a larger sample. If we have a difference of 0.085 with a very large sample, say one million people, then we should be almost certain that the true rates differ (i.e. the data are highly incompatiable with the hypothesis that \(p=q\) ). If we have the same difference in rates of 0.085, but with a small sample, say 50 people per age group, then there would be almost no evidence for a true difference in the rates (i.e. the data are compatiable with the hypothesis \(p=q\) ).
To address this issue, we need to consider the uncertainty in the estimated rate difference, which is 0.085. Recall that the estimated rate difference is obtained from the sample and therefore is almost certain to deviate somewhat from the true rate difference in the population (which is unknown). Recall from our study of standard errors that the standard error for an estimated proportion is \(\sqrt{p(1-p)/n}\) , where \(p\) is the outcome probability (here the outcome is that a person perceives a high risk of dying), and \(n\) is the sample size.
In the present analysis, we are comparing two proportions, so we have two standard errors. The estimated standard error for the younger people is \(\sqrt{0.11\cdot 0.89/227} \approx 0.021\) . The estimated standard error for the older people is \(\sqrt{0.195\cdot 0.805/154} \approx 0.032\) . Note that both standard errors are estimated, rather than exact, because we are plugging in estimates of the rates (0.11 and 0.195). Also note that the standard error for the rate among older people is greater than that for younger people. This is because the sample size for older people is smaller, and also because the estimated rate for older people is closer to 1/2.
In our previous discussion of standard errors, we saw how standard errors for independent quantities \(A\) and \(B\) can be used to obtain the standard error for the difference \(A-B\) . Applying that result here, we see that the standard error for the estimated difference in rates 0.195-0.11=0.085 is \(\sqrt{0.021^2 + 0.032^2} \approx 0.038\) .
The final step in constructing our test statistic is to construct a Z-score from the estimated difference in rates. As with all Z-scores, we proceed by taking the estimated difference in rates, and then divide it by its standard error. Thus, we get a test statistic value of \(0.085 / 0.038 \approx 2.24\) .
A test statistic value of 2.24 is not very close to zero, so there is some evidence against the null hypothesis. But the strength of this evidence remains unclear. Thus, we must consider how to calibrate this evidence in a way that makes it more interpretable.
By the central limit theorem (CLT), a Z-score approximately follows a normal distribution. When the null hypothesis holds, the Z-score approximately follows the standard normal distribution (recall that a standard normal distribution is a normal distribution with expected value equal to 0 and variance equal to 1). If the null hypothesis does not hold, then the test statistic continues to approximately follow a normal distribution, but it is not the standard normal distribution.
A test statistic of zero represents the least possible evidence against the null hypothesis. Here, we will obtain a test statistic of zero when the two proportions being compared are identical, i.e. exactly the same proportions of younger and older people perceive a substantial risk of dying from a disease. Even if the test statistic is exactly zero, this does not guarantee that the null hypothesis is true. However it is the least amount of evidence that the data can present against the null hypothesis.
In a hypothesis testing setting using normally-distrbuted Z-scores, as is the case here (due to the CLT), the standard normal distribution is the reference distribution for our test statistic. If the Z-score falls in the center of the reference distribution, there is no evidence against the null hypothesis. If the Z-score falls into either tail of the reference distribution, then there is evidence against the null distribution, and the further into the tails of the reference distribution the Z-score falls, the greater the evidence.
The most conventional way to quantify the evidence in our test statistic is through a probability called the p-value . The p-value has a somewhat complex definition that many people find difficult to grasp. It is the probability of observing as much or more evidence against the null hypothesis as we actually observe, calculated when the null hypothesis is assumed to be true. We will discuss some ways to think about this more intuitively below.
For our purposes, “evidence against the null hypothesis” is reflected in how far into the tails of the reference distribution the Z-score (test statistic) falls. We observed a test statistic of 2.24 in our COVID risk perception analysis. Recall that due to the “empirical rule”, 95% of the time, a draw from a standard normal distribution falls between -2 and 2. Thus, the p-value must be less than 0.05, since 2.24 falls outside this interval. The p-value can be calculated using a computer, in this case it happens to be approximately 0.025.
As stated above, the p-value tells us how likely it would be for us to obtain as much evidence against the the null hypothesis as we observed in our actual data analysis, if we were certain that the null hypothesis were true. When the null hypothesis holds, any evidence against the null hypothesis is spurious. Thus, we will want to see stronger evidence against the null from our actual analysis than we would see if we know that the null hypothesis were true. A smaller p-value therefore reflects more evidence against the null hypothesis than a larger p-value.
By convention, p-values of 0.05 or smaller are considered to represent sufficiently strong evidence against the null hypothesis to make a finding “statistically significant”. This threshold of 0.05 was chosen arbitrarily 100 years ago, and there is no objective reason for it. In recent years, people have argued that either a lesser or a greater p-value threshold should be used. But largely due to convention, the practice of deeming p-values smaller than 0.05 to be statistically significant continues.
Here is a restatement of the above discussion, using slightly different language. In our analysis of COVID risk perceptions, we found a difference in proportions of 0.085 between younger and older subjects, with younger people perceiving a lower risk of dying. This is a difference based on the sample of data that we observed, but what we really want to know is whether there is a difference in COVID risk perception in the population (say, all US adults).
Suppose that in fact there is no difference in risk perception between younger and older people. For instance, suppose that in the population, 15% of people believe that they have a substantial risk of dying should they become infected with the novel coronavirus, regardless of their age. Even though the rates are equal in this imaginary population (both being 15%), the rates in our sample would typically not be equal. Around 3% of the time (0.024=2.4% to be exact), if the rates are actually equal in the population, we would see a test statistic that is 2.4 or larger. Since 3% represents a fairly rare event, we can conclude that our observed data are not compatible with the null hypothesis. We can also say that there is statistically significant evidence against the null hypothesis, and that we have “rejected” the null hypothesis at the 3% level.
In this data analysis, as in any data analysis, we cannot confirm definitively that the alternative hypothesis is true. But based on our data and the analysis performed above, we can claim that there is substantial evidence against the null hypothesis, using standard criteria for what is considered to be “substantial evidence”.
A very common setting where hypothesis testing is used arises when we wish to compare the means of a quantitative measurement obtained for two populations. Imagine, for example, that we have two ways of manufacturing a battery, and we wish to assess which approach yields batteries that are longer-lasting in actual use. To do this, suppose we obtain data that tells us the number of charge cycles that were completed in 200 batteries of type A, and in 300 batteries of type B. For the test developed below to be meaningful, the data must be independent and identically distributed samples.
The raw data for this study consists of 500 numbers, but it turns out that the most relevant information from the data is contained in the sample means and sample standard deviations computed within each battery type. Note that this is a huge reduction in complexity, since we started with 500 measurements and are able to summarize this down to just four numbers.
Suppose the summary statistics are as follows, where \(\bar{x}\) , \(\hat{\sigma}_x\) , and \(n\) denote the sample mean, sample standard deviation, and sample size, respectively.
Type | \(\bar{x}\) | \(\hat{\sigma}_x\) | \(n\) |
---|---|---|---|
420 | 70 | 200 | |
403 | 90 | 300 |
The simplest measure comparing the two manufacturing approaches is the difference 420 - 403 = 17. That is, batteries of type A tend to have 17 more charge cycles compared to batteries of type B. This difference is present in our sample, but is it also true that the entire population of type A batteries has more charge cycles than the entire population of type B batteries? That is the goal of conducting a hypothesis test.
The next step in the present analysis is to divide the mean difference, which is 17, by its standard error. As we have seen, the standard error of the mean, or SEM, is \(\sigma/n\) , where \(\sigma\) is the standard deviation and \(n\) is the sample size. Since \(\sigma\) is almost never known, we plug in its estimate \(\hat{\sigma}\) . For the type A batteries, the estimated SEM is thus \(70/\sqrt{200} \approx 4.95\) , and for the type B batteries the estimated SEM is \(90/\sqrt{300} \approx 5.2\) .
Since we are comparing two estimated means that are obtained from independent samples, we can pool the standard deviations to obtain an overall standard deviation of \(\sqrt{4.95^2 + 5.2^2} \approx 7.18\) . We can now obtain our test statistic \(17/7.18 \approx 2.37\) .
The test statistic can be calibrated against a standard normal reference distribution. The probability of observing a standard normal value that is greater in magnitude than 2.37 is 0.018 (this can be obtained from a computer). This is the p-value, and since it is smaller than the conventional threshold of 0.05, we can claim that there is a statistically significant difference between the average number of charge cycles for the two types of batteries, with the A batteries having more charge cycles on average.
The analysis illustrated here is called a two independent samples Z-test , or just a two sample Z-test . It may be the most commonly employed of all statistical tests. It is also common to see the very similar two sample t-test , which is different only in that it uses the Student t distribution rather than the normal (Gaussian) distribution to calculate the p-values. In fact, there are quite a few minor variations on this testing framework, including “one sided” and “two sided” tests, and tests based on different ways of pooling the variance. Due to the CLT, if the sample size is modestly large (which is the case here), the results of all of these tests will be almost identical. For simplicity, we only cover the Z-test in this course.
The tests for comparing proportions and means presented above are quite similar in many ways. To provide one more example of a hypothesis test that is somewhat different, we consider a test for a correlation coefficient.
Recall that the sample correlation coefficient \(\hat{r}\) is used to assess the relationship, or association, between two quantities X and Y that are measured on the same units. For example, we may ask whether two biomarkers, serum creatinine and D-dimer, are correlated with each other. These biomarkers are both commonly used in medical settings and are obtained using blood tests. D-dimer is used to assess whether a person has blood clots, and serum creatinine is used to measure kidney performance.
Suppose we are interested in whether there is a correlation in the population between D-dimer and serum creatinine. The population correlation coefficient between these two quantitites can be denoted \(r\) . Our null hypothesis is \(r=0\) . Suppose that we observe a sample correlation coefficient of \(\hat{r}=0.15\) , using an independent and identically distributed sample of pairs \((x, y)\) , where \(x\) is a D-dimer measurement and \(y\) is a serum creatinine measurement. Are these data consistent with the null hypothesis?
As above, we proceed by constructing a test statistic by taking the estimated statistic and dividing it by its standard error. The approximate standard error for \(\hat{r}\) is \(1/\sqrt{n}\) , where \(n\) is the sample size. The test statistic is therefore \(\sqrt{n}\cdot \hat{r} \approx 1.48\) .
We now calibrate this test statistic by comparing it to a standard normal reference distribution. Recall from the empirical rule that 5% of the time, a standard normal value falls outside the interval (-2, 2). Therefore, if the test statistic is smaller than 2 in magnitude, as is the case here, its p-value is greater than 0.05. Thus, in this case we know that the p-value will exceed 0.05 without calculating it, and therefore there is no basis for claiming that D-dimer and serum creatinine levels are correlated in this population.
A p-value is the most common way of calibrating evidence. Smaller p-values indicate stronger evidence against a null hypothesis. By convention, if the p-value is smaller than some threshold, usually 0.05, we reject the null hypothesis and declare a finding to be “statistically significant”. How can we understand more deeply what this means? One major concern should be obtaining a small p-value when the null hypothesis is true. If the null hypothesis is true, then it is incorrect to reject it. If we reject the null hypothesis, we are making a false claim. This can never be prevented with complete certainty, but we would like to have a very clear understanding of how likely it is to reject the null hypothesis when the null hypothesis is in fact true.
P-values have a special property that when the null distribution is true, the probability of observing a p-value smaller than 0.05 is 0.05 (5%). In fact, the probability of observing a p-value smaller than \(t\) is equal to \(t\) , for any threshold \(t\) . For example, the probability of observing a p-value smaller than 0.1, when the null hypothesis is true, is 10%.
This fact gives a more concrete understanding of how strong the evidence is for a particular p-value. If we always reject the null hypothesis when the p-value is 0.1 or smaller, then over the long run we will reject the null hypothesis 10% of the time when the null hypothesis is true. If we always reject the null hypothesis when the p-value is 0.05 or smaller, then over the long run we will reject the null hypothesis 5% of the time when the null hypothesis is true.
The approach to hypothesis testing discussed above largely follows the framework developed by RA Fisher around 1925. Note that although we mentioned the alternative hypothesis above, we never actually used it. A more elaborate approach to hypothesis testing was developed somewhat later by Egon Pearson and Jerzy Neyman. The “Neyman-Pearson” approach to hypothesis testing is even more formal than Fisher’s approach, and is most suited to highly planned research efforts in which the study is carefully designed, then executed. While ideally all research projects should be carried out this way, in reality we often conduct research using data that are already available, rather than using data that are specifically collected to address the research question.
Neyman-Pearson hypothesis testing involves specifying an alternative hypothesis that we anticipate encountering. Usually this alternative hypothesis represents a realistic guess about what we might find once the data are collected. In each of the three examples above, imagine that the data are not yet collected, and we are asked to specify an alternative hypothesis. We may arrive at the following:
In comparing risk perceptions for COVID, we may anticipate that older people will perceive a 30% risk of dying, and younger people will anticipate a 5% risk of dying.
In comparing the number of charge cycles for two types of batteries, we may anticipate that batter type A will have on average 500 charge cycles, and battery type B will have on average 400 charge cycles.
In assessing the correlation between D-dimer and serum creatinine levels, we may anticipate a correlation of 0.3.
Note that none of the numbers stated here are data-driven – they are specified before any data are collected, so they do not match the results from the data, which were collected only later. These alternative hypotheses are all essentially speculations, based perhaps on related data or theoretical considerations.
There are several benefits of specifying an explicit alternative hypothesis, as done here, even though it is not strictly necessary and can be avoided entirely by adopting Fisher’s approach to hypothesis testing. One benefit of specifying an alternative hypothesis is that we can use it to assess the power of our planned study, which can in turn inform the design of the study, in particular the sample size. The power is the probability of rejecting the null hypothesis when the alternative hypothesis is true. That is, it is the probability of discovering something real. The power should be contrasted with the level of a hypothesis test, which is the probability of rejecting the null hypothesis when the null hypothesis is true. That is, the level is the probability of “discovering” something that is not real.
To calculate the power, recall that for many of the test statistics that we are considering here, the test statistic has the form \(\hat{\theta}/{\rm SE}(\hat{\theta})\) , where \(\hat{\theta}\) is an estimate. For example, \(\hat{\theta}\) ) may be the correlation coefficient between D-dimer and serum creatinine levels. As stated above, the power is the probability of rejecting the null hypothesis when the alternative hypothesis is true. Suppose we decide to reject the null hypothesis when the test statistic is greater than 2, which is approximately equivalent to rejecting the null hypothesis when the p-value is less than 0.05. The following calculation tells us how to obtain the power in this setting:
Under the alternative hypothesis, \(\sqrt{n}(\hat{r} - r)\) approximately follows a standard normal distribution. Therefore, if \(r\) and \(n\) are given, we can easily use the computer to obtain the probability of observing a value greater than \(2 - \sqrt{n}r\) . This gives us the power of the test. For example, if we anticipate \(r=0.3\) and plan to collect data for \(n=100\) observations, the power is 0.84. This is generally considered to be good power – if the true value of \(r\) is in fact 0.3, we would reject the null hypothesis 84% of the time.
A study usually has poor power because it has too small of a sample size. Poorly powered studies can be very misleading, but since large sample sizes are expensive to collect, a lot of research is conducted using sample sizes that yield moderate or even low power. If a study has low power, it is unlikely to reject the null hypothesis even when the alternative hypothesis is true, but it remains possible to reject the null hypothesis when the null hypothesis is true (usually this probability is 5%). Therefore the most likely outcome of a poorly powered study may be an incorrectly rejected null hypothesis.
Table of Contents
As per the definition from Oxford languages, a hypothesis is a supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation. As per the Dictionary page on Hypothesis , Hypothesis means a proposition or set of propositions, set forth as an explanation for the occurrence of some specified group of phenomena, either asserted merely as a provisional conjecture to guide investigation (working hypothesis) or accepted as highly probable in the light of established facts.
The hypothesis can be defined as the claim that can either be related to the truth about something that exists in the world, or, truth about something that’s needs to be established a fresh . In simple words, another word for the hypothesis is the “claim” . Until the claim is proven to be true, it is called the hypothesis. Once the claim is proved, it becomes the new truth or new knowledge about the thing. For example , let’s say that a claim is made that students studying for more than 6 hours a day gets more than 90% of marks in their examination. Now, this is just a claim or a hypothesis and not the truth in the real world. However, in order for the claim to become the truth for widespread adoption, it needs to be proved using pieces of evidence, e.g., data. In order to reject this claim or otherwise, one needs to do some empirical analysis by gathering data samples and evaluating the claim. The process of gathering data and evaluating the claims or hypotheses with the goal to reject or otherwise (failing to reject) can be called as hypothesis testing . Note the wordings – “failing to reject”. It means that we don’t have enough evidence to reject the claim. Thus, until the time that new evidence comes up, the claim can be considered the truth. There are different techniques to test the hypothesis in order to reach the conclusion of whether the hypothesis can be used to represent the truth of the world.
One must note that the hypothesis testing never constitutes a proof that the hypothesis is absolute truth based on the observations. It only provides added support to consider the hypothesis as truth until the time that new evidences can against the hypotheses can be gathered. We can never be 100% sure about truth related to those hypotheses based on the hypothesis testing.
Simply speaking, hypothesis testing is a framework that can be used to assert whether the claim or the hypothesis made about a real-world/real-life event can be seen as the truth or otherwise based on the given data (evidences).
Before we get ahead and start understanding more details about hypothesis and hypothesis testing steps, lets take a look at some real-world examples of how to think about hypothesis and hypothesis testing when dealing with real-world problems :
You may note different hypotheses which are listed above. The next step would be validate some of these hypotheses. This is where data scientists will come into picture. One or more data scientists may be asked to work on different hypotheses. This would result in these data scientists looking for appropriate data related to the hypothesis they are working. This section will be detailed out in near future.
The first step to hypothesis testing is defining or stating a hypothesis. Before the hypothesis can be tested, we need to formulate the hypothesis in terms of mathematical expressions. There are two important aspects to pay attention to, prior to the formulation of the hypothesis. The following represents different types of hypothesis that could be put to hypothesis testing:
Based on the above considerations, the following hypothesis can be stated for doing hypothesis testing.
Once the hypothesis is defined or stated, the next step is to formulate the null and alternate hypothesis in order to begin hypothesis testing as described above.
In the case where the given statement is a well-established fact or default state of being in the real world, one can call it a null hypothesis (in the simpler word, nothing new). Well-established facts don’t need any hypothesis testing and hence can be called the null hypothesis. In cases, when there are any new claims made which is not well established in the real world, the null hypothesis can be thought of as the default state or opposite state of that claim. For example , in the previous section, the claim or hypothesis is made that the students studying for more than 6 hours a day gets more than 90% of marks in their examination. The null hypothesis, in this case, will be that the claim is not true or real. The null hypothesis can be stated that there is no relationship or association between the students reading more than 6 hours a day and they getting 90% of the marks. Any occurrence is only a chance occurrence. Another example of hypothesis is when somebody is alleged that they have performed a crime.
Null hypothesis is denoted by letter H with 0, e.g., [latex]H_0[/latex]
When the given statement is a claim (unexpected event in the real world) and not yet proven, one can call/formulate it as an alternate hypothesis and accordingly define a null hypothesis which is the opposite state of the hypothesis. The alternate hypothesis is a new knowledge or truth that needs to be established. In simple words, the hypothesis or claim that needs to be tested against reality in the real world can be termed the alternate hypothesis. In order to reach a conclusion that the claim (alternate hypothesis) can be considered the new knowledge or truth (based on the available evidence), it would be important to reject the null hypothesis. It should be noted that null and alternate hypotheses are mutually exclusive and at the same time asymmetric. In the example given in the previous section, the claim that the students studying for more than 6 hours get more than 90% of marks can be termed as the alternate hypothesis.
Alternate hypothesis is denoted with H subscript a, e.g., [latex]H_a[/latex]
Once the hypothesis is formulated as null([latex]H_0[/latex]) and alternate hypothesis ([latex]H_a[/latex]), there are two possible outcomes that can happen from hypothesis testing. These outcomes are the following:
The following are some examples of the null and alternate hypothesis.
The weight of the sugar packet is 500 gm. (A well-established fact) | |
The weight of the sugar packet is 500 gm. |
Running 5 miles a day result in the reduction of 10 kg of weight within a month. | |
Running 5 miles a day results in the reduction of 10 kg of weight within a month. |
The housing price depend upon the average income of people staying in the locality. | |
The housing price depends upon the average income of people staying in the locality. |
Here is the diagram which represents the workflow of Hypothesis Testing.
Figure 1. Hypothesis Testing Steps
Based on the above, the following are some of the steps to be taken when doing hypothesis testing:
Once you formulate the hypotheses, there is the need to test those hypotheses. Meaning, say that the null hypothesis is stated as the statement that housing price does not depend upon the average income of people staying in the locality, it would be required to be tested by taking samples of housing prices and, based on the test results, this Null hypothesis could either be rejected or failed to be rejected . In hypothesis testing, the following two are the outcomes:
Take the above example of the sugar packet weighing 500 gm. The Null hypothesis is set as the statement that the sugar packet weighs 500 gm. After taking a sample of 20 sugar packets and testing/taking its weight, it was found that the average weight of the sugar packets came to 495 gm. The test statistics (t-statistics) were calculated for this sample and the P-value was determined. Let’s say the P-value was found to be 15%. Assuming that the level of significance is selected to be 5%, the test statistic is not statistically significant (P-value > 5%) and thus, the null hypothesis fails to get rejected. Thus, one could safely conclude that the sugar packet does weigh 500 gm. However, if the average weight of canned sauce would have found to be 465 gm, this is way beyond/away from the mean value of 500 gm and one could have ended up rejecting the Null Hypothesis based on the P-value .
Hypothesis testing can be applied in both problem analysis and solution implementation. The following represents method on how you can apply hypothesis testing technique for both problem and solution space:
The claim that needs to be established is set as ____________, the outcome of hypothesis testing is _________.
Please select 2 correct answers
There is a claim that doing pranayama yoga results in reversing diabetes. which of the following is true about null hypothesis.
In this post, you learned about hypothesis testing and related nuances such as the null and alternate hypothesis formulation techniques, ways to go about doing hypothesis testing etc. In data science, one of the reasons why one needs to understand the concepts of hypothesis testing is the need to verify the relationship between the dependent (response) and independent (predictor) variables. One would, thus, need to understand the related concepts such as hypothesis formulation into null and alternate hypothesis, level of significance, test statistics calculation, P-value, etc. Given that the relationship between dependent and independent variables is a sort of hypothesis or claim , the null hypothesis could be set as the scenario where there is no relationship between dependent and independent variables.
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I found it very helpful. However the differences are not too understandable for me
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in your case E respresent Member or Oraganization which include on e or more peers?
Such a informative post. Keep it up
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The bottom line.
Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.
Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population or a data-generating process. The word "population" will be used for both of these cases in the following descriptions.
In hypothesis testing, an analyst tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.
The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.
The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.
If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.
A random sample of 100 coin flips is taken, and the null hypothesis is tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.
If there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."
Some statisticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”
Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.
Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.
Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.
Sage. " Introduction to Hypothesis Testing ," Page 4.
Elder Research. " Who Invented the Null Hypothesis? "
Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."
Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.
A hypothesis is an assumption or idea, specifically a statistical claim about an unknown population parameter. For example, a judge assumes a person is innocent and verifies this by reviewing evidence and hearing testimony before reaching a verdict.
Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.
To test the validity of the claim or assumption about the population parameter:
Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.
Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing.
One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.
There are two types of one-tailed test:
A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.
Example: H 0 : [Tex]\mu = [/Tex] 50 and H 1 : [Tex]\mu \neq 50 [/Tex]
To delve deeper into differences into both types of test: Refer to link
In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.
Null Hypothesis is True | Null Hypothesis is False | |
---|---|---|
Null Hypothesis is True (Accept) | Correct Decision | Type II Error (False Negative) |
Alternative Hypothesis is True (Reject) | Type I Error (False Positive) | Correct Decision |
Step 1: define null and alternative hypothesis.
State the null hypothesis ( [Tex]H_0 [/Tex] ), representing no effect, and the alternative hypothesis ( [Tex]H_1 [/Tex] ), suggesting an effect or difference.
We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.
Select a significance level ( [Tex]\alpha [/Tex] ), typically 0.05, to determine the threshold for rejecting the null hypothesis. It provides validity to our hypothesis test, ensuring that we have sufficient data to back up our claims. Usually, we determine our significance level beforehand of the test. The p-value is the criterion used to calculate our significance value.
Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.
The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.
There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.
We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.
T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.
In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.
Comparing the test statistic and tabulated critical value we have,
Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.
We can also come to an conclusion using the p-value,
Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.
At last, we can conclude our experiment using method A or B.
To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .
When population means and standard deviations are known.
[Tex]z = \frac{\bar{x} – \mu}{\frac{\sigma}{\sqrt{n}}}[/Tex]
T test is used when n<30,
t-statistic calculation is given by:
[Tex]t=\frac{x̄-μ}{s/\sqrt{n}} [/Tex]
Chi-Square Test for Independence categorical Data (Non-normally distributed) using:
[Tex]\chi^2 = \sum \frac{(O_{ij} – E_{ij})^2}{E_{ij}}[/Tex]
Let’s examine hypothesis testing using two real life situations,
Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.
Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.
If the evidence suggests less than a 5% chance of observing the results due to random variation.
Using paired T-test analyze the data to obtain a test statistic and a p-value.
The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.
t = m/(s/√n)
then, m= -3.9, s= 1.8 and n= 10
we, calculate the , T-statistic = -9 based on the formula for paired t test
The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.
thus, p-value = 8.538051223166285e-06
Step 5: Result
Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.
Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.
Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.
We will implement our first real life problem via python,
import numpy as np from scipy import stats # Data before_treatment = np . array ([ 120 , 122 , 118 , 130 , 125 , 128 , 115 , 121 , 123 , 119 ]) after_treatment = np . array ([ 115 , 120 , 112 , 128 , 122 , 125 , 110 , 117 , 119 , 114 ]) # Step 1: Null and Alternate Hypotheses # Null Hypothesis: The new drug has no effect on blood pressure. # Alternate Hypothesis: The new drug has an effect on blood pressure. null_hypothesis = "The new drug has no effect on blood pressure." alternate_hypothesis = "The new drug has an effect on blood pressure." # Step 2: Significance Level alpha = 0.05 # Step 3: Paired T-test t_statistic , p_value = stats . ttest_rel ( after_treatment , before_treatment ) # Step 4: Calculate T-statistic manually m = np . mean ( after_treatment - before_treatment ) s = np . std ( after_treatment - before_treatment , ddof = 1 ) # using ddof=1 for sample standard deviation n = len ( before_treatment ) t_statistic_manual = m / ( s / np . sqrt ( n )) # Step 5: Decision if p_value <= alpha : decision = "Reject" else : decision = "Fail to reject" # Conclusion if decision == "Reject" : conclusion = "There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different." else : conclusion = "There is insufficient evidence to claim a significant difference in average blood pressure before and after treatment with the new drug." # Display results print ( "T-statistic (from scipy):" , t_statistic ) print ( "P-value (from scipy):" , p_value ) print ( "T-statistic (calculated manually):" , t_statistic_manual ) print ( f "Decision: { decision } the null hypothesis at alpha= { alpha } ." ) print ( "Conclusion:" , conclusion )
T-statistic (from scipy): -9.0 P-value (from scipy): 8.538051223166285e-06 T-statistic (calculated manually): -9.0 Decision: Reject the null hypothesis at alpha=0.05. Conclusion: There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.
In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05.
Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.
Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.
Populations Mean = 200
Population Standard Deviation (σ): 5 mg/dL(given for this problem)
As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.
The test statistic is calculated by using the z formula Z = [Tex](203.8 – 200) / (5 \div \sqrt{25}) [/Tex] and we get accordingly , Z =2.039999999999992.
Step 4: Result
Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL
import scipy.stats as stats import math import numpy as np # Given data sample_data = np . array ( [ 205 , 198 , 210 , 190 , 215 , 205 , 200 , 192 , 198 , 205 , 198 , 202 , 208 , 200 , 205 , 198 , 205 , 210 , 192 , 205 , 198 , 205 , 210 , 192 , 205 ]) population_std_dev = 5 population_mean = 200 sample_size = len ( sample_data ) # Step 1: Define the Hypotheses # Null Hypothesis (H0): The average cholesterol level in a population is 200 mg/dL. # Alternate Hypothesis (H1): The average cholesterol level in a population is different from 200 mg/dL. # Step 2: Define the Significance Level alpha = 0.05 # Two-tailed test # Critical values for a significance level of 0.05 (two-tailed) critical_value_left = stats . norm . ppf ( alpha / 2 ) critical_value_right = - critical_value_left # Step 3: Compute the test statistic sample_mean = sample_data . mean () z_score = ( sample_mean - population_mean ) / \ ( population_std_dev / math . sqrt ( sample_size )) # Step 4: Result # Check if the absolute value of the test statistic is greater than the critical values if abs ( z_score ) > max ( abs ( critical_value_left ), abs ( critical_value_right )): print ( "Reject the null hypothesis." ) print ( "There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL." ) else : print ( "Fail to reject the null hypothesis." ) print ( "There is not enough evidence to conclude that the average cholesterol level in the population is different from 200 mg/dL." )
Reject the null hypothesis. There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL.
Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.
1. what are the 3 types of hypothesis test.
There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.
Null Hypothesis ( [Tex]H_o [/Tex] ): No effect or difference exists. Alternative Hypothesis ( [Tex]H_1 [/Tex] ): An effect or difference exists. Significance Level ( [Tex]\alpha [/Tex] ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.
Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.
Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.
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Aline alves soares.
1 Postgraduating Program in Health Sciences, Federal University of Rio Grande do Norte, Natal, RN, Brazil
2 Department of Nutrition, Liga Contra o Câncer, Natal, RN, Brazil
Camila xavier alves, kleyton santos de medeiros.
3 Pesquisa e Inovação, Liga Contra o Câncer, Instituto de Ensino, Natal, RN, Brazil
4 Department of Health Sciences, Federal University of Rio Grande do Norte, Natal, RN, Brazil
5 Department of Nutrition.Federal University of Rio Grande do Norte, Natal, RN, Brazil
6 Department of Internal Medicine, Federal University of Rio Grande do Norte, Natal, RN, Brazil
No datasets were generated or analysed during the current study. All relevant data from this study will be made available upon study completion.
The thyroid cancer has the ninth larger incidence of cancer in the world. Investigations related to the exposure to metals have become important due to the sensibility of the thyroid gland to them. Studies reveal that carcinogenic progressions are associated to the deficiency of the essential trace elements. In this context, the zinc is highlighted, essential for the metabolism of the thyroidal hormone and has a potential relation with the pathogenesis of the thyroid cancer. The objective of this systematic review and meta-analysis is to evaluate the low serum zinc as a risk factor for thyroid cancer in adults.
PubMed/MEDLINE, Scopus, Embase and LILACS databases will be searched for observational studies investigating the low serum zinc as a risk factor for thyroid cancer in adults. No language or publication period restrictions will be imposed. The primary outcome will be that the low serum zinc is a risk factor for thyroid cancer. Three independent reviewers will select the studies and extract data from the original publications. The risk-of-bias will be assessed by using the Newcastle-Ottawa Quality Assessment Scale (NOS). Data synthesis will be performed using the R software (V.4.3.1) and to assess heterogeneity, we will compute the I2 statistic and the results will be based on either random-effects or fixed-effects models, depending on the heterogeneity. The Grading of Recommendations, Development, and Evaluation (GRADE) system will be used to evaluate the reliability and quality of evidence.
International Prospective Register of Systematic Reviews (PROSPERO) CRD42023463747 .
The thyroid cancer (TC) has the ninth larger incidence of cancer in the whole world [ 1 , 2 ]. And if the recent tendencies are maintained, it can become the fourth most common cancer until 2030 in the United States [ 3 ].
There is a number of reasons responsible for this high incidence, as the enhancement of access to diagnostic procedures more intensive and sensitive. Nevertheless, it has been suggested that diagnostic technologies may not totally explain the growth in TC frequency, arguing that the environmental factors, lifestyle and comorbidities may contribute with this phenomenon [ 4 – 6 ]. The previous irradiation in the head/neck, history of benign thyroid nodules, goiter and family history of proliferative thyroid disease are risk factors established for TC [ 7 , 8 ].
In addition, investigations related to exposure to metals have been becoming more important due to the sensibility of the thyroid gland to them. Studies reveal that carcinogenic progressions are associated to the excess of toxic metals (such as nickel, lead, cadmium), whereas the majority of the essential elements (selenium, zinc, magnesium) shows deficiency. This imbalance is capable of affecting the thyroid homeostasis because many of these trace elements are part of the metabolism of the thyroidal hormones, being an important risk factor in the development of TC [ 9 , 10 ].
In this context, considering the health of the thyroid gland, among the essential trace elements, zinc (Zn) is highlighted, defined as a regulator metal in a number of aspects concerning the cellular function and metabolism. With Zn deficiency, multiple nonspecific general changes in metabolism and function occur, including reductions in growth, as well as the impairment of reproductive function and neurobehavioral development [ 11 ]. In addition, Zn is essential for the metabolism of the thyroidal hormone and has a potential relation with the pathogenesis of the TC [ 12 ]. Studies reveal that the Zn serum concentration is significantly reduced in many malignant tumors [ 13 ], including the TC. Specifically in the papillary thyroid carcinoma (PTC) and medullary thyroid carcinoma (MTC), the levels of serum Zn are lower than the ones found in healthy individuals [ 13 , 14 ].
However, the results of studies concerning the Zn deficiency and TC are still inconsistent [ 13 , 15 , 16 ], showing that little is known about the role of Zn and the risk of progression of TC [ 9 ], preventing definitive recommendations.
In addition to the growing number of patients with TC 1 and the inconclusive results of studies on Zn deficiency and TC risk [ 13 , 15 , 16 ], a study exploring the serum status of this trace element with greater depth is useful, as it is considered a vital component for the proper functioning of thyroid hormone metabolism and its deficiency can have a detrimental effect on thyroid activity [ 17 ].
Research with this objective may help understand the possible biological mechanisms involved in the deficiency of Zn and the thyroid carcinogenesis, helping the diagnosis and handling of patients with the worst prognoses. With that said, the objective of this systematic review and meta-analysis is to evaluate the low serum Zn as a risk factor for TC in adults.
The systematic review and meta-analysis will be conducted following the Meta-analysis of Observational Studies in Epidemiology (MOOSE) guidelines [ 18 ] and reported following the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) statement [ 19 , 20 ]. This protocol is listed in the International Prospective Registry of Systematic Reviews (PROSPERO) (CRD42023463747).
This systematic review and meta-analysis will include the following studies: observational studies (cohort, case-control, transversal) that evaluated the serum Zn a risk factor for TC; studies involving patients (age>18); with an apparently healthy population (in the controls for the case-control studies); studies without time restriction and studies published in any language.
The studies will be excluded if they are case reports, meeting abstracts, review papers and commentaries. Children and adolescents under 18 years of age will be excluded.
The following databases will be used: PubMed/MEDLINE, Scopus, Embase and LILACS. No language or publication period restrictions will be imposed.
The Medical Subject Headings (MeSH) terms will be: ((Zinc) AND (Thyroid Neoplasm OR Neoplasm, Thyroid OR Thyroid Carcinoma OR Carcinoma, Thyroid OR Cancer of Thyroid OR Thyroid Cancer OR Cancer, Thyroid OR Thyroid Adenoma OR Adenoma, Thyroid) AND (Observational Study OR Cohort Study OR Retrospective Study)) ( Table 1 ). The librarian participated in the development of the search strategy. The search strategy is shown in the S1 File .
Pubmed/MEDLINE search strategy | |
---|---|
1 | Zinc |
2 | 1/AND |
3 | Thyroid Neoplasm |
4 | Neoplasm, Thyroid |
5 | Thyroid Carcinoma |
6 | Carcinoma, Thyroid |
7 | Cancer of Thyroid |
8 | Thyroid Cancer |
9 | Cancer, Thyroid |
10 | Thyroid Adenoma |
11 | Adenoma, Thyroid |
12 | 3–11/OR |
13 | 1 AND 12 |
The reference lists of the retrieved papers may also be used to choose appropriate research. In other words, the reference lists of the articles that were retrieved may allow the computerized literature search to be expanded. Identical strategies will be applied to other databases S1 File .
With Rayyan ( https://www.rayyan.ai ), two authors, AAS and YGN, will independently filter the search results based on titles and abstracts. Reviews and duplicate entries will be eliminated from the database. There will be an Excel table with the articles in it (Google Drive). To ascertain whether the research satisfy the inclusion criteria, the same authors will examine the entire text. Any differences will be resolved by CXA, the third reviewer. A PRISMA flow diagram will be used to summarize the chosen studies Fig 1 .
In accordance with the Cochrane tool, a standardized data extraction form will be created and evaluated. Two reviewers (AAS and YGN) will extract data separately from each included study and any inconsistencies will be discussed and addressed with a third reviewer (CXA). The data extracted will include information as the name of the first author; year of publication; country; sample size; gender and age of participants; number of participants in the case group (if case-control study); number of participants in the control group (if case-control study); kind of study; follow-up period; eligibility criteria; serum zinc levels; zinc measurement methods; quality control procedure of the serum Zn measurement; quantitative method of variable analysis. Likewise, we will extract the odds ratio (OR) and the 95% confidence interval (CI) for TC risk.
Reviewers (AAS and YGN) will contact the authors or co-authors of the article if there are studies with missing, suppressed, or incomplete data. Communication will be via email. Additionally, supplementary documents related to the studies will be reviewed. If it is not feasible to obtain the necessary information, these studies will be addressed in the discussion section and excluded from the analysis.
The bias risks of the included researches will evaluated independently by two investigators (AAS and YGN). The Newcastle-Ottawa Quality Assessment Scale (NOS) [ 22 ] will be utilized to evaluate the methodological quality of the studies. This evaluation tool comprises eight criteria that are grouped into three overarching perspectives: choosing the study groups, group comparability, and exposures or outcomes of interest. All things on the scale are given one point, or one star, with the exception of the item "Comparability", which has a score between zero and two stars. A study that is considered high quality will receive a rating of at least six stars; a study that is considered moderate quality will receive four or five stars; and a study that is considered low quality will receive less than four stars [ 22 ].
A standard χ 2 test will assess the heterogeneity between the study outcomes at a significance threshold of p<0.1. We intended to compute the I2 statistic, a quantitative indicator of study inconsistency, to evaluate heterogeneity. Heterogeneity will only be assessed if a meta-analysis is warranted [ 23 ].
The I2 statistics <25% represented low heterogeneity, 25%-50%, moderate heterogeneity and >50%, high heterogeneity. In cases where there was substantial heterogeneity in the included studies (I2>50%), the random-effect model will be used, and when low heterogeneity exists in included studies the fixed-effects model will be used.
The R Software V.4.3.1 will be used to enter the data. The user can enter protocols, finish reviews, add text, research features, comparison tables, and study data, as well as carry out meta-analyses, with this software. The OR and 95% CI for each research will be extracted or computed for dichotomous data. The studies will be combined using the random-effects model in the event of heterogeneity (I2>50%), and the DerSimonian-Laird method will be used to get the OR and 95% CI. The robustness of the findings in relation to study quality and sample size will be investigated using sensitivity analysis. Only in the event that a meta-analysis is successful will this be feasible. In a summary table, the sensitivity analysis will be shown.
Considering the subgroup analyses, the assessment of serum Zn as a TC risk may be handled differently in the result analysis. The decision to perform subgroup analysis will take into account the heterogeneity and quantity of available studies. If a meta-analysis includes at least ten papers, we will attempt to perform subgroup analyses to account for any found heterogeneity among studies in order to provide for statistical power in these types of investigations. The nation, research type, age, gender, TC type, and Zn measuring techniques are the factors that will be taken into account.
If it is not possible to do a meta-analysis for all or part of the included studies, other research features and results will be narratively presented.
The Grading of Recommendations Assessment, Development and Evaluation (GRADE) [ 24 ] method or a comparable approach that is properly stated and documented will be used to assess the degree of certainty in the evidence. The quality of evidence will be defined as “high”, “moderate”, “low,” and “very low” [ 24 ].
Since this review will rely on publicly available scientific literature, ethical approval is not necessary. The results of this systematic review and meta-analysis will be published in a peer-reviewed publication and if sufficient new evidence becomes available to warrant a revision in the review’s conclusions, updates will be carried out. Any modifications to the protocol made while the review was being conducted will be noted in the manuscript.
Considering that the metal ions assemble in the thyroid and some play an important part in the function and homeostatic mechanisms of the thyroid gland, Zhou et al . [ 12 ] explain that alterations in some serums may be related to the pathogenesis of the TC.
Zn is a crucial trace element in the link of triiodothyronine (T3) with the nuclear receptor and is involved in the conversion of the thyrotropin-releasing hormone (TRH) to produce TRH via proteolytic conversion by a carboxypeptidase enzyme. The most important way towards the metabolism of thyroxine (T4) is through monodeiodination to produce the active thyroidal hormone, T3. This reaction is catalyzed by deiodinases type I and II (DI and DII) that need Zn as cofactor [ 25 ]. Therefore, the decrease of the Zn serum level may have a harmful effect over the thyroid activity that may be involved in the carcinogenic activity [ 17 ].
In this case, to help understand the biological mechanisms involved in the thyroidal carcinogenesis, this study was based in the evaluation of Zn serums in patients with TC.
Findings in Stojsavljević A. et al . [ 15 ] studies have indicated that the Zn (1613 ng/g) concentration average was significantly reduced (p<0.05) in blood samples of patients with TC when compared to the ones of the control group (5147 ng/g), result that may have an important role from the clinical point of view, for the purposes of diagnostics and traces. Analyzing other studies, similar outcomes support hypothesis that low Zn serums are associated to TC [ 16 , 17 ].
The results of H. Al-Sayer et al . [ 16 ] and of Baltaci et al . [ 13 ] have discovered that the content of pre-operative Zn serum in patients with TC was significantly reduced when compared to a healthy one and that the surgical excision of the malignant thyroidal tissue has resulted in the restauration of the Zn content in regular amounts. Also, in the study by Baltaci et al . [ 13 ], measurements made immediately after the thyroid surgery have also shown lower levels of Zn serum in these patients (p<0.05). The surgical tissue though, indicated high average amounts of Zn. The fact that the same patients have presented lower zinc amounts in the serum samples indicates that this element is excessively withheld in the thyroidal tissue and can be related to the thyroid pathogenesis.
On the contrary, Rezaei M. et al . [ 26 ] couldn’t show any significant association between the Zn serum level and the risk of developing TC. The A. Emami et al . [ 14 ] study that sought to evaluate the status of micronutrients in Iranian patients with MTC before the thyroidectomy, has shown that the low Zn serum levels were not a risk factor for MTC.
Among types of TC, Bibi K and Shah MH [ 17 ] have compared the average Zn levels measured in the blood of various types of TC patients (anaplastic, follicular, medullary and papillary), identifying higher levels in anaplastic TC.
The results evidence the presence of altered Zn content in pathological blood samples in comparison to the control, indicating that the relation between Zn serum and TC is still controversial [ 13 , 15 , 16 ].
A systematic review and meta-analysis will help us to identify and synthesize the evidence of the association between Zn serum and TC. The results will also help us better understand the risk differences depending on gender, age, geographical location and types of TC. Also, a systematic review and meta-analysis about the matter will provide data about the methodology of different studies and the important points in published literature, which may help in the development of new experimental drawings, identifying the reasons of the discrepancies or contradictions between the results of the different investigations, encouraging the redrawing of the studies to improve the existing research methods.
The limitations of this review may involve the quality of primary studies, due to high methodological, clinical, and statistical heterogeneity among them. Especially, there is heterogeneity among the studies regarding Zn results and thyroid cancer risk, stemming from differences in social, demographic, and environmental factors, as well as variations in the types of TC among participants and characteristics of the measurement methods.
S1 checklist, acknowledgments.
The authors acknowledge the assistance provided by the Graduate Program in Health Sciences of the Federal University of Rio Grande do Norte (UFRN), the Liga Norte Riograndense Contra o Câncer and the librarian Rafaela Carla Melo de Paiva for the assistance with literary research.
The author(s) received no specific funding for this work.
The data genie is well and truly out of the bottle .
The presence of data-driven consultancies, the rise of public data websites and the growth of its use in the media (cough, cough) highlight how integral statistics have become in the way we view and analyse the game.
Knowledge sharing has been a crucial catalyst for the creation and development of many metrics and statistical models. However, the curtain every football fan wants to peek behind is the use of analytics within professional clubs. Understandably, these in-house data departments will maintain a high degree of confidentiality to maintain a competitive advantage over their rivals, but what does this landscape look like?
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Having a ‘Moneyball’ approach remains the in-vogue term used to explain the data-led methods adopted by clubs such as Brentford , Brighton & Hove Albion and Liverpool . But any club that has had success with data knows it is not as simple as Oakland As baseball general manager Brad Pitt clicking his fingers and pointing at numbers guy Jonah Hill in the Moneyball movie.
Talking to Michael Lewis on the 20th anniversary of 'Moneyball'
Analytics departments must focus their energy wisely. The simplicity of the message delivered by Dr Ian Graham , Liverpool’s director of research until 2023 , was notable during StatsBomb’s 2021 conference, declaring that “player recruitment and retention is the most important work — by a factor of 10”.
Buy-in is also crucial. You might have the best statistical models and machine learning algorithms in the world, but aligning and integrating such work with key decision-makers is where the impact of analytics can be maximised at club level.
Brighton ’s owner/chairman Tony Bloom ensures club staff use data provided by his company Starlizard, which has helped turn lesser-known players into Premier League stars, including Kaoru Mitoma , Moises Caicedo , Alexis Mac Allister and Julio Enciso .
Similarly, his Brentford counterpart, Matthew Benham, is the founder of statistical research company Smartodds — primarily designed for professional gamblers but crucial in helping Thomas Frank’s side find value in the player recruitment market.
Premier League: How to find the edge in data analytics - examining trends and what is to come
If a club’s owners or sporting directors are less data-minded, a communication gap can often develop between analysts and the powers that be. Recently, companies such as Soccerment and SentientSports have used Generative AI to help bridge that by condensing complex statistical analysis into simple football language — think ChatGPT for player scouting — but challenges can still exist.
“Best-practice analytics is not creating the most ‘complex’ model or algorithm, it is analyses that are trusted and adopted by decision-makers that ultimately have an impact on their processes,” says Dan Pelchen, founder of analytics company Traits Insights. “Trust and understanding can empower more experts to use data daily, helping avoid biases and mitigating risk.”
There has been a lot of commentary on the growing world of football analytics in recent years, but — aside from a recent research paper published in April — there has rarely been an objective, statistically-led depiction of the data ecosystems across the leagues.
Traits Insights collected information on approximately 500 staff members from more than 90 clubs in the top four divisions of English football — categorised into data analysts (a catch-all statistically-based role), recruitment analysts, first-team analysts (for example, performance/technical/opposition analysis), and overarching heads of analysis — to better understand the challenges facing clubs to build “best-practice” analytics processes, which The Athletic can now exclusively share.
Outlining best practices is one thing, but implementing them is another.
Setting up a coherent, self-sustaining analytics department requires a significant investment from board level, and making a business case for its long-term utility can be challenging.
Traits Insights’ analysis showed the ‘traditional’ top six Premier League clubs ( Manchester City , Arsenal , Liverpool, Manchester United , Tottenham Hotspur and Chelsea ) have approximately 14 analysis-based staff members on average — which is double the average among clubs in the bottom half of the same division.
Unsurprisingly, those numbers dwindle as you descend into the second-tier Championship , League One and League Two , the fourth level of the English game.
For some, limited staff capacity can mean some analysts will often be asked to have a Jack-of-all-trades role — data engineer (collecting and managing large datasets), data analyst (interpreting the information and presenting to colleagues), and data scientist (building statistical models to provide insight) all rolled into one, for example.
“Data analysis is still a relatively new department within clubs,” said one data scientist at a Premier League club, speaking anonymously to protect relationships. “People from different backgrounds are often enthusiastic about introducing data into their workflows, but a club typically begins by dipping their toe with a small investment — for example, one junior data role and one data provider subscription.
“Those who allocate this initial investment typically don’t come from a data background and understandably don’t know the different skill sets required between data analysts, scientists and engineers. When the first junior hire begins work, they can quickly become overrun with demands that cannot be met without the structure in place to produce quick and valuable insights.
“This can quickly lead to frustration on both sides. It is no coincidence that the clubs with the most successful data departments have people at the very top of the club who have come from quantitive backgrounds.”
This is a sentiment shared among other staff members throughout the English football pyramid.
“A good data engineer is crucial for productivity and enabling other roles to succeed. It is a role that is often the hardest to fill and is frequently overlooked because it’s not flashy or particularly visible to day-to-day practitioners,” said a data scientist at a Championship club , also speaking anonymously to protect relationships . “Data scientists are predominantly responsible for model generation and delivery of these insights, and data analysts are the most people-facing — responsible for the development of tools and delivering clear visualisations and presentations.
“Each of these three roles have specific responsibilities and skills that are essential for fulfilling their tasks. Without one, the others would face increasing challenges. If a team member leaves, our skill sets are all deemed as the same when, in reality, they are very different disciplines.”
The desire to use analytics has grown exponentially in recent years but it is important to note that specialised expertise is required to manage and interpret data, build statistical models, and create interfaces (for example, dashboards and visualisations) that allow the analysis to be understood by others at the club.
This requires specific education and technical training to create such advanced models (for example, neural networks and machine learning algorithms) — stemming from backgrounds in hard sciences such as data science, economics, computer science, engineering and mathematics.
Many staff members will have qualifications in sport-and-exercise science, performance analysis or similar — which requires a lot of technical training — but the statistical qualifications among staff are scarce within football. Traits Insights’ analysis found that 46 per cent of data analysts in the sample had a technical statistical education, with approximately five per cent of the remaining analysis staff having such a background.
The limited number of support staff with expertise in data and statistical insight can put strain on specific individuals when internally building technical systems, with the core goal being that all team members can develop such systems and extract insight at all stages along the “production line” of a club’s workflow — from junior data analysts up to senior staff members, including sporting directors.
Approximately 75 per cent of the 20 Premier League clubs have specialised data analysts, with 50 per cent having multiple. By contrast, only half of the Championship’s 24 clubs have a dedicated data analyst, which similarly dwindles when reaching the 24 sides in both League One (25 per cent) and League Two (less than 10 per cent).
However cliched you might think it is, football is a results-based business. Sporting directors will often have a long-term view of the club, but that may not always be as stable when going down the three tiers of the Football League.
For support staff, being afforded the time to build statistical models and generate tangible insight can be easier said than done. At clubs with a higher turnover of coaching staff, these workflows and systems can naturally break down if a new manager or head coach has a different method of operating. If this does occur, it can stifle the progression of an analysis department.
Similarly, analysts working at lower-division teams may also want to work at other clubs and climb up through the leagues, making staff turnover more likely further down the pyramid. This was reflected in Traits Insights’ analysis, which showed analysts at top-six Premier League sides had an average tenure of 4.7 years, compared with 2.5 years or less in League One and League Two.
Broken down by role, a club’s head of analysis is most commonly in their role for the longest period. Notably, data analysts have been in their positions with the team concerned for 2.5 years by comparison, which speaks to the infancy and potential transcience of the job compared with other support staff at a club.
“These results are not surprising when you think how new data is within football, but also how valued these skill sets are within other industries,” said the Premier League data scientist quoted earlier in this article.
“Within clubs, a data analyst role often begins as a junior role, but the skill set required at a club is more in line with a senior role within other industries. If you can meet all of the requirements to work at a club, you will be in huge demand outside of football, so it is understandable why people may move on quicker than other roles.”
When building an analytics department, there is neither a single path to success nor an established method for clubs to develop their infrastructure. Best-practice is difficult to come by without stability, strong technical skills, and investment — and the complexity of such work means the Moneyball method is often idealised beyond reality.
Naturally, clubs with bigger budgets can invest more in their analysis departments but work that influences player recruitment, player retention and talent development is where data analysis can find its best outcomes, and establishing clear lines of communication between departments is crucial.
Whether outsourcing work to third-party consultancies or developing your own data team within the club, ample opportunities remain to gain a competitive advantage — at any level of the game.
(Top photo: Nick Potts/PA Images via Getty Images)
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Mark Carey is a Data Analyst for The Athletic. With his background in research and analytics, he will look to provide data-driven insight across the football world. Follow Mark on Twitter @ MarkCarey93
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Statistical analysis means investigating trends, patterns, and relationships using quantitative data . It is an important research tool used by scientists, governments, businesses, and other organizations.
To draw valid conclusions, statistical analysis requires careful planning from the very start of the research process . You need to specify your hypotheses and make decisions about your research design, sample size, and sampling procedure.
After collecting data from your sample, you can organize and summarize the data using descriptive statistics . Then, you can use inferential statistics to formally test hypotheses and make estimates about the population. Finally, you can interpret and generalize your findings.
This article is a practical introduction to statistical analysis for students and researchers. We’ll walk you through the steps using two research examples. The first investigates a potential cause-and-effect relationship, while the second investigates a potential correlation between variables.
Step 1: write your hypotheses and plan your research design, step 2: collect data from a sample, step 3: summarize your data with descriptive statistics, step 4: test hypotheses or make estimates with inferential statistics, step 5: interpret your results, other interesting articles.
To collect valid data for statistical analysis, you first need to specify your hypotheses and plan out your research design.
The goal of research is often to investigate a relationship between variables within a population . You start with a prediction, and use statistical analysis to test that prediction.
A statistical hypothesis is a formal way of writing a prediction about a population. Every research prediction is rephrased into null and alternative hypotheses that can be tested using sample data.
While the null hypothesis always predicts no effect or no relationship between variables, the alternative hypothesis states your research prediction of an effect or relationship.
A research design is your overall strategy for data collection and analysis. It determines the statistical tests you can use to test your hypothesis later on.
First, decide whether your research will use a descriptive, correlational, or experimental design. Experiments directly influence variables, whereas descriptive and correlational studies only measure variables.
Your research design also concerns whether you’ll compare participants at the group level or individual level, or both.
First, you’ll take baseline test scores from participants. Then, your participants will undergo a 5-minute meditation exercise. Finally, you’ll record participants’ scores from a second math test.
In this experiment, the independent variable is the 5-minute meditation exercise, and the dependent variable is the math test score from before and after the intervention. Example: Correlational research design In a correlational study, you test whether there is a relationship between parental income and GPA in graduating college students. To collect your data, you will ask participants to fill in a survey and self-report their parents’ incomes and their own GPA.
When planning a research design, you should operationalize your variables and decide exactly how you will measure them.
For statistical analysis, it’s important to consider the level of measurement of your variables, which tells you what kind of data they contain:
Many variables can be measured at different levels of precision. For example, age data can be quantitative (8 years old) or categorical (young). If a variable is coded numerically (e.g., level of agreement from 1–5), it doesn’t automatically mean that it’s quantitative instead of categorical.
Identifying the measurement level is important for choosing appropriate statistics and hypothesis tests. For example, you can calculate a mean score with quantitative data, but not with categorical data.
In a research study, along with measures of your variables of interest, you’ll often collect data on relevant participant characteristics.
Variable | Type of data |
---|---|
Age | Quantitative (ratio) |
Gender | Categorical (nominal) |
Race or ethnicity | Categorical (nominal) |
Baseline test scores | Quantitative (interval) |
Final test scores | Quantitative (interval) |
Parental income | Quantitative (ratio) |
---|---|
GPA | Quantitative (interval) |
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In most cases, it’s too difficult or expensive to collect data from every member of the population you’re interested in studying. Instead, you’ll collect data from a sample.
Statistical analysis allows you to apply your findings beyond your own sample as long as you use appropriate sampling procedures . You should aim for a sample that is representative of the population.
There are two main approaches to selecting a sample.
In theory, for highly generalizable findings, you should use a probability sampling method. Random selection reduces several types of research bias , like sampling bias , and ensures that data from your sample is actually typical of the population. Parametric tests can be used to make strong statistical inferences when data are collected using probability sampling.
But in practice, it’s rarely possible to gather the ideal sample. While non-probability samples are more likely to at risk for biases like self-selection bias , they are much easier to recruit and collect data from. Non-parametric tests are more appropriate for non-probability samples, but they result in weaker inferences about the population.
If you want to use parametric tests for non-probability samples, you have to make the case that:
Keep in mind that external validity means that you can only generalize your conclusions to others who share the characteristics of your sample. For instance, results from Western, Educated, Industrialized, Rich and Democratic samples (e.g., college students in the US) aren’t automatically applicable to all non-WEIRD populations.
If you apply parametric tests to data from non-probability samples, be sure to elaborate on the limitations of how far your results can be generalized in your discussion section .
Based on the resources available for your research, decide on how you’ll recruit participants.
Your participants are self-selected by their schools. Although you’re using a non-probability sample, you aim for a diverse and representative sample. Example: Sampling (correlational study) Your main population of interest is male college students in the US. Using social media advertising, you recruit senior-year male college students from a smaller subpopulation: seven universities in the Boston area.
Before recruiting participants, decide on your sample size either by looking at other studies in your field or using statistics. A sample that’s too small may be unrepresentative of the sample, while a sample that’s too large will be more costly than necessary.
There are many sample size calculators online. Different formulas are used depending on whether you have subgroups or how rigorous your study should be (e.g., in clinical research). As a rule of thumb, a minimum of 30 units or more per subgroup is necessary.
To use these calculators, you have to understand and input these key components:
Once you’ve collected all of your data, you can inspect them and calculate descriptive statistics that summarize them.
There are various ways to inspect your data, including the following:
By visualizing your data in tables and graphs, you can assess whether your data follow a skewed or normal distribution and whether there are any outliers or missing data.
A normal distribution means that your data are symmetrically distributed around a center where most values lie, with the values tapering off at the tail ends.
In contrast, a skewed distribution is asymmetric and has more values on one end than the other. The shape of the distribution is important to keep in mind because only some descriptive statistics should be used with skewed distributions.
Extreme outliers can also produce misleading statistics, so you may need a systematic approach to dealing with these values.
Measures of central tendency describe where most of the values in a data set lie. Three main measures of central tendency are often reported:
However, depending on the shape of the distribution and level of measurement, only one or two of these measures may be appropriate. For example, many demographic characteristics can only be described using the mode or proportions, while a variable like reaction time may not have a mode at all.
Measures of variability tell you how spread out the values in a data set are. Four main measures of variability are often reported:
Once again, the shape of the distribution and level of measurement should guide your choice of variability statistics. The interquartile range is the best measure for skewed distributions, while standard deviation and variance provide the best information for normal distributions.
Using your table, you should check whether the units of the descriptive statistics are comparable for pretest and posttest scores. For example, are the variance levels similar across the groups? Are there any extreme values? If there are, you may need to identify and remove extreme outliers in your data set or transform your data before performing a statistical test.
Pretest scores | Posttest scores | |
---|---|---|
Mean | 68.44 | 75.25 |
Standard deviation | 9.43 | 9.88 |
Variance | 88.96 | 97.96 |
Range | 36.25 | 45.12 |
30 |
From this table, we can see that the mean score increased after the meditation exercise, and the variances of the two scores are comparable. Next, we can perform a statistical test to find out if this improvement in test scores is statistically significant in the population. Example: Descriptive statistics (correlational study) After collecting data from 653 students, you tabulate descriptive statistics for annual parental income and GPA.
It’s important to check whether you have a broad range of data points. If you don’t, your data may be skewed towards some groups more than others (e.g., high academic achievers), and only limited inferences can be made about a relationship.
Parental income (USD) | GPA | |
---|---|---|
Mean | 62,100 | 3.12 |
Standard deviation | 15,000 | 0.45 |
Variance | 225,000,000 | 0.16 |
Range | 8,000–378,000 | 2.64–4.00 |
653 |
A number that describes a sample is called a statistic , while a number describing a population is called a parameter . Using inferential statistics , you can make conclusions about population parameters based on sample statistics.
Researchers often use two main methods (simultaneously) to make inferences in statistics.
You can make two types of estimates of population parameters from sample statistics:
If your aim is to infer and report population characteristics from sample data, it’s best to use both point and interval estimates in your paper.
You can consider a sample statistic a point estimate for the population parameter when you have a representative sample (e.g., in a wide public opinion poll, the proportion of a sample that supports the current government is taken as the population proportion of government supporters).
There’s always error involved in estimation, so you should also provide a confidence interval as an interval estimate to show the variability around a point estimate.
A confidence interval uses the standard error and the z score from the standard normal distribution to convey where you’d generally expect to find the population parameter most of the time.
Using data from a sample, you can test hypotheses about relationships between variables in the population. Hypothesis testing starts with the assumption that the null hypothesis is true in the population, and you use statistical tests to assess whether the null hypothesis can be rejected or not.
Statistical tests determine where your sample data would lie on an expected distribution of sample data if the null hypothesis were true. These tests give two main outputs:
Statistical tests come in three main varieties:
Your choice of statistical test depends on your research questions, research design, sampling method, and data characteristics.
Parametric tests make powerful inferences about the population based on sample data. But to use them, some assumptions must be met, and only some types of variables can be used. If your data violate these assumptions, you can perform appropriate data transformations or use alternative non-parametric tests instead.
A regression models the extent to which changes in a predictor variable results in changes in outcome variable(s).
Comparison tests usually compare the means of groups. These may be the means of different groups within a sample (e.g., a treatment and control group), the means of one sample group taken at different times (e.g., pretest and posttest scores), or a sample mean and a population mean.
The z and t tests have subtypes based on the number and types of samples and the hypotheses:
The only parametric correlation test is Pearson’s r . The correlation coefficient ( r ) tells you the strength of a linear relationship between two quantitative variables.
However, to test whether the correlation in the sample is strong enough to be important in the population, you also need to perform a significance test of the correlation coefficient, usually a t test, to obtain a p value. This test uses your sample size to calculate how much the correlation coefficient differs from zero in the population.
You use a dependent-samples, one-tailed t test to assess whether the meditation exercise significantly improved math test scores. The test gives you:
Although Pearson’s r is a test statistic, it doesn’t tell you anything about how significant the correlation is in the population. You also need to test whether this sample correlation coefficient is large enough to demonstrate a correlation in the population.
A t test can also determine how significantly a correlation coefficient differs from zero based on sample size. Since you expect a positive correlation between parental income and GPA, you use a one-sample, one-tailed t test. The t test gives you:
The final step of statistical analysis is interpreting your results.
In hypothesis testing, statistical significance is the main criterion for forming conclusions. You compare your p value to a set significance level (usually 0.05) to decide whether your results are statistically significant or non-significant.
Statistically significant results are considered unlikely to have arisen solely due to chance. There is only a very low chance of such a result occurring if the null hypothesis is true in the population.
This means that you believe the meditation intervention, rather than random factors, directly caused the increase in test scores. Example: Interpret your results (correlational study) You compare your p value of 0.001 to your significance threshold of 0.05. With a p value under this threshold, you can reject the null hypothesis. This indicates a statistically significant correlation between parental income and GPA in male college students.
Note that correlation doesn’t always mean causation, because there are often many underlying factors contributing to a complex variable like GPA. Even if one variable is related to another, this may be because of a third variable influencing both of them, or indirect links between the two variables.
A statistically significant result doesn’t necessarily mean that there are important real life applications or clinical outcomes for a finding.
In contrast, the effect size indicates the practical significance of your results. It’s important to report effect sizes along with your inferential statistics for a complete picture of your results. You should also report interval estimates of effect sizes if you’re writing an APA style paper .
With a Cohen’s d of 0.72, there’s medium to high practical significance to your finding that the meditation exercise improved test scores. Example: Effect size (correlational study) To determine the effect size of the correlation coefficient, you compare your Pearson’s r value to Cohen’s effect size criteria.
Type I and Type II errors are mistakes made in research conclusions. A Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s false.
You can aim to minimize the risk of these errors by selecting an optimal significance level and ensuring high power . However, there’s a trade-off between the two errors, so a fine balance is necessary.
Traditionally, frequentist statistics emphasizes null hypothesis significance testing and always starts with the assumption of a true null hypothesis.
However, Bayesian statistics has grown in popularity as an alternative approach in the last few decades. In this approach, you use previous research to continually update your hypotheses based on your expectations and observations.
Bayes factor compares the relative strength of evidence for the null versus the alternative hypothesis rather than making a conclusion about rejecting the null hypothesis or not.
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
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Last updated on Fri Aug 23 2024
Imagine spending months or even years developing a new feature only to find out it doesn’t resonate with your users, argh! This kind of situation could be any worst Product manager’s nightmare.
There's a way to fix this problem called the Value Hypothesis . This idea helps builders to validate whether the ideas they’re working on are worth pursuing and useful to the people they want to sell to.
This guide will teach you what you need to know about Value Hypothesis and a step-by-step process on how to create a strong one. At the end of this post, you’ll learn how to create a product that satisfies your users.
Are you ready? Let’s get to it!
Scrutinizing this hypothesis helps you as a developer to come up with a product that your customers like and love to use.
Product managers use the Value Hypothesis as a north star, ensuring focus on client needs and avoiding wasted resources. For more on this, read about the product management process .
Let's get into the step-by-step process, but first, we need to understand the basics of the Value Hypothesis:
A Value Hypothesis is like a smart guess you can test to see if your product truly solves a problem for your customers. It’s your way of predicting how well your product will address a particular issue for the people you’re trying to help.
You need to know what a Value Hypothesis is, what it covers, and its key parts before you use it. To learn more about finding out what customers need, take a look at our guide on discovering features .
The Value Hypothesis does more than just help with the initial launch, it guides the whole development process. This keeps teams focused on what their users care about helping them choose features that their audience will like.
A strong Value Hypothesis rests on three key components:
Value Proposition: The Value Proposition spells out the main advantage your product gives to customers. It explains the "what" and "why" of your product showing how it eases a particular pain point.
This proposition targets a specific group of consumers. To learn more, check out our guide on roadmapping .
Customer Segmentation: Knowing and grasping your target audience is essential. This involves studying their demographics, needs, behaviors, and problems. By dividing your market, you can shape your value proposition to address the unique needs of each group.
Customer feedback surveys can prove priceless in this process. Find out more about this in our customer feedback surveys guide.
Problem Statement : The Problem Statement defines the exact issue your product aims to fix. It should zero in on a real fixable pain point your target users face. For hands-on applications, see our product launch communication plan .
Here are some key questions to guide you:
What are the primary challenges and obstacles faced by your target users?
What existing solutions are available, and where do they fall short?
What unmet needs or desires does your target audience have?
For a structured approach to prioritizing features based on customer needs, consider using a feature prioritization matrix .
Now that we've covered the basics, let's look at how to build a convincing Value Hypothesis. Here's a two-step method, along with value hypothesis templates, to point you in the right direction:
To start with, you need to carry out market research. By carrying out proper market research, you will have an understanding of existing solutions and identify areas in which customers' needs are yet to be met. This is integral to effective idea tracking .
Next, use customer interviews, surveys, and support data to understand your target audience's problems and what they want. Check out our list of tools for getting customer feedback to help with this.
Once you've completed your research, it's crucial to identify your customers' needs. By merging insights from market research with direct user feedback, you can pinpoint the key requirements of your customers.
Here are some key questions to think about:
What are the most significant challenges that your target users encounter daily?
Which current solutions are available to them, and how do these solutions fail to fully address their needs?
What specific pain points are your target users struggling with that aren't being resolved?
Are there any gaps or shortcomings in the existing products or services that your customers use?
What unfulfilled needs or desires does your target audience express that aren't currently met by the market?
To prioritize features based on customer needs in a structured way, think about using a feature prioritization matrix .
Once you've created your Value Hypothesis with a template, you need to check if it holds up. Here's how you can do this:
Build a minimum viable product (MVP)—a basic version of your product with essential functions. This lets you test your value proposition with actual users and get feedback without spending too much. To achieve the best outcomes, look into the best practices for customer feedback software .
Build mock-ups to show your product idea. Use these mock-ups to get user input on the user experience and overall value offer.
After you've gathered data about your hypothesis, it's time to examine it. Here are some metrics you can use:
User Engagement : Monitor stats like time on the platform, feature use, and return visits to see how much users interact with your MVP or mock-up.
Conversion Rates : Check conversion rates for key actions like sign-ups, buys, or feature adoption. These numbers help you judge if your value offer clicks with users. To learn more, read our article on SaaS growth benchmarks .
The Value Hypothesis framework shines because you can keep making it better. Here's how to fine-tune your hypothesis:
Set up an ongoing system to gather user data as you develop your product.
Look at what users say to spot areas that need work then update your value proposition based on what you learn.
Read about managing product updates to keep your hypotheses current.
The market keeps changing, and your Value Hypothesis should too. Stay up to date on what's happening in your industry and watch how users' habits change. Tweak your value proposition to stay useful and ahead of the competition.
Here are some ways to keep your Value Hypothesis fresh:
Do market research often to keep up with what's happening in your industry and what your competitors are up to.
Keep an eye on what users are saying to spot new problems or things they need but don't have yet.
Try out different value statements and features to see which ones your audience likes best.
To keep your guesses up-to-date, check out our guide on handling product changes .
While the Value Hypothesis approach is powerful, it's key to steer clear of these common traps:
Avoid Confirmation Bias : People tend to focus on data that backs up their initial guesses. But it's key to look at feedback that goes against your ideas and stay open to different views.
Watch out for Shiny Object Syndrome : Don't let the newest fads sway you unless they solve a main customer problem. Your value proposition should fix actual issues for your users.
Don't Cling to Your First Hypothesis : As the market changes, your value proposition should too. Be ready to shift your hypothesis when new evidence and user feedback comes in.
Don't Mix Up Busywork with Real Progress : Getting user feedback is key, but making sense of it brings real value. Look at the data to find useful insights that can shape your product. To learn more about this, check out our guide on handling customer feedback .
To build a product that succeeds, you need to know your target users inside out and understand how you help them. The Value Hypothesis framework gives you a step-by-step way to do this.
If you follow the steps in this guide, you can create a strong value proposition, check if it works, and keep improving it to ensure your product stays useful and important to your customers.
Keep in mind, a good Value Hypothesis changes as your product and market change. When you use data and put customers first, you're on the right track to create a product that works.
Want to put the Value Hypothesis framework into action? Check out our top templates for creating product roadmaps to streamline your process. Think about using featureOS to manage customer feedback. This tool makes it easier to collect, examine, and put user feedback to work.
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Berlin, September 1, 2024 – The FINE-HEART prespecified pooled analysis of the three completed pivotal Phase III clinical trials with finerenone (namely FINEARTS-HF, FIDELIO-DKD, and FIGARO-DKD), showed that the incidence for the primary endpoint of cardiovascular (CV) death was numerically lower in patients treated with finerenone versus placebo, but narrowly missed statistical significance (11% relative risk reduction, HR 0.89 [95% CI, 0.78-1.01; p=0.076]). Importantly, in a prespecified sensitivity analysis for the primary endpoint in FINE-HEART that included both cardiovascular deaths and undetermined deaths, finerenone significantly reduced the risk to develop these events by 12% (relative risk reduction, HR 0.88 [95% CI, 0.79-0.98; p=0.025]). The effects of finerenone on CV death were generally consistent across the 16 subgroups examined in FINE-HEART. Results also indicate significant reductions of finerenone versus placebo of all-cause death as well as CV and kidney outcomes. The overall findings of FINE-HEART suggest cardio-kidney benefits of finerenone across a broad range of high-risk patient populations encompassing cardiovascular, kidney, and metabolic conditions. The FINE-HEART findings were presented today during a Hot Line session at ESC Congress 2024, and simultaneously published in Nature Medicine .
“Given the strong epidemiological overlap and shared mechanistic pathways of cardio-kidney-metabolic conditions, these data are welcome news for clinicians. It is great to see that finerenone addresses fundamental drivers of heart and kidney pathophysiology,” said Muthiah Vaduganathan, MD, MPHD, cardiologist and co-director of the Center for Cardiometabolic Implementation Science at Brigham and Women’s Hospital and faculty at Harvard Medical School. “While the individual Phase III studies with finerenone were not powered to evaluate CV mortality or efficacy in key subgroups, the high number of patients in FINE-HEART allowed us to explore these outcomes, and provided important, encouraging insights for clinicians for the treatment of these multimorbid patients, confirming efficacy is consistent across key subgroups.”
While the primary endpoint CV death did not reach statistical significance, the results of the secondary endpoints in FINE-HEART all suggest benefits of finerenone versus placebo. Most notably, finerenone reduced all-cause mortality by 9% (HR 0.91 [95% CI, 0.84-0.99; p=0.027]); the composite kidney endpoint of time to first onset of kidney failure, sustained ≥50% decrease in eGFR from baseline over ≥4 weeks, or renal death was reduced by 20% with finerenone (HR 0.80 [95% CI, 0.72-0.90; p<0.001]), and the incidence of HF hospitalizations was lowered by 17% (HR 0.83 [95% CI, 0.75-0.92; p<0.001]).
FINE-HEART is the largest analysis of efficacy and safety of finerenone in patients across a broad range of cardio-kidney-metabolic (CKM) conditions. The pooled analysis included around 19,000 patients with heart failure (HF) and/or chronic kidney disease (CKD) and type 2 diabetes (T2D) from the Phase III studies FINEARTS-HF, FIDELIO-DKD, and FIGARO-DKD. The FINE-HEART analysis was designed to explore the effects of finerenone (Kerendia™/Firialta™) on cardiovascular and kidney outcomes in patients with HF and/or CKD and T2D, including patients with a high burden of comorbid conditions – a key characteristic of patients with HF and a left ventricular ejection fraction (LVEF) of ≥40%.
“Heart failure, chronic kidney disease, and type 2 diabetes have shared disease drivers, and FINE-HEART, including around 19,000 patients from three Phase studies, complements and confirms the positive results seen so far with finerenone,” said Dr. Christian Rommel, Head of Research and Development at Bayer’s Pharmaceuticals Division. “These findings are highly relevant for clinicians as they demonstrate that finerenone can improve outcomes in these patients with a high unmet medical need."
Finerenone is a non-steroidal, selective mineralocorticoid receptor (MR) antagonist. By targeting MR / renin-angiotensin-aldosterone system (RAAS) overactivation, finerenone addresses chronic and progressive inflammatory and fibrotic drivers, known to be strongly associated with HF and CKD.
Finerenone was well-tolerated in the FINE-HEART pooled analysis, which is consistent with the well-established safety profile of finerenone.
About FINE-HEART Since there is a strong epidemiological overlap and shared mechanistic drivers of cardio-kidney-metabolic conditions, the prespecified pooled analysis FINE-HEART was designed to explore the effects of finerenone (Kerendia™ / Firialta™) on cardio-kidney outcomes in patients with heart failure and/or chronic kidney disease and type 2 diabetes, including patients with a high burden of a broad range of cardio-kidney-metabolic conditions. FINE-HEART had increased statistical power to assess CV and all-cause death, alongside other CV and kidney outcomes. Given the unmet need in these patients, the FINE-HEART analysis studied the effect of finerenone use in patients with a high burden of multimorbidity across three completed Phase III studies.
FINE-HEART is a protocol prespecified, participant-level pooled analysis which includes around 19,000 patients with heart failure (HF) and/or chronic kidney disease (CKD) and type 2 diabetes (T2D) from three Phase III studies, namely FINEARTS-HF, FIDELIO-DKD, and FIGARO-DKD. FIDELIO-DKD and FIGARO-DKD trials together randomized around 13,000 patients with CKD and T2D with albuminuria (UACR≥30mg/g) across 48 countries. FINEARTS-HF included around 6,000 patients with symptomatic HF and a LVEF of ≥40%, elevated natriuretic peptides, and evidence of structural heart disease across 37 countries.
Baseline characteristics of FINE-HEART show that the prevalence of cardio-kidney-metabolic (CKM) comorbidities was high with a history of heart failure in 37%, T2D in 81%, and CKD in 84% of patients. Among recruited patients, 10% had one condition (HF), 78% had two conditions (HF and CKD, HF and T2D or CKD and T2D), while 12% presented with all three.
Over a median follow-up in the pooled patient population of 2.9 years, the incidence of CV death was numerically lower in patients treated with finerenone, with a 11% relative risk reduction versus placebo, which narrowly missed statistical significance (HR 0.89 [95% CI, 0.78-1.01; p=0.076]). A prespecified sensitivity analysis for the primary endpoint in FINE-HEART included both cardiovascular deaths and undetermined deaths; here, the relative risk reduction with finerenone was 12% (HR 0.88 [95% CI, 0.79-0.98; p=0.025]). The effects of finerenone on CV death were generally consistent across all 16 subgroups examined in FINE-HEART.
As shown in the secondary endpoints in FINE-HEART, finerenone showed significant reductions for deaths from any cause, CV and kidney events. Secondary endpoints included a kidney composite endpoint including a ≥50% sustained decline in eGFR, heart failure (HF) hospitalization, the composite of CV death or HF hospitalization, new-onset atrial fibrillation, major adverse CV events, all-cause death, all-cause hospitalization, and the composite of all-cause death or all-cause hospitalization. All-cause mortality was significantly reduced with finerenone versus placebo (HR 0.91 [95% CI, 0.84-0.99; p=0.027]); finerenone reduced the risk of the composite kidney endpoint (HR 0.80 [95% CI, 0.72-0.90; p<0.001]), as well as HF hospitalizations (HR 0.83 [95% CI, 0.75-0.92; p<0.001]), the composite of cardiovascular death or HF hospitalization (HR 0.85 [95% CI, 0.78-0.93; p<0.001]), new-onset atrial fibrillation (HR 0.83 [95% CI, 0.71-0.97; p=0.018]), major adverse cardiovascular events (HR 0.95 [95% CI, 0.85-0.98; p=0.010]), hospitalizations of any cause (HR 0.95 [95% CI, 0.91-0.99; p=0.025]), and the composite of all-cause death or all-cause hospitalization (HR 0.94 [95% CI, 0.91-0.98; p=0.007]).
About Kerendia ™ / Firialta ™ (finerenone) Kerendia™ and Firialta™ are globally protected trademarks for finerenone. Finerenone is a non-steroidal, selective mineralocorticoid receptor (MR) antagonist that has been shown to block harmful effects of MR overactivation. MR overactivation contributes to chronic kidney disease (CKD) progression and cardiovascular damage which can be driven by metabolic, hemodynamic, or inflammatory and fibrotic factors.
Finerenone is marketed as Kerendia™ or, in some countries, as Firialta™, and approved for the treatment of adult patients with CKD associated with type 2 diabetes (T2D) in more than 90 countries worldwide, including in China, Europe, Japan, and the U.S.
The study program with finerenone, FINEOVATE, currently comprises ten Phase III studies with dedicated programs in HF and CKD respectively. The MOONRAKER program includes FINEARTS-HF, as well as the ongoing collaborative, investigator-sponsored studies REDEFINE-HF, CONFIRMATION-HF, and FINALITY-HF. The THUNDERBALL CKD program consists of the completed studies FIDELIO-DKD and FIGARO-DKD, as well as the ongoing studies FIND-CKD, FIONA, FIONA-OLE, FINE-ONE, and the Phase II study CONFIDENCE.
About Bayer’s Commitment in Cardiovascular and Kidney Diseases Bayer is an innovation leader in the area of cardiovascular diseases, with a long-standing commitment to delivering science for a better life by advancing a portfolio of innovative treatments. The heart and the kidneys are closely linked in health and disease, and Bayer is working in a wide range of therapeutic areas on new treatment approaches for cardiovascular and kidney diseases with high unmet medical needs. The cardiology franchise at Bayer already includes a number of products and several other compounds in various stages of preclinical and clinical development. Together, these products reflect the company’s approach to research, which prioritizes targets and pathways with the potential to impact the way that cardiovascular diseases are treated.
About Bayer Bayer is a global enterprise with core competencies in the life science fields of health care and nutrition. In line with its mission, “Health for all, Hunger for none,” the company’s products and services are designed to help people and the planet thrive by supporting efforts to master the major challenges presented by a growing and aging global population. Bayer is committed to driving sustainable development and generating a positive impact with its businesses. At the same time, the Group aims to increase its earning power and create value through innovation and growth. The Bayer brand stands for trust, reliability and quality throughout the world. In fiscal 2023, the Group employed around 100,000 people and had sales of 47.6 billion euros. R&D expenses before special items amounted to 5.8 billion euros. For more information, go to www.bayer.com .
Forward-Looking Statements This release may contain forward-looking statements based on current assumptions and forecasts made by Bayer management. Various known and unknown risks, uncertainties and other factors could lead to material differences between the actual future results, financial situation, development or performance of the company and the estimates given here. These factors include those discussed in Bayer’s public reports which are available on the Bayer website at www.bayer.com . The company assumes no liability whatsoever to update these forward-looking statements or to conform them to future events or developments.
In FINE-HEART, the incidence for the primary endpoint of cardiovascular (CV) death was numerically lower in patients treated with finerenone versus placebo, but narrowly missed statistical significance / FINE-HEART is a prespecified pooled analysis of all completed finerenone Phase III studies in around 19,000 high-risk patients across a broad range of cardio-kidney-metabolic (CKM) conditions / FINE-HEART results indicate significant reductions with finerenone versus placebo for all-cause death, CV and kidney outcomes / Results from FINE-HEART were simultaneously published in Nature Medicine
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Climate change is an ethical and moral challenge of a global scale due to its potentially catastrophic implications for human welfare. Understanding forces that drive corporate adaptation to climate change is an important research topic in business ethics. In this paper, we propose that shareholder climate-related proposals could be a catalyst for corporate innovations in technologies mitigating climate change. Our results, based on the analysis of US firms, indicate that corporations respond positively to these proposals by producing more climate-related patents and citations. We also uncover potential casual channels of influence. Further, we find that corporate governance moderates the documented effects. These proposals lead to a more efficient and valuable innovation output, but lower firm performance in the short term. The real effect that shareholder proposals have on innovation gains clarity in the context of climate change, contributing to the discussion of investor “voice.”
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Xiao and Shailer ( 2022 ) provide a novel systematic investigation of factors influencing stakeholders’ perceptions of the credibility of corporate sustainability reports.
What are shareholder proposals, and what makes them interesting? Established in 1942 (and amended several times), Rule 14a-8 was designed to give small shareholders a voice and managers ample opportunity to listen before being heard at annual meetings. The Rule now permits a shareholder to make a proposal of 500 words or less, if any of the following ownership amount and time requirements are met: 1) at least $2,000 in market value for at least three years; 2) or at least $15,000 for at least two years; 3) or at least $25,000 for at least one year. The proposal must be received at the company’s principal executive offices not less than 120 calendar days before the release of company's annual proxy statement, with shareholder intent to maintain the requisite interest through the annual meeting. For more formation, please see the Code of Federal Regulations, (Title 17, Volume 3, Sect. 240.14a-8, www.govinfo.gov ).
Theoretical perspectives on management’s response to stakeholder demands are influenced by corporate purpose.
Literature presents opposing views: Friedman’s ( 1970 ) profit-focused shareholder priority versus Stout’s ( 2013 )
inclusive stakeholder approach considering broader goals. See discussion on the subject in Clarke ( 2020 ).
The climate-related proposals to Chevron reflect this shift in emphasis toward a direct assessment of financial risk, from one of simple emission disclosure. From 1999 to 2009, requests for a “Report on Greenhouse Gas Emissions” were recurrent. Beginning in 2010, Chevron saw “Stockholder Proposals Regarding Financial Risks from Climate Change.”
Two examples from the 2016 proxy season highlight shareholder demands for innovation. Shareholders of Ameren Corp proposed “ITEM (4): SHAREHOLDER PROPOSAL RELATING TO A REPORT ON AGGRESSIVE RENEWABLE ENERGY ADOPTION.” Shareholders in AES Corp sponsored “PROPOSAL 4: A REPORT ON COMPANY POLICIES AND TECHNOLOGICAL ADVANCES” targeting the firm’s energy policies and emphasis on renewable sources.
In 2010, St. Joseph of the Capuchin Order requested a study “on how ExxonMobil, within a reasonable timeframe, can become the recognized industry leader in developing and making available the necessary technology (such as enhanced sequestration, engineered geothermal and the development of other renewable energy sources) to enable the U.S.A. to become energy independent in an environmentally sustainable way. By 2017, The New York State Common Retirement Fund sponsored the climate proposal that gained substantial press coverage, which essentially made a similar request: “…an annual assessment of the long-term portfolio impacts of technological advances and global climate change policies…” Further, the Board for Fluor Corporation has stated its opposition to repeated proposals from 2016 to 2018 requesting GHG reduction goals, by “Creating Technology to Reduce Greenhouse Gas Emissions,” more specifically, by investing in NuScale Power, LLC along with Rolls-Royce.
We emphasize that climate-friendly boards and heightened managerial perceptions of climate risk are potential mechanisms. We argue that shareholder proposals positively influence these factors. However, we acknowledge without direct demonstration that these mechanisms, in turn, enhance innovations, considering them as established facts based on prior research (Homroy and Slechten, 2019 ; Sautner et al., 2023 ).
We considered using alternate terms such as “greenhouse gases” or “carbon emissions,” but due to the content of the DEF14A filing, it is not possible to ensure that a term appears directly within a shareholder proposal or management’s response to one without visual inspection, thus hand-collection. Often, the proposals are only a small portion of the DEF14A which often presents year-end results at the annual meeting. Further, word lists invariably subject samples to gaming. “Climate Change” has fairly unambiguous meaning to management and is the phrase used by both the SEC and USPTO.
We also consider that firm innovation may not have a perfect memory of a pressure over the past 25 years of all proposals related to climate change. For robustness, we construct the same three-year, backward average but for only the last three years as well as the last five years. The results that follow remain unchanged. We also use lagged proposals as a proxy for shareholder pressure on climate-related issues for additional robustness, and our main findings are qualitatively similar. These results are not reported for brevity but are presented in online Appendix 1 .
In fact, of the 1.9 million patents we examine from 1994 to 2019, only 8 begin with the Y02 classification, even though 105,737 patents contain the Y02 classification in the CPC coding scheme. For example, patent 5,426,677 appears to be primarily concerned with Physics, the G classification, (G21C1/09; G21C17/00; G21Y2002/202; G21Y2002/204; G21Y2004/304; Y02E30/40), but also has a Climate Mitigation (Y02) component. Disentangling truncation bias by year-technology for the Y02 classification is not feasible for this paper. Further, from our discussions with the USPTO, the first classification tends to be more dominant than the last.
In unreported results, we also construct dependent variables looking forward five years to allow more time for the stockholder pressure to influence innovative behavior.
As Wooldridge ( 2012 ) explains, “sometimes log(1 + y) is used, but interpretation of the coefficients is difficult.” (p. 216) However, this practice is commonplace in corporate finance settings. For robustness, the inverse hyperbolic sine (IHS), as suggested by Burbidge et al. ( 1988 ) and proposed by Johnson ( 1949 ), for zero-value observations is used to log transform both the logged dependent variables and the independent variable of interest, Pressure . The IHS transformation is sinh-1(x) = log(x + (× 2 + 1)1/2). The results using IHS for OLS regressions suggest that the coefficients tend to overstate the economic impact of models (3) and (6) of Table 2 as well as models with Y02 Counts pct and Y02 Cites pct as dependent variables, while understate the coefficients of models with Y02 Top 1 pct and Y02 Top 10 pct as dependent variables (Appendix B ), but the statistical inference remains unchanged in sign or significance.
The Pope’s sentiment also intuitively satisfies the exclusion restriction as it is unlikely to directly influence corporate innovations. To gain some reassurance on the (notorious) exclusion restriction, we divide the sample along the lines of Religious Social Capital considered by Rupasingha et al. ( 2006 ) and obtained from the U.S. Census Bureau’s number of establishments in religious organizations (NAICS 813110), also examined by Grennan ( 2022 ) along with other donor-advised funds. In splitting the sample between More and Less Religious at the county level, we find that firms headquartered in less religious counties have a more acute influence on climate innovations when the Pope serves as an instrument. We would expect the Pope to have a stronger influence in more religious counties, if the Pope were directly influencing management to develop climate technologies and bypassing proposals made by shareholders who are not concentrated near headquarters. Since we find the opposite, we feel better about the exclusion restriction, instead of relying only on our (notorious) intuitions for justification.
We implement causal mediation analysis using the ivmediate command in Stata (e.g., Dippel, Ferrara, and Heblich, 2020 ), allowing us to estimate the treatment effect and determine the proportion attributable to a mediator. The primary advantage, as noted, is that despite both the treatment and mediator being endogenous, a single instrument can accurately detect both causal treatment and mediation effects. However, the method does not produce the first-stage result of the IV regression. Instead, it reports the F-test of excluded instruments directly from the first stage to assess instrument strength, which suffices to establish validity. In our models, detailed in Table 4 , the F-tests from the first stage across all models greatly exceed the conventional cutoff value of 10, ensuring the validity of the instrument. Nevertheless, we manually performed IV regressions and confirmed that our instrument, PopeUS, significantly and positively affects both Pressure and mediators.
In the results, not tabulated for brevity, we re-estimate the same model as in Panel A but with firm fixed effects. We find significant causal mediation effects of Pressure on Y02 Counts that pass through Ind Dir Exp. In parallel to Panel B, we re-estimated the same model with firm fixed effects using CC Bigrams as a mediator and found nearly full mediation. Additionally, we detected marginal mediation in the model with Y02 Cites as a dependent variable using CC Bigrams as a mediator, but not Ind Dir Exp. Thus, the results of firm fixed effects analysis are more suggestive in this case.
We also perform robustness checks of our mediation analysis using alternative measures of shareholder proposals (three-year backward averages for the last three and five years, and lagged proposals). We find statistically significant mediation in all cases, with the mediated effect ranging from 0.54 to 0.91 of the total effect. We also limit the sample to firms that have ever received a proposal related to climate change during our sample period and find the proportion of the total effect mediated varies from 0.62 to 0.74 of the total effect. Finally, using the percentage of votes at the annual meetings in favor of a climate-related proposal collected by ISS (ISS Vote For), the mediated effect ranges from 0.83 to 0.90 of the total effect. We estimate these models using industry fixed effects, with industries identified using 3-digit SIC codes. Overall, our results are in line with our main findings.
To ensure our results are not due to selection of matching estimator, we also employ entropy balancing, nearest neighbor, propensity score, and the CEM (Blackwell et al., 2009 ) and find our results to be robust. The main advantage of EBCT, of course, is that it allows us to match on our continuous treatment variable ( Pressure ), instead of a binary one required for the other estimators.
We note that, following the approach of Faleye et al., ( 2014 ), we also examined the short-term performance implications of the change in patent counts attributable to shareholder climate-related proposals. That is, we regress our performance metrics on predicted patent counts as well as patent cites, where the predicted values are from the regression of innovation variables in our shareholder proposal measures. Our findings remain consistent.
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Greg Tindall
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Rebel A. Cole
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Appendix a: description of variables and sources.
Variables | Description | Source |
---|---|---|
Innovation | ||
Y02 counts | The average, from t + 1 to t + 3, of the natural log of one plus the number of patents with the Y02 classification for each firm by the date the patent is filed, adjusted for truncation bias |
|
Y02 cites | The average, from t + 1 to t + 3, of the natural log of one plus the number of patent citation with the Y02 classification for each firm by the date the patent is filed, adjusted for truncation bias |
|
Climate-related proposals | ||
Pressure | The average, from t to t-2, of the natural log of one plus running total of the number of climate-related proposals that a firm receives over entire sample period: (1) by allowing the running total to equal zero in years where no climate proposals appear at an annual meeting and (2) by resuming the running total when proposals resurface at subsequent annual meetings | SEC’s Edgar website and SeekEdgar cloud technology |
Controls | ||
Size | The average, from t to t-2, of the natural log of one plus total revenues | Compustat |
R&D/assets | The average, from t to t-2, of Research and development expense divided by beginning assets | Compustat |
Tobin’s Q | The average, from t to t-2, of Tobin’s Q, calculated as the Market Value of Equity minus the Book Value of Equity plus Book Value of Assets divided by Book Value of Assets | Perfect & Wiles, ; Baker, Wurgler and Stein, 2003 |
Firm Age | The average, from t to t-2, of the natural log of one plus the number of years that a firm is listed in Compustat | Compustat |
Revenue growth | The average, from t to t-2, of the change in revenues from the end of each year | Compustat |
Stock return | The average, from t to t-2, of the annual change in the adjusted stock price | Compustat |
Leverage | The average, from t to t-2, of total Liabilities divided by total Assets | Compustat |
Cash surplus | The average, from t to t-2, of Cash Surplus, calculated as the net cash from operations minus depreciation plus research and development scaled by total assets | Compustat |
This table shows the results of ordinary least square regressions with Innovation as the dependent variable based on the patent data by date filed with the US Patent Office containing the Y02 (climate change). In Columns (1)–(4), dependent variables are Y02 Count Pct —the percent of a firm’s Y02 patents in a given year relative to all of that firm’s patents filed in the same year, Y02 Cite Pct —the percent of a firm’s Y02 patent citations in a given year relative to all of that firm’s patent citations filed in the same year, Y02 Top 1—the natural log of one plus the number of Y02 patents whose citations were in the top 1 percent of all Y02 patents in a given year, Y02 Top 10 —the natural log of one plus the number of Y02 patents whose citations were in the top 10 percent of all Y02 patents in a given year, respectively. Pressure is the natural log of one plus a three-year, backward average of an accumulated total of the climate-related shareholder proposals that a firm has received from 1994 to 2019. The control variables are also averaged over three years and include Size, R&D, Tobin’s Q, Age, Revenue Growth, Stock Returns, Leverage and Cash Surplus, as defined in Appendix A. t-statistic, based on robust standard errors, adjusted for heteroskedasticity and clustered at the industry-year level, are reported in brackets below the coefficients. ***, **, and * indicate significance at the 1%, 5%, and 10% level, respectively
(1) | (2) | (3) | (4) | |
---|---|---|---|---|
Y02 counts pct | Y02 cites pct | Y02 top 1 pct | Y02 top 10 pct | |
Pressure | 0.028*** | 0.025** | 0.04** | 0.084** |
(2.808) | (2.294) | (2.421) | (2.497) | |
Size | 0.008*** | 0.009*** | 0.013*** | 0.024*** |
(4.676) | (4.791) | (2.719) | (2.619) | |
R&D/Assets | − 0.055** | − 0.043 | − 0.147 | 0.467* |
(− 2.213) | (− 1.482) | (− 1.25) | (1.876) | |
Tobin's Q | 0.001** | − 0.001 | − 0.001 | − 0.004 |
(2.448) | (− 1.121) | (− 0.48) | (− 0.754) | |
Age | 0.007 | 0.016*** | 0.024** | 0.149*** |
(1.33) | (2.628) | (2.183) | (4.091) | |
Sales Growth | 0.002** | 0.002** | 0.002* | 0.005** |
(2.215) | (2.219) | (1.683) | (2.172) | |
Stock Return | 0.002 | 0.003* | 0.005 | 0.007 |
(1.077) | (1.697) | (1.552) | (0.996) | |
Leverage | − 0.003 | 0.000 | − 0.015* | − 0.067*** |
(− 0.761) | (− 0.049) | (− 1.862) | (− 2.826) | |
Cash Surplus | − 0.014 | − 0.012 | − 0.014 | − 0.078 |
(− 1.149) | (− 0.823) | (− 0.473) | (− 1.119) | |
Obs | 13,527 | 13,527 | 13,527 | 13,527 |
R-squared | 0.666 | 0.644 | 0.663 | 0.845 |
Firm FE | Yes | Yes | Yes | Yes |
Year FE | Yes | Yes | Yes | Yes |
Industry-year FE | Yes | Yes | Yes | Yes |
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