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A Gentle Introduction to Statistical Hypothesis Testing

Final Updated on Apr 10, 2020

Data must be interpreted in order to add meaning.

We can interpret information by assuming a specific structure our upshot and utilise statistical methods to ostend or turn down the assumption. The supposition is called a hypothesis and the statistical tests used for this purpose are called statistical hypothesis tests.

Whenever we want to make claims about the distribution of data or whether one set of results are different from another set of results in applied car learning, we must rely on statistical hypothesis tests.

In this tutorial, you will discover statistical hypothesis testing and how to interpret and carefully country the results from statistical tests.

After completing this tutorial, you will know:

Statistical hypothesis tests are of import for quantifying answers to questions about samples of data.
The interpretation of a statistical hypothesis test requires a correct understanding of p-values and critical values.
Regardless of the significance level, the finding of hypothesis tests may still contain errors.

Kick-kickoff your project with my new book Statistics for Car Learning, including step-past-step tutorials and the Python source code files for all examples.

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Update May/2018: Added note virtually "reject" vs "failure to reject", improved language on this issue.
Update Jun/2018: Fixed typo in the explanation of blazon I and blazon II errors.
Update Jun/2019: Added examples of tests and links to Python tutorials.

A Gentle Introduction to Statistical Hypothesis Tests
Photo by Kevin Verbeem, some rights reserved.

Tutorial Overview

This tutorial is divided into five parts; they are:

Statistical Hypothesis Testing
Statistical Test Interpretation
Errors in Statistical Tests
Examples of Hypothesis Tests
Python Tutorials

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Statistical Hypothesis Testing

Data lonely is not interesting. It is the estimation of the data that we are really interested in.

In statistics, when we wish to kickoff request questions nigh the data and translate the results, we use statistical methods that provide a confidence or likelihood about the answers. In general, this class of methods is called statistical hypothesis testing, or significance tests.

The term "hypothesis" may make you think about science, where nosotros investigate a hypothesis. This is forth the right track.

In statistics, a hypothesis test calculates some quantity nether a given assumption. The event of the examination allows united states of america to interpret whether the assumption holds or whether the supposition has been violated.

Two concrete examples that we will utilize a lot in machine learning are:

A test that assumes that data has a normal distribution.
A test that assumes that two samples were drawn from the same underlying population distribution.

The assumption of a statistical test is called the null hypothesis, or hypothesis 0 (H0 for short). It is often called the default supposition, or the assumption that nothing has changed.

A violation of the test's assumption is ofttimes chosen the first hypothesis, hypothesis i or H1 for short. H1 is really a short hand for "some other hypothesis," as all we know is that the evidence suggests that the H0 can exist rejected.

Hypothesis 0 (H0): Assumption of the test holds and is failed to be rejected at some level of significance.
Hypothesis 1 (H1): Assumption of the test does not hold and is rejected at some level of significance.

Earlier we can refuse or fail to reject the zippo hypothesis, we must interpret the event of the test.

Statistical Test Interpretation

The results of a statistical hypothesis test must be interpreted for us to start making claims.

This is a bespeak that may cause a lot of confusion for beginners and experienced practitioners alike.

At that place are two common forms that a result from a statistical hypothesis test may accept, and they must be interpreted in different ways. They are the p-value and critical values.

Interpret the p-value

We describe a finding as statistically meaning by interpreting the p-value.

For example, we may perform a normality test on a data sample and detect that it is unlikely that sample of data deviates from a Gaussian distribution, failing to pass up the nada hypothesis.

A statistical hypothesis examination may return a value called p or the p-value. This is a quantity that we can use to interpret or quantify the result of the test and either reject or fail to turn down the null hypothesis. This is washed past comparing the p-value to a threshold value chosen beforehand called the significance level.

The significance level is ofttimes referred to by the Greek lower case letter of the alphabet blastoff.

A common value used for alpha is 5% or 0.05. A smaller alpha value suggests a more robust interpretation of the zippo hypothesis, such as i% or 0.1%.

The p-value is compared to the pre-called alpha value. A event is statistically pregnant when the p-value is less than blastoff. This signifies a alter was detected: that the default hypothesis tin can be rejected.

If p-value > alpha: Fail to refuse the nix hypothesis (i.e. not significant result).
If p-value <= alpha: Decline the null hypothesis (i.eastward. significant result).

For case, if nosotros were performing a examination of whether a data sample was normal and nosotros calculated a p-value of .07, we could state something like:

The test establish that the data sample was normal, failing to pass up the null hypothesis at a 5% significance level.

The significance level can be inverted past subtracting it from i to give a confidence level of the hypothesis given the observed sample data.

confidence level = one - significance level

Therefore, statements such every bit the post-obit tin can also be made:

The test institute that the data was normal, failing to reject the nil hypothesis at a 95% confidence level.

"Reject" vs "Failure to Turn down"

The p-value is probabilistic.

This means that when we interpret the effect of a statistical exam, we do not know what is true or false, only what is likely.

Rejecting the null hypothesis means that in that location is sufficient statistical evidence that the naught hypothesis does not look likely. Otherwise, it ways that there is not sufficient statistical testify to reject the zip hypothesis.

Nosotros may think about the statistical exam in terms of the dichotomy of rejecting and accepting the cipher hypothesis. The danger is that if we say that we "have" the null hypothesis, the language suggests that the null hypothesis is true. Instead, it is safer to say that nosotros "fail to reject" the nix hypothesis, equally in, there is insufficient statistical evidence to reject information technology.

When reading "reject" vs "fail to reject" for the first fourth dimension, it is confusing to beginners. Y'all tin think of information technology equally "reject" vs "take" in your heed, as long as you remind yourself that the effect is probabilistic and that even an "accepted" goose egg hypothesis all the same has a small-scale probability of existence wrong.

Mutual p-value Misinterpretations

This section highlights some common misinterpretations of the p-value in the results of statistical tests.

True or Fake Null Hypothesis

The estimation of the p-value does not mean that the null hypothesis is truthful or false.

It does mean that we have called to reject or fail to refuse the zero hypothesis at a specific statistical significance level based on empirical evidence and the chosen statistical test.

You are express to making probabilistic claims, not crisp binary or true/false claims about the result.

p-value every bit Probability

A mutual misunderstanding is that the p-value is a probability of the null hypothesis being true or simulated given the information.

In probability, this would be written equally follows:

This is wrong.

Instead, the p-value can exist idea of as the probability of the data given the pre-specified supposition embedded in the statistical examination.

Once again, using probability notation, this would be written equally:

It allows us to reason about whether or not the data fits the hypothesis. Not the other way around.

The p-value is a measure of how likely the data sample would be observed if the null hypothesis were true.

Post-Hoc Tuning

Information technology does not mean that you tin re-sample your domain or tune your data sample and re-run the statistical test until y'all attain a desired result.

It also does non mean that you lot can choose your p-value later on you lot run the test.

This is called p-hacking or hill climbing and will mean that the upshot you present will be fragile and not representative. In scientific discipline, this is at all-time unethical, and at worst fraud.

Interpret Disquisitional Values

Some tests do non return a p-value.

Instead, they might return a listing of critical values and their associated significance levels, as well every bit a test statistic.

These are usually nonparametric or distribution-free statistical hypothesis tests.

The option of returning a p-value or a list of critical values is actually an implementation choice.

The results are interpreted in a similar way. Instead of comparison a single p-value to a pre-specified significance level, the examination statistic is compared to the critical value at a chosen significance level.

If exam statistic < critical value: Fail to decline the null hypothesis.
If test statistic >= critical value: Reject the goose egg hypothesis.

Again, the meaning of the result is similar in that the chosen significance level is a probabilistic conclusion on rejection or fail to turn down the base assumption of the test given the information.

Results are presented in the same mode as with a p-value, as either significance level or confidence level. For example, if a normality test was calculated and the test statistic was compared to the critical value at the 5% significance level, results could be stated as:

The exam constitute that the data sample was normal, failing to reject the null hypothesis at a v% significance level.

Or:

The test institute that the information was normal, failing to reject the nada hypothesis at a 95% conviction level.

Errors in Statistical Tests

The interpretation of a statistical hypothesis exam is probabilistic.

That means that the evidence of the test may suggest an outcome and be mistaken.

For instance, if alpha was 5%, information technology suggests that (at most) ane time in 20 that the null hypothesis would be mistakenly rejected or failed to be rejected because of the statistical racket in the data sample.

Given a small-scale p-value (reject the zilch hypothesis) either ways that the null hypothesis false (we got information technology right) or it is true and some rare and unlikely consequence has been observed (nosotros fabricated a error). If this blazon of error is made, it is chosen a simulated positive. Nosotros falsely believe the rejection of the null hypothesis.

Alternately, given a big p-value (fail to turn down the zilch hypothesis), it may mean that the aught hypothesis is truthful (nosotros got it right) or that the null hypothesis is false and some unlikely event occurred (we made a mistake). If this type of error is made, it is chosen a false negative. Nosotros falsely believe the null hypothesis or assumption of the statistical test.

Each of these 2 types of error has a specific proper name.

Type I Error: The incorrect rejection of a true naught hypothesis or a false positive.
Type II Error: The incorrect failure of rejection of a false zero hypothesis or a false negative.

All statistical hypothesis tests have a take chances of making either of these types of errors. False findings or false disoveries are more than possible; they are probable.

Ideally, we want to choose a significance level that minimizes the likelihood of one of these errors. E.thousand. a very small significance level. Although significance levels such as 0.05 and 0.01 are common in many fields of science, harder sciences, such every bit physics, are more than aggressive.

It is common to use a significance level of 3 * ten^-seven or 0.0000003, oftentimes referred to as five-sigma. This means that the finding was due to chance with a probability of 1 in iii.5 million independent repeats of the experiments. To use a threshold similar this may require a much large data sample.

Nevertheless, these types of errors are ever present and must be kept in mind when presenting and interpreting the results of statistical tests. It is as well a reason why it is important to have findings independently verified.

Examples of Hypothesis Tests

There are many types of statistical hypothesis tests.

This department lists some common examples of statistical hypothesis tests and the types of problems that they are used to address:

Variable Distribution Type Tests (Gaussian)

Shapiro-Wilk Exam
D'Agostino'south K^ii Test
Anderson-Darling Test

Variable Human relationship Tests (correlation)

Pearson's Correlation Coefficient
Spearman'due south Rank Correlation
Kendall's Rank Correlation
Chi-Squared Test

Compare Sample Means (parametric)

Student's t-test
Paired Student's t-test
Assay of Variance Test (ANOVA)
Repeated Measures ANOVA Test

Compare Sample Ways (nonparametric)

Isle of mann-Whitney U Test
Wilcoxon Signed-Rank Test
Kruskal-Wallis H Exam
Friedman Exam

For example Python lawmaking on how to use each of these tests, see the adjacent section.

Python Tutorials

This department provides links to Python tutorials on statistical hypothesis testing:

Examples of many tests:

15 Statistical Hypothesis Tests in Python (Crook Sheet)

Variable distribution tests:

A Gentle Introduction to Normality Tests in Python

Evaluating variable relationships:

How to Calculate Correlation Between Variables in Python
How to Calculate Nonparametric Rank Correlation in Python

Comparing sample means:

How to Summate Parametric Statistical Hypothesis Tests in Python
How to Summate Nonparametric Statistical Hypothesis Tests in Python

Extensions

This section lists some ideas for extending the tutorial that you may wish to explore.

Find an case of a enquiry paper that does non nowadays results using p-values.
Find an case of a research newspaper that presents results with statistical significance, just makes one of the mutual misinterpretations of p-values.
Detect an example of a research newspaper that presents results with statistical significance and correctly interprets and presents the p-value and findings.

If you explore any of these extensions, I'd honey to know.

Summary

In this tutorial, yous discovered statistical hypothesis testing and how to translate and advisedly state the results from statistical tests.

Specifically, yous learned:

Statistical hypothesis tests are important for quantifying answers to questions most samples of data.
The interpretation of a statistical hypothesis test requires a correct understanding of p-values.
Regardless of the significance level, the finding of hypothesis tests may notwithstanding comprise errors.

Do you take any questions?
Enquire your questions in the comments below and I will do my best to answer.

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