what is a hypothesis test in r

The Complete Guide: Hypothesis Testing in R

A hypothesis test is a formal statistical test we use to reject or fail to reject some statistical hypothesis.

This tutorial explains how to perform the following hypothesis tests in R:

One sample t-test
Two sample t-test
Paired samples t-test

We can use the t.test() function in R to perform each type of test:

x, y: The two samples of data.
alternative: The alternative hypothesis of the test.
mu: The true value of the mean.
paired: Whether to perform a paired t-test or not.
var.equal: Whether to assume the variances are equal between the samples.
conf.level: The confidence level to use.

The following examples show how to use this function in practice.

Example 1: One Sample t-test in R

A one sample t-test is used to test whether or not the mean of a population is equal to some value.

For example, suppose we want to know whether or not the mean weight of a certain species of some turtle is equal to 310 pounds. We go out and collect a simple random sample of turtles with the following weights:

Weights : 300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303

The following code shows how to perform this one sample t-test in R:

From the output we can see:

t-test statistic: -1.5848
degrees of freedom: 12
p-value: 0.139
95% confidence interval for true mean: [303.4236, 311.0379]
mean of turtle weights: 307.230

Since the p-value of the test (0.139) is not less than .05, we fail to reject the null hypothesis.

This means we do not have sufficient evidence to say that the mean weight of this species of turtle is different from 310 pounds.

Example 2: Two Sample t-test in R

A two sample t-test is used to test whether or not the means of two populations are equal.

For example, suppose we want to know whether or not the mean weight between two different species of turtles is equal. To test this, we collect a simple random sample of turtles from each species with the following weights:

Sample 1 : 300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303

Sample 2 : 335, 329, 322, 321, 324, 319, 304, 308, 305, 311, 307, 300, 305

The following code shows how to perform this two sample t-test in R:

t-test statistic: -2.1009
degrees of freedom: 19.112
p-value: 0.04914
95% confidence interval for true mean difference: [-14.74, -0.03]
mean of sample 1 weights: 307.2308
mean of sample 2 weights: 314.6154

Since the p-value of the test (0.04914) is less than .05, we reject the null hypothesis.

This means we have sufficient evidence to say that the mean weight between the two species is not equal.

Example 3: Paired Samples t-test in R

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

For example, suppose we want to know whether or not a certain training program is able to increase the max vertical jump (in inches) of basketball players.

To test this, we may recruit a simple random sample of 12 college basketball players and measure each of their max vertical jumps. Then, we may have each player use the training program for one month and then measure their max vertical jump again at the end of the month.

The following data shows the max jump height (in inches) before and after using the training program for each player:

Before : 22, 24, 20, 19, 19, 20, 22, 25, 24, 23, 22, 21

After : 23, 25, 20, 24, 18, 22, 23, 28, 24, 25, 24, 20

The following code shows how to perform this paired samples t-test in R:

t-test statistic: -2.5289
degrees of freedom: 11
p-value: 0.02803
95% confidence interval for true mean difference: [-2.34, -0.16]
mean difference between before and after: -1.25

Since the p-value of the test (0.02803) is less than .05, we reject the null hypothesis.

This means we have sufficient evidence to say that the mean jump height before and after using the training program is not equal.

Additional Resources

Use the following online calculators to automatically perform various t-tests:

One Sample t-test Calculator Two Sample t-test Calculator Paired Samples t-test Calculator

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Hypothesis testing in r programming.

A hypothesis is made by the researchers about the data collected for any experiment or data set. A hypothesis is an assumption made by the researchers that are not mandatory true. In simple words, a hypothesis is a decision taken by the researchers based on the data of the population collected. Hypothesis Testing in R Programming is a process of testing the hypothesis made by the researcher or to validate the hypothesis. To perform hypothesis testing, a random sample of data from the population is taken and testing is performed. Based on the results of the testing, the hypothesis is either selected or rejected. This concept is known as Statistical Inference . In this article, we’ll discuss the four-step process of hypothesis testing, One sample T-Testing, Two-sample T-Testing, Directional Hypothesis, one sample [Tex]\mu [/Tex] -test, two samples [Tex]\mu [/Tex] -test and correlation test in R programming.

Four Step Process of Hypothesis Testing

There are 4 major steps in hypothesis testing:

State the hypothesis- This step is started by stating null and alternative hypothesis which is presumed as true.
Formulate an analysis plan and set the criteria for decision- In this step, a significance level of test is set. The significance level is the probability of a false rejection in a hypothesis test.
Analyze sample data- In this, a test statistic is used to formulate the statistical comparison between the sample mean and the mean of the population or standard deviation of the sample and standard deviation of the population.
Interpret decision- The value of the test statistic is used to make the decision based on the significance level. For example, if the significance level is set to 0.1 probability, then the sample mean less than 10% will be rejected. Otherwise, the hypothesis is retained to be true.

One Sample T-Testing

One sample T-Testing approach collects a huge amount of data and tests it on random samples. To perform T-Test in R, normally distributed data is required. This test is used to test the mean of the sample with the population. For example, the height of persons living in an area is different or identical to other persons living in other areas.

Syntax: t.test(x, mu) Parameters: x: represents numeric vector of data mu: represents true value of the mean

To know about more optional parameters of t.test() , try the below command:

help("t.test")

Example:

One Sample t-test data: x t = -49.504, df = 99, p-value < 2.2e-16 alternative hypothesis: true mean is not equal to 5 95 percent confidence interval: -0.1910645 0.2090349 sample estimates: mean of x 0.008985172

Data: The dataset ‘x’ was used for the test.
The determined t-value is -49.504.
Degrees of Freedom (df): The t-test has 99 degrees of freedom.
The p-value is 2.2e-16, which indicates that there is substantial evidence refuting the null hypothesis.
Alternative hypothesis: The true mean is not equal to five, according to the alternative hypothesis.
95 percent confidence interval: (-0.1910645, 0.2090349) is the confidence interval’s value. This range denotes the values that, with 95% confidence, correspond to the genuine population mean.

Two Sample T-Testing

In two sample T-Testing, the sample vectors are compared. If var. equal = TRUE, the test assumes that the variances of both the samples are equal.

Syntax: t.test(x, y) Parameters: x and y: Numeric vectors

Welch Two Sample t-test data: x and y t = -1.0601, df = 197.86, p-value = 0.2904 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -0.4362140 0.1311918 sample estimates: mean of x mean of y -0.05075633 0.10175478

Directional Hypothesis

Using the directional hypothesis, the direction of the hypothesis can be specified like, if the user wants to know the sample mean is lower or greater than another mean sample of the data.

Syntax: t.test(x, mu, alternative) Parameters: x: represents numeric vector data mu: represents mean against which sample data has to be tested alternative: sets the alternative hypothesis

One Sample t-test data: x t = -20.708, df = 99, p-value = 1 alternative hypothesis: true mean is greater than 2 95 percent confidence interval: -0.2307534 Inf sample estimates: mean of x -0.0651628

One Sample [Tex]\mu [/Tex] -Test

This type of test is used when comparison has to be computed on one sample and the data is non-parametric. It is performed using wilcox.test() function in R programming.

Syntax: wilcox.test(x, y, exact = NULL) Parameters: x and y: represents numeric vector exact: represents logical value which indicates whether p-value be computed

To know about more optional parameters of wilcox.test() , use below command:

help("wilcox.test")

Wilcoxon signed rank test with continuity correction data: x V = 2555, p-value = 0.9192 alternative hypothesis: true location is not equal to 0

The calculated test statistic or V value is 2555.
P-value: The null hypothesis is weakly supported by the p-value of 0.9192.
The alternative hypothesis asserts that the real location is not equal to 0. This indicates that there is a reasonable suspicion that the distribution’s median or location parameter is different from 0.

Two Sample [Tex]\mu [/Tex] -Test

This test is performed to compare two samples of data. Example:

Wilcoxon rank sum test with continuity correction data: x and y W = 5300, p-value = 0.4643 alternative hypothesis: true location shift is not equal to 0

Correlation Test

This test is used to compare the correlation of the two vectors provided in the function call or to test for the association between the paired samples.

Syntax: cor.test(x, y) Parameters: x and y: represents numeric data vectors

To know about more optional parameters in cor.test() function, use below command:

help("cor.test")

Pearson's product-moment correlation data: mtcars$mpg and mtcars$hp t = -6.7424, df = 30, p-value = 1.788e-07 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: -0.8852686 -0.5860994 sample estimates: cor -0.7761684

Data: The variables’mtcars$mpg’ and’mtcars$hp’ from the ‘mtcars’ dataset were subjected to a correlation test.
t-value: The t-value that was determined is -6.7424.
Degrees of Freedom (df): The test has 30 degrees of freedom.
The p-value is 1.788e-07, indicating that there is substantial evidence that rules out the null hypothesis.
The alternative hypothesis asserts that the true correlation is not equal to 0, indicating that “mtcars$mpg” and “mtcars$hp” are significantly correlated.
95 percent confidence interval: (-0.8852686, -0.5860994) is the confidence interval. This range denotes the values that, with a 95% level of confidence, represent the genuine population correlation coefficient.
Correlation coefficient sample estimate: The correlation coefficient sample estimate is -0.7761684.

Introduction

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The Complete Guide: Hypothesis Testing in R

A hypothesis test is a formal statistical test we use to reject or fail to reject some statistical hypothesis.

This tutorial explains how to perform the following hypothesis tests in R:

One sample t-test
Two sample t-test
Paired samples t-test

We can use the t.test() function in R to perform each type of test:

x, y: The two samples of data.
alternative: The alternative hypothesis of the test.
mu: The true value of the mean.
paired: Whether to perform a paired t-test or not.
var.equal: Whether to assume the variances are equal between the samples.
conf.level: The confidence level to use.

The following examples show how to use this function in practice.

Example 1: One Sample t-test in R

A one sample t-test is used to test whether or not the mean of a population is equal to some value.

Weights : 300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303

The following code shows how to perform this one sample t-test in R:

From the output we can see:

t-test statistic: -1.5848
degrees of freedom: 12
p-value: 0.139
95% confidence interval for true mean: [303.4236, 311.0379]
mean of turtle weights: 307.230

Since the p-value of the test (0.139) is not less than .05, we fail to reject the null hypothesis.

This means we do not have sufficient evidence to say that the mean weight of this species of turtle is different from 310 pounds.

Example 2: Two Sample t-test in R

A two sample t-test is used to test whether or not the means of two populations are equal.

Sample 1 : 300, 315, 320, 311, 314, 309, 300, 308, 305, 303, 305, 301, 303

Sample 2 : 335, 329, 322, 321, 324, 319, 304, 308, 305, 311, 307, 300, 305

The following code shows how to perform this two sample t-test in R:

t-test statistic: -2.1009
degrees of freedom: 19.112
p-value: 0.04914
95% confidence interval for true mean difference: [-14.74, -0.03]
mean of sample 1 weights: 307.2308
mean of sample 2 weights: 314.6154

Since the p-value of the test (0.04914) is less than .05, we reject the null hypothesis.

This means we have sufficient evidence to say that the mean weight between the two species is not equal.

Example 3: Paired Samples t-test in R

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

For example, suppose we want to know whether or not a certain training program is able to increase the max vertical jump (in inches) of basketball players.

The following data shows the max jump height (in inches) before and after using the training program for each player:

Before : 22, 24, 20, 19, 19, 20, 22, 25, 24, 23, 22, 21

After : 23, 25, 20, 24, 18, 22, 23, 28, 24, 25, 24, 20

The following code shows how to perform this paired samples t-test in R:

t-test statistic: -2.5289
degrees of freedom: 11
p-value: 0.02803
95% confidence interval for true mean difference: [-2.34, -0.16]
mean difference between before and after: -1.25

Since the p-value of the test (0.02803) is less than .05, we reject the null hypothesis.

This means we have sufficient evidence to say that the mean jump height before and after using the training program is not equal.

Additional Resources

Use the following online calculators to automatically perform various t-tests:

One Sample t-test Calculator Two Sample t-test Calculator Paired Samples t-test Calculator

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Statistics with R
R Objects, Numbers, Attributes, Vectors, Coercion
Matrices, Lists, Factors
Data Frames in R
Control Structures in R
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Data Basics: Compute Summary Statistics in R
Central Tendency and Spread in R Programming
Data Basics: Plotting – Charts and Graphs
Normal Distribution in R
Skewness of statistical data
Bernoulli Distribution in R
Binomial Distribution in R Programming
Compute Randomly Drawn Negative Binomial Density in R Programming
Poisson Functions in R Programming
How to Use the Multinomial Distribution in R
Beta Distribution in R
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Log Normal Distribution in R
Continuous Uniform Distribution in R
Understanding the t-distribution in R
Gamma Distribution in R Programming
How to Calculate Conditional Probability in R?

How to Plot a Weibull Distribution in R

Hypothesis testing in r programming.

One Sample T-test in R
Two sample T-test in R
Paired Sample T-test in R
Type I Error in R
Type II Error in R
Confidence Intervals in R
Covariance and Correlation in R
Covariance Matrix in R
Pearson Correlation in R
Normal Probability Plot in R

Hypothesis testing is a statistical method used to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In R programming, you can perform various types of hypothesis tests, such as t-tests, chi-squared tests, and ANOVA tests, among others.

In R programming, you can perform hypothesis testing using various built-in functions. Here’s an overview of some commonly used hypothesis testing methods in R:

T-test (one-sample, paired, and independent two-sample)
Chi-square test
ANOVA (Analysis of Variance)
Wilcoxon signed-rank test
Mann-Whitney U test

1. One-sample t-test:

The one-sample t-test is used to compare the mean of a sample to a known value (usually a population mean) to see if there is a significant difference.

2. Two-sample t-test:

The two-sample t-test is used to compare the means of two independent samples to see if there is a significant difference.

3. Paired t-test:

The paired t-test is used to compare the means of two dependent samples, usually to test the effect of a treatment or intervention.

4. Chi-squared test:

The chi-squared test is used to test the association between two categorical variables.

5. One-way ANOVA

For a one-way ANOVA, use the aov() and summary() functions:

6. Wilcoxon signed-rank test

7. mann-whitney u test.

For a Mann-Whitney U test, use the wilcox.test() function with the paired argument set to FALSE :

Steps for conducting a Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In R programming, you can perform various types of hypothesis tests, such as t-tests, chi-squared tests, and ANOVA, depending on the nature of your data and research question.

Here, I’ll walk you through the steps for conducting a t-test (one of the most common hypothesis tests) in R. A t-test is used to compare the means of two groups, often in order to determine whether there’s a significant difference between them.

1. Prepare your data:

First, you’ll need to have your data in R. You can either read data from a file (e.g., using read.csv() ), or you can create vectors directly in R. For this example, I’ll create two sample vectors for Group 1 and Group 2:

2. State your null and alternative hypotheses:

In hypothesis testing, we start with a null hypothesis (H0) and an alternative hypothesis (H1). For a t-test, the null hypothesis is typically that there’s no difference between the means of the two groups, while the alternative hypothesis is that there is a difference. In this example:

H0: μ1 = μ2 (the means of Group 1 and Group 2 are equal)
H1: μ1 ≠ μ2 (the means of Group 1 and Group 2 are not equal)

3. Perform the t-test:

Use the t.test() function to perform the t-test on your data. You can specify the type of t-test (independent samples, paired, or one-sample) with the appropriate arguments. In this case, we’ll perform an independent samples t-test:

4. Interpret the results:

The t-test result will include the t-value, degrees of freedom, and the p-value, among other information. The p-value is particularly important, as it helps you determine whether to accept or reject the null hypothesis. A common significance level (alpha) is 0.05. If the p-value is less than alpha, you can reject the null hypothesis, otherwise you fail to reject it.

5. Make a decision:

Based on the p-value and your chosen significance level, make a decision about whether to reject or fail to reject the null hypothesis. If the p-value is less than 0.05, you would reject the null hypothesis and conclude that there is a significant difference between the means of the two groups.

Keep in mind that this example demonstrates the basic process of hypothesis testing using a t-test in R. Different tests and data may require additional steps, arguments, or functions. Be sure to consult R documentation and resources to ensure you’re using the appropriate test and interpreting the results correctly.

Few more examples of hypothesis tests using R

1. one-sample t-test: compares the mean of a sample to a known value., 2. two-sample t-test: compares the means of two independent samples., 3. paired t-test: compares the means of two paired samples., 4. chi-squared test: tests the independence between two categorical variables., 5. anova: compares the means of three or more independent samples..

Remember to interpret the results (p-value) according to the significance level (commonly 0.05). If the p-value is less than the significance level, you can reject the null hypothesis in favor of the alternative hypothesis.

T-Test in R Programming

Hypothesis Tests in R

This tutorial covers basic hypothesis testing in R.

Normality tests
Shapiro-Wilk normality test
Kolmogorov-Smirnov test
Comparing central tendencies: Tests with continuous / discrete data
One-sample t-test : Normally-distributed sample vs. expected mean
Two-sample t-test : Two normally-distributed samples
Wilcoxen rank sum : Two non-normally-distributed samples
Weighted two-sample t-test : Two continuous samples with weights
Comparing proportions: Tests with categorical data
Chi-squared goodness of fit test : Sampled frequencies of categorical values vs. expected frequencies
Chi-squared independence test : Two sampled frequencies of categorical values
Weighted chi-squared independence test : Two weighted sampled frequencies of categorical values
Comparing multiple groups: Tests with categorical and continuous / discrete data
Analysis of Variation (ANOVA) : Normally-distributed samples in groups defined by categorical variable(s)
Kruskal-Wallace One-Way Analysis of Variance : Nonparametric test of the significance of differences between two or more groups

Hypothesis Testing

Science is "knowledge or a system of knowledge covering general truths or the operation of general laws especially as obtained and tested through scientific method" (Merriam-Webster 2022) .

The idealized world of the scientific method is question-driven , with the collection and analysis of data determined by the formulation of research questions and the testing of hypotheses. Hypotheses are tentative assumptions about what the answers to your research questions may be.

Formulate questions: How can I understand some phenomenon?
Literature review: What does existing research say about my questions?
Formulate hypotheses: What do I think the answers to my questions will be?
Collect data: What data can I gather to test my hypothesis?
Test hypotheses: Does the data support my hypothesis?
Communicate results: Who else needs to know about this?
Formulate questions: Frame missing knowledge about a phenomenon as research question(s).
Literature review: A literature review is an investigation of what existing research says about the phenomenon you are studying. A thorough literature review is essential to identify gaps in existing knowledge you can fill, and to avoid unnecessarily duplicating existing research.
Formulate hypotheses: Develop possible answers to your research questions.
Collect data: Acquire data that supports or refutes the hypothesis.
Test hypotheses: Run tools to determine if the data corroborates the hypothesis.
Communicate results: Share your findings with the broader community that might find them useful.

While the process of knowledge production is, in practice, often more iterative than this waterfall model, the testing of hypotheses is usually a fundamental element of scientific endeavors involving quantitative data.

The Problem of Induction

The scientific method looks to the past or present to build a model that can be used to infer what will happen in the future. General knowledge asserts that given a particular set of conditions, a particular outcome will or is likely to occur.

The problem of induction is that we cannot be 100% certain that what we are assuming is a general principle is not, in fact, specific to the particular set of conditions when we made our empirical observations. We cannot prove that that such principles will hold true under future conditions or different locations that we have not yet experienced (Vickers 2014) .

The problem of induction is often associated with the 18th-century British philosopher David Hume . This problem is especially vexing in the study of human beings, where behaviors are a function of complex social interactions that vary over both space and time.

Falsification

One way of addressing the problem of induction was proposed by the 20th-century Viennese philosopher Karl Popper .

Rather than try to prove a hypothesis is true, which we cannot do because we cannot know all possible situations that will arise in the future, we should instead concentrate on falsification , where we try to find situations where a hypothesis is false. While you cannot prove your hypothesis will always be true, you only need to find one situation where the hypothesis is false to demonstrate that the hypothesis can be false (Popper 1962) .

If a hypothesis is not demonstrated to be false by a particular test, we have corroborated that hypothesis. While corroboration does not "prove" anything with 100% certainty, by subjecting a hypothesis to multiple tests that fail to demonstrate that it is false, we can have increasing confidence that our hypothesis reflects reality.

Null and Alternative Hypotheses

In scientific inquiry, we are often concerned with whether a factor we are considering (such as taking a specific drug) results in a specific effect (such as reduced recovery time).

To evaluate whether a factor results in an effect, we will perform an experiment and / or gather data. For example, in a clinical drug trial, half of the test subjects will be given the drug, and half will be given a placebo (something that appears to be the drug but is actually a neutral substance).

Because the data we gather will usually only be a portion (sample) of total possible people or places that could be affected (population), there is a possibility that the sample is unrepresentative of the population. We use a statistical test that considers that uncertainty when assessing whether an effect is associated with a factor.

Statistical testing begins with an alternative hypothesis (H 1 ) that states that the factor we are considering results in a particular effect. The alternative hypothesis is based on the research question and the type of statistical test being used.
Because of the problem of induction , we cannot prove our alternative hypothesis. However, under the concept of falsification , we can evaluate the data to see if there is a significant probability that our data falsifies our alternative hypothesis (Wilkinson 2012) .
The null hypothesis (H 0 ) states that the factor has no effect. The null hypothesis is the opposite of the alternative hypothesis. The null hypothesis is what we are testing when we perform a hypothesis test.

The output of a statistical test like the t-test is a p -value. A p -value is the probability that any effects we see in the sampled data are the result of random sampling error (chance).

If a p -value is greater than the significance level (0.05 for 5% significance) we fail to reject the null hypothesis since there is a significant possibility that our results falsify our alternative hypothesis.
If a p -value is lower than the significance level (0.05 for 5% significance) we reject the null hypothesis and have corroborated (provided evidence for) our alternative hypothesis.

The calculation and interpretation of the p -value goes back to the central limit theorem , which states that random sampling error has a normal distribution.

Using our example of a clinical drug trial, if the mean recovery times for the two groups are close enough together that there is a significant possibility ( p > 0.05) that the recovery times are the same (falsification), we fail to reject the null hypothesis.

However, if the mean recovery times for the two groups are far enough apart that the probability they are the same is under the level of significance ( p < 0.05), we reject the null hypothesis and have corroborated our alternative hypothesis.

Significance means that an effect is "probably caused by something other than mere chance" (Merriam-Webster 2022) .

The significance level (α) is the threshold for significance and, by convention, is usually 5%, 10%, or 1%, which corresponds to 95% confidence, 90% confidence, or 99% confidence, respectively.
A factor is considered statistically significant if the probability that the effect we see in the data is a result of random sampling error (the p -value) is below the chosen significance level.
A statistical test is used to evaluate whether a factor being considered is statistically significant (Gallo 2016) .

Type I vs. Type II Errors

Although we are making a binary choice between rejecting and failing to reject the null hypothesis, because we are using sampled data, there is always the possibility that the choice we have made is an error.

There are two types of errors that can occur in hypothesis testing.

Type I error (false positive) occurs when a low p -value causes us to reject the null hypothesis, but the factor does not actually result in the effect.
Type II error (false negative) occurs when a high p -value causes us to fail to reject the null hypothesis, but the factor does actually result in the effect.

The numbering of the errors reflects the predisposition of the scientific method to be fundamentally skeptical . Accepting a fact about the world as true when it is not true is considered worse than rejecting a fact about the world that actually is true.

Statistical Significance vs. Importance

When we fail to reject the null hypothesis, we have found information that is commonly called statistically significant . But there are multiple challenges with this terminology.

First, statistical significance is distinct from importance (NIST 2012) . For example, if sampled data reveals a statistically significant difference in cancer rates, that does not mean that the increased risk is important enough to justify expensive mitigation measures. All statistical results require critical interpretation within the context of the phenomenon being observed. People with different values and incentives can have different interpretations of whether statistically significant results are important.

Second, the use of 95% probability for defining confidence intervals is an arbitrary convention. This creates a good vs. bad binary that suggests a "finality and certitude that are rarely justified." Alternative approaches like Beyesian statistics that express results as probabilities can offer more nuanced ways of dealing with complexity and uncertainty (Clayton 2022) .

Science vs. Non-science

Not all ideas can be falsified, and Popper uses the distinction between falsifiable and non-falsifiable ideas to make a distinction between science and non-science. In order for an idea to be science it must be an idea that can be demonstrated to be false.

While Popper asserts there is still value in ideas that are not falsifiable, such ideas are not science in his conception of what science is. Such non-science ideas often involve questions of subjective values or unseen forces that are complex, amorphous, or difficult to objectively observe.

Example Data

As example data, this tutorial will use a table of anonymized individual responses from the CDC's Behavioral Risk Factor Surveillance System . The BRFSS is a "system of health-related telephone surveys that collect state data about U.S. residents regarding their health-related risk behaviors, chronic health conditions, and use of preventive services" (CDC 2019) .

A CSV file with the selected variables used in this tutorial is available here and can be imported into R with read.csv() .

Guidance on how to download and process this data directly from the CDC website is available here...

Variable Types

The publicly-available BRFSS data contains a wide variety of discrete, ordinal, and categorical variables. Variables often contain special codes for non-responsiveness or missing (NA) values. Examples of how to clean these variables are given here...

The BRFSS has a codebook that gives the survey questions associated with each variable, and the way that responses are encoded in the variable values.

Normality Tests

Tests are commonly divided into two groups depending on whether they are built on the assumption that the continuous variable has a normal distribution.

Parametric tests presume a normal distribution.
Non-parametric tests can work with normal and non-normal distributions.

The distinction between parametric and non-parametric techniques is especially important when working with small numbers of samples (less than 40 or so) from a larger population.

The normality tests given below do not work with large numbers of values, but with many statistical techniques, violations of normality assumptions do not cause major problems when large sample sizes are used. (Ghasemi and Sahediasi 2012) .

The Shapiro-Wilk Normality Test

Data: A continuous or discrete sampled variable
R Function: shapiro.test()
Null hypothesis (H 0 ): The population distribution from which the sample is drawn is not normal
History: Samuel Sanford Shapiro and Martin Wilk (1965)

This is an example with random values from a normal distribution.

This is an example with random values from a uniform (non-normal) distribution.

The Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov is a more-generalized test than the Shapiro-Wilks test that can be used to test whether a sample is drawn from any type of distribution.

Data: A continuous or discrete sampled variable and a reference probability distribution
R Function: ks.test()
Null hypothesis (H 0 ): The population distribution from which the sample is drawn does not match the reference distribution
History: Andrey Kolmogorov (1933) and Nikolai Smirnov (1948)
pearson.test() The Pearson Chi-square Normality Test from the nortest library. Lower p-values (closer to 0) means to reject the reject the null hypothesis that the distribution IS normal.

Modality Tests of Samples

Comparing two central tendencies: tests with continuous / discrete data, one sample t-test (two-sided).

The one-sample t-test tests the significance of the difference between the mean of a sample and an expected mean.

Data: A continuous or discrete sampled variable and a single expected mean (μ)
Parametric (normal distributions)
R Function: t.test()
Null hypothesis (H 0 ): The means of the sampled distribution matches the expected mean.
History: William Sealy Gosset (1908)

t = ( Χ - μ) / (σ̂ / √ n )

t : The value of t used to find the p-value
Χ : The sample mean
μ: The population mean
σ̂: The estimate of the standard deviation of the population (usually the stdev of the sample
n : The sample size

T-tests should only be used when the population is at least 20 times larger than its respective sample. If the sample size is too large, the low p-value makes the insignificant look significant. .

For example, we test a hypothesis that the mean weight in IL in 2020 is different than the 2005 continental mean weight.

Walpole et al. (2012) estimated that the average adult weight in North America in 2005 was 178 pounds. We could presume that Illinois is a comparatively normal North American state that would follow the trend of both increased age and increased weight (CDC 2021) .

The low p-value leads us to reject the null hypothesis and corroborate our alternative hypothesis that mean weight changed between 2005 and 2020 in Illinois.

One Sample T-Test (One-Sided)

Because we were expecting an increase, we can modify our hypothesis that the mean weight in 2020 is higher than the continental weight in 2005. We can perform a one-sided t-test using the alternative="greater" parameter.

The low p-value leads us to again reject the null hypothesis and corroborate our alternative hypothesis that mean weight in 2020 is higher than the continental weight in 2005.

Note that this does not clearly evaluate whether weight increased specifically in Illinois, or, if it did, whether that was caused by an aging population or decreasingly healthy diets. Hypotheses based on such questions would require more detailed analysis of individual data.

Although we can see that the mean cancer incidence rate is higher for counties near nuclear plants, there is the possiblity that the difference in means happened by accident and the nuclear plants have nothing to do with those higher rates.

The t-test allows us to test a hypothesis. Note that a t-test does not "prove" or "disprove" anything. It only gives the probability that the differences we see between two areas happened by chance. It also does not evaluate whether there are other problems with the data, such as a third variable, or inaccurate cancer incidence rate estimates.

Note that this does not prove that nuclear power plants present a higher cancer risk to their neighbors. It simply says that the slightly higher risk is probably not due to chance alone. But there are a wide variety of other other related or unrelated social, environmental, or economic factors that could contribute to this difference.

Box-and-Whisker Chart

One visualization commonly used when comparing distributions (collections of numbers) is a box-and-whisker chart. The boxes show the range of values in the middle 25% to 50% to 75% of the distribution and the whiskers show the extreme high and low values.

Although Google Sheets does not provide the capability to create box-and-whisker charts, Google Sheets does have candlestick charts , which are similar to box-and-whisker charts, and which are normally used to display the range of stock price changes over a period of time.

This video shows how to create a candlestick chart comparing the distributions of cancer incidence rates. The QUARTILE() function gets the values that divide the distribution into four equally-sized parts. This shows that while the range of incidence rates in the non-nuclear counties are wider, the bulk of the rates are below the rates in nuclear counties, giving a visual demonstration of the numeric output of our t-test.

While categorical data can often be reduced to dichotomous data and used with proportions tests or t-tests, there are situations where you are sampling data that falls into more than two categories and you would like to make hypothesis tests about those categories. This tutorial describes a group of tests that can be used with that type of data.

Two-Sample T-Test

When comparing means of values from two different groups in your sample, a two-sample t-test is in order.

The two-sample t-test tests the significance of the difference between the means of two different samples.

Two normally-distributed, continuous or discrete sampled variables, OR
A normally-distributed continuous or sampled variable and a parallel dichotomous variable indicating what group each of the values in the first variable belong to
Null hypothesis (H 0 ): The means of the two sampled distributions are equal.

For example, given the low incomes and delicious foods prevalent in Mississippi, we might presume that average weight in Mississippi would be higher than in Illinois.

We test a hypothesis that the mean weight in IL in 2020 is less than the 2020 mean weight in Mississippi.

The low p-value leads us to reject the null hypothesis and corroborate our alternative hypothesis that mean weight in Illinois is less than in Mississippi.

While the difference in means is statistically significant, it is small (182 vs. 187), which should lead to caution in interpretation that you avoid using your analysis simply to reinforce unhelpful stigmatization.

Wilcoxen Rank Sum Test (Mann-Whitney U-Test)

The Wilcoxen rank sum test tests the significance of the difference between the means of two different samples. This is a non-parametric alternative to the t-test.

Data: Two continuous sampled variables
Non-parametric (normal or non-normal distributions)
R Function: wilcox.test()
Null hypothesis (H 0 ): For randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X.
History: Frank Wilcoxon (1945) and Henry Mann and Donald Whitney (1947)

The test is is implemented with the wilcox.test() function.

When the test is performed on one sample in comparison to an expected value around which the distribution is symmetrical (μ), the test is known as a Mann-Whitney U test .
When the test is performed to compare two samples, the test is known as a Wilcoxon rank sum test .

For this example, we will use AVEDRNK3: During the past 30 days, on the days when you drank, about how many drinks did you drink on the average?

1 - 76: Number of drinks
77: Don’t know/Not sure
99: Refused
NA: Not asked or Missing

The histogram clearly shows this to be a non-normal distribution.

Continuing the comparison of Illinois and Mississippi from above, we might presume that with all that warm weather and excellent food in Mississippi, they might be inclined to drink more. The means of average number of drinks per month seem to suggest that Mississippians do drink more than Illinoians.

We can test use wilcox.test() to test a hypothesis that the average amount of drinking in Illinois is different than in Mississippi. Like the t-test, the alternative can be specified as two-sided or one-sided, and for this example we will test whether the sampled Illinois value is indeed less than the Mississippi value.

The low p-value leads us to reject the null hypothesis and corroborates our hypothesis that average drinking is lower in Illinois than in Mississippi. As before, this tells us nothing about why this is the case.

Weighted Two-Sample T-Test

The downloadable BRFSS data is raw, anonymized survey data that is biased by uneven geographic coverage of survey administration (noncoverage) and lack of responsiveness from some segments of the population (nonresponse). The X_LLCPWT field (landline, cellphone weighting) is a weighting factor added by the CDC that can be assigned to each response to compensate for these biases.

The wtd.t.test() function from the weights library has a weights parameter that can be used to include a weighting factor as part of the t-test.

Comparing Proportions: Tests with Categorical Data

Chi-squared goodness of fit.

Tests the significance of the difference between sampled frequencies of different values and expected frequencies of those values
Data: A categorical sampled variable and a table of expected frequencies for each of the categories
R Function: chisq.test()
Null hypothesis (H 0 ): The relative proportions of categories in one variable are different from the expected proportions
History: Karl Pearson (1900)
Example Question: Are the voting preferences of voters in my district significantly different from the current national polls?

For example, we test a hypothesis that smoking rates changed between 2000 and 2020.

In 2000, the estimated rate of adult smoking in Illinois was 22.3% (Illinois Department of Public Health 2004) .

The variable we will use is SMOKDAY2: Do you now smoke cigarettes every day, some days, or not at all?

1: Current smoker - now smokes every day
2: Current smoker - now smokes some days
3: Not at all
7: Don't know
NA: Not asked or missing - NA is used for people who have never smoked

We subset only yes/no responses in Illinois and convert into a dummy variable (yes = 1, no = 0).

The listing of the table as percentages indicates that smoking rates were halved between 2000 and 2020, but since this is sampled data, we need to run a chi-squared test to make sure the difference can't be explained by the randomness of sampling.

In this case, the very low p-value leads us to reject the null hypothesis and corroborates the alternative hypothesis that smoking rates changed between 2000 and 2020.

Chi-Squared Contingency Analysis / Test of Independence

Tests the significance of the difference between frequencies between two different groups
Data: Two categorical sampled variables
Null hypothesis (H 0 ): The relative proportions of one variable are independent of the second variable.

We can also compare categorical proportions between two sets of sampled categorical variables.

The chi-squared test can is used to determine if two categorical variables are independent. What is passed as the parameter is a contingency table created with the table() function that cross-classifies the number of rows that are in the categories specified by the two categorical variables.

The null hypothesis with this test is that the two categories are independent. The alternative hypothesis is that there is some dependency between the two categories.

For this example, we can compare the three categories of smokers (daily = 1, occasionally = 2, never = 3) across the two categories of states (Illinois and Mississippi).

The low p-value leads us to reject the null hypotheses that the categories are independent and corroborates our hypotheses that smoking behaviors in the two states are indeed different.

p-value = 1.516e-09

Weighted Chi-Squared Contingency Analysis

As with the weighted t-test above, the weights library contains the wtd.chi.sq() function for incorporating weighting into chi-squared contingency analysis.

As above, the even lower p-value leads us to again reject the null hypothesis that smoking behaviors are independent in the two states.

Suppose that the Macrander campaign would like to know how partisan this election is. If people are largely choosing to vote along party lines, the campaign will seek to get their base voters out to the polls. If people are splitting their ticket, the campaign may focus their efforts more broadly.

In the example below, the Macrander campaign took a small poll of 30 people asking who they wished to vote for AND what party they most strongly affiliate with.

The output of table() shows fairly strong relationship between party affiliation and candidates. Democrats tend to vote for Macrander, while Republicans tend to vote for Stewart, while independents all vote for Miller.

This is reflected in the very low p-value from the chi-squared test. This indicates that there is a very low probability that the two categories are independent. Therefore we reject the null hypothesis.

In contrast, suppose that the poll results had showed there were a number of people crossing party lines to vote for candidates outside their party. The simulated data below uses the runif() function to randomly choose 50 party names.

The contingency table() shows no clear relationship between party affiliation and candidate. This is validated quantitatively by the chi-squared test. The fairly high p-value of 0.4018 indicates a 40% chance that the two categories are independent. Therefore, we fail to reject the null hypothesis and the campaign should focus their efforts on the broader electorate.

The warning message given by the chisq.test() function indicates that the sample size is too small to make an accurate analysis. The simulate.p.value = T parameter adds Monte Carlo simulation to the test to improve the estimation and get rid of the warning message. However, the best way to get rid of this message is to get a larger sample.

Comparing Categorical and Continuous Variables

Analysis of variation (anova).

Analysis of Variance (ANOVA) is a test that you can use when you have a categorical variable and a continuous variable. It is a test that considers variability between means for different categories as well as the variability of observations within groups.

There are a wide variety of different extensions of ANOVA that deal with covariance (ANCOVA), multiple variables (MANOVA), and both of those together (MANCOVA). These techniques can become quite complicated and also assume that the values in the continuous variables have a normal distribution.

Data: One or more categorical (independent) variables and one continuous (dependent) sampled variable
R Function: aov()
Null hypothesis (H 0 ): There is no difference in means of the groups defined by each level of the categorical (independent) variable
History: Ronald Fisher (1921)
Example Question: Do low-, middle- and high-income people vary in the amount of time they spend watching TV?

As an example, we look at the continuous weight variable (WEIGHT2) split into groups by the eight income categories in INCOME2: Is your annual household income from all sources?

1: Less than $10,000
2: $10,000 to less than $15,000
3: $15,000 to less than $20,000
4: $20,000 to less than $25,000
5: $25,000 to less than $35,000
6: $35,000 to less than $50,000
7: $50,000 to less than $75,000)
8: $75,000 or more

The barplot() of means does show variation among groups, although there is no clear linear relationship between income and weight.

To test whether this variation could be explained by randomness in the sample, we run the ANOVA test.

The low p-value leads us to reject the null hypothesis that there is no difference in the means of the different groups, and corroborates the alternative hypothesis that mean weights differ based on income group.

However, it gives us no clear model for describing that relationship and offers no insights into why income would affect weight, especially in such a nonlinear manner.

Suppose you are performing research into obesity in your city. You take a sample of 30 people in three different neighborhoods (90 people total), collecting information on health and lifestyle. Two variables you collect are height and weight so you can calculate body mass index . Although this index can be misleading for some populations (notably very athletic people), ordinary sedentary people can be classified according to BMI:

Average BMI in the US from 2007-2010 was around 28.6 and rising, standard deviation of around 5 .

You would like to know if there is a difference in BMI between different neighborhoods so you can know whether to target specific neighborhoods or make broader city-wide efforts. Since you have more than two groups, you cannot use a t-test().

Kruskal-Wallace One-Way Analysis of Variance

A somewhat simpler test is the Kruskal-Wallace test which is a nonparametric analogue to ANOVA for testing the significance of differences between two or more groups.

R Function: kruskal.test()
Null hypothesis (H 0 ): The samples come from the same distribution.
History: William Kruskal and W. Allen Wallis (1952)

For this example, we will investigate whether mean weight varies between the three major US urban states: New York, Illinois, and California.

To test whether this variation could be explained by randomness in the sample, we run the Kruskal-Wallace test.

The low p-value leads us to reject the null hypothesis that the samples come from the same distribution. This corroborates the alternative hypothesis that mean weights differ based on state.

A convienent way of visualizing a comparison between continuous and categorical data is with a box plot , which shows the distribution of a continuous variable across different groups:

A percentile is the level at which a given percentage of the values in the distribution are below: the 5th percentile means that five percent of the numbers are below that value.

The quartiles divide the distribution into four parts. 25% of the numbers are below the first quartile. 75% are below the third quartile. 50% are below the second quartile, making it the median.

Box plots can be used with both sampled data and population data.

The first parameter to the box plot is a formula: the continuous variable as a function of (the tilde) the second variable. A data= parameter can be added if you are using variables in a data frame.

The chi-squared test can be used to determine if two categorical variables are independent of each other.

Hypothesis Testing in R Programming

Hypothesis testing in R programming involves making statistical inferences about data sets. It helps to assess the validity of assumptions and draw conclusions based on sample data. Key steps include formulating null and alternative hypotheses, choosing an appropriate test, calculating test statistics, and determining p-values. R offers a range of functions like t.test(), chisq.test(), and others to perform hypothesis tests. By comparing results with significance levels, researchers can accept or reject hypotheses, providing valuable insights into the population from which the data was collected.

Introduction

Hypothesis testing is a fundamental concept in statistical analysis that allows researchers to make informed decisions based on sample data. In the context of R programming, it becomes a powerful tool to draw meaningful conclusions about populations from which the data is collected.

The process of hypothesis testing involves two competing statements: the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis represents the status quo or the assumption that there is no significant difference or relationship between variables, while the alternative hypothesis suggests otherwise. The goal of hypothesis testing is to either support or refute the null hypothesis based on the evidence in the data.

R, as a popular programming language for statistical computing and data analysis, provides a wide range of functions and packages to conduct various hypothesis tests. Whether dealing with means, proportions, variances, or relationships between categorical variables, R offers a diverse set of statistical tests, including t-tests, chi-square tests, ANOVA, regression analysis, and more.

The process of hypothesis testing in R generally involves the following steps: formulating the null and alternative hypotheses, selecting an appropriate test based on data type and assumptions, calculating the test statistic, determining the p-value (the probability of observing the data under the null hypothesis), and comparing the p-value to a pre-defined significance level (alpha). If the p-value is less than alpha, the null hypothesis is rejected in favor of the alternative hypothesis.

What is Hypothesis Testing in R ?

Hypothesis testing in R is a statistical method used to draw conclusions about populations based on sample data. It involves testing a hypothesis or a claim made about a population parameter, such as the population mean, proportion, variance, or correlation. The process of hypothesis testing in R follows a systematic approach to determine if there is enough evidence in the data to support or reject a particular claim.

The two main hypotheses involved in hypothesis testing are the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis represents the default assumption, suggesting that there is no significant difference or effect in the population. The alternative hypothesis, on the other hand, proposes that there is a meaningful relationship or effect in the population.

Types of Statistical Hypothesis testing

Null hypothesis.

The null hypothesis, often denoted as H0, is a fundamental concept in hypothesis testing. It represents the default assumption or status quo about a population parameter, such as the population mean, proportion, variance, or correlation. In simple terms, it suggests that there is no significant difference, effect, or relationship between variables under investigation.

When conducting a hypothesis test, researchers or analysts start by assuming the null hypothesis is true. They then collect sample data and perform statistical tests to determine if there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis (Ha). The alternative hypothesis represents the claim or the proposition that contradicts the null hypothesis.

The decision to accept or reject the null hypothesis is based on the results of the statistical test and the calculation of a p-value. The p-value represents the probability of obtaining the observed data, or more extreme data, assuming that the null hypothesis is true. If the p-value is lower than a pre-defined significance level (alpha), typically 0.05, then there is enough evidence to reject the null hypothesis and accept the alternative hypothesis.

If the p-value is higher than the significance level, there is insufficient evidence to reject the null hypothesis, and researchers must maintain the default assumption that there is no significant effect or difference in the population.

Alternative Hypothesis

In R, the alternative hypothesis, often denoted as Ha or H1, is a complementary statement to the null hypothesis (H0) in hypothesis testing. While the null hypothesis assumes that there is no significant effect, difference, or relationship between variables in the population, the alternative hypothesis proposes otherwise. It represents the claim or hypothesis that researchers or analysts are trying to find evidence for.

The alternative hypothesis can take different forms, depending on the nature of the research question and the statistical test being performed. There are three main types of alternative hypotheses:

Two-tailed (or two-sided) alternative hypothesis: This form of the alternative hypothesis states that there is a significant difference between groups or a relationship between variables, without specifying the direction of the effect. It is often used in tests such as t-tests or correlation analysis when researchers are interested in detecting any kind of difference or relationship.
One-tailed (or one-sided) alternative hypothesis: This form of the alternative hypothesis specifies the direction of the effect. It indicates that there is either a positive or negative effect, but not both. One-tailed tests are used when researchers have a specific directional expectation or hypothesis.
Non-directional (or two-directional) alternative hypothesis: This form of the alternative hypothesis is similar to the two-tailed alternative but is used in non-parametric tests or situations where a direction cannot be determined.

Error Types

In the context of hypothesis testing and statistical analysis in R, there are two main types of errors that can occur: Type I error (False Positive) and Type II error (False Negative). These errors are associated with the acceptance or rejection of the null hypothesis based on the results of a hypothesis test.

Type I Error (False Positive): A Type I error occurs when the null hypothesis (H0) is wrongly rejected when it is actually true. In other words, it is the incorrect conclusion that there is a significant effect or difference in the population when, in reality, there is no such effect. The probability of committing a Type I error is denoted by the significance level (alpha) of the test, typically set at 0.05 or 5%. A lower significance level reduces the chances of Type I errors but increases the risk of Type II errors.
Type II Error (False Negative): A Type II error occurs when the null hypothesis (H0) is erroneously accepted when it is actually false. It means that the test fails to detect a significant effect or difference that exists in the population. The probability of committing a Type II error is denoted by the symbol beta (β). The power of a statistical test is equal to 1 - β and represents the test's ability to correctly reject a false null hypothesis.

The trade-off between Type I and Type II errors is common in hypothesis testing. Lowering the significance level (alpha) to reduce the risk of Type I errors often leads to an increase in the risk of Type II errors. Finding an appropriate balance between these error types depends on the research question and the consequences of making each type of error.

Processes in Hypothesis Testing

Hypothesis testing is a crucial statistical method used to draw meaningful conclusions from sample data about a larger population. In the context of R programming, hypothesis testing involves a systematic set of processes that guide researchers or data analysts through the evaluation of hypotheses and making data-driven decisions. Four Step Process of Hypothesis Testing

State the hypothesis The first step is to clearly state the null hypothesis (H0) and the alternative hypothesis (Ha) based on the research question or problem. The null hypothesis represents the status quo or the assumption of no significant effect or difference, while the alternative hypothesis proposes a specific effect, relationship, or difference that researchers want to investigate.

For example: H0: There is no significant difference in the mean weight of apples from two different orchards. Ha: There is a significant difference in the mean weight of apples from two different orchards.

Formulate an Analysis Plan and Set the Criteria for Decision In this step, you need to choose an appropriate statistical test based on the data type, research question, and assumptions. You also set the significance level (alpha), which determines the probability of committing a Type I error (rejecting a true null hypothesis).

For example: Test: We will use a two-sample t-test to compare the mean weights of apples from two orchards. Significance level (alpha): α = 0.05 (commonly used)

Analyze Sample Data Using R, you collect and input the sample data for analysis. In this case, you would have data on the weights of apples from both orchards. Next, you use the appropriate function to conduct the chosen statistical test.

Interpret Decision After conducting the test in R, you will obtain the test statistic, degrees of freedom, and the p-value. The p-value represents the probability of obtaining the observed data (or more extreme data) under the assumption that the null hypothesis is true.

If the p-value is less than or equal to the significance level (alpha), which in this case is 0.05, you reject the null hypothesis in favor of the alternative hypothesis. It indicates that there is enough evidence to conclude that there is a significant difference in the mean weights of apples from the two orchards.

If the p-value is greater than the significance level, you fail to reject the null hypothesis. It suggests that there is insufficient evidence to conclude that there is a significant difference in the mean weights of apples from the two orchards.

Interpreting the results correctly is crucial to making informed decisions based on the data and drawing meaningful conclusions.

One Sample T-Testing

One-sample t-test is a type of hypothesis test in R used to compare the mean of a single sample to a known value or a hypothesized population mean. It is typically employed when you have a single sample and want to determine if the sample mean is significantly different from a specific value or a theoretical mean. In the context of hypothesis testing in R, the one-sample t-test assumes a normal distribution of the sample data. This assumption is crucial for accurate interpretation of the results. The one-sample t-test evaluates whether the mean of a single sample is significantly different from a hypothesized population mean. The underlying assumption of normality ensures that the sampling distribution of the sample mean follows a bell-shaped curve, which is a prerequisite for valid t-test results. If the data deviates substantially from normality, the reliability of the test's outcomes may be compromised, and alternative methods might be more appropriate for analysis. Therefore, when conducting a one-sample t-test in R, it's important to consider the normality assumption and, if necessary, explore the data distribution and potentially apply appropriate transformations or alternative tests if the assumption is not met.

In R, you can perform a one-sample t-test using the t.test() function. Here's the basic syntax of the one-sample t-test in R:

sample_vector: This is the numeric vector containing the sample data for which you want to conduct the t-test.
mu: This is the hypothesized population mean. It represents the value you are comparing the sample mean against. The default value is 0, which implies a test for a sample mean of zero (i.e., testing if the sample is significantly different from zero).

Let's look at an example of a one-sample t-test in R:

The output will provide information such as the sample mean, the hypothesized mean, the test statistic (t-value), the degrees of freedom, and the p-value. The p-value is the key factor in determining the significance of the test. If the p-value is less than the chosen significance level (commonly 0.05), you can reject the null hypothesis and conclude that there is a significant difference between the sample mean and the hypothesized mean. If the p-value is greater than the significance level, you fail to reject the null hypothesis, suggesting that there is no significant difference between the sample mean and the hypothesized mean.

One-sample t-tests are useful when you want to examine if a sample is significantly different from a specific value or when comparing the sample to a theoretical value based on prior knowledge or established standards.

Two Sample T-Testing

Two-sample t-test is a statistical method used in hypothesis testing to compare the means of two independent samples and determine if they come from populations with different average values. It is commonly employed when you want to assess whether there is a significant difference between two groups or conditions.

In R, you can perform a two-sample t-test using the t.test() function. There are two types of two-sample t-tests, depending on whether the variances of the two samples are assumed to be equal or not:

Two-sample t-test with equal variances (also known as the "pooled" t-test):

2.Two-sample t-test with unequal variances (also known as the "Welch's" t-test):

sample1 and sample2: These are the numeric vectors containing the data for the two independent samples that you want to compare.
var.equal: This argument determines whether the variances of the two samples are assumed to be equal (TRUE) or not (FALSE). If var.equal = TRUE, the pooled t-test is performed, and if var.equal = FALSE, the Welch t-test is used. Here's an example of performing a two-sample t-test in R:

The output will include information such as the sample means, the test statistic (t-value), the degrees of freedom, and the p-value. The p-value is essential in determining the significance of the test. If the p-value is less than the chosen significance level (commonly 0.05), you can reject the null hypothesis and conclude that there is a significant difference between the means of the two groups. If the p-value is greater than the significance level, you fail to reject the null hypothesis, suggesting that there is no significant difference between the means of the two groups.

The choice between the pooled t-test and the Welch t-test depends on the assumption of equal or unequal variances between the two groups. If you are unsure about the equality of variances, it is safer to use the Welch t-test, as it provides a more robust and accurate approach when variances differ between the groups.

Directional Hypothesis In hypothesis testing, a directional hypothesis (also known as one-tailed hypothesis) is a type of alternative hypothesis (Ha) that specifies the direction of the effect or difference between groups. It is used when researchers have a specific expectation or prediction about the relationship between variables, and they want to test whether the effect occurs in a particular direction.

There are two types of directional hypotheses:

One-tailed hypothesis with a greater-than sign (>) indicates an expectation of a positive effect or a difference in one direction. Example: Ha: The mean score of Group A is greater than the mean score of Group B.
One-tailed hypothesis with a less-than sign (<) indicates an expectation of a negative effect or a difference in the opposite direction. Example: Ha: The mean score of Group A is less than the mean score of Group B.

In R, when conducting a hypothesis test with a directional hypothesis, you need to specify the alternative hypothesis accordingly in the t.test() function (or other relevant functions for different tests). The alternative hypothesis is set using the alternative argument.

Here's an example of performing a one-tailed t-test in R with a directional hypothesis:

The output will include information such as the test statistic (t-value), degrees of freedom, and the p-value. If the p-value is less than the chosen significance level (commonly 0.05) and the direction specified in the alternative hypothesis is consistent with the results, you can reject the null hypothesis in favor of the directional hypothesis. If the p-value is greater than the significance level, or if the direction specified in the alternative hypothesis does not match the results, you fail to reject the null hypothesis.

Directional hypotheses are useful when researchers have a specific expectation about the outcome of the study and want to test that particular expectation. However, it is essential to have a strong theoretical or empirical basis for formulating directional hypotheses, as it reduces the scope of the test and may lead to Type I or Type II errors if the direction is chosen arbitrarily.

Directional Hypothesis

In R, when performing hypothesis tests with a directional hypothesis (one-tailed hypothesis), you can specify the alternative hypothesis using the alternative argument in the relevant statistical test function. Let's go through an example using the t.test() function for a one-tailed t-test.

Suppose we have two groups of exam scores: Group A and Group B. We want to test whether the mean score of Group A is greater than the mean score of Group B.

Here's how you can conduct a one-tailed t-test in R:

In this code, we set alternative = "greater" to specify the directional hypothesis. The output will include information such as the test statistic (t-value), degrees of freedom, and the p-value.

The interpretation of the result is as follows:

If the p-value is less than the chosen significance level (e.g., 0.05), and the direction specified in the alternative hypothesis (mean of groupA is greater than groupB) is consistent with the results, you can reject the null hypothesis in favor of the directional hypothesis.
If the p-value is greater than the significance level, or if the direction specified in the alternative hypothesis does not match the results, you fail to reject the null hypothesis.

Remember that when using a directional hypothesis, you are testing a specific expectation, and the choice of direction should be based on strong theoretical or empirical reasoning. Using directional hypotheses should be done thoughtfully and not arbitrarily, as it narrows the scope of the test and may lead to incorrect conclusions if not supported by solid evidence.

One Sample µ-Test

In hypothesis testing, a one-sample µ-test (mu-test) is used to compare the mean of a single sample to a known value or a hypothesized population mean (µ). It is commonly employed when you have a single sample and want to determine if the sample mean is significantly different from a specific value or a theoretical mean.

In R, you can perform a one-sample µ-test using the t.test() function. The argument mu is used to specify the hypothesized population mean (µ) that you want to compare the sample mean against. The default value of mu is 0, which implies a test for a sample mean of zero (i.e., testing if the sample mean is significantly different from zero).

Here's the basic syntax of the one-sample µ-test in R:

sample_vector: This is the numeric vector containing the sample data for which you want to conduct the one-sample µ-test.
mu: This is the hypothesized population mean (µ) that you want to compare the sample mean against. Let's look at an example of performing a one-sample µ-test in R:

The output will provide information such as the sample mean, the hypothesized mean (µ), the test statistic (t-value), the degrees of freedom, and the p-value. The p-value is crucial in determining the significance of the test. If the p-value is less than the chosen significance level (commonly 0.05), you can reject the null hypothesis and conclude that the sample mean is significantly different from the hypothesized mean. If the p-value is greater than the significance level, you fail to reject the null hypothesis, suggesting that there is no significant difference between the sample mean and the hypothesized mean.

The one-sample µ-test is useful when you want to examine if a sample mean is significantly different from a specific value or when comparing the sample to a theoretical value based on prior knowledge or established standards.

Two Sample µ-Test

In hypothesis testing, a two-sample µ-test (mu-test) is used to compare the means of two independent samples and determine if they come from populations with different average values (µ). It is commonly employed when you want to assess whether there is a significant difference between two groups or conditions.

In R, you can perform a two-sample µ-test using the t.test() function. The function allows you to compare the means of two groups, assuming their variances are either equal or unequal, depending on the var.equal argument.

Here's the basic syntax of the two-sample µ-test in R:

var.equal: This argument determines whether the variances of the two samples are assumed to be equal (TRUE) or not (FALSE). If var.equal = TRUE, the pooled t-test is performed, and if var.equal = FALSE, the Welch t-test (unequal variance t-test) is used. Let's look at an example of performing a two-sample µ-test in R:

The output will include information such as the sample means, the test statistic (t-value), degrees of freedom, and the p-value. The p-value is essential in determining the significance of the test. If the p-value is less than the chosen significance level (commonly 0.05), you can reject the null hypothesis and conclude that there is a significant difference between the means of the two groups. If the p-value is greater than the significance level, you fail to reject the null hypothesis, suggesting that there is no significant difference between the means of the two groups.

Correlation Test

In R, you can perform correlation tests to measure the strength and direction of the linear relationship between two numeric variables. The most common correlation test is the Pearson correlation coefficient, which quantifies the degree of linear association between two variables. The correlation coefficient can take values between -1 and 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no linear correlation.

To perform a correlation test in R, you can use the cor.test() function. Here's the basic syntax:

x and y: These are the numeric vectors representing the two variables for which you want to calculate the correlation coefficient.
method: This argument specifies the correlation method to use. The default is "pearson," but you can also choose other methods like "spearman" for Spearman's rank correlation or "kendall" for Kendall's rank correlation. Here's an example of performing a Pearson correlation test in R:

The output will include information such as the correlation coefficient, the test statistic (t-value), degrees of freedom, and the p-value. The p-value is essential in determining the significance of the correlation. If the p-value is less than the chosen significance level (commonly 0.05), you can conclude that there is a significant linear correlation between the two variables. If the p-value is greater than the significance level, you fail to reject the null hypothesis, suggesting that there is no significant linear correlation.

Keep in mind that correlation does not imply causation. A significant correlation between two variables means they are associated, but it does not necessarily mean one variable causes the other.

Correlation tests are valuable tools for exploring relationships between variables and understanding the strength and direction of their associations in data analysis and research studies.

Hypothesis testing is a fundamental statistical method used in R to draw meaningful conclusions about populations based on sample data.
R provides a wide range of functions and packages to perform various hypothesis tests, including t-tests, chi-square tests, ANOVA, correlation tests, and more.
The hypothesis testing process involves formulating null and alternative hypotheses, selecting an appropriate test, calculating test statistics, and determining p-values.
The p-value is a critical measure in hypothesis testing, representing the probability of obtaining the observed data under the null hypothesis.
Researchers set a significance level (alpha) to determine the threshold for accepting or rejecting the null hypothesis based on the p-value.
The choice of one-tailed or two-tailed tests depends on the research question and whether directional expectations exist.
Hypothesis testing allows researchers to make data-driven decisions, validate assumptions, and draw conclusions about populations from sample data.
It is important to interpret results in the context of the research question and consider the potential impact of Type I and Type II errors.
Assumes equal variances between two samples and Suitable when there's confidence in equal variability.
Accounts for potentially different variances between two samples and More robust when variances differ significantly.

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Hypothesis Testing in R: A Comprehensive Guide with Code and Examples

Hypothesis testing is a fundamental statistical technique used to make inferences about population parameters based on sample data. It helps us determine whether an observed effect or difference is statistically significant or if it could have occurred by random chance. In this guide, we’ll explore hypothesis testing in R, a powerful statistical programming language, through practical examples and code snippets.

Understanding the Hypothesis Testing Process

Before diving into code examples, let’s grasp the key concepts of hypothesis testing:

Null Hypothesis (H0): This is the default assumption that there is no significant effect, difference, or relationship in the population. It’s often denoted as H0.
Alternative Hypothesis (Ha): This is the statement we want to test; it asserts that there is a significant effect, difference, or relationship in the population. It’s often denoted as Ha.
Significance Level (α): This is the predetermined threshold that defines when we reject the null hypothesis. Common values are 0.05 or 0.01, representing a 5% or 1% chance of making a Type I error (false positive), respectively.
Test Statistic: A statistic calculated from the sample data that measures the strength of evidence against the null hypothesis.
P-value: The probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data under the null hypothesis. A smaller p-value suggests stronger evidence against the null hypothesis.
Decision Rule: Based on the p-value, we decide whether to reject the null hypothesis. If the p-value is less than α, we reject H0; otherwise, we fail to reject it.

Now, let’s explore some practical examples of hypothesis testing in R.

Example 1: One-Sample T-Test

Suppose we have a dataset of exam scores, and we want to test if the average score is significantly different from 75.

In this example, we perform a one-sample t-test to determine if the sample mean is significantly different from 75. The resulting p-value will help us make the decision.

Example 2: Two-Sample T-Test

Let’s say we want to compare the exam scores of two different classes (Class A and Class B) to see if there is a significant difference between their average scores.

Here, we perform a two-sample t-test to compare the means of two independent samples (Class A and Class B).

Example 3: Chi-Square Test of Independence

Suppose we have data on the preferred mode of transportation for two groups of people (Group X and Group Y), and we want to test if there is an association between the groups and their transportation preferences.

In this example, we use a chi-square test to determine if there is an association between the groups and their transportation preferences.

Hypothesis testing is a powerful tool for making data-driven decisions in various fields, from medicine to business. In R, you can conduct a wide range of hypothesis tests using built-in functions and libraries like t.test() and chisq.test() . Remember to set your significance level appropriately, and interpret the results cautiously based on the p-value.

By mastering hypothesis testing in R, you’ll be better equipped to draw meaningful conclusions from your data and make informed decisions in your research and analysis.

By Benard Mbithi

A statistics graduate with a knack for crafting data-powered business solutions. I assist businesses in overcoming challenges and achieving their goals through strategic data analysis and problem-solving expertise.

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The Complete Guide: Hypothesis Testing in R - Statology
Jun 8, 2021 · A hypothesis test is a formal statistical test we use to reject or fail to reject some statistical hypothesis. This tutorial explains how to perform the following hypothesis tests in R: One sample t-test; Two sample t-test; Paired samples t-test; We can use the t.test() function in R to perform each type of test:
Hypothesis Testing in R Programming - GeeksforGeeks
Aug 8, 2024 · A hypothesis is an assumption made by the researchers that are not mandatory true. In simple words, a hypothesis is a decision taken by the researchers based on the data of the population collected. Hypothesis Testing in R Programming is a process of testing the hypothesis made by the researcher or to validate the hypothesis. To perform ...
Introduction to Hypothesis Testing in R - Learn every concept ...
Hypothesis Testing in R. Statisticians use hypothesis testing to formally check whether the hypothesis is accepted or rejected. Hypothesis testing is conducted in the following manner: State the Hypotheses – Stating the null and alternative hypotheses. Formulate an Analysis Plan – The formulation of an analysis plan is a crucial step in ...
The Complete Guide: Hypothesis Testing in R | Online ...
Jan 17, 2023 · A hypothesis test is a formal statistical test we use to reject or fail to reject some statistical hypothesis. This tutorial explains how to perform the following hypothesis tests in R: One sample t-test; Two sample t-test; Paired samples t-test; We can use the t.test() function in R to perform each type of test:
Hypothesis Testing in R Programming - MAKE ME ANALYST
Hypothesis testing is a statistical method used to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In R programming, you can perform various types of hypothesis tests, such as t-tests, chi-squared tests, and ANOVA tests, among others. In R programming, you can perform hypothesis ...
Hypothesis Tests in R - michaelminn.net
The null hypothesis (H 0) states that the factor has no effect. The null hypothesis is the opposite of the alternative hypothesis. The null hypothesis is what we are testing when we perform a hypothesis test. Hypothesis test flow chart. The output of a statistical test like the t-test is a p-value.
Hypothesis Testing. With examples in R | by Md Sohel Mahmood ...
Mar 27, 2024 · Hypothesis testing is a fundamental concept in statistics used to make inferences about populations based on sample data. It involves formulating a hypothesis about the population parameter ...
Hypothesis Testing in R Programming - Scaler Topics
Sep 6, 2023 · Hypothesis testing is a crucial statistical method used to draw meaningful conclusions from sample data about a larger population. In the context of R programming, hypothesis testing involves a systematic set of processes that guide researchers or data analysts through the evaluation of hypotheses and making data-driven decisions.
How to Perform Hypothesis Testing in R using T-tests and μ ...
Steps Involved in Hypothesis Testing Process. Hypothesis testing involves the following steps: 1. Stating the hypothesis. The first step in hypothesis testing is to state the null as well as an alternative hypothesis. We have to come up with a hypothesis that gives us suitable information about the data.
Hypothesis Testing in R: A Comprehensive Guide with Code and ...
Aug 21, 2023 · Hypothesis testing is a powerful tool for making data-driven decisions in various fields, from medicine to business. In R, you can conduct a wide range of hypothesis tests using built-in functions and libraries like t.test() and chisq.test(). Remember to set your significance level appropriately, and interpret the results cautiously based on ...

The Complete Guide: Hypothesis Testing in R

Example 1: One Sample t-test in R

Example 2: Two Sample t-test in R

Example 3: Paired Samples t-test in R

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The Complete Guide: Hypothesis Testing in R

Example 1: One Sample t-test in R

Example 2: Two Sample t-test in R

Example 3: Paired Samples t-test in R

Additional Resources

How to Calculate Mode from Frequency Table (With Examples)

How to Plot a Weibull Distribution in R

1. One-sample t-test:

2. Two-sample t-test:

3. Paired t-test:

4. Chi-squared test:

5. One-way ANOVA

6. Wilcoxon signed-rank test

Steps for conducting a Hypothesis Testing

1. Prepare your data:

2. State your null and alternative hypotheses:

3. Perform the t-test:

4. Interpret the results:

5. Make a decision:

Few more examples of hypothesis tests using R

T-Test in R Programming

Hypothesis Tests in R

Hypothesis Testing

The Problem of Induction

Falsification

Null and Alternative Hypotheses

Type I vs. Type II Errors

Statistical Significance vs. Importance

Science vs. Non-science

Example Data

Variable Types

Normality Tests

The Shapiro-Wilk Normality Test

The Kolmogorov-Smirnov Test

Modality Tests of Samples

One Sample T-Test (One-Sided)

Box-and-Whisker Chart

Two-Sample T-Test

Wilcoxen Rank Sum Test (Mann-Whitney U-Test)

Weighted Two-Sample T-Test

Comparing Proportions: Tests with Categorical Data

Chi-Squared Contingency Analysis / Test of Independence

Weighted Chi-Squared Contingency Analysis

Comparing Categorical and Continuous Variables

Kruskal-Wallace One-Way Analysis of Variance

Hypothesis Testing in R Programming

Introduction

What is Hypothesis Testing in R ?

Types of Statistical Hypothesis testing

Alternative Hypothesis