How to perform a T-test in R: A comprehensive guide

Statistical tests are an indispensable tool in data analysis for testing hypotheses and drawing conclusions from data sets. One of the most commonly used tests is the T-test, which helps to assess whether the mean values of two groups differ statistically significantly from each other. R, a language and environment for statistical computing and graphics, offers extensive possibilities for performing different types of T-tests. In this blog post, we will explore the different forms of T-test in R, discuss the prerequisites for their use and show alternatives.

Types of T-tests in R

In R, you can perform different types of T-tests depending on the structure of your data and the hypothesis you want to test. The main types are:

One-sample T-testChecks whether the mean value of a single sample deviates significantly from a known or hypothetical population mean value. Syntax: t.test(x, mu = 0) This is x a vector with data values and mu the hypothetical mean value of the population.

Independent two-sample T-testCompares the mean values of two independent groups to determine whether there is a significant difference between them. Syntax: t.test(x, y, paired = FALSE) x and y are vectors with data values of the two groups, assuming we have data from two groups (Group A and Group B), each representing the test scores of different students. We want to know if there is a significant difference between the mean test scores of the two groups. In this example, we have the data in wide format, i.e. we have a column/column/variable with the values for each of the two groups.

# Daten für Gruppe A und Gruppe B
daten$gruppeA <- c(88, 92, 94, 78, 88, 95)
daten$gruppeB <- c(75, 80, 79, 88, 85, 92)

# Unabhängiger Zweistichproben-T-Test
ergebnis <- t.test(gruppeA, gruppeB, data = daten)

# Ergebnis anzeigen
print(ergebnis)

It is also possible that we have data in long format, i.e. one column/column/variable with the independent grouping variable and one column/column/variable with the dependent numeric variable.

# Daten vorbereiten
daten <- data.frame(
  gruppe = c("A", "A", "A", "A","A", "A", "B", "B", "B", "B", "B", "B"),
  ergebnis = c(88, 92, 94, 78, 88, 95,75, 80, 79, 88, 85, 92)
)
# Unabhängigen Zweistichproben-T-Test durchführen
ergebnis <- t.test(ergebnis ~ gruppe, data = daten)

# Ergebnis anzeigen
print(ergebnis)

Paired T-testIs used when the data values are available in pairs, for example before and after measurements for the same subjects. Syntax: t.test(x, y, paired = TRUE). Here we normally want the data in wide format (it is also possible in long format, but then the order of the observations in the data set must be exactly right)

# Vorher- und Nachher-Daten
vorher <- c(120, 112, 123, 132, 115, 127)
nachher <- c(112, 118, 121, 128, 122, 130)

# Gepaarter T-Test
ergebnis <- t.test(vorher, nachher, paired = TRUE)

# Ergebnis anzeigen
print(ergebnis)

Requirements for the T-test

Certain requirements must be met before a T-test can be carried out:

Normal distribution: The data should be normally distributed. This is particularly important for small sample sizes. For larger samples, the T-test is more robust to deviations from the normal distribution due to the central limit theorem. We can test this using the Shapiro-Wilk test for small samples and the Kolmogorov-Smirnov test for larger samples. We can also use QQ plots or histograms to visually check the distribution.

Variance homogeneityThe variances of the groups should be equal, especially in the independent two-sample t-test. If this assumption is violated, the option var.equal = FALSE can be used to perform a Welch t-test that does not assume equality of variances.

Independence of the observations: The data values must be independent of each other, which applies in particular to the independent two-sample T-test and the one-sample T-test.

Alternatives to the T-test

If the requirements for a T-test are not met, the following alternatives can be considered:

Wilcoxon test: A non-parametric test used when the normal distribution assumption is violated. For paired samples, use wilcox.test(x, y, paired = TRUE)and for two independent samples let paired = FALSE away.

Mann-Whitney U-test: Also a non-parametric test for two independent samples, which is used instead of the independent two-sample T-test if the data are not normally distributed.

BootstrappingAnother method that can be used to assess the significance of differences between groups without requiring the assumptions of the T-test. R offers packages such as bootwhich facilitate bootstrapping.

Summary

The T-test is a flexible and powerful tool in R for analyzing differences between groups. However, it is important to understand the prerequisites for its application and to consider alternative methods if necessary. By using the T-test and its alternatives correctly, you can extract valid and meaningful results from your data.