11 Independent Samples T - Test

The table below provides a comprehensive comparison between dependent and independent samples, including their definitions, examples, hypothesis testing, assumptions, statistical tests, effect size measures, and R functions. By understanding these differences, we can choose the appropriate statistical test for their data and interpret the results correctly.

Feature	Dependent Samples	Independent Samples
Definition	Dependent samples are paired or related observations collected from the same individuals or matched pairs.	Independent samples consist of observations collected from two separate and unrelated groups.
Examples	- Pre-test and post-test scores from the same individuals. <br> - Scores from matched pairs of individuals (e.g., siblings, twins).	- Test scores from two different groups of students taught using different teaching methods. <br> - Scores from two groups of participants exposed to different conditions in an experiment.
Hypothesis Testing	The null hypothesis states that there is no significant difference between the means of the paired differences. <br> H₀: μ₁ - μ₂ = 0	The null hypothesis states that there is no significant difference between the means of the two independent groups. <br> H₀: μ₁ = μ₂
Assumptions	- The differences between the paired observations should be approximately normally distributed. <br> - The observations within each pair should be related or matched.	- The observations in each group must be independent of each other. <br> - The data in each group should be approximately normally distributed. <br> - The variances of the two groups should be approximately equal.
Statistical Test	Paired samples t-test	Independent samples t-test (or Welch’s t-test if equal variances assumption is not met)
Effect Size	Cohen’s d or the point-biserial correlation for paired samples	Cohen’s d or the point-biserial correlation for independent samples
R Function	t.test(x, y, paired = TRUE)	t.test(x, y) (or t.test(x, y, var.equal = FALSE) for Welch’s t-test)

11.1 Independent Samples t-test

The independent samples t-test is used to compare the means of two independent groups to determine if there is a significant difference between them.

The independent samples t-test is based on the following null (H₀) and alternative (H₁) hypotheses:

H₀: μ₁ = μ₂ (There is no significant difference between the means of the two groups.) H₁: μ₁ ≠ μ₂ (There is a significant difference between the means of the two groups.) The test statistic for the independent samples t-test is the t-value, which is calculated using the following formula:

t = (M₁ - M₂) / sqrt((s₁²/n₁) + (s₂²/n₂))

where:

M₁ and M₂ are the means of the two groups s₁² and s₂² are the variances of the two groups n₁ and n₂ are the sample sizes of the two groups The t-value follows a t-distribution with degrees of freedom (df) approximated by the following formula:

df = min(n₁ - 1, n₂ - 1)

Once the t-value and degrees of freedom are calculated, the p-value can be determined by comparing the t-value to the t-distribution with the appropriate degrees of freedom. If the p-value is less than the chosen significance level (e.g., 0.05), the null hypothesis can be rejected, indicating a significant difference between the means of the two groups.

11.1.1 Independent t-test using R

You will need data from two independent groups, typically stored in a data frame with one variable representing the group membership and another variable representing the outcome of interest.

# Example data
group <- c("A", "A", "A", "A", "A", "B", "B", "B", "B", "B")
outcome <- c(10, 12, 14, 16, 18, 20, 22, 24, 26, 28)

# Create a data frame
data <- data.frame(group, outcome)

Perform the independent samples t-test: Use the t.test() function in R, specifying the formula and the data frame as arguments.

# Perform the independent samples t-test
t_test_result <- t.test(outcome ~ group, data = data)

# Print the test result
print(t_test_result)
#> 
#>  Welch Two Sample t-test
#> 
#> data:  outcome by group
#> t = -5, df = 8, p-value = 0.001053
#> alternative hypothesis: true difference in means between group A and group B is not equal to 0
#> 95 percent confidence interval:
#>  -14.612008  -5.387992
#> sample estimates:
#> mean in group A mean in group B 
#>              14              24

The output of the t.test() function will include the t-value, degrees of freedom, p-value, and confidence interval for the difference in means. If the p-value is less than the chosen significance level (e.g., 0.05), you can reject the null hypothesis, concluding that there is a significant difference between the means of the two groups.