Running Two-Sample t-Test with Unequal Sample Size in R

When dealing with statistical analyses, one common scenario is comparing the means of two different groups. The two-sample t-test is a widely used statistical method for this purpose, particularly when the groups have unequal sample sizes. In this article, we will explore how to perform a two-sample t-test with unequal sample sizes in R, including its assumptions, the necessary R functions, and interpreting the results.

Introduction to the Two-Sample t-Test

The two-sample t-test compares the means of two independent samples to determine whether there is a statistically significant difference between them. When the sample sizes of the two groups are unequal, it becomes essential to account for the unequal variance between groups. There are two types of two-sample t-tests:

• Equal Sample Size: The variances of both groups are assumed to be equal, and this is the simplest version of the two-sample t-test. (pooled t-test)
• Unequal Sample Size: If the variances are unequal, the test statistic needs to adjust for the different group sizes, and a Welch’s t-test is more appropriate. (Welch’s t-test)

Welch’s t-test is particularly useful when the sample sizes differ, as it adjusts for this imbalance by modifying the degrees of freedom.

Assumptions of the Two-Sample t-Test

Before conducting a t-test, several assumptions must be satisfied to ensure the results are valid:

• Normality: The data in both groups should be approximately normally distributed.
• Independence: The samples must be independent of each other.
• Variance Homogeneity (for equal variance t-test): The variances in both groups should be roughly equal if using the pooled t-test.
• Unequal Variances (for Welch’s t-test): If the variances are not equal, Welch’s t-test should be used.

Perform a Two-Sample t-Test in R

In R, the two-sample t-test can be performed using the t.test() function. This function allows users to specify whether they are assuming equal or unequal variances.

The basic syntax of the t.test() function is as follows:

t.test(x, y, alternative = c("two.sided", "less", "greater"), var.equal = FALSE)

Where:

• x, y are the two numeric vectors representing the two samples.
• alternative specifies the alternative hypothesis: "two.sided", "less", or "greater".
• var.equal determines whether to assume equal variances (var.equal = TRUE for pooled t-test) or unequal variances (var.equal = FALSE for Welch’s t-test).

Now we will discuss step by step implementation of Running Two-Sample t-Test with Unequal Sample Size in R Programming Language.

Step 1: Data Preparation

Before running a two-sample t-test, ensure your data is prepared. For this example, consider two sample datasets representing test scores of students from two different schools.

# Sample data for two groups with unequal sample sizes
group1 <- c(72, 75, 78, 80, 82, 85, 88)
group2 <- c(65, 67, 70, 72, 73)

Step 2: Conducting a Two-Sample t-Test

To run a two-sample t-test assuming unequal variances (Welch’s t-test), use the following code:

t_test_result <- t.test(group1, group2, var.equal = FALSE)
print(t_test_result)

Output:

	Welch Two Sample t-test

data:  group1 and group2
t = 4.0986, df = 9.8418, p-value = 0.002222
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
  4.824947 16.375053
sample estimates:
mean of x mean of y 
     80.0      69.4 

• In this case, the p-value is 0.00981. If the p-value is less than the chosen significance level (commonly 0.05), you reject the null hypothesis. This means that there is a statistically significant difference between the two groups.
• The 95% confidence interval provides a range of values within which the true difference between the group means lies. In this example, the interval is [4.194123, 19.663734]. Since 0 is not within this range, it further supports the conclusion that the group means are significantly different.
• The t-statistic (t = 3.517) measures the magnitude of difference relative to the variation in your sample data. The degrees of freedom (df = 7.251) are calculated using Welch’s formula, which adjusts for unequal sample sizes.

Step 3: Choosing Between Equal and Unequal Variance t-Tests

To decide whether to use the equal or unequal variance t-test, you can perform Levene’s Test for equality of variances. If the p-value from Levene’s Test is less than 0.05, it indicates that the variances are significantly different, and you should use Welch’s t-test.

In R, you can use the car package to conduct Levene’s Test:

install.packages("car")
library(car)
leveneTest(c(group1, group2), factor(c(rep(1, length(group1)), rep(2, length(group2)))))

Output:

Levene's Test for Homogeneity of Variance (center = median)
      Df F value Pr(>F)
group  1  1.1735 0.3041
      10 

Handling Unequal Sample Sizes

Unequal sample sizes are common in real-world data. For example, in clinical trials, it is often challenging to recruit equal numbers of participants for different treatment groups. When faced with unequal sample sizes, you must choose the correct version of the t-test to ensure valid results.

Example 1: Handling Unequal Sample Sizes on Medical Research data

Suppose you are comparing the blood pressure levels of patients who underwent two different treatments, but the number of patients in each group differs due to patient dropouts. Here’s how you can apply Welch’s t-test in such a scenario to determine if the treatments resulted in different average blood pressure levels.

# Simulated blood pressure levels for two groups
treatment1 <- c(120, 115, 118, 123, 121, 119, 116, 122)  # 8 patients
treatment2 <- c(125, 127, 130, 128, 126)  # 5 patients
# Perform Welch's t-test
t_test_result <- t.test(treatment1, treatment2, var.equal = FALSE)

# Display the results
print(t_test_result)

Output:

	Welch Two Sample t-test

data:  treatment1 and treatment2
t = -6.0424, df = 10.81, p-value = 9.036e-05
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -10.852076  -5.047924
sample estimates:
mean of x mean of y 
   119.25    127.20 

The Welch’s t-test indicates a significant difference in the average blood pressure levels between the two treatment groups. The results suggest that the second treatment (with a mean blood pressure of 127.20) led to higher blood pressure levels compared to the first treatment (with a mean blood pressure of 119.25).

Example 2: Comparing Exam Scores

Consider a scenario where you want to compare exam scores from two different teaching methods:

# Sample data for exam scores
program1 <- rnorm(500, mean=80, sd=5)
program2 <- rnorm(20, mean=85, sd=5)

# Perform Welch's t-test due to unequal sample sizes and potential variance differences
t.test(program1, program2, var.equal = FALSE)

Output:

	Welch Two Sample t-test

data:  program1 and program2
t = -4.5571, df = 20.846, p-value = 0.0001744
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -6.769173 -2.525611
sample estimates:
mean of x mean of y 
 80.14914  84.79653 

There is a statistically significant difference between the two programs, with program2 showing a higher mean value than program1.The confidence interval indicates that the true difference in means could be as large as 7.93 points or as small as 3.09 points, but it is consistently in favor of program2.

Conclusion

The two-sample t-test is a powerful tool for comparing means between two independent groups. When sample sizes are unequal, Welch’s t-test provides a more accurate result by adjusting the degrees of freedom. Understanding the assumptions and properly interpreting the test results are crucial steps in ensuring that your statistical analysis is valid.