Welch’s t-test in R

In statistical analysis, comparing the means of two groups is a common task. However, traditional methods like the Student's t-test assume equal variances between groups, which may not hold true in real-world data. Welch’s t-test, named after its developer B. L. Welch, provides a robust solution for comparing means when dealing with unequal variances or sample sizes between groups in the R Programming Language.

What is Welch’s t-test?

Welch's t-test is a statistical tool used to compare the averages of two groups. It's helpful when the groups have different amounts of variability (like how spread out their scores are). Traditional methods assume the variability is the same, but Welch's t-test adjusts for these differences, making comparisons fairer and more accurate.

[Tex]t = \frac{{\bar{x}_1 - \bar{x}_2}}{{\sqrt{\frac{{s_1^2}}{{n_1}} + \frac{{s_2^2}}{{n_2}}}}} [/Tex]

Where,

• t is the t-statistic that follows a t-distribution with degrees of freedom calculated using the Welch-Satterthwaite equation.
• x₁ and x₂ bar are the sample means of two independent groups.
• s²₁ and s²₂ are the sample variances of the two groups.
• n₁ and n₂ are the sample sizes of the two groups.

Syntax:

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

Here ,

• x and y are the vectors containing the sample data for the two groups being compared.
• alternative specifies the alternative hypothesis and can be set to "two.sided" for a two-tailed test, "greater" for a one-tailed test where the alternative hypothesis is that the mean of x is greater than the mean of y, or "less" for a one-tailed test where the alternative hypothesis is that the mean of x is less than the mean of y.
• var.equal is a logical argument that indicates whether to assume equal variances between the groups (TRUE) or not (FALSE). In Welch's t-test, this should be set to FALSE to account for unequal variances.

Implementing Welch’s t-test in R

We will use a hypothetical example involving two groups of students and their exam scores. Let's assume Group A consists of 30 students, while Group B consists of 25 students.

R

# Generate sample data
set.seed(123)  # for reproducibility
group_a <- rnorm(30, mean = 75, sd = 10)  # Group A scores
group_b <- rnorm(25, mean = 80, sd = 12)  # Group B scores
# Perform Welch's t-test
result <- t.test(group_a, group_b, alternative = "two.sided", var.equal = FALSE)
result

Output:

    Welch Two Sample t-test

data:  group_a and group_b
t = -2.9025, df = 51.646, p-value = 0.005432
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -12.873584  -2.348193
sample estimates:
mean of x mean of y 
 74.52896  82.13985 

The t-value is -2.9025, which indicates the size of the difference between the means of the two groups relative to the variability within the groups.

• The degrees of freedom (df) are approximately 51.646, adjusted for unequal variances.
• The p-value is 0.005432, indicating that the difference in means between the groups is statistically significant at a 95% confidence level (since the p-value is less than 0.05).
• The alternative hypothesis suggests that the true difference in means is not equal to zero, meaning there is a real difference between Group A and Group B.
• The 95% confidence interval for the difference in means is from -12.873584 to -2.348193, providing a range of plausible values for the true difference.
• The sample estimates show that the mean score for Group A (mean of x) is approximately 74.53, while the mean score for Group B (mean of y) is around 82.14.

These results indicate that there is a significant difference in the means of the two groups, with Group B having a higher average score compared to Group A.

Implementing Welch’s t-test on mtcars dataset

R

# Load the dataset
data(mtcars)

# Subset the data for automatic and manual transmission cars
automatic <- subset(mtcars, am == 0)$mpg
manual <- subset(mtcars, am == 1)$mpg

# Perform Welch's t-test
result <- t.test(automatic, manual, var.equal = FALSE)

# Print the test result
print(result)

Output:

        Welch Two Sample t-test

data:  automatic and manual
t = -3.7671, df = 18.332, p-value = 0.001374
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -11.280194  -3.209684
sample estimates:
mean of x mean of y 
 17.14737  24.39231 

The t-statistic is -3.7671, indicating a difference in means between automatic and manual transmission cars.

• The degrees of freedom (df) are approximately 18.332, calculated using the Welch-Satterthwaite equation.
• The p-value is 0.001374, which is less than the typical significance level of 0.05, suggesting that there is a statistically significant difference in the mean mpg between automatic and manual transmission cars.
• The 95% confidence interval for the difference in means ranges from -11.280194 to -3.209684. This interval gives a range of plausible values for the true difference in mean mpg between the two transmission types.
• The sample estimates show that the mean mpg for automatic transmission cars is 17.14737, while the mean mpg for manual transmission cars is 24.39231.

Conclusion

Welch’s t-test is a powerful tool in statistics for comparing averages of two groups, especially when they have different spreads or sizes. It helps us figure out if the differences we see in averages are real or just due to chance.Using Welch’s t-test in R is easy with the t.test function. This makes it useful for researchers and analysts who need to compare groups in various studies, ensuring more accurate and reliable results.