How to Perform an Easy Welch’s t-Test in SAS

Understanding When to Use Welch’s t-Test

The Welch’s t-Test is essential when comparing the means of two distinct groups where we suspect or confirm that the variability (variance) within those groups differs significantly. Unlike the traditional Student’s t-test, which assumes homogeneity of variances, the Welch test provides a more accurate and conservative assessment of mean differences under conditions of unequal variances, thereby preventing inflated Type I error rates.

This tutorial focuses specifically on the practical execution of the Welch’s t-Test within the SAS environment. By following the steps outlined below, users can efficiently generate, run, and interpret the necessary statistical output to determine if a significant difference exists between their group means, especially in scenarios where sample sizes or variances are dissimilar.

Example Scenario: Comparing Exam Preparation Methods

Consider a practical scenario in educational statistics: a teacher wishes to evaluate the effectiveness of an exam preparation booklet. The objective is to compare the exam scores of two independent cohorts of students. The first group comprises 12 students who utilized the prep booklet, while the second group consists of 12 students who did not use the booklet.

The following data represents the raw exam scores for the students in each respective group. We aim to use Welch’s t-Test to formally determine if the mean exam score is statistically equal or different between these two groups, accounting for potential differences in score variability.

• Booklet Group Scores: 90, 85, 88, 89, 94, 91, 79, 83, 87, 88, 91, 90
• No Booklet Group Scores: 67, 90, 71, 95, 88, 83, 72, 66, 75, 86, 93, 84

The subsequent steps detail the process required to implement this analysis in SAS, starting with data creation and culminating in the interpretation of the statistical results.

Step 1: Data Preparation and Creation in SAS

The first critical step involves structuring the raw data into a format that SAS can process effectively. We use the familiar DATA step to create a dataset named exam_scores. This dataset will contain two primary variables: group (a character variable indicating whether the booklet was used) and score (a numeric variable representing the student’s exam score).

The following SAS code block demonstrates the necessary syntax for defining the dataset structure and inputting the scores listed above using the DATALINES statement. It is essential that the data is correctly entered to ensure the validity of the subsequent statistical testing.

/*create dataset*/
data exam_scores;
    input group $ score;
    datalines;
booklet 90
booklet 85
booklet 88
booklet 89
booklet 94
booklet 91
booklet 79
booklet 83
booklet 87
booklet 88
booklet 91
booklet 90
no_booklet 67
no_booklet 90
no_booklet 71
no_booklet 95
no_booklet 88
no_booklet 83
no_booklet 72
no_booklet 66
no_booklet 75
no_booklet 86
no_booklet 93
no_booklet 84
;
run;

Step 2: Executing Welch’s t-Test using PROC TTEST

Once the data set is successfully created and loaded into the SAS session, we proceed directly to statistical analysis using the PROC TTEST procedure. This procedure is specifically designed for t-tests, and it automatically incorporates the necessary calculations for Welch’s t-Test when variances are found to be unequal.

In the syntax below, we specify the input data (data=exam_scores), set the significance level (alpha=0.05), and crucially define the variables: the grouping variable using the CLASS statement (group), and the measurement variable using the VAR statement (score). Executing this code will produce detailed output tables necessary for drawing conclusions about the mean scores.

/*perform Welch's t-test*/
proc ttest data=exam_scores alpha=0.05;
    class group;
    var score;
run;

The resulting output from the PROC TTEST execution includes several key tables, starting with descriptive statistics for each group and culminating in the actual t-test results and tests for variance equality. We must carefully examine these sections to arrive at a statistically sound conclusion.

Interpreting the Equality of Variances Output

The first crucial step in analyzing the PROC TTEST results is to examine the table titled “Equality of Variances.” This table presents the results of an F-test, designed to rigorously determine if the population variances are equal between the two independent samples (Booklet vs. No Booklet). The outcome of this preliminary test dictates which row of the subsequent t-test table we should rely upon.

The F-test operates under the following formalized null and alternative hypotheses:

• H₀: The population variances are equal (σ₁² = σ₂²).
• H_A: The population variances are not equal (σ₁² ≠ σ₂²).

In our example, inspecting the output reveals a p-value of .0046 for the F-test. Since this p-value is significantly less than our chosen alpha level of 0.05, we must reject the null hypothesis (H₀). This statistical evidence confirms that the variances of the exam scores are indeed unequal between the students who used the booklet and those who did not. Consequently, we are justified in using the adjusted statistics provided by Welch’s t-Test.

Drawing Conclusions from the Unequal Variance Test

Because the test for the Equality of Variances indicated a significant difference (p < 0.05), we must now focus on the row titled “Unequal” in the primary t-test output table. This row provides the t-statistic and the corresponding p-value calculated using the Satterthwaite approximation, which is the core of Welch’s t-Test. The relevant metrics from the output are:

• t Value: 2.24
• p-value: .0417

We now recall the fundamental null and alternative hypotheses for the t-test itself, which addresses the equality of the population means:

• H₀: μ₁ = μ₂ (The mean scores are equal between the two groups.)
• H_A: μ₁ ≠ μ₂ (The mean scores are not equal between the two groups.)

Since the calculated p-value of .0417 is less than the predetermined significance level (α = 0.05), we decisively reject the null hypothesis (H₀). This outcome provides statistically sufficient evidence to conclude that there is a significant difference in the mean exam scores between the students who used the preparation booklet and those who did not. The preparation method appears to have had a tangible impact on performance.

Further Exploration and Related Statistical Procedures

For those interested in conducting quick calculations or verifying results, several online calculators are available that automatically perform Welch’s t-Test for any two samples. Using these tools can be a valuable supplement to statistical software outputs, particularly for understanding the input variables and resulting degrees of freedom.

In addition to the Welch test, SAS offers extensive capabilities for various statistical methods. Analysts frequently utilize other procedures for different research designs.

The following tutorials explain how to perform other common statistical tests in SAS: