17 Two-Sample t-Test and ANOVA in SAS

Learning Objectives

In this hands-on session, we will practice applying two-sample t-tests and one-way ANOVA in SAS using a built-in dataset from SASHELP. The goal is to help you develop a systematic workflow for comparing group means and interpreting the results.

By the end of this activity, you should be able to:

Understand when to use a two-sample t-test.

Understand when to use one-way ANOVA.

Identify the response variable and grouping variable in SAS code.

Write SAS code for two-sample t-tests and ANOVA.

Interpret p-values, assumptions, and post-hoc comparisons.

17.1 Guideline: How to Analyze Group Mean Comparisons in SAS

Based on experience, a reliable way to solve mean-comparison problems is to follow three steps:

Step 1: Understand the data and the question

Identify the response variable
Identify the grouping variable
Determine how many groups are being compared

Step 2: Choose the correct inferential method

If there are two groups, use a two-sample t-test
If there are three or more groups, use one-way ANOVA

Step 3: Write SAS code and interpret the output

Run the statistical procedure
Check assumptions when appropriate
Interpret the p-value in context
For ANOVA, if the global test is significant, run a multiple comparison procedure

💡 This same workflow is also useful when analyzing grouped data in R or Python.

17.2 Practice Examples in this Activity

To practice these steps, in this activity, we will:

Use a two-sample t-test to compare city fuel efficiency between cars from Asia and the USA
Use one-way ANOVA to compare horsepower across different vehicle types

Eventually, you will see that the two-sample t-test is a special case of ANOVA.

17.3 Dataset for this exercise: `SASHELP.CARS`

In this activity, we use the built-in SAS dataset SASHELP.CARS.

This dataset contains information on many car models and includes variables such as:

Origin: region where the car was produced
Type: vehicle type
MPG_City: city miles per gallon
MPG_Highway: highway miles per gallon
Horsepower: engine horsepower
Weight: vehicle weight

Since SASHELP.CARS is built into SAS OnDemand / SAS Studio, no data import is required.

17.4 First Practice: Two-Sample t-Test

In this first example, we compare the city fuel efficiency (MPG_City) of cars from Asia and the USA.

17.4.1 Step 1: Explore the Dataset

First, look at the structure of the dataset and preview a few rows.

PROC CONTENTS DATA=SASHELP.CARS;
RUN;

PROC PRINT DATA=SASHELP.CARS (OBS=10);
RUN;

At this stage:

Focus on identifying useful variables
Decide which variable is the response
Decide which variable defines the groups

17.4.2 Step 2: Create a Dataset with Two Groups

Since we only want to compare Asia and USA, we subset the built-in dataset.

DATA CARS_SUB;
  SET SASHELP.CARS;

  IF ORIGIN IN ("Asia", "USA");
RUN;

Key idea:

MPG_City is the response variable
Origin is the grouping variable
There are two groups, so the correct method is a two-sample t-test

### Step 3: Run the Two-Sample t-Test

The hypotheses are

\[ H_0: \mu_{\text{Asia}} = \mu_{\text{USA}} \]

and

\[ H_1: \mu_{\text{Asia}} \neq \mu_{\text{USA}}. \]

Use the following code:

PROC TTEST DATA=CARS_SUB;
CLASS ORIGIN;
VAR MPG_CITY;
RUN;

17.4.3 Code Description

CLASS ORIGIN defines the grouping variable
VAR MPG_CITY defines the response variable
PROC TTEST performs the two-sample t-test

17.4.4 Step 4: Visualize the Two Groups

It is often helpful to examine the two groups visually.

PROC SGPLOT DATA=CARS_SUB;
TITLE "City MPG by Car Origin";
VBOX MPG_CITY / CATEGORY=ORIGIN;
RUN;

17.4.5 In-Class Questions

Which group appears to have the larger mean city MPG?
What is the null hypothesis being tested?
If the p-value is smaller than 0.05, what conclusion should we make?
Does SAS report only one t-test result, or more than one method?

17.5 Second Practice: One-Way ANOVA

In this second example, we compare horsepower across different vehicle types.

17.5.1 Step 1: Understand the Problem

Now the response variable is

Horsepower

and the grouping variable is

Type

Since Type has more than two levels, we use one-way ANOVA.

17.5.2 Step 2: Compute Summary Statistics

Before fitting the ANOVA model, compute summary statistics by group.

PROC MEANS DATA=SASHELP.CARS N MEAN STD;
CLASS TYPE;
VAR HORSEPOWER;
RUN;

This code helps you inspect:

sample size in each group
mean horsepower in each group
standard deviation in each group

17.5.3 Step 3: Visualize the Groups

PROC SGPLOT DATA=SASHELP.CARS;
TITLE "Horsepower by Vehicle Type";
VBOX HORSEPOWER / CATEGORY=TYPE;
RUN;

17.5.4 Step 4: Write the Global Hypotheses

The one-way ANOVA hypotheses are

\[ H_0: \mu_1 = \mu_2 = \cdots = \mu_k \]

versus

\[ H_1: \text{At least one group mean differs.} \]

### Step 5: Fit the ANOVA Model

PROC ANOVA DATA=SASHELP.CARS;
CLASS TYPE;
MODEL HORSEPOWER = TYPE;
RUN;

17.5.5 Code Description

CLASS TYPE defines the grouping variable
MODEL HORSEPOWER = TYPE tests whether mean horsepower differs across vehicle types
The global ANOVA test is based on the F-statistic

### In-Class Questions

What does the global ANOVA test tell us?
If the ANOVA p-value is smaller than 0.05, what can we conclude?
If the ANOVA test is significant, do we immediately know which groups differ?

17.6 Assumption Checking for ANOVA

One important ANOVA assumption is homogeneity of variances.

We can check this using Levene’s test.

PROC ANOVA DATA=SASHELP.CARS;
CLASS TYPE;
MODEL HORSEPOWER = TYPE;
MEANS TYPE / HOVTEST=LEVENE WELCH;
RUN;

17.6.1 Code Description

HOVTEST=LEVENE requests Levene’s test for equal variances
WELCH requests Welch’s ANOVA, which is useful when the equal variance assumption is questionable

17.6.2 In-Class Questions

What does Levene’s test check?
If Levene’s test has a p-value below 0.05, what does that suggest?
Why might Welch’s ANOVA be preferable in that case?

17.7 Multiple Comparison After ANOVA

If the global ANOVA test is significant, we often want to know:

Which groups are different?

We can do this with a multiple comparison procedure.

17.7.1 Example: Tukey Test

PROC ANOVA DATA=SASHELP.CARS;
CLASS TYPE;
MODEL HORSEPOWER = TYPE;
MEANS TYPE / TUKEY;
RUN;

17.7.2 Code Description

TUKEY requests Tukey’s multiple comparison procedure
This compares all pairs of group means while controlling the overall error rate

17.7.3 In-Class Questions

Why should we only interpret Tukey results after the global ANOVA test is significant?
What is the purpose of a post-hoc test?
How is this different from simply running many separate t-tests?

17.8 Compare the Two Methods

At this point, compare the two procedures used in this activity.

17.8.1 Two-Sample t-Test

Used when:

there are exactly two groups
the goal is to compare two group means

Example in this activity:

MPG_City for Asia vs USA

17.8.2 One-Way ANOVA

Used when:

there are three or more groups
the goal is to determine whether at least one mean differs

Example in this activity:

Horsepower across vehicle types

17.8.3 Important Connection

When there are only two groups, the two-sample t-test and one-way ANOVA test the same null hypothesis.

So:

The two-sample t-test is a special case of ANOVA.

17.9 Suggested Practice Tasks

Work through the following during class.

17.9.1 Task 1

Compare MPG_Highway between Asia and Europe cars.

Suggested steps:

Create a subset dataset
Identify the response and grouping variables
Run a two-sample t-test
Draw a boxplot
Interpret the p-value

17.9.2 Task 2

Test whether vehicle weight differs by Origin.