Comparing Group Means

The TTEST Procedure

Comparing Group Means

If you want to compare values obtained from two different groups, and if the groups are independent of each other and the data are normally distributed in each group, then a group t test can be used. Examples of such group comparisons include

test scores for two third-grade classes, where one of the classes receives tutoring
fuel efficiency readings of two automobile nameplates, where each nameplate uses the same fuel
sunburn scores for two sunblock lotions, each applied to a different group of people
political attitude scores of males and females

In the following example, the golf scores for males and females in a physical education class are compared. The sample sizes from each population are equal, but this is not required for further analysis. The data are read by the following statements:

   title 'Comparing Group Means';
   data scores;
      input Gender $ Score @@;
      datalines;
   f 75  f 76  f 80  f 77  f 80  f 77  f 73
   m 82  m 80  m 85  m 85  m 78  m 87  m 82
   ;
   run;

The dollar sign ($) following Gender in the INPUT statement indicates that Gender is a character variable. The trailing at signs (@@) enable the procedure to read more than one observation per line.

You can use a group t test to determine if the mean golf score for the men in the class differs significantly from the mean score for the women. If you also suspect that the distributions of the golf scores of males and females have unequal variances, then submitting the following statements invokes PROC TTEST with options to deal with the unequal variance case.

 
   proc ttest cochran ci=equal umpu;
      class Gender;
      var Score;
   run;

The CLASS statement contains the variable that distinguishes the groups being compared, and the VAR statement specifies the response variable to be used in calculations. The COCHRAN option produces p-values for the unequal variance situation using the Cochran and Cox(1950) approximation. Equal tailed and uniformly most powerful unbiased (UMPU) confidence intervals for $\sigma$ are requested by the CI= option. Output from these statements is displayed in Figure 67.2 through Figure 67.4.

Comparing Group Means

The TTEST Procedure

Statistics
Variable	Class	N	Lower CL Mean	Mean	Upper CL Mean	Lower CL Std Dev	UMPU Lower CL Std Dev	Std Dev	UMPU Upper CL Std Dev	Upper CL Std Dev	Std Err	Minimum	Maximum
Score	f	7	74.504	76.857	79.211	1.6399	1.5634	2.5448	5.2219	5.6039	0.9619	73	80
Score	m	7	79.804	82.714	85.625	2.028	1.9335	3.1472	6.4579	6.9303	1.1895	78	87
Score	Diff (1-2)		-9.19	-5.857	-2.524	2.0522	2.0019	2.8619	4.5727	4.7242	1.5298

Figure 67.2: Simple Statistics

Simple statistics for the two populations being compared, as well as for the difference of the means between the populations, are displayed in Figure 67.2. The Variable column denotes the response variable, while the Class column indicates the population corresponding to the statistics in that row. The sample size (N) for each population, the sample means (Mean), and lower and upper confidence bounds for the means (Lower CL Mean and Upper CL Mean) are displayed next. The standard deviations (Std Dev) are displayed as well, with equal tailed confidence bounds in the Lower CL Std Dev and Upper CL Std Dev columns and UMPU confidence bounds in the UMPU Upper CL Std Dev and UMPU Lower CL Std Dev columns. In addition, standard error of the mean and the minimum and maximum data values are displayed.

Comparing Group Means

The TTEST Procedure

T-Tests
Variable	Method	Variances	DF	t Value	Pr > \|t\|
Score	Pooled	Equal	12	-3.83	0.0024
Score	Satterthwaite	Unequal	11.5	-3.83	0.0026
Score	Cochran	Unequal	6	-3.83	0.0087

Figure 67.3: t Tests

The test statistics, associated degrees of freedom, and p-values are displayed in Figure 67.3. The Method column denotes which t test is being used for that row, and the Variances column indicates what assumption about variances is being made. The pooled test assumes that the two populations have equal variances and uses degrees of freedom n₁ + n₂ - 2, where n₁ and n₂ are the sample sizes for the two populations. The remaining two tests do not assume that the populations have equal variances. The Satterthwaite test uses the Satterthwaite approximation for degrees of freedom, while the Cochran test uses the Cochran and Cox approximation for the p-value.

Comparing Group Means

The TTEST Procedure

Equality of Variances
Variable	Method	Num DF	Den DF	F Value	Pr > F
Score	Folded F	6	6	1.53	0.6189

Figure 67.4: Tests of Equality of Variances

Examine the output in Figure 67.4 to determine which t test is appropriate. The "Equality of Variances" test results show that the assumption of equal variances is reasonable for these data (the Folded F statistic F' = 1.53, with p=0.6189). If the assumption of normality is also reasonable, the appropriate test is the usual pooled t test, which shows that the average golf scores for men and women are significantly different (t=-3.83, p=0.0024). If the assumption of equality of variances is not reasonable, then either the Satterthwaite or the Cochran test should be used.

The assumption of normality can be checked using PROC UNIVARIATE; if the assumption of normality is not reasonable, you should analyze the data with the nonparametric Wilcoxon Rank Sum test using PROC NPAR1WAY.

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