Statistical Analysis for Education and Psychology Researchers

significant so a post hoc test would not usually be appropriate. For illustrative purposes,
however, both an a priori test to estimate the differences in means between Group 1 and
Group 2, and a post hoc test for all pairwise comparisons among subgroup means are
illustrated. There are a number of multiple comparison tests and the reader is referred to
an informative text by Toothaker (1991) on choice of appropriate multiple comparison
test procedures.

Preplanned comparisons

When variances are homogeneous and sample sizes are equal then a planned t-test can be
computed by substituting the standard error of the difference in the usual t-test with the
MSerror value from the ANOVA output. The obtained t-statistic is evaluated against
degrees of freedom for MSerror (in the ANOVA output). Consider for example a
preplanned comparison of the difference in means between Group 1 (Christian) and
Group 2 (Muslim), the observed difference is −0.875, an investigator wants to know
whether this difference is significant. As there are equal sample sizes and variances are
not drastically different, the t-statistic is given by

where n is the number in the group, here 8. A 95 per cent interval for the difference could
be estimated.
When sample sizes are unequal or when variances are heterogeneous, individual
variances and a Satterthwaite correction for degrees of freedom should be used (see
section on t-tests). Preplanned comparisons can be handled easily in SAS. If an estimate
of the difference in means is required then the statements LSMEANS and ESTIMATE
are used. Least square means (LSMEANS) adjust for unequal sample sizes and when we
have a balanced design are the same as the ordinary means. To output an estimate of the
difference in means between Group 1 and Group 2, an orthogonal contrast because Group
3 is not involved, the following SAS code is entered after the model statement:

lsmeans religion /stderr pdiff;

estimate '1–2' religion 1–1 0;

The label 1–2 is given to the selected contrast in the SAS output—see Figure 8.16.
General Linear Models Procedure’
Dependent Variable: ATTRIB1
Parameter Estimate T for H0: Parameter=0 Pr>|T| Std. Error. of Eastimate
1–2 −0.87500000 −0.68 0.5064 1.29444785

Figure 8.16: Estimate of the difference

between the means of Group 1 and

Group 2

Statistical analysis for education and psychology researchers 324