Statistical Analysis for Education and Psychology Researchers

The author did not report any tests of the ANOVA assumptions even though sample
sizes were small, cells were unbalanced and the variance in one cell is much smaller than
the variances in the other three cells.

Computer Analysis

When analysing any two-factor design it is important to distinguish between fixed and
random effects. The error term, the denominator in the F-test, is dependent upon whether
a factor is fixed or random. For example, in a two-factor design where both factors are
random (unusual in education), both main factors would be tested against the interaction
term, for example FF1=MS(F1)/MS(F1×F2), however, the interaction effect would be tested
against the error term (e.g., FFl×F2= MS(F1×F2)/MS(error). With a fixed effects design the F-
statistic would be found for all effects by dividing by MS(error) in the usual way. In a
mixed design with both fixed and random effects (common in education) both the
random factor and the interaction term are tested against the usual error term, but the
fixed factor is tested against the MS(interaction) term.
Data for the following analysis is taken from the study on pupils’ understanding of
probability and cultural background that was described briefly in section 8.7. The
additional factor here is sex of respondent. One of the religious groups has also been
changed so we now have Christian and Sikh as the two levels of the other factor. The data
is shown in Table 8.14.

Table 8.14: Data for 2×2 factorial ANOVA

Variable Religion (REL) Variable (Sex)

Male (code=0) Female (code=1)

Christian 17 19 22 18 15 19

(code=2) 19 18 13 14 12 15

Sikh 18 17 22 16 17 15

(code=5) 19 18 20 14 17 18

The following SAS code is used to perform a 2×2 factorial analysis of variance on the
data shown in Table 8.13. Pre-planned significance tests as well as estimates of the
differences in means between: i) males and females, (ESTIMATE ‘0–1’ SEX 1–1) and ii)
Christians and Sikhs (ESTIMATE ‘5–2’ REL–1 1) are also output.

proc glm data=a;

class sex rel;

model attrib1 = sex rel sex*rel;

lsmeans sex rel/ stderr pdiff;

means sex rel/ deponly;

estimate '0–1' sex 1 −1;

estimate '5−2' rel −1 1;

output out = new r = res1 p = pred1;

run;

Inferences involving continuous data 335