Encyclopedia of Sociology

ANALYSIS OF VARIANCE AND COVARIANCE

said to be extremely strong (i.e., able to overshad-
ow any other source of influence). If, on the other
hand, this assumption is not true, then the SSBETWEEN
will be small relative to the SSTOTAL because other
influences randomly dispersed across groups are
generating differences in scores not reflected in
mean score differences. This situation indicates
that group differences in experimental conditions
add little to our ability to predict or explain differ-
ences in outcome scores. In the extreme case,
when there is no difference in the group means,
the SSBETWEEN will equal 0, indicating no effect.

The within-groups sums of squares is a meas-
ure of how much variation exists in the outcome
scores within the groups. Using the same example
as above, adolescents in a given type of school
might differ in their self-esteem in spite of the
esteem-inflating or -depressing influence of their
school environment. For example, elementary
school students might feel good about themselves
because of the supportive and secure nature of
their school environment, but some of these stu-
dents will feel worse than others because of other
factors such as their home environments (e.g., the
effects of parental conflict and divorce) or their
neighborhood conditions (e.g., wealth vs. pover-
ty). The procedure for calculating the within-groups
sums of squares is represented by the following
equation:

∑i∑j(Yij – Yj)^2 (^3 )

where Σi Σj indicates to sum across all individuals
(i) in all groups (j), and (Yij − .j)^2 is the squared
difference between the individual scores (Yij) and
their respective group mean scores (Yj).

Both in terms of comparing means and ex-
plaining variance, the within-groups sums of squares
represents the variance due to other factors or
‘‘error’’; it is the degree of variation in scores
despite the fact that individuals in a given group
were exposed to the same influences or stimuli.
This component of variance can also serve as an
estimate of how much variability in outcome scores
occurs in the population from which each group of
respondents was drawn. If the within-groups sums
of squares is high relative to the between-groups
sums of squares (or the difference between the
means), then less confidence can be placed in the
conclusion that any group differences are meaningful.

SUMMARY MEASURES

Analysis of variance procedures produce two sum-

mary statistics. The first of these—ETA^2 —is a

measure of how much effect the predictor variable

or factor has on the outcome variable. The second

statistic—F—tests the null hypothesis that there is

no difference between group means in the larger

population from which the sample data was ran-

domly selected.

ETA^2. As noted above, a large between-groups

sums of squares is indicative of a large difference

in the mean scores between groups. The meaning-

fulness of this difference, however, can only be

judged against the overall variation in the scores. If

there is a large amount of variation in scores

relative to the variation due exclusively to be-

tween-groups differences, then group effects can

only explain a small proportion of the total varia-

tion in scores (i.e., a weak effect). ETA^2 takes into

account the difference between means and the

total variation in scores. The general equation for

computing ETA^2 is as follows:

SSBETWEEN ( 4 )

SSTOTAL

ETA^2 =

As can be seen from this equation, ETA^2 is the

proportion of the total sums of squares explained

by group differences. When all the variance is

explained, there will be no within-group variance,

leaving SSTOTAL= SSBETWEEN (SSTOTAL= SSBETWEEN

+ 0). Thus, ETA^2 will be equal to 1, indicating a

perfect relationship. When there is no effect, there

will be no difference in the group means (SSBETWEEN =

0) and ETA^2 will be equal to 0.

F Tests. Even if ETA^2 indicates that a sizable

proportion of the total variance in the sample

scores is explained by group differences, the possi-

bility exists that the sample results do not reflect

true differences in the larger population from

which the samples were selected. For example, in a

study of the effects of cohabitation on marital

stability, a researcher might select a sample of the

population and find that, among those in his or

her sample, previous cohabitors have lower mari-

tal satisfaction than those without a history of

cohabitation. Before concluding that cohabitation

has a negative effect on marriage in the broader

population, however, the researcher must assess

the probability that, by chance, the sample used in