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demands, and adequate pilot research should reduce participant
attrition. Such loss could be random across treatment levels or the
loss could be disproportionate across conditions. It is important that
researchers pay attention to where their losses are occurring.
Unequal samples caused by demographic variables: The relative
frequency of subpopulations (e.g., racial/ethnic, socioeconomic
groups) will almost always produce unequal samples when these
populations are sampled. Comparison of the means of these subpop-
ulations may therefore become distorted because of the inequality
of group sizes. For example, the precision of the sample mean as an
estimate of the population mean (e.g., standard error of the mean) is
a function of sample size, and different sample sizes will therefore be
associated with different levels of precision. One way to overcome
this bias is to conduct a weighted means ANOVA (or analysis of
weighted means). This approach is accomplished by weighting each
mean in proportion to its sample size. Researchers employ a special
average called a “harmonic mean,” which is computed by dividing
the number of groups by the sum of the reciprocals of the group
sample sizes:
a
1
n
i
.
(17.1)
This harmonic mean is then used to produce the sums of squares
for the weighted means ANOVA (Keppel, 1991).
For more discussion of this topic, as it relates to ANOVA procedures,
see Anderson, (2001), Keppel (1991), Keppel and Wickens (2004), and
Tabachnick and Fidell (2001, 2007).
17.6 MULTIVARIATE ANALYSIS OF VARIANCE (MANOVA)
To this point, the focus of the present topic has been to assess the effects
of the independent variable(s) on a single dependent variable. In the
context of ANOVA, this is referred to as
univariate
ANOVA. Sometimes
research questions can be framed to include the simultaneous assessment
of two or more dependent variables. Such an approach is characterized as
multivariate
ANOVA (MANOVA).
In a MANOVA design, the dependent variables are combined into a
weighted linear composite or
variate
. The weights are created to maximally
differentiate between the groups or levels of the independent variable(s).
As in ANOVA, researchers employing MANOVA designs can examine the
effects of a single independent variable (i.e., a one-way MANOVA) or
two or more independent variables (i.e., two-way or
k
-way MANOVA).
The difference between the univariate and multivariate approaches is
that researchers using MANOVA are concerned with assessing the effects
of the independent variable(s) on the combined dependent variate - a
multivariate effect.
