Practica/ Assessment, Research & Evaluation, Vol 19, No 17 Page 2

Warne, MANOVA

behavioral scientists fallows the explanation to make then the null hypothesis is rejected (as would be the
the arride more applicable. case far vectors A, C, and Din Figure 2). However, if a
vector <loes not extend into the rejection regían (e.g.,
What is MANOVA? vector B in Figure 2), then the null hyp othesis is
MANOVA is a member of the General linear retained.
Model-a family of statistical procedures that are often
used to quantify the strength between variables
(Zientek & Thompson, 2009). Many of the procedures
in the General Llnear Model are hierarchically
organized-i.e., more specifi.c procedures are often
special cases of general procedures (Zientek &
Thompson, 2009). MANOVA, specifi.cally, is an
analysis of variance (ANOVA) that has two or more
dependent variables (Fish, 1988). Because most
behavioral scientists already have a fi.rm grasp of
ANOVA (Aiken, West, & Millsap, 2008), I will use it as
the starting point in this arride far explaining Figure 1. Example of the logíc of univariate
MANOVA. hypothesis testing.
In an ANOVA the independent variable is a The shaded area is the rejection regían. If the
nominal variable that has two or more values, and the test statistic is in the rejection regían (shown
dependent variable is intervally or ratio scaled.1 The both on the histogram and on the number line
null hypothesis is that the mean score on the dependent below), then the null hypothesis is rejected.
variable will be statistically equal far every group. As
with any null hypothesis statistical significance testing
procedure, an observed statistic is calculated and then
compared to a sampling distribution. If the observed
statistic is faund to be a more extreme value in the
sampling distribution than the critical value (as shown
in Figure 1), then the null hypothesis will be rejected;
otherwise, the null hyp othesis is retained. Afterwards,
an effect size-usually r¡2 in an ANOVA-quantifi.es
the relationship between the independent and
dependent variables (Thompson, 2006).
MANOVA is merely an ANOVA that has been
mathematically extended to apply to situations where
there are two or more dependent variables (Stevens,
2002). This necessitates several changes to the logíc of
ANOVA. The first is displayed in Figure 2. In contrast
to Figure 1, which is a one dimensional number line, Figure 2. Example of the logíc of multivariate
Figure 2 shows the rejection regían of a MANOVA as analysis of variance hypothesis testing far
a regían outside of a cirde on a two-dimensional perfectly uncorrelated variables.
Cartesian plane. Therefare, instead of the observed The shaded area is the rejection regían. If the
statistic being expressed as a point on a number line, it vector far the test ends in the rejection regían,
can be graphically expressed as a vector (Thomas, then the null hypothesis is rejected. If the dotted
1992). If the vector extends into the rejection regían, lines origínating at the end of a vector meet an
axis in the shaded region, then the ANOVA far
1 Please note that a one-way ANOVA simplifies to a !-test if the that axis's dependent variable would also be
independent nominal variable has only two groups (Thompson, rejected.
2006).
Practica/ Assessment, Research & Evaluation, Vol 19, No 17 Page 3
Warne, MANOVA

same two dependent variables would be only .OS. The

severity of Typ e I error inflation in multiple ANOVAs
Figure 2 also shows how the results of a
depends on how correlated the dependent variables are
MANOVA can vary from the results of two ANOVAs.
with one another, with Typ e I error inflation being
Each vector in the figure can be decomposed into two
most severe when dependent variables are uncorrelated
pieces (which are represented in the figure by the
(Hummel & Sligo, 1971). Readers will recognize the
dotted lines), one for each dependent variable (Saville
Type I error inflation that occurs with multiple
& Wood, 1986). Notice how vector A extends beyond
ANOVAs because experiment-wise Typ e I error
the point on both axes where the unshaded region
inflation also occurs when conducting multiple
ends. This means that if an ANOVA were conducted
independent sample t-tests. In fact, one of the main
on either dependent variable (which visually would
rationales for conducting ANOVA is to avoid
collapse the two-dimensional plane into a number line
conducting multiple t-tests, which inflates the Type I
resembling the lower portian of Figure 1), the null
error that occurs when researchers conduct many t­
hypothesis of the ANOVA would be rejected.
tests (Thompson, 2006).2
Likewise, analyzing vector B would produce a non­
statistically significant MANOVA and two ANOVAs Figure 2 represents an ANOVA with two perfectly
that were both also non-statistically significant. On the uncorrelated dependent variables. However, if
other hand, vector C would produce a statistically dependent variables are correlated, then the MANOVA
significant MANOVA and one statistically significant is better represented in Figure 3. Notice how the axes
ANOVA (for the vertical axis). Finally, vector D would are no longer perpendicular with each other. Rather,
produce a statistically significant MANOVA, but in the correlation between the dependent variables is
both ANOVAs the null hyp otheses would be retained. represented by the angle of the axes, with a more
oblique angle indicating a higher correlation between
Why Use MANOVA? dependent variables (Saville & Wood, 1986). Although
Given MANOVA's more complicated nature, the axes in Figures 3 look strange, it is important for
sorne researchers may question whether it is worth the readers to remember that in statistics ali axes and scales
added complexity. The alternative to using MANOVA are arbitrary. Therefore, having nonperpendicular axes
is to conduct an ANOVA for each dependent variable. is acceptable. In exploratory factor analysis, for
However, this approach is not advantageous because example, factor solutions are almost always rotated so
(a) conducting multiple ANOVAs increases the that the pattern of factor loadings is more interpretable
likelihood of committing a Typ e I error, and (b) (Costello & Osborne, 2005). Indeed, in oblique
multiple ANOVAs cannot determine whether rotation methods, such as promax rotation, the axes are
independent variable(s) are related to combinations of not only rotated, but can be nonperpendicular-just
dependent variables, which is often more useful like the axes in Figure 3 (Thompson, 2004).
information for behavioral scientists who study Although the examples in Figures 2 and 3
correlated dependent variables. represent MANOVAs with two dependent variables,
Avoiding Type I Error Inflation an extension to three dependent variables can be
imagined without difficulty. For three perfectly
When multiple dependent variables are present, uncorrelated dependent variables, the graph would be
conducting a MANOVA is beneficia! because it three dimensional, with the unshaded region being a
reduces the likelihood of Typ e I error (Fish, 1988;
Haase & Ellis, 1987; Huberty & Mortis, 1989). The 2
Sorne researchers who choose to conduct multiple t-tests instead
probability of Typ e I error at least once in the series of of an ANOVA control Typ e I error inflation with a Bonferroni
ANOVAs (called experiment-wise error) can be as high correction (fhompson, 2006). The analogous relationship between
as 1 - (1 - .OS)\ where k is the number of ANOVAs t-tests and ANOVA and between ANOVA and MANOVA
extend even to this point, and applying a Bonferroni correction
conducted. Therefore, if a researcher chooses the would also control Typ e I error inflation in a group of ANOVAs.
traditional value of .OS for two ANOVAs, then the
O(
However, Bonferroni methods are overly conservative, especially
experiment-wise Type I error can be as high as .0975- with a large number of tests (Stevens, 2002) or correlated variables
not .05-even though the for each ANOVA is .OS.
O( (Hummel & Sligo, 1971). Moreover, a Bonferroni correction to a
However, the Type I error for a MANOVA on the series of ANOVAs would not provide the benefits of MANOVA
described elsewhere in the article.