Profile Analysis
Marcelo Macedo and Tyler Waterson
Introduction
Profile analysis is the
multivariate equivalent of repeated measures or mixed ANOVA. Profile analysis is most commonly used in two
cases:
1) Comparing the same dependent variables between
groups over several time-points.
2) When there are several measures of the same
dependent variable (Ex. several different psychological tests that all measure
depression).
Profile analysis uses plots
of the data to visually compare across groups.
Following this, specific equations can be used to test for the
significance of the various patterns or effects.
Applying Profile Analysis
In profile analysis, the
data are usually plotted with time points, observations, tests, etc. on the
x-axis, with the response, score, etc. on the y-axis. These plots are then made into
profiles—lines—representing the score across time points or tests for each
group.
Profile analysis asks three
basic questions about the data plots:
1) Are the groups parallel between time points or
observations?
2) Are the groups at equal levels across time points or observations?
3) Do the profiles exhibit flatness across time points or observations?
If the answer to any of
these questions is no (i.e. that specific null hypothesis is rejected) then
there is a significant effect. The type
of effect depends on which of these null hypotheses is rejected.
Equal Levels
Whether or not the profiles
have equal levels is the most straightforward test in profile analysis. The test is basically asking does one group
score higher on average across all measures or time points?
To evaluate this, the grand
mean of all time points or measures is calculated for each group. Since all of the time points or scores are collapsed
into a group mean, this a univariate test.
Essentially, this is equivalent to a between groups main effect.
Here are a couple of graphs
to help with visualization of equal levels:
Graph 1. Equal levels
(coincident)—no between group main effect.
Graph 2. Unequal
levels (non-coincident)—between group main effect.
Graph 3. Equal
levels—no between group main effect. Although these profiles are not coincident,
the average response for each group is
the same, which is an important concept to remember here.
Mathematically, we are
simply measuring the relative contributions of between-group and within-group
contributions to the total sum of squared errors (the left side of the
equation). This should look familiar, as
it is the basis behind simple ANOVAs.
For i
groups measured on j dependent variables:
If the group “levels” are
significantly different, then the equal levels null hypothesis is rejected.
Flatness
Flatness and parallelism are
both multivariate tests which compare the multiple segments of the
profile. A segment in this context is simply the difference in the response
between time points or dependent variables.
Therefore, the segment is equivalent to the slope of the line between
two points on the x-axis.
The flatness null hypothesis
is that the segments are 0, i.e. the slope of each line segment is zero and the
profile is flat. This is evaluated
independently for each group, making this a within-subjects test. If the line is not flat (any of the segments
vary significantly from 0 then there is a within groups main effect of
time-point, dependent variable, measure, etc.
MANOVA is used to test the
difference of the zero-matrix and the segmented data for each group. Usually Hotelling’s
T² is used here:
Where N is the number of
segments, GM is the grand mean of segments, and Swg is the within-group
variance-covariance matrix.
Wilks’ λ can then be calculated from T using the following
equation:
Parallelism
Parallelism is usually the
main test of interest in profile analysis.
The test for parallelism asks whether each segment is the same across
all groups.
Here are some graphs to
illustrate the concept of parallelism as it is used here:
Graph 1. Parallel—no
within group/between group interaction
Graph 2.
Non-parallel—within group/between group interaction.
To test whether or not there
is significant non-parallelism between groups, a MANOVA is used. The within-group variance comes from
subtracting the segment matrix for each individual in the group from the group
mean. The between groups variance is
obtained by subtracting each group mean segment matrix from the grand mean
segment matrix (see example). If the
parallelism null hypothesis is rejected, there is a significant group by DV
interaction effect.
Limitations
The data used in profile
analysis must be on the same scale. This
is not an issue for repeated measures since the same dependent variable is used
at multiple time points. However, scales may differ if your profile analysis
uses multiple DVs.
In this case, a z-score or other transformation may be necessary. If responses are all on the same scale, no
transformation is necessary.
Sample Size and Power
There must be more subjects
in the smallest cell than the number of dependent variables as a rule of
thumb. Small sample size can affect
power and the homogeneity of variance/covariance test. However, missing data can be replaced.
Profile analysis is often
used when univariate assumptions are not met.
Profile analysis generally has more power than a corrected univariate
test.
Assumptions
The assumptions made in
profile analysis are similar to those made when using MANOVA.
Multivariate normality
- Not important if there are more subjects in
the smallest cell than number of DVs and there are
equal overall sample sizes
- Otherwise, check for skewness and kurtosis of
DVs and perform transformation if needed
- All DVs should be
checked for univariate and multivariate outliers
Homogeneity of Variance-Covariance matrices
- If sample sizes are equal, this is usually
not an issue
- If sample sizes are unequal, then you need to
test for homogeneity (ex. Box’s M)
Linearity
- It is assumed that the DVs
are linearly related to one another
- Scatter plots of the DVs
can be used to assess linearity
- When DVs are normal
and sample size is large this is not an issue
Example
The following example comes
from Tabachnik & Fidell
(1996).
Here is a data-set of
leisure activity rankings for three different groups: politicians,
administrators and belly-dancers:
When you plot these data,
you get the following profiles:
To test the equal levels
hypothesis, you use the equation presented above. Applied to these data, the calculations look
like this:
To prepare the data for
multivariate analyses, you need to “segment” the data. For this example, you would get the
following:
This
data can then be used to in a MANOVA to test for parallelism or flatness.
For
example, to test for parallelism you would start by calculating the within
group variance for the first belly dancer and it would look like this:
Then
you would square this (variance²):
We repeat this for all
individuals to calculate within group SS and also repeat the process using
group means to calculate between group SS.
Flatness is calculated in
the same way, except that we are calculating the significance of the difference
of each segment and zero. For example,
if we wanted to test the overall flatness in this example we would start by
subtracting the zero matrix from the segment matrix:
Then
we would use the customary equation for Hotelling’s
T²:
With this example, it would
look something like this:
And then the F statistic can
easily be calculated as follows:
Where
N is the total number of subjects, k is the number of groups, and p is the
number of dependent variables.