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for each participant in our numerical example. For the sake of conve-
nience, we have spaced the conditions equally on the
x
axis. Each line or
function represents the data for a single subject. Differences in the heights
of the functions represent different degrees of symptom intensity.
The display is a little cluttered with all eight subjects depicted, but
the important feature to note in Figure 10.2 is that the lines are not
parallel. This indicates that symptom intensity across the time periods
presents a different pattern for the different patients in the study and is
the key to understanding the error term in the ANOVA. Consider for
a moment the unlikely situation where the lines for each patient were
actually parallel but of different elevations in the graph. What would this
suggest? It would suggest that although the patients differ in the inten-
sity of their symptoms, the “progression” or change in symptom inten-
sity over the period during which our measurements occurred would
be the same for everyone. Thus, knowing the level of their initial inten-
sity (or the intensity at any one point), the symptom intensity of the
patients at every measurement occasion would be perfectly predictable.
With such perfect predictability, there would be no variability in the way
that patients exhibited symptom intensity over the year or so of our mea-
surement. Such consistency would in turn suggest that there is no unac-
counted for variability or error variance concerning this progression over
time.
The data shown in Figure 10.2 are more typical of what would be
observed in an empirical study in that they are far from being error free.
The degree of inconsistency of the study participants is visually represented
by their different patterns of symptom intensity across time. To the extent
that the functions are not parallel, we can say that we have unaccounted
for variance, that is, we can say that we have
error variance
observed in
the data with respect to the consistency of patients reporting symptom
intensity over time. Specifically, despite the fact that all of the individuals
were exposed to the same treatment and were measured at the same
time, we observe quite different patterns in their reports of symptom
intensity. Note that this effect is not the same as different levels of symptom
intensity observed between the patients. Although it is true that some
patients experienced more intense symptoms than others, our focus here
is on the differences in the pattern of symptom intensity for the different
patients.
At this point, some of you might be thinking, “It is not terribly surpris-
ing that different people are exhibiting different patterns, so why make so
much out of it? Different people have different body chemistries, different
reactions to having illness, and so on.” We agree that this finding is not
surprising. The key issue is that all of these factors that produce differ-
ent patterns are
unknown
in the context of the research study. Without
having measured these factors (variables), we have no way to examine
the way in which they covary with the dependent measure. And so these
different patterns exhibited by the subjects in the study represent, from a
measurement perspective,
error variance.
