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CHAPTER 13 Analysis partialling out effects of a covariate 431
differ on the statistics test because of the teaching method they were under, so we would like
to get rid of (partial out) the effects due to IQ.
This is the ideal situation in which to use ANCOVA because ANCOVA gets rid of the
effects due to the covariate (in this case IQ): that is, it reduces error variance, which, as we
have said previously, leads to a larger F-value. Therefore the fi rst purpose of ANCOVA is to
reduce error variance.
Assume that we have a situation where the means on the covariate differ signifi cantly.
ANCOVA is still useful in that it adjusts the means on y (the statistics marks) to what they
would be, had the groups had exactly the same means on IQ. So this is the second purpose:
ANCOVA adjusts the means on the covariate for all of the groups, which leads to an adjustment
in the means of the y variable – in this case, statistics marks.
We will now explain this further, using the example of groups that are pre-existing: that is,
we have not randomly allocated participants to groups. These are also called non-equivalent
or intact groups. In such cases, we may fi nd that the groups differ signifi cantly on the
covariate.
Activity 13.1
A covariate is a variable that has a:
(a) Curvilinear relationship with the dependent variable
(b) Linear relationship with the dependent variable
(c) Curvilinear relationship with the independent variable
(d) Linear relationship with the independent variable
13.1 Pre-existing groups
Imagine a case where there are three groups of women (nightclub hostesses, part-time secretaries
and full-time high-powered scientists). These are naturally occurring groups (i.e. we
cannot allot participants to these groups; they are in them already). We wish to test the hypothesis
that the more complex the occupation, the higher the testosterone level. Testosterone is
known as a ‘male’ hormone, but although men do have a much higher level of testosterone,
women produce testosterone too. There has, in fact, been research that shows a weak association
between occupational level and testosterone. Can you think of other variables that might
be related to the dependent variable (testosterone level)? Remember, these variables are called
the covariates. You can probably think of several – the timing of the menstrual cycle for one:
hormones fl uctuate according to the day of the cycle. If we were measuring testosterone in the
groups, we would like them to be measured on the same day of the cycle. Age is another. But
in order to keep it simple, we will stick to one covariate – age. Assume that age is positively
related to testosterone levels.
The scattergram in Figure 13.3 (fi ctional data) shows this relationship for all three groups
combined.
Now think of your three groups. Is it likely that the mean age of the three groups would be
the same? Why not?
It is not likely, of course. It is more likely that the high-powered scientists would be signifi
cantly older than the nightclub hostesses. So now if we use ANCOVA, we are using it in a
slightly different way. Not only does ANCOVA reduce the error variance by removing the
variance due to the relationship between age (covariate) and the DV (testosterone) (the fi rst
purpose), it also adjusts the means on the covariate for all of the groups, leading to the adjustment
of the y means (testosterone).