David A. Kenny
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Moderator Variables: Introduction
Categorical Moderator and
Causal Variables
Categorical Moderator and a
Continuous Causal Variable
Continuous Moderator and a
Categorical Causal Variable
Continuous Moderator and Causal
Variables
Moderator Variables: Introduction
Considered is the case in which a variable M is presumed to change the X to Y causal relationship. So for instance, a certain form of psychotherapy
may reduce depression more for men than for women, and so we would say that
gender (M) moderates the causal
effect of psychotherapy (X) on
depression (Y). Although classically, moderation implies a
weakening of a causal effect, a moderator can amplify or even reverse that
effect. Complete moderation would occur
in the case in which the causal effect of X
on Y would go to zero when M took on a particular value.
The reader might consult Kraemer et al. (2001; 2002) for a related but somewhat different approach to defining and testing of moderators.
Causal Assumptions
A key question is whether X a randomized variable. Much of what follows is based on this
presumption. Uncertainties arise when X is not randomized. If X
is not manipulated, then the direction of causation must be assumed. It is possible that the moderator effect can
reverse if the direction of causation is flipped (presuming that Y causes X instead of vice versa).
Timing of Measurement
Ideally the moderator should be measured
prior to variable X being
measured. So if X is manipulated, then M
should be measured prior to X being
manipulated. Of course, if M is a variable that does not change (e.g., race), the timing of
its measurement is less problematic.
Level of Measurement of the Variables
This page is
largely organized around the levels of measurement of the moderator and the
causal variable. The causal variable, X, can either be categorical (typically
a dichotomy) or a continuous variable.
So for instance, it might be psychotherapy versus no psychotherapy (a
dichotomy) or it might be the amount of psychotherapy (none, one month, two
months, or six months; a continuous variable).
Much in the same way, the moderator or M can be either categorical (e.g., gender) or continuous (e.g.,
age).
Moderator and Causal Variable
Relationship
If X is a manipulated variable, in principal, there should be no relationship between X and M. The two variable are said to be independent. If X is not randomized, it might be correlated with M. Unlike mediation, there is no need for X and M to be correlated and then correlation has no special interpretation.
Statistical Measurement of Moderation
Generally, moderator effects are indicated by
the interaction of X and M in explaining Y. So the following
regression equation is estimated:
Y = d + aX + bM + cXM + E
The interaction of X and M measures the moderation effect.
As will be seen, the test of moderation is not always operationalized by
the product term XM, but it often
is.
Categorical
Moderator and Causal Variables
This is the prototypical case. When both variables are dichotomous, we have
a 2 x 2 design. So for instance,
psychotherapy (therapy versus no therapy), might be more effective for women than
for men. To estimate the above
regression equation, we need to dummy code the moderator and the causal
variable. So for instance, if we use
codes of zero and one, then we have the following interpretations of the
coefficients in the above multiple regression equation:
a – the effect of X when M is zero
b – the effect of M when X is zero
c – how the effect of X varies as M varies
I focus on c because it captures the moderator effect. If c
is positive, then it indicates that the effect of X on Y increases as M goes from 0 to 1. If c
is negative, then it indicates that the effect of X on Y decreases as M goes from 0 to 1.
If effect coding (one value of X and M is 1
and the other value is –1), the interpretation of the coefficients is as
follows:
a – the effect of X averaged across M
b – the effect of M averaged across X
c – half of how much the effect
of X changes as M changes
While coding affects the coefficients, it
does not affect the inferential statistic for the test of the interaction, the
multiple correlation, the predicted values, and the residuals.
It is generally not advised to trim out of
the multiple regression equation main effects if the interaction is present in
the equation.
There might be an interest in the effect of
the causal variable or X for each of
the levels of the moderator, something called the simple effects of X. To estimate the simple effects, a different
regression equation is run and in each we recode the moderator so that a given
level is set to zero. So, if we want to
test the effect of X when the
moderator or M = 1, the equation is
run but M is not used but M׳ = M – 1.
If X
or M have more than two levels, then
multiple dummy variables are needed (the number of levels less one), and
moderation is tested by a set of product variables.
If there are covariates (variables that cause
Y and measured prior to Y), they can be entered into the equation
and of course they themselves can be moderators.
Because the key results in this case are a
set of means (or adjusted means), one can test whether there are differences
between them using post hoc tests.
Categorical
Moderator and Continuous Causal Variables
An example of this case, M might be race, X might be a personnel test, and Y might be some job performance
score. Generally, it is assumed that the
effect of X on Y is linear. It is also assumed
(but it can be tested, see below) that the moderation is linear. That is, as M varies, the linear effect of X
on Y might vary. Thus, the linear relationship increases or
decreases as M increases.
It is almost always preferable
to measure the linear effect by using a regression coefficient and not a
correlation coefficient.
More Complex Specification
The typical way to estimate
nonlinear moderation would be to estimate the following equation:
Y = d + a1X + a2X2 +
bM + c1XM + c2X2M + E
Nonlinear mediation can be
tested by determining if c2 is different from zero.
Baron and Kenny (1986, page
1175) discuss alternative specifications of moderation. For instance, the moderator might act as a threshold
variable and there would be no effect of the causal variable when the moderator is low, but at a certain
value of the moderator the effect emerges.
Power
Aguinas (2004) has shown that the power of this test can be very low, typically below 50%. One needs very large sample sizes over 200 to have reasonable power to detetect moderator effects when one of the variables is continuous.
Simple Effects
There are two ways to determine simple
effects. The first and relatively
simpler way is to estimate the simple effects within the regression
equation. The second way is to estimate
separate regression equations for each level of the moderator. The latter strategy is preferable if there
are differences in error variance for the different levels of the moderator.
Continuous
Moderator and Categorical Causal Variable
One example might be that the socio-economic
status moderates the effect of some intervention. One key issue is to center the variable of
socio-economic status; i.e., make sure that zero is a meaningful value.
We may want to determine the effect of X for various levels of the moderator, M.
One idea is to determine the effect of X for different values of M. Ideally, these values would be chosen
logically. Alternatively, you might pick
values at the mean of M, one
standard deviation above the mean of M,
and one standard deviation below the mean of M.
Continuous Moderator and Causal Variables
This is the most complicated
case. One key issue has to do with the
type of moderation that is assumed.
One must assume that both X and M are measured without error, an often dubious assumption. Aiken and West (1991) describe methods that do allow
researchers to have measurement error but such procedures are rather
complicated.
Centering of both X and M is necessary if
neither have zero as a meaningful value.
We would then measure the effect of X at various levels of M. Ideally, the levels of M would be logically
chosen. If not possible, one might use M
at the mean and at plus and minus one standard deviation from the mean.
Two additional issues
that are discussed here are repeated measures and multilevel modeling.
Repeated Measures
All of the above discussion presumes that the
design is between participants. In some
cases, the design is repeated measures.
Judd,
Kenny, and McClelland (2001) describe moderator analyses in this case. In essence moderation is indicating by
computing a difference score and seeing if the moderator predicts that
difference.
Multilevel Modeling
In multilevel modeling, there are two levels.
For instance there might be students in classrooms. Sometimes, there are level 1 moderators;
these being moderators that vary within the classroom. More typically there are level 2 moderators;
these being moderators that vary between classrooms. One can also determine a generic moderator, that is,
measure the extent to which there is variation in the X-Y relationship.
.
Aguinis, H. (2004).
Moderated regression.
Aiken, L. S., &
West, S. G. (1991). Multiple
regression: Testing and interpreting interactions.
Baron, R. M., &
Kenny, D. A. (1986). The
moderator-mediator variable distinction in social psychological research:
Conceptual, strategic and statistical considerations. Journal
of Personality and Social Psychology, 51, 1173-1182.
Judd, C. M., Kenny,
D. A., & McClelland, G. H.
(2001). Estimating and testing
mediation and moderation in within-participant designs. Psychological
Methods, 6, 115-134.
Kraemer, H. C.,
Stice E., Kazdin A., Offord D., & Kupfer D.
(2001). How do risk factors work together? Moderators, mediators, independent,
overlapping, and proxy risk factors. American Journal of Psychiatry, 158, 848-856.
Kraemer H. C., Wilson G. T., Fairburn C. G., & Agras W. S. (2002). Mediators and moderators of treatment effects in randomized clinical trials. Arch Gen Psychiatry, 59, 877-883.
