**David A. Kenny
September 15, 2018
**

**Moderator Variables: Introduction**

A key part of
moderation is the measurement of **X** to **Y** causal relationship for
different values of **M**. We refer to the effect of **X** on **Y** for a given value of **M** as the *simple effect* **X** on **Y**.

Deciding which variable is the moderator depends in large part on the researcher's interest. For the earlier example in which gender moderates the effect of psychotherapy, if one was a gender researcher, one might say that psychotherapy moderates the effect of gender.

**Causal Assumptions**

If **X** is a randomized variable, there is no causal ambiguity. Uncertainties arise when **X** is not randomized. If **X** is not manipulated, then the direction of causation must be assumed based on
theory or common sense. As shown in Judd and Kenny (2010), it is even possible
that the moderator effect can reverse if the direction of causation is flipped
(presuming that **Y** causes **X** instead of vice versa).
In a moderator analysis, if **X** is not manipulated, the researcher
needs to justify the choice of causal direction.

**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.
It is possible, but quite complicated, but **M** can be both a mediator and a
moderator (see Kraemer et al. (2001) for a different point of view.

**Moderator and Causal Variable
Relationship****
**If

**Measurement of Moderation****
**Generally, moderator effects are
indicated by the interaction of

** Y = i + aX
+ bM + cXM + E
(1)**

The interaction of **X** and **M** or coefficient **c** measures the moderation effect. Note that path **a** measures the *simple
effect* of **X**, sometimes
called the *main effect* of X, when **M** equals zero. As will be seen, the test of moderation is
not always operationalized by the product term **XM**.
Given
Equation 1, the effect of **X** on **Y** is **a + cM.** Thus, the effect of **X** on **Y** depends on the
value of **M**. It is noted that the effect of **X** on **Y** equals zero when **M** equals **–a/c**, which may or may not be
a plausible value of **M**.

**Alternative Interpretations of Moderator
Effects**

Finding
that **c** is statistically significant
does not prove moderator effects. One
major worry is non-additivity. Consider
the case in which the relationship between **X** and **Y** is nonlinear. For instance, **X** is income and **Y** is
work motivation. Imagine that the
relationship between the two is nonlinear such that if **X** is small the relationship is larger than when **X** is large. If age were tested as a moderator the
income-motivation relationship, then because younger workers make less money,
we would find the “moderator” effect, that the income-motivation relationship
is stronger for younger than for older workers.

Another worry is the actual moderator may not be the moderator but some other variable with which the moderator correlates. For instance, if we find that gender is a moderator, the real moderator might be height, masculinity-femininity, expectations of others, or income. Unless the moderator is a manipulated variable, we do not know whether it is a “true” moderator or just a “proxy” moderator.

**Level of Measurement of the Variables****
**It is
presumed here and throughout that the outcome variable is measured at the
interval level of measurement. Should
the outcome be a dichotomy, logistic regression would need to be used (Hayes & Matthes, 2009).

The remainder of
the page is 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, **X** 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).
Readers are
encouraged to read the next two sections, even if they are more interested in
one of the other cases, as many key concepts in mediation are discussed there.

**Categorical Moderator and Causal Variables**

When both **X** and **M** are dichotomous
(i.e., each have two levels), we have a 2 x 2 design. So for instance,
psychotherapy (therapy versus no therapy), might be more effective for women
than for men. We denote the four cells as **X**_{1}**M**_{1}, **X**_{1}**M**_{2},** X**_{2}**M**_{1},and **X**_{2}**M**_{2}. 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 (Equation 1):

**a** – the effect of **X** when **M** is zero
(the simple effect of **X** when **M** is zero)

**b** – the effect of **M** when **X** is zero

**c** – how much the effect of **X** changes as **M** goes from 0 to 1

The 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.
Obviously the interpretation of moderator depends very much on how **X** and **M** are coded.

If effect coding (one value of **X** and **M** is 1 and the other value is –1) is used, the interpretation of the coefficients
is as follows:

**a** – the effect of **X** averaged across **M**
(i.e., when **M** = 0)

**b** – the effect of **M** averaged across **X**
(i.e., when **X** = 0)

**c** – half of how much the effect of **X** changes as **M** goes from -1 to 1

Which particular coding method that is used is largely a matter of personal preference. The important thing is to know what coding system is used and interpret coefficients accordingly. Although coding affects the coefficients, it does not affect the inferential statistic for the test of the interaction (but it does affect the tests of main effects), the multiple correlation, the predicted values, and the residuals. It is generally inadvisable to trim out of the multiple regression equation the main effects if the interaction is present in the equation.

Regardless which coding system is used, there
are four means because the design is 2 x 2. If effect coding were used, the means would equal (where **i** is the intercept in the regression
equation):

__Cell Coding Predicted Mean __

**X**_{1}**M**_{1} **X** = -1; **M** = -1 **i **–** a – b + c**

**X**_{2}**M**_{1} **X** = 1; **M** = -1 **i **+** a – b – c **

**X**_{1}**M**_{2} **X** = -1; **M** = 1 **i – a + b – c **

**X**_{2}**M**_{2} **X** = 1; **M** = 1 **i +
a + b + c**

There might be an interest in the effect of
the causal variable or **X** for each of
the levels of the moderator or 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
(Aiken & West, 1991). If we want to test the effect of **X** when the **M** = 1, the
equation is run but **M** is not used
but **M**׳ = **M** – 1.
Coefficient **b** is now the simple effect of **X** on **Y** when **M** is 1, because when **M** =
1, **M**׳is zero.

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. If the covariates are themselves considered to be moderators, then
they would be allowed to interact with **X**.
Note that predicted values for the four cells
would no longer exactly equal the mean for the cell and so they should be
referred to as *least squares means*.

**Effect Size Measurement of Moderator
Effects and Power Analysis****
**One
can use traditional measures of effect size in measuring moderator effects

**Categorical Moderator and Continuous Causal Variable**

An example of this case, **M** is race, **X** is a personnel test, and **Y** is 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****
**Nonlinear
moderation refers to effect of

** Y = d + a _{1}X + b_{1}M + b_{2}M^{2} + c_{1}XM
+ c_{2}XM^{2} + E
(2)**

Nonlinear moderation can be
tested by determining if **c _{2}** is different from zero. (Note that

If **c _{1}** were positive and

If **c _{1}** were positive and

If **c _{1}** were negative and

If **c _{1}** were negative and

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.
In this case, the moderator is no longer continuous, but rather it is
dichotomized at the point of the threshold. The difficulty is that threshold point must be known a priori and cannot
be obtained by a simple median split. [The value of **M** at which the
effect of **X** on **Y** changes might be empirically determined by adapting an approach
described by Hamaker, Grasman, and
Kamphuis (2010)].

**Effect Size and Power
**The most common measure of
effect size in tests of moderation is

If **f ^{2 }**is known, one can conduct a power analysis using a power analysis
program. For instance, if

**Simple Effects****
**There are three methods to
determine

The second method is to re-estimate separate
regression equation but transform **M** by subtracting 2 or **M' = M – 2**. For this new equation, the effect of **X** refers to the case in which M is
2. This second method should result in
the same answer as the first.

The third method requires that **M** take on a few values. Separate regression equations would be
estimated for each value of **M**. This method does not assume homogeneity of
error variances and so it would likely produce estimates different from the
previous two.

**Continuous Moderator and Categorical Causal Variable**

An example is that the socioeconomic status moderates the effect of some intervention. One key issue is to center the variable of socioeconomic status; i.e., make sure that zero is a meaningful value for the moderator.

We may want to determine the effect of **X** for various levels of the moderator, **M, **i.e., simple effects. In principal,
the values of **M** would be chosen
using some sort conceptual rationale. For instance, if IQ were the moderator,
we might use 140 (genius level) and 100 (average level) to compute the effects
of **X** on **Y**. More commonly, the values
are one standard deviation above the mean of **M** and one standard deviation below the mean of **M**.
To obtain
these estimates we use either the first or second method described above.

**Continuous Moderator and Causal Variable**

One key question
is the assumption of how the moderator changes the causal relationship between **X** and **Y**. Normally, the assumption
is made that the change is linear: As **M** goes up or down by a fixed amount, the effect of **X** on **Y** changes by a
constant amount. Alternatively, M may
have a different type of effect: Threshold – The effect of **X** on **Y** changes when **M** is greater than a certain value;
Discrepancy – When **X** and **M** are measured using the same units,
the absolute difference between **X** and **M** is what matters (see also
Edwards, 1995). The key point is that
moderation is not always best captured by a product term.

If a product term is used, one
must assume that both **X** and **M** are measured without error, an often dubious assumption. Latent variables are
discussed below.

Centering of both **X** and **M** is
necessary if neither have zero as a meaningful value. To interpret the
results and determine simple effects, the effect of **X** at various levels of **M **would be measured. Ideally,
the levels of **M** would be theoretically motivated. If not possible,
one might use **M** at the mean and at plus and minus one standard deviation
from the mean.

Power of tests of moderation with two continuous variables is particularly low (McClelland & Judd, 1993).

**Latent Variables****
**In this case one latent variable interacts with another latent
variable. This is the most complicated case. Kenny and Judd (1984)
have developed a solution using product indicators of

Three additional issues that are discussed here briefly are repeated measures, multilevel modeling, meta-analysis, moderated mediation or mediated moderation, and mixture 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 indicated
by computing a difference score across conditions and determining whether the
moderator predicts that difference: Because the difference score measures the
effect of

**Multilevel Modeling****
**In some situations
the data are said to be

If **X** is measured at level 1, one can determine a *generic moderator*, that is, measure the extent to which there is
variation in the **X-Y** relationship. Evidence of generic moderation would
be obtained if there was variation in the **X**-**Y** slopes.

**Meta-analysis****
**Much of
meta-analysis involves the study of moderation. If a variable predicts effect sizes, that variable is moderator. Moreover, as with multilevel modeling, one
can test for a generic moderator by determining if effect sizes vary more than
would be expected by sampling error. One
of the key tasks in meta-analysis is the understanding or what are the
moderators of the effect.

**Mediated Moderation and Moderated
Mediation****
**In mediated
moderation, the moderation disappears when the mediator is introduced. In moderated mediation, the pattern of
mediation varies as a function of the moderator. See my mediation page for more
information.

Papers by Muller, Judd, and Yzerbyt (2005) and Edwards and Lambert (2007) discuss the relationship between mediated moderation and moderated mediation. They also present examples of each.

**Mixture Modeling****
**We can use mixture
modeling to search for a "latent" moderator. In such a case, we measure

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