David A. Kenny
November 14, 2009

 

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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 Variable

Other Issues
Bibliography

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My Mediation Page

DataToText Macro ModText that Performs a Moderation Analysis and Writes Text To Describe the Results

Andrew Hayes Macro ModProbe

Moderator Variables: Introduction

Basic Definitions

We begin with a linear causal relationship in which the variable X is presumed to cause the variable Y. A moderator variable M is a variable that alters the strength of the causal relationship.  So for instance, 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).  Most moderator analysis measure the causal relationship between X and Y by using a regression coefficient. 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 papers by Kraemer and colleagues (2001; 2002) for a related but somewhat different approach to defining and testing of moderators.  Frazier, Tix, and Barron (2004) provide a very good introduction to the topic of moderation.

A moderator analysis is an exercise of external validity in that the question is how universal is the causal effect.

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.

Causal Assumptions

If X is a randomized variable, there are fewer causal ambiguities. 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. 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.

Moderator and Causal Variable Relationship

If X is a manipulated variable, in principal, there should be no relationship between X and M.  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 that correlation has no special interpretation.  However, if X and M are too highly correlated, there can be estimation problems.

Measurement of Moderation

Generally, moderator effects are indicated by the interaction of X and M in explaining Y. The following multiple regression equation is estimated:

Y = d + aX + bM + cXM + E

The interaction of X and M or coefficient c measures the moderation effect. Note that path a measures the simple effect of X when M equals zero.  As will be seen, the test of moderation is not always operationalized by the product term XM, but it often is.

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, 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).  Readers are encouraged to read the next section, 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 X1M1, X1M2, X2M1,and X2M2. 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 (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

          X1M1           X = -1; M = -1                 i – a – b + c

          X2M1           X = 1; M = -1                  i + a – b – c

          X1M2           X = -1; M = 1                  i – a + b – c

          X2M2           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.  However, what seems preferable is to use a d change.  That is, we measure the d for the two levels of M and compare them.  In computing the two d’s, we should use the same standard deviation.  For instance, we might state:  The effect of psychotherapy on depression yields a d of 0.4 for men and a d of 0.7 for women or 0.3.  This d change measure is itself a d. 
Because the design is 2 X 2, the estimate of the moderator effects can be viewed as a difference between two means (X1M1 and X2M2 vs. X2M1 and X1M2).  Using these two means a d can be computed and a power analysis can be undertaken.

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

The typical way to estimate nonlinear moderation would be to estimate the following equation:

Y = d + a1X + a2X2 + bM + c1XM + c2X2M + E

Nonlinear moderation 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.  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.

Power and Effect Size

Aguinis (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 detect moderator effects when one of the variables is continuous.

The most common measure of effect size in tests of moderation is f2 (Aiken & West, 2001).  This measure represents the proportion of variance explained by the interaction after the effects of the other variables are controlled.  So far as I know, there are no universally accepted standards for small, medium and large effect sizes for interaction.  It should be realized that tests of interactions almost always have much less power than tests of main effects (McClelland & Judd, 1993).

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 is 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 for that variable.

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. In principal, these values 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.

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, but it quite complicated with many nonlinear constraints and it requires a relatively large sample size to have sufficient power.  Ping (1996) has developed a computational strategy that does not require nonlinear constraints.  Also Klein and Moosbrugger (2000) have developed a method of estimation that does not require product indicators or nonlinear constraints.  Their procedure is implement in Mplus.

Latent Moderation

In this approach indicators of the latent product are created.  So if there are two X indicator and two M indicators, four product indicators would be created X1M1, X1M2, X2M1, and X2M2.  The loading of these four variables are a function of the indicators' loading, i.e., their product.

Other Issues

Three additional issues that are discussed here briefly are repeated measures, multilevel modeling, meta-analysis, and moderated mediation or mediated moderation.

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 X on Y for each person, using it as the outcome variable gives an ideographic measure the causal effect and it is then determined if the moderator predicts that causal effect.

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. 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.

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.  Also Preacher, Rucker, and Hayes have developed a macro for estimating moderated mediation (click here).

Bibliography

Aguinis, H. (2004). Moderated regression. New York: Guilford.

Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park, CA: Sage.

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.

Edwards, J. R. (1995). Alternatives to difference scores as dependent variables in the study of congruence in organizational research. Organizational Behavior and Human Decision Processes, 64, 307-324.

Edwards, J. R., & Lambert L. S. (2007). Methods for integrating moderation and mediation: A general analytical framework using moderated path analysis. Psychological Methods, 12, 1-22.

Frazier, P. A., Tix, A. P. & Barron, K. E. (2004). Testing moderator and mediator effects in counseling psychology research. Journal of Counseling Psychology, 51, 115-134.

Hayes, A. F., & Matthes, J. (2009). Computational procedures for probing interactions in OLS and logistic regression: SPSS and SAS implementations.  Behavior Research Methods, 41, 924-936.

Judd, C. M., & Kenny, D. A. (2010). Data analysis. In S. Fiske, D. Gilbert, G. Lindsay (Eds.), The handbook of social psychology, in press.

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.

Kenny, D. A., & Judd, C. M. (1984). Estimating the nonlinear and interactive effects of latent variables. Psychological Bulletin, 96, 201-210.

Klein, A. G., & Moosbrugger H. (2000). Maximum likelihood estimation of latent interaction effects with the LMS method. Psychometrika, 65, 457-474.

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. Archives of General Psychiatry, 59, 877-883.

McClelland, G. H., & Judd, C. M. (1993). Statistical difficulties of detecting interactions and moderator effects. Psychological Bulletin, 114, 376-390.

Muller, D., Judd, C. M., & Yzerbyt, V. Y. (2005). When moderation is mediated and mediation is moderated. Journal of Personality and Social Psychology, 89,, 852-863.

Ping, R. A. (1996). Latent variable interaction and quadratic effect estimation: A two-step technique using structural equation analysis. Psychological Bulletin, 119, 166-175.


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