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Typically theoretical and empirical models in the behavioral sciences posit that independent variables aﬀect dependent variables. A moderator variable is a variable, which is thought to temper or modulate the magnitude of the eﬀect of an independent variable on a dependent one. Conceptually it is important to diﬀerentiate between a moderator and a mediator. A moderator is a variable that aﬀects the magnitude of the relationship between the independent and dependent variables. It identiﬁes the conditions under which, or the type of participant for whom, the eﬀect is likely to be particularly large or particularly small. A mediator is a variable that describes the process that is responsible for the eﬀect of the independent variable on the dependent one. It is a variable that is aﬀected by the independent variable and in turn aﬀects the dependent one, thus being responsible for the eﬀect. (See Baron and Kenny 1986).

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## 1. Examples of Hypotheses Involving Moderation

Hypotheses about moderation are ubiquitous in the social and behavioral sciences. The following examples, taken from a variety of ﬁelds, provide illustrations. Each example consists of two sentences. The ﬁrst hypothesizes an eﬀect of an independent variable on a dependent one. The second sentence identiﬁes a hypothesized moderator.

(a) Level of education attained aﬀects lifelong earnings. This relationship is less strong for females than it is for males.

(b) Stressful life events tend to produce psychological problems. These eﬀects are lessened if one has an extensive social support network.

(c) Some forms of instruction lead to better retention than others. These eﬀects are larger among more able students.

(d) Exercise has pronounced health beneﬁts. These beneﬁts are more pronounced among older people.

Note in these examples that moderator variables can take on a variety of forms. They can refer to characteristics of the participants in the research (i.e., their gender, ability, or age); equally plausibly they may characterize the situations or environments which moderate the relationship between the independent and dependent variables. Additionally, their scale of measurement may be nominal (e.g., gender) or may vary more or less continuously (e.g., ability levels).

## 2. Conceptual and Analytic Issues in Testing Moderation

To say that one variable moderates the eﬀect of another is equivalent to saying that there is a statistical interaction between the two variables. Accordingly, moderation implies interaction, although not all inter-

actions involve moderating variables. The distinction is as follows: moderation means that there is a main eﬀect of the independent variable plus an interaction between the independent variable and the moderator. Overall there is an eﬀect of the independent variable, but the magnitude of that eﬀect depends on the moderator. An interaction, more generally, makes no claims about the presence or absence of the main eﬀect of the independent variable. It simply indicates that the eﬀect of one variable depends on another.

Because moderation implies a main eﬀect of the independent variable plus an interaction of that variable with the moderator, typically interactions in the presence of moderators are not what are called crossover or disordinal interactions. That is, from one level of the moderator to others, the eﬀect of the independent variable varies in magnitude but not in direction. To illustrate, consider the ﬁrst of the examples given above, that gender moderates the relationship between educational attainment and lifetime earnings. For both genders, there is presumably a positive eﬀect of educational attainment on earnings, but that positive eﬀect is less strong among females than among males. For no level of the moderator does the relationship between educational attainment and earnings reverse.

Because moderation implies an interaction, it is necessary that the moderator and the independent variable be at least partially independent of each other. Ideally, the two variables would be fully crossed as in an experimental design where both variables are manipulated independently of each other. More typically, while the independent variable may be manipulated, the moderator may be a measured variable that varies more or less continuously. Again, however, to assess moderation it is important to insure that observations from all levels of the moderator occur at every level of the independent variable.

Analytically, moderation is typically assessed via either analysis of variance or multiple regression procedures, with the choice between the procedures traditionally depending on the scale of measurement of the independent and moderator variables. Two eﬀects are of interest. First, the overall main eﬀect of the independent variable is expected. Second, moderation of that eﬀect implies that the interaction between the independent variable and the moderator is statistically signiﬁcant. The use of analysis of variance to assess moderation is perfectly appropriate when both the independent variable and the moderator are measured on a nominal scale, with a few discrete levels of each variable. With moderators that vary more or less continuously, regression procedures are more appropriate, since substantial loss of power can occur if continuously varying moderators are broken up into discrete levels (e.g., using a median split) to accommodate them in analysis of variance. In general, multiple regression can be used to assess moderation, regardless of whether independent and moderator variables are scaled discretely or more or less continuously. In the case of discrete or nominal scaling, one simply must compute dummy coded or contrast coded variables to be included as predictors. The general model involves including as predictors the independent variable, the moderator variable, and a variable that is the product of the two. The test of the product variable, controlling for the two components of that product, provides a test of the statistical interaction (Cohen 1978). In this analysis, one must be cautious in interpreting the coeﬃcients (and their associated statistical tests) of the component variables (i.e., the independent and moderator variables in a model that includes their product). Unless these variables have been centered on their means (and the product computed from those centered variables), their coeﬃcients will not be readily interpretable as the average or main eﬀect of each variable (Aiken and West 1991, Judd and McClelland 1989).

In some situations, the independent variable is likely to vary within units rather than between them. For instance, the eﬀects of mood as an independent variable may be assessed by naturally occurring variations in mood within individuals over time. In these cases, traditional tests of statistical signiﬁcance of both the main eﬀects of the independent variable and its interaction with a moderator are likely to be biased unless dependencies in the data are appropriately handled. Repeated measures analysis of variance and, more generally, hierarchical linear models can be used to assess moderation in these sorts of designs.

Tests of moderation with naturally varying, as opposed to manipulated, independent and moderator variables are likely to have relatively low statistical power. Compared to the experimental situation where both variables are manipulated, there is likely to be less variation in the independent and moderator variables in the case where these variables are measured as they naturally occur. As a result the residual variation in their product is likely to be particularly small compared with the situation where both variables are manipulated. Tests of moderation (and interaction more generally) might then be most powerfully conducted by selecting units with relatively extreme values on the two variables (McClelland and Judd 1993).

## 3. Interpretive Ambiguities

As the examples in Sect. 1 illustrated, moderation hypotheses are ubiquitous in the social and behavioral sciences and analytic procedures for testing interactions, as described in the previous paragraphs, are relatively well worked out. However, interpretive diﬃculties can arise in describing moderation unless strong assumptions can be made about the metric of the dependent variable in any analysis. Nonlinear transformations of the dependent variable may eliminate interactions, especially when those interactions involve moderation of a main eﬀect of an independent variable, rather than a crossover interaction.

Consider a simple example from cognitive psychology. Participants are timed while identifying letter strings as words or not (a lexical decision task). Words are either preceded by a semantically related prime or not. The main hypothesis is that primes should decrease the latency of the lexical response. Target words vary in their familiarity or frequency in the language. The eﬀect of primes is expected to be moderated by word familiarity, such that prime eﬀects are larger for less familiar target words. Hypothetical latency means (in seconds) are given in the left-hand portion of Table 1. Evidence is found for moderation: the prime versus no prime diﬀerence is more than twice as large for unfamiliar words (0.033 seconds) as it is for familiar ones (0.015 seconds). But now consider what happens if we convert the means to represent speed of response (rather than its latency) by taking the inverse of the latencies. In the speed metric, there is absolutely no evidence for moderation due to familiarity.

Which is the correct analysis? What conclusion about moderation is correct? Unfortunately, since there is no way to deﬁne the more appropriate response metric, there is no way to answer this question. The lesson is that many interactions in the social and behavioral sciences, where the metrics of the variables are somewhat arbitrary, disappear under nonlinear scale transformations. This is particularly likely to be the case when interactions are not crossover, as in the case of moderation.

To be conﬁdent about the presence of moderation, given arbitrarily deﬁned metrics of most dependent variables in the social and behavioral sciences, analyses should be conducted using a variety of diﬀerent metrics. If interactions prove to be signiﬁcant under a variety of diﬀerent nonlinear transformations, then conﬁdence concerning the presence of moderation increases.

**Bibliography:**

- Aiken L S, West S G 1991 Multiple regression: Testing and interpreting interactions. Sage, Newbury Park, CA
- Baron R M, Kenny D A 1986 The moderator-mediator variable distinction in social psychological research. Journal of Personality and Social Psychology 51: 1173–82
- Cohen J 1978 Partialed products are interactions; partialed powers are curvecomponents. Psychological Bulletin 85: 858–66
- Judd C M, McClelland G H 1989 Data Analysis: A Model Comparison Approach. Harcourt, Brace, Jovanovich, San Diego, CA
- McClelland G H, Judd C M 1993 Statistical diﬃculties of detecting interactions and moderator eﬀects. Psychological Bulletin 114: 376–90