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15.1 ♦^ Moderation Versus Mediation
Chapter 10 described various ways in which including a third variable ( X 2 ) in an analysis can change our understanding of the nature of the relationship between a predictor ( X 1 ) and an outcome ( Y ). These included moderation or interaction between X 1 and X 2 as predictors of Y and mediation of the effect of X 1 on Y through X 2. When moderation or interaction is present, the slope to predict Y from X 1 differs across scores on the X 2 control variable; in other words, the nature of the X 1 , Y relationship differs depending on scores on X 2. This chapter describes tests for the statistical significance of moderation or interaction between predictor variables in a regression analysis. Chapter 10 introduced path models as a way to describe these patterns of association. There are two conventional ways to represent moderation or interaction between predictor variables in path models, as shown in Figure 15.1. The top panel has an arrow from X 2 that points toward the X 1 , Y path. This path diagram represents the idea that the coefficient for the X 1 , Y path is modified by X 2. A second way to represent moderation appears in the lower panel of Figure 15.1. Moderation, or interaction between X 1 and X 2 as predictors of Y , can be assessed by including the product of X 1 and X 2 as an additional predictor variable in a regression model. The unidirectional arrows toward the outcome variable Y represent a regression model in which Y is predicted from X 1 , X 2 and from a product term that represents an interaction between X 1 and X 2. The three predictors are correlated with each other (these correlations are represented by the double-headed arrows). Moderation should not be confused with mediation (see Baron & Kenny, 1986). In a mediated causal model, the path model (as shown in Figure 15.2) represents a hypothesized causal sequence. When X 1 is the initial cause, Y is the outcome, and X 2 is the hypothesized mediating variable, a mediation model includes a unidirectional arrow from X 1 to X 2 (to represent the hypothesis that X 1 causes X 2 ) and a unidirectional arrow from X 2 to Y (the hypothesis that X 2 causes Y ). In addition, a mediation model
MODERATION Tests for Interaction in Multiple Regression
612 ——CHAPTER 15 X 1 X 2 Y X 1 * X 2 may include a direct path from X 1 to Y , as shown in Figure 15.2. Although the terms moderation and mediation sound similar, they imply completely different hypotheses about the nature of association among variables. This chapter describes methods for tests of moderation or interaction; Chapter 16 discusses tests of mediated causal models.
15.2 ♦^ Situations in Which Researchers Test Interactions
15.2.1 ♦^ Factorial ANOVA Designs The most familiar situation in which interactions are examined is the factorial design discussed in Chapter 13. In a factorial design, the independent variables are categorical, and the levels or values of these variables correspond to different types or amounts of treatment (or exposure to some independent variable). Factorial designs are more common in experiments (in which a researcher may manipulate one or more variables), but they can also include categorical predictor variables that are not manipulated. The example of a factorial design presented in Chapter 13 included two factors that were observed rather than manipulated (social support and stress). As another example, Lyon and Greenberg (1991) conducted a 2 × 2 factorial study in which the first factor (family background of participant, i.e., whether the father was diagnosed with alcoholism) was Figure 15.1 ♦ Two Path Models That Represent Moderation of the X 1 , Y Relationship by X 2 X 2 X 1 Y Figure 15.2 ♦ Path Model That Represents Partial Mediation of the X 1 , Y Relationship by X 2 NOTE: See Chapter 16 for discussion of mediation analysis. X 2 X 1 Y
614 ——CHAPTER 15 where Y ′ is the predicted score on a quantitative Y outcome variable, and X 1 and X 2 are predictor variables. If the b 3 coefficient for the X 1 × X 2 product term in this regression is statistically significant, this is interpreted as a statistically significant interaction between X 1 and X 2 as predictors of Y. Note that there are almost always correlations among these predictor variables ( X 1 , X 2 , and X 1 × X 2 ). As discussed in Chapter 14, information from regression analysis, such as the squared semipartial correlation sr^2 , provides information about the proportion of variance that is uniquely predictable from each independent variable. It is undesirable to have very high correlations (or multicollinearity) among predictor variables in regression analysis because this makes it difficult to distinguish their unique contributions as predictors; high correlations among predictors also create other problems such as larger standard errors for regression coefficients. SOURCE: Graph based on data presented in Lyon and Greenberg (1991). Figure 15.3 ♦ Plot of Cell Means for the Significant Interaction Between Type of Person Needing Help and Family Background of Female Participant as Predictors of Amount of Time Offered to Help 0 Exploitive Nurturant Person Described as Needing Help 25 50 75 Mean Time Offered 100 125 Family Background Alcoholic Not Alcoholic
Moderation—— 615
15.3 ♦^ When Should Interaction Terms Be
Included in Regression Analysis?
When researchers use factorial ANOVA design (as discussed Chapter 13), the default analysis usually includes interactions between all factors; this makes it unlikely that researchers will overlook interactions. In multiple regression analysis, the default model does not automatically include interactions between predictor variables. Unless the data analyst creates and adds one or more interaction terms to the analysis, interactions may be overlooked. When an interaction is present but is not included in the regression analysis, the model is not correctly specified. This is a problem for two reasons. First, an interaction that may be of theoretical interest is missed. Second, estimates of coefficients and explained variance associated with other predictor variables will be incorrect. There are two reasons why researchers may choose to include interaction terms in regression analyses. First, prior theory may suggest the existence of interactions. For example, in the nonexperimental factorial ANOVA example presented in Chapter 13, the first factor was level of social support (low vs. high), and the second factor was exposure to stress (low vs. high). Theories suggest that social support buffers people from effects of stress; people who have low levels of social support tend to have more physical illness symptoms as stress increases, while people who have high levels of social support show much smaller increases in physical illness symptoms in response to stress. This buffering hypothesis can be tested by looking for a significant interaction between social support and stress as predictors of symptoms. Second, there may be empirical evidence of interaction, either from prior research or based on patterns that appear in preliminary data screening. Chapter 10 provided examples to show that sometimes, when data files are split into separate groups (such as male and female), the nature of the relationship between other variables appears to differ across groups. In the Chapter 10 example, the correlation between emotional intelligence (EI) and drug use (DU) was significant and negative within the male group; these variables were not significantly correlated within the female group. Apparent “interactions” that are seen during data screening may arise due to Type I error, of course. (Particularly if an interaction does not seem to make any sense, researchers should not concoct post hoc explanations for interactions detected during preliminary data screening.) Looking for evidence of interactions in the absence of a clear theoretical rationale is a form of data snooping. To limit the risk of Type I error, it is important to conduct follow-up studies to verify that interactions can be replicated with new data, whether the interaction was predicted from theory or noticed during data screening.
15.4 ♦^ Types of Predictor Variables Included in Interactions
The methods used to assess interaction differ somewhat depending on the type of predictor variables. Predictor variables in regression may be either quantitative or categorical variables. 15.4.1 ♦^ Interaction Between Two Categorical Predictor Variables In order for categorical variables to be used as predictors in regression, group membership should be represented by dummy variables when there are more than two
Moderation—— 617 large enough to obtain estimates of sample means that have relatively narrow confidence intervals (as discussed in Chapters 5 and 6). Detailed empirical examples of preliminary data screening for both quantitative and categorical variables have been presented in earlier chapters, and that information is not repeated here. An additional assumption in multiple regression models that do not include interaction terms (e.g., Y ′ = b 0 + b 1 X 1 + b 2 X 2 , as discussed in Chapter 11) is that the partial slope to predict Y from predictor variable X 1 is the same across all values of predictor variable X 2. This is the assumption of homogeneity of regression slopes; in other words, this is an assumption that there is no interaction between X 1 and X 2 as predictors of Y. It is important to screen for possible interactions whether these are expected or not, because the presence of interaction is a violation of this assumption. When interaction is present, the regression model is not correctly specified unless the interaction is included. Simple ways to look for evidence of interactions in preliminary data screening were discussed in Chapter 10. The split files command can be used to examine bivariate associations between quantitative variables separately within groups. For instance, a split file command that set up different groups based on sex ( X 2 ), followed by a scatter plot between emotional intelligence ( X 1 ) and drug use ( Y ), provided preliminary information about a possible interaction between sex and EI as predictors of drug use. Preliminary data screening should include inspection of scatter plots to look for possible interactions among predictors.
15.6 ♦^ Issues in Designing a Study
The same considerations involved in planning multiple regression studies in general still apply (see Chapter 14). These include the following. The scores on the outcome variable Y and all quantitative X predictor variables should cover the range of values that is of interest to the researcher and should show sufficient variability to yield effects that are large enough to be detectable. Consider age as a predictor variable, for example. If a researcher wants to know how happiness ( Y ) varies in relation to age ( X 1 ) across the adult life span, the sample should include people whose ages cover the range of interest (e.g., from 21 to 85), and there should be reasonably large numbers of people at all age levels. Results of regression are generalizable only to people who are within the range of scores for X predictor variables and Y outcome scores that are represented in adequate numbers in the sample. Whisman and McClelland (2005) suggest that it may be useful to oversample cases with relatively extreme values on both predictor variables X 1 and X 2 , particularly when sample size is low, to improve statistical power to detect an interaction; however, this strategy may result in overestimation of the effect size for interactions.
15.7 ♦^ Sample Size and Statistical Power
in Tests of Moderation or Interaction
Statistical power for detection of interaction depends on the proportion of variance in Y that is predictable from X 1 and X 2 , in addition to the proportion of additional variance in
618 ——CHAPTER 15 Y that can be predicted from an interaction between X 1 and X 2. As in earlier discussions of statistical power, educated guesses (based on results from similar past research, when available) about effect size can be used to guide decisions about sample size. Table 15. provides estimates of minimum sample sizes required for power of .80 to detect interaction in regression using a = .05 as the criterion for statistical significance. To use this table, the researcher needs to be able to estimate two effect sizes: a. What proportion of variance in Y is explained in a “main effects”^1 –only model? b. What proportion of variance in Y is explained in a main effects plus interaction model? If X 1 and X 2 are the predictors and Y is the outcome variable, the main effects–only model is Y ′ = b 0 + b 1 X 1 + b 2 X 2. (15.2) The main effects plus interaction model is Y ′ = b 0 + b 1 X 1 + b 2 X 2 + b 3 ( X 1 × X 2 ). (15.3) To use Table 15.1, make an educated guess about the R^2 values for these two equations. (The value of R^2 for the main effects plus interaction model must be equal to or larger than the value of R^2 for the main effects–only model.) Find the row that corresponds to R^2 for the main effects–only model and the column that corresponds to the main effects plus interaction R^2. If R^2 for the main effects–only model = .15 and the R^2 for main effects plus interaction = .20, the sample size required for power of. would be 127. In Section 12 of Chapter 14, additional recommendations about power were made; Tabachnick and Fidell (2007) suggested that the ratio of N (number of cases) to k Table 15.1 ♦^ Sample Size Required for Statistical Power of .80 to Detect Interaction in Regression Using a =. R^2 for Model With Main Effects and Interaction R^2 for Main Effects Only .05 .10 .15 .20 .25 .30. .05 143 68 43 32 24 19 .10 135 65 41 29 22 .15 127 60 39 27 .20 119 57 36 .25 111 53 .30 103 SOURCE: Adapted from Aiken and West (1991, Table 8.2). NOTE: Sample sizes given in this table should provide power of .80 if the variables are quantitative, multivariate normal in distribution, and measured with perfect reliability. In practice, violations of these assumptions are likely; categorical variables may require larger sample sizes.
620 ——CHAPTER 15 from sex, years, and a new variable that was obtained by forming the product of sex × years; this product represents the interaction between sex and years as predictors of salary. Salary′ = b 0 + b 1 Sex + b 2 Years + b 3 (Sex × Years). (15.4) Part of the regression results appear in Figure 15.4. In this example, the inclusion of a sex × years product term makes it possible to assess whether the annual increase in salary for each 1 additional year of experience differs significantly between the male versus female groups. The raw score regression coefficient for this product term was b = .258, with t (46) = .808, p = .423. There was not a statistically significant interaction in this preliminary example. When there is not a significant interaction, the slope to predict salary from years is the same for the male and female groups, and the regression lines to predict salary from years are parallel. When there is not a significant interaction, a data analyst has two choices about the way to report results. If a test for an interaction is theoretically based or of practical importance, it may be informative to retain the interaction term in the model and report that the interaction was not found to be statistically significant. Of course, failure to find statistical significance for an interaction term may occur because of lack of statistical power, unreliability of measures, nonlinear forms of interaction, or sampling error. On the other hand, if an interaction term is not statistically significant and/or not predicted from theory, does not make sense, and/or accounts for a very small proportion of the variance in the regression, the researcher might choose to drop the interaction term from the regression model when the final analysis is reported, as shown in Equation 15.5. In any case, researchers should provide a clear and honest account about the nature of decisions about which variables to include (and not include) in the final analysis. Salary′ = b 0 + b 1 Sex + b 2 Years. (15.5) Figure 15.4 ♦ Part of Regression Results for Data in Table 12.1 and in the SPSS File Named sexyears.sav NOTE: Sex was dummy coded with 1 = male, 0 = female. This regression model included an interaction term named genyears; this interaction was not significant, t (46) = .808, p = .423. Coefficientsa Model Unstandardized Coefficients Standardized Coefficients B Std. Error Beta t Sig. 1 (Constant) 35.117 1.675 20.970. Sex 1.936 2.289 .098 .846. years 1.236 .282 .692 4.383. sexyears .258 .320 .164 .808. a. Dependent Variable: salary
Moderation—— 621 Results of the regression for the model given by Equation 15.5 appear in Figure 15.5. The raw score or unstandardized equation to predict salary for this model was Salary′ = 34.193+ 3.355Sex + 1.436Years. (15.6) By substituting in values of 1 for male and 0 for female, two prediction equations are obtained: For males: Salary′ = 34.193 + 3.355 × 1 + 1.436 Years. Salary′ = 37.548 + 1.436 Years. For females: Salary′ = 34.193 + 1.436 Years. Figure 15.5 ♦ Second Regression Analysis for Data in Table 12.1 and in the SPSS File Named sexyears.sav NOTES: The nonsignificant interaction term was dropped from this model. Sex was dummy coded with 1 = male, 0 = female. Equation to predict salary from sex and years: Salary′ = 34.193 + 3.355 × Sex + 1.436 × Years. Model Summary Model R R Square Square Adjusted R Std. Error of the Estimate 1 .880a^ .775 .765 4. a. Predictors: (Constant), years, sex ANOVAb Model Sum of Squares df Mean Square F Sig. 1 Regression 3609.469 2 1804.734 80. Residual 1048.151 47 22. Total 4657.620 49 a. Predictors: (Constant), years, sex b. Dependent Variable: salary Coefficientsa Model B Std. Error t Sig. 1 (Constant) 34.193 1.220 28.038. sex 3.355 1.463 .17 2.293. a. Dependent Variable: salary .000a Model Unstandardized Coefficients B Std. Error t Sig. Coefficients Standardized Beta 1 (Constant) 34.193 1.220 28.038. sex 3.355 1.463 .170 2.293. years 1.436 .133 .804 10.830. a. Dependent Variable: salary
Moderation—— 623
15.11 ♦^ Example 1: Significant Interaction Between
One Categorical and One Quantitative Predictor Variable
The next hypothetical example involves an interaction between sex and years as predictors of annual salary in thousands of dollars per year ( Y ). Sex is dummy coded (0 = female, 1 = male). The data for this example^2 are in an SPSS file named sexyearswithinteraction. sav. Because preliminary data screening for regression was discussed in earlier chapters, it is omitted here. Before running the regression, the data analyst needs to compute a new variable to represent the interaction; this is called sexbyyear; it is obtained by forming the product sex × year.^3 A linear regression was performed using sex, year, and sexbyyear as predictors of salary, as shown in Equation 15.7; SPSS results appear in Figure 15.7. Salary′ = b 0 + b 1 Sex + b 2 Years + b 3 SexbyYear. (15.7) Figure 15.7 ♦ Regression Results for Data in an SPSS File Named sexyearswithinteraction.sav Variables Entered/Removedb Model Variables Entered Variables Removed Method 1 sexbyyear, years, sex a
. Enter a. All requested variables entered. b. Dependent Variable: salary Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate 1 .941a^ .886 .880 7. a. Predictors: (Constant), sexbyyear, years, sex ANOVA b Model Sum of Squares df Mean Square F Sig. 1 Regression 26756.614 3 8918.871 144.586 .000 a Residual 3454.386 56 61. Total 30211.000 59 a. Predictors: (Constant), sexbyyear, years, sex Coefficients a Model Unstandardized Coefficients Standardized Coefficients t Sig. Correlations B Std. Error Beta Zero-order Partial Part 1 (Constant) 34.582 2.773 12.470. sex -1.500 3.895 -.033 -.385 .702 .248 -.051 -. years 2.346 .211 .658 11.101 .000 .808 .829. sexbyyear 2.002 .335 .530 5.980 .000 .684 .624. a. Dependent Variable: salary
624 ——CHAPTER 15 To decide whether the interaction is statistically significant, examine the coefficient and t test for the predictor sexbyyear. For this product term, the unstandardized regression slope was b = 2.002, t (56) = 5.98, p < .001. The coefficient for the interaction term was statistically significant; this implies that the slope that predicts the change in salary as years increase differs significantly between the male and female groups. The overall regression equation to predict salary was as follows: Salary′ = 34.582 – 1.5 × Sex + 2.346 × Year + 2.002 × (Sex × Year). The nature of this interaction can be understood by substituting the dummy variable score values into this regression equation. For females, sex was coded 0. To obtain the predictive equation for females, substitute this score value of 0 for sex into the prediction equation above and simplify the expression. Females: Salary′ = 34.582 – 1.5 × 0 + 2.346 × Year + 2.002 × (0 × Year). Salary′ = 34.582 + 2.346 × Year. In words: A female with 0 years of experience would be predicted to earn a (starting) salary of $34,582. For each additional year on the job, the average predicted salary increase for females is $2,346. For males, sex was coded 1: To obtain the predictive equation for males, substitute the score of 1 for sex into the prediction equation above and then simplify the expression. Males: Salary′ = 34.582 – 1.5 × 1 + 2.346 × Years + 2.002 × (1 × Years). Salary′ = 34.582 – 1.5 + 2.346 × Years + 2.002 × Years. Collecting terms and simplifying the expression, the equation becomes Males: Salary′ = (34.582 – 1.5) + (2.346 + 2.002) × Years. Salary′ = 33.082 + 4.348 × Years. In words, the predicted salary for males with 0 years of experience is $33,082, and the average increase (or slope) for salary for each 1-year increase in experience is $4,348. An equation of the form Y ′ = b 0 + b 1 X 1 + b 2 X 2 + b 3 ( X 1 × X 2 ) makes it possible to generate lines that have different intercepts and different slopes. As discussed in Chapter 12, the b 1 coefficient is the difference in the intercepts of the regression lines for males versus females. The unstandardized b 1 coefficient provides the following information: How much does the intercept for the male group (coded 1) differ from the female group (coded 0)? For these data, b 1 = –1.500; with X 1 coded 1 for males and 0 for females, this implies that the starting salary (for years = 0) was $1,500 lower for the male group than for the female group. The t ratio associated with this b 1 coefficient, t (56) = –.385, p = .702, was not statistically significant. The unstandardized b 3 coefficient provides the following information: What was the difference between the male slope to predict salary from years and the female slope? For these data, b 3 corresponded to a $2,002 difference in predicted salary (with a higher predicted salary for males); this difference was statistically significant. For these
626 ——CHAPTER 15 To see the regression lines for females and males superimposed on this plot, we need to edit the scatter plot. To start the SPSS Chart Editor, right click on the scatter plot and select → <In Separate Window > , as shown in Figure 15.9. Next, from the top-level menu in the Chart Editor Dialog window, select ; from the pull-down menu, choose “Fit Line at Subgroups,” as shown in Figure 15.10. The separate regression lines for males (top line) and females (bottom line) appear in Figure 15.11. Further editing makes the lines and case markers easier to distinguish. Click on the lower regression line to select it for editing; it will be highlighted as shown in Figure 15.12. (In the SPSS chart editor, the highlighted object is surrounded by a pale yellow border that can be somewhat difficult to see.) Then open the Line Properties window (as shown in Figure 15.13). Within the properties window, it is possible to change the line style (from solid to dashed), as well as its weight and color. The results of choosing a heavier, dashed, black line for the female group appear in Figure 15.14. Case markers for one group can be selected by clicking on them to highlight them; within the Marker Properties window (shown in Figure 15.15), the case markers can be changed in size, type, and color. The final editing results (not all steps were shown) appear in Figure 15.16. The upper/ solid line represents the regression to predict salary from years for the male subgroup; the lower/dashed line represents the regression to predict salary from years for the female subgroup. In this example, the two groups have approximately equal intercepts but different slopes. Figure 15.9 ♦^ SPSS Scatter Plot Showing Menu to Open Chart Editor
Moderation—— 627 Figure 15.10 ♦^ Pull-Down Menu in Chart Editor Dialog Window: Fit Line at Subgroups Figure 15.11 ♦^ Output Showing Separate Fit Lines for Male and Female Subgroups
Moderation—— 629 Figure 15.15 ♦^ Marker Properties Window to Edit Case Marker Properties Including Marker Style, Size, Fill, and Color Figure 15.14 ♦^ Selection of Case Markers for Group 1 for Editing
630 ——CHAPTER 15 Figure 15.16 ♦^ Final Edited Line Graph NOTE: Upper/solid line shows the regression to predict salary for the male group (sex = 1). Lower/dashed line shows regression to predict salary for the female group (sex = 0).
15.13 ♦^ Interaction With a Categorical Predictor With
More Than Two Categories
If a set of two or more dummy-coded dummy predictor variables is needed to represent group membership (as shown in Section 12.6.2), a similar approach can be used. For example, suppose that the problem involved predicting salary from college membership. In the hypothetical data in the SPSS file genderyears.sav introduced in Chapter 12, there were k = 3 colleges (Liberal Arts, Science, and Business). College membership was represented by two dummy-coded variables, C 1 and C 2. C 1 was coded 1 for members of the Liberal Arts college; C 2 was coded 1 for members of the Science college. Suppose that the goal of the analysis is to evaluate whether there is an interaction between college membership and years of job experience as predictors of salary. Two product terms ( C 1 multiplied by years and also C 2 multiplied by years) would be needed to present the college-by-year interaction. The overall regression equation to test for an interaction between college and years as predictors of salary appears in Equation 15.8: Salary′ = b 0 + b 1 C 1 + b 2 C 2 + b 3 Years + b 4 ( C 1 × Years) + b 5 (C 2 × Years). (15.8)
