Interaction Lyytinen & Gaskin

Interaction – Definition In factorial designs, interaction effects are the joint effects of two predictor variables in addition to the individual main effects. This is another form of moderation (along with multi-grouping) – i.e., the XY relationship changes form (gets stronger, weaker, changes signs) depending on the value of another explanatory variable (the moderator) Hair et al 2010 pg. 347

Interaction Effects Interactions represent non-additive effects i.e., when the joint effects X and Z on Y are more or less than their additive effects. e.g., Diet * Exercise = greater Weight loss e.g., Chocolate * Cheese = Yucky!

Additive versus Interaction effects Negative Interaction Positive Interaction X=0 X=1 Z=0 1 3 Z=1 5 Diff 2 X=0 X=1 Z=0 1 3 Z=1 4 Diff 2 X=0 X=1 Z=0 1 3 Z=1 6 Diff 2 Y Y Y = > <

Simple interaction example Diet X Diet x Exercise Weight Loss The interaction term is used for testing the moderating effect of Exercise Y Exercise Z Exercise Diet Weight Loss

Diet Weight loss Exercise Weight loss Diet x Exercise Weight loss H2O low high Exercise Weight loss low high Diet x Exercise Weight loss low high H2O

Weight loss example Additive Effect Negative Interaction Positive Diet=0 Diet=1 Exer=0 1 3 Exer=1 5 Diff 2 Diet=0 Diet=1 Exer=0 1 3 Exer=1 4 Diff 2 Diet=0 Diet=1 Exer=0 1 3 Exer=1 6 Diff 2 WL WL WL = > < i.e., Exercising does not alter the effectiveness of dieting i.e., Exercising makes dieting less effective i.e., Exercising makes dieting more effective

Other examples Interaction between adding sugar to coffee and stirring the coffee. Neither of the two individual variables has much effect on sweetness but a combination of the two does. Interaction between adding carbon to steel and quenching. Neither of the two individually has much effect on strength but a combination of the two has a dramatic effect. Interaction between smoking and inhaling asbestos fibers: Both raise lung carcinoma risk, but exposure to asbestos multiplies the cancer risk in smokers and non-smokers. Interaction between genetic risk factors for type 2 diabetes and diet (specifically, a "western" diet). The western dietary pattern has been shown to increase diabetes risk for subjects with a high "genetic risk score", but not for other subjects.

Interaction in literature Interaction (a form of moderation) is central to research in the organizational and social sciences. Interaction is involved in research demonstrating: the effects of motivation on job performance are stronger among employees with high abilities (Locke & Latham, 1990), the effects of distributive justice on employee reactions are greater when procedural justice is low (Brockner & Wiesenfeld, 1996), the effects of job demands on illness are weaker when employees have control in their work environment (Karasek, 1979; Karasek & Theorell, 1990). Edwards 2009

Why interaction Interactions enable more precise explanation of causal effects by providing a method for explaining not only how X effects Y, but also under what circumstances the effect of X changes depending on the moderating variable of Z.

Interaction vs. Multi-group Literature makes little distinction between interaction and multi-group. Interaction is often treated like multi-group High vs Low values of age, income, size etc. The interpretation is much the same, but the method is different. Multi-group: categorical variables, split dataset, constrained paths Interaction: continuous variables, whole dataset, interaction variables

The statistical side of it A regular regression equation involving two independent variables: Y = b0 + b1X + b2Z + e In ordinary least squares (OLS) regression, the product of two variables can be used to represent the interactive effect: Y = b0 + b1X + b2Z + b3XZ + e where, XZ is the product term that represents the interaction effect, and b3 is the change in the slope of the regression of XY when Z changes by one unit.

The statistical side of it (cont) Essentially, the interaction regression equation specifies that the slope of the line relating X to Y changes at different levels of Z, or equivalently, that the slope of the line relating Z to Y changes at different levels of X. Saunders (1956) first demonstrated that a product term accurately reflects a continuous variable interaction. Similarly, natural polynomial or powered variables (X2, X3, etc.) can be used to represent higher order nonlinear effects of a variable such as a quadratic or cubic trend of age or time.

Significance of interaction effects Are slopes of regression lines for XY significantly different at differing values of Z? e.g., is the slope of the relationship between diet and weight loss significantly different between those who exercise very little and those who exercise a lot? The way to determine the significance is to calculate a p-value for the regression of XZY (AMOS handles this)

Significance of interaction part 2 One may also desire to know whether the change in the XY relationship with and without the interaction effect is significant. This can be done through a rather complex method available at: http://www.people.ku.edu/~preacher/interact/mlr2.htm He has a tool for it.

Range of Significance The region of significance defines the specific values of z at which the regression of y on x moves from non-significance to significance (see Preacher 2007). There are lower and upper bounds to the region. In many cases, the regression of y on the focal predictor is significant at values of the moderator that are less than the lower bound and greater than the upper bound, and the regression is non-significant at values of the moderator falling within the region. However, there are some cases in which the opposite holds (e.g., the significant slopes fall within the region). We will not calculate this, but there are ways to do so (see Preacher 2007).

Statistical Interaction Considerations Multicollinearity Interaction terms can be highly collinear with constituent IVs – this can be addressed by centering the means (Edwards 2009) Non-normal distribution handling If IVs are not normally distributed, the product term (interaction term) will likely result in biased estimations Neither is it true that if the IVs are normally distributed their product will be normally distributed This can sometimes be fixed through transformation of either or both the IVs and the interaction term

Statistical Interaction Considerations Reliability low reliability of either the IVs or the moderator (especially) is likely to increase either type I or type II errors (Edwards 2009) Isolating unique effects If the effect of XZY is significant, both XY and ZY must remain in the path model, even if non-significant, in order to isolate the unique effects of the interaction term

Mean Centering One way to fix multicollinearity issues inherent in the use of interaction terms is to mean-center all involved variables. This involves subtracting the variable mean from each response, thus placing the new mean at zero. Similarly, you can standardize the variable, which is simply replacing the variable values with their corresponding z-scores (mean=0, sd=1)

Benefits of Centering Centering can make otherwise uninterpretable regression coefficients meaningful, and Centering reduces multicollinearity among predictor variables. Centering has no effect on linear regression coefficients (except b0)

Mean Centering vs. Standardizing Original value Mean Centered Standardized Mean 5 Std Dev 2.73 1 A -4 -1.46 B 2 -3 -1.10 C 3 -2 -0.73 D 4 -1 -0.37 E 0.00 F 6 0.37 G 7 0.73 H 8 1.10 I 9 1.46

Higher order interactions We work with 2-way interactions, but interactions are not limited to X and Z. A three-way interaction looks like this: Y = b0 + b1X + b2Z + b3W + b4XZ + b5XW + b6ZW + b7XZW Where W is a second interaction term There are no mathematical limits on the number of interacting terms, but there are certainly practical limitations. One challenge is testing significance, as there are approaches to do this with 3 variables but not with more Another challenge is the interpretation of the interactions and what they truly mean theoretically

How To In SPSS create new variables by standardizing all variables in the model (except categorical ones) and then computing a product variable Use like an IV in AMOS. Trim model, starting with interaction effects Don’t trim paths from constituent IVs unless the parent interaction is deleted due to insignificance (this is because all the paths and their coefficients need to be interpreted together with the interaction term) Adjust per Model Fit issues If interactions are significant, plot them (there is software for it) Interpret the interaction/moderation effects

Standardize all variables in the model (unless categorical) 1a. Standardize Standardize all variables in the model (unless categorical) Result

1b. Compute product variable Go here Type new name (no spaces or mathematical symbols allowed) Type or click the product expression Hit OK Result

2. Put in AMOS

3. Trim Start here

4. Attend to Model Fit Just normal model fit stuff Don’t forget about: SRMR Chi-square/df CFI RMSEA Modification indices

5. Plot interaction (if the interaction is significant)

6. Interpret atrust is a stronger predictor of vallong (i.e., has a larger slope value – Beta) for cases of high ctrust. Thus, ctrust positively moderates (amplifies) the effect of atrust on vallong. For small differences like this one, the moderation is significant if the ZY Beta is significant (in AMOS output)

Plotting approach To ease interpretation of interaction, we treat them somewhat like multi-group variables… Select high and low values of moderator High: one standard deviation above the mean Low: one standard deviation below the mean -1 +1

More on Interpretation Exercise positively moderates (amplifies) the relationship between diet and weight loss. Exercise does not moderate the relationship between diet and weight loss. i.e., Exercising does not alter the effectiveness of dieting i.e., Exercising makes dieting more effective Exercise negatively moderates (dampens) the relationship between diet and weight loss. Exercise inversely moderates the relationship between diet and weight loss. i.e., Exercising makes dieting less effective i.e., All or nothing moderation, do both or do neither

Additional Resources This is a site hosted by Kristopher Preacher (as in the Preacher and Hayes articles), and is very informative regarding interactions: http://www.people.ku.edu/~preacher/interact/interactions.htm If you are interested in calculating the range of significance interaction values, refer to this somewhat complex (yet simplified) tool: http://www.people.ku.edu/~preacher/interact/mlr2.htm