2Interaction – 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 XY relationship changes form (gets stronger, weaker, changes signs) depending on the value of another explanatory variable (the moderator)Hair et al 2010 pg. 347
3Interaction 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 losse.g., Chocolate * Cheese = Yucky!
4Additive versus Interaction effects NegativeInteractionPositiveInteractionX=0X=1Z=013Z=15Diff2X=0X=1Z=013Z=14Diff2X=0X=1Z=013Z=16Diff2YYY=><
5Simple interaction example DietXDiet x ExerciseWeight LossThe interaction term is used for testing the moderating effect of ExerciseYExerciseZExerciseDietWeight Loss
6Diet Weight loss Exercise Weight loss Diet x Exercise Weight loss H2O lowhighExerciseWeightlosslowhighDiet x ExerciseWeightlosslowhighH2O
7Weight loss example Additive Effect Negative Interaction Positive Diet=0Diet=1Exer=013Exer=15Diff2Diet=0Diet=1Exer=013Exer=14Diff2Diet=0Diet=1Exer=013Exer=16Diff2WLWLWL=><i.e., Exercising does not alter the effectiveness of dietingi.e., Exercising makes dieting less effectivei.e., Exercising makes dieting more effective
8Other examplesInteraction 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.
9Interaction 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
10Why interactionInteractions 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.
11Interaction vs. Multi-group Literature makes little distinction between interaction and multi-group.Interaction is often treated like multi-groupHigh 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 pathsInteraction: continuous variables, whole dataset, interaction variables
12The statistical side of it A regular regression equation involving two independent variables:Y = b0 + b1X + b2Z + eIn ordinary least squares (OLS) regression, the product of two variables can be used to represent the interactive effect:Y = b0 + b1X + b2Z + b3XZ + ewhere, XZ is the product term that represents the interaction effect, and b3 is the change in the slope of the regression of XY when Z changes by one unit.
13The 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.
14Significance of interaction effects Are slopes of regression lines for XY 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 XZY (AMOS handles this)
15Significance of interaction part 2 One may also desire to know whether the change in the XY relationship with and without the interaction effect is significant.This can be done through a rather complex method available at:He has a tool for it.
16Range of SignificanceThe 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).
17Statistical Interaction Considerations MulticollinearityInteraction terms can be highly collinear with constituent IVs – this can be addressed by centering the means (Edwards 2009)Non-normal distribution handlingIf IVs are not normally distributed, the product term (interaction term) will likely result in biased estimationsNeither is it true that if the IVs are normally distributed their product will be normally distributedThis can sometimes be fixed through transformation of either or both the IVs and the interaction term
18Statistical Interaction Considerations Reliabilitylow reliability of either the IVs or the moderator (especially) is likely to increase either type I or type II errors (Edwards 2009)Isolating unique effectsIf the effect of XZY is significant, both XY and ZY must remain in the path model, even if non-significant, in order to isolate the unique effects of the interaction term
19Mean CenteringOne 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)
20Benefits of CenteringCentering can make otherwise uninterpretable regression coefficients meaningful, andCentering reduces multicollinearity among predictor variables.Centering has no effect on linear regression coefficients (except b0)
21Mean Centering vs. Standardizing Original valueMean CenteredStandardizedMean5Std Dev2.731A-4-1.46B2-3-1.10C3-2-0.73D4-1-0.37E0.00F60.37G70.73H81.10I91.46
22Higher 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 + b7XZWWhere W is a second interaction termThere 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 moreAnother challenge is the interpretation of the interactions and what they truly mean theoretically
23How ToIn SPSS create new variables by standardizing all variables in the model (except categorical ones) and then computing a product variableUse like an IV in AMOS.Trim model, starting with interaction effectsDon’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 issuesIf interactions are significant, plot them (there is software for it)Interpret the interaction/moderation effects
24Standardize all variables in the model (unless categorical) 1a. StandardizeStandardize all variables in the model (unless categorical)Result
251b. Compute product variable Go hereType new name (no spaces or mathematical symbols allowed)Type or click the product expressionHit OKResult
284. Attend to Model Fit Just normal model fit stuff Don’t forget about: SRMRChi-square/dfCFIRMSEAModification indices
295. Plot interaction (if the interaction is significant)
306. Interpretatrust 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 ZY Beta is significant (in AMOS output)
31Plotting approachTo ease interpretation of interaction, we treat them somewhat like multi-group variables…Select high and low values of moderatorHigh: one standard deviation above the meanLow: one standard deviation below the mean-1+1
32More 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 dietingi.e., Exercising makes dieting more effectiveExercise 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 effectivei.e., All or nothing moderation, do both or do neither
33Additional ResourcesThis is a site hosted by Kristopher Preacher (as in the Preacher and Hayes articles), and is very informative regarding interactions:If you are interested in calculating the range of significance interaction values, refer to this somewhat complex (yet simplified) tool: