Interaction

Lyytinen & Gaskin

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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 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

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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!

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Additive versus Interaction effects

X=0 X=1Z=0 1 3Z=1 3 5Diff 2 2

X=0 X=1Z=0 1 3Z=1 3 4Diff 2 1

X=0 X=1Z=0 1 3Z=1 3 6Diff 2 3

Additive Effect

Negative Interaction

PositiveInteraction

= > <

Y Y Y

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Simple interaction example

Diet

Weight Loss

Exercise

Diet x Exercise

X

Y

Z

The interaction term is used for testing the moderating effect of Exercise

Diet Weight Loss

Exercise

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Diet

Weight loss

low high

high

Exercise

Weight loss

low high

high

Diet x Exercise

Weight loss

low high

high

H2O

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Weight loss example

Diet=0 Diet=1Exer=0 1 3Exer=1 3 5

Diff 2 2

Diet=0 Diet=1Exer=0 1 3Exer=1 3 4

Diff 2 1

Diet=0 Diet=1Exer=0 1 3Exer=1 3 6

Diff 2 3

Additive Effect

Negative Interaction

PositiveInteraction

i.e., Exercising makes dieting more effective

i.e., Exercising makes dieting less effective

i.e., Exercising does not alter the effectiveness of dieting

= > <

WL WL WL

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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.

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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

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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.

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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

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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 XY when Z changes by one unit.

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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.

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Significance 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)

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Significance 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: http://www.people.ku.edu/~preacher/interact/mlr2.htm

• He has a tool for it.

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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).

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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

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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 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

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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)

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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)

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Mean Centering vs. StandardizingOriginal value Mean Centered Standardized

Mean 5 0 0Std Dev 2.73 2.73 1

A 1 -4 -1.46B 2 -3 -1.10C 3 -2 -0.73D 4 -1 -0.37E 5 0 0.00F 6 1 0.37G 7 2 0.73H 8 3 1.10I 9 4 1.46

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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

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How To1. In SPSS create new variables by standardizing all variables

in the model (except categorical ones) and then computing a product variable

2. Use like an IV in AMOS.3. Trim model, starting with interaction effects

1. 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)

4. Adjust per Model Fit issues5. If interactions are significant, plot them (there is software

for it)6. Interpret the interaction/moderation effects

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1a. Standardize

Result

Standardize all variables in the model (unless categorical)

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1b. Compute product variable

Type new name (no spaces or mathematical symbols allowed)

Type or click the product expression

Hit OK

Go here

Result

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2. Put in AMOS

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3. Trim

Start here

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4. Attend to Model Fit

• Just normal model fit stuff• Don’t forget about:– SRMR– Chi-square/df– CFI– RMSEA– Modification indices

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5. Plot interaction (if the interaction is significant)

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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 ZY Beta is significant(in AMOS output)

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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

0 +1-1

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More on InterpretationExercise positively moderates (amplifies) the relationship between diet and weight loss.

Exercise does not moderate the relationship between diet and weight loss.

Exercise inversely moderates the relationship between diet and weight loss.

i.e., Exercising makes dieting more effective

i.e., Exercising does not alter the effectiveness of dieting

i.e., All or nothing moderation, do both or do neither

Exercise negatively moderates (dampens) the relationship between diet and weight loss.

i.e., Exercising makes dieting less effective

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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/ml

r2.htm