INTERACTIONS

 

I.  Types of models

 

            A.  All variables are nominal

                        1. Main effects

                        2. Factorial ANOVA

            B.  All variables are quantitative

                        1. Multiple regression

            C.  Both quantitative and nominal variables

                        1. Multiple regression

                        2. ANCOVA: no interaction between nominal and quantitative

 

II.  Interactions between nominal variables

 

            A.  See categorical variables notes. 

B.  Create dummy variables that are the result of multiplying the main effect dummy variables.

C.  Could also create dummy variables that capture both traditional main effects and interactions (e.g. orthogonal trend analysis contrasts).

 

III.  Interactions between quantitative variables

 

A.  Create new variable(s) which is the product of main effect independent variables

            B.  Example

                        f: authoritarianism

                        g: intelligence

                        Y: cognitive style

                        fg: interaction between f and g

 

                        Y= a + b1f + b2g + b3fg

 

In a model with an interaction, the slopes of each independent variable depend on the value of the other variables:

 

                        Y= (b1 +b3g)f + (b2g + a)                                             [1]

                        Y= (b2 + b3f)g + (b1f + a)                                             [2]

 

                                                            correlations                              

                                    Y                     f                       g                      fg

            f                       .115                 1

            g                      -.483                -.274                1

            fg                     -.455                -.759                .740                 1

 

            Mean               30.44   39.33   45.33   1987.8

            sd                     11.50   28.25   26.47   2261.9

            N=108

 

           
Results of hierarchical regression on data above:

 

            Y=a + b1f +b2g             R2y.12=   .233                            F(2,105)=15.978**  [3]

                        b1= -.008                     t(105)=   .209

                        b2=-.212                      t(105)= -5.491 **

                        a= 40.3

 

            Y=a + b1f +b2g + b3fg                                                  R2y.123=.332     F(3,104)=17.264**  [4]

                                                                                                Change R2=.099, F(1,104)=15.21 **

                        b1= -.232         t(104)= -3.493 **

                        b2= .0172         t(104)=   .251

                        b3= -.00466     t(104)= -3.932 **

                        a= 48.1

 

Note that the bj change greatly when the interaction term is added because of the high correlation between fg and main effect variables.

 

Interpretation:  Although g would appear to have a significant main effect if one stopped at the first step, the presence of a significant interaction means that one must examine the quantitative variable equivalent of "simple main effects" -- changes in slope due to the presence of the other variable.

 

            Using  [1] above:  Y=(-.232 - .00466g)f + (.0172g + 48.1)

                          [2]          Y=(.0172 - .00466f)g + (-.232f + 48.1)

 

            select values of f or g and observe equations:

 

            e.g. a low value of f, say 11.1                Y=-.035g + 45.5

                   a middle value of f, say 39.3           Y=-.166g + 39.0

                   a high value of f, say 67.6  Y=-.294g + 32.4

 

Conclusion: the effect of g on Y grows with size of f.  To see the effect of adding the interaction term, compare these equations to equation [3]: Y=-.008f -.212g +40.3.  This equation underestimates the effect of g for high values and overestimates the effect of g for low values.  If one selects values of f as above, equation [3] would yield:

 

            for   f = 11.1                             Y=-.212g + 40.21

                   f = 39.3                             Y=-.212g + 39.98

                   f = 67.6                             Y=-.212g + 39.76

 

These are the equations of parallel lines that are very close together, i.e. equation [3] suggests that f has very little effect on Y.

 

To help interpret the interaction terms, one might also set one factor equal to zero:

 

                        f = 0:                Y=48.1 + .0172g

                        g = 0:               Y=48.1 - .232 f

This is not meaningful in this example, unless f and g are rescaled to deviation scores so that the zero value is in the middle of the range of the independent variables.

 

Moral:  Main effects are uninterpretable in the presence of a significant interaction without reference to that interaction.

 

IV.  Interactions between nominal and quantitative variables

 

            A.  Steps in the analysis

1.  Does the full model (with the interaction) account for a significant proportion of the variance in Y (is the R2 significant)?  If NO, stop; if YES go to 2.

2.  Is there a significant interaction between the nominal and quantitative variables?

 

                                    a)  calculate  (R2Y.abc - R2Y.ab)/(kfm-krm)

                                                                (1-R2Y.abc)/(N-kfm-1)

 

If YES, interpret the model as one would a model with interactions between quantitative variables (i.e. calculate separate regression lines for each level of the nominal variable); if NO, go to 3.

3.  Is the bj for the continuous variable(s) in the reduced model (without the interaction) significant (can use change in R2 measure if one has multiple correlated quantitative variables)?  If NO, then reduce the model to a regression on categorical variables; if YES, go to 4.

4.  Are the bj for the dummy variables representing the categorical factors significant (Can ask whether the change in R2 due to deleting these variables is significant)?  If YES, then one has separate parallel regression lines; if NO, then there is only one regression line and the categorical factor can be deleted.

B.  The test of the bj for a categorical variable in a model with a significant quantitative variable and non-significant interactions is an ANCOVA.

 

V.  When not to report data in classic ANCOVA format

 

A.  Do not use ANCOVA format with non-overlapping groups or when "adjustment of means" is meaningless for some other reason.

B.  Example:  From an ANCOVA of the data in the graph below, one might be tempted to conclude that controlling for SES, whites would still be more intelligent than blacks.  However, since there are no whites in the data at that level of SES, this statement is misleading.  The appropriate conclusion is that SES affects IQ scores in both blacks and whites.