SPSS Data Management
SPSS Data Management EDLD 610
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Interaction Effects in Regression This handout is designed to provide some background and information on the analysis and interpretation of interaction effects in Multiple Regression MR This is a complex topic and the handout is necessarily incomplete In practice be sure to consult other references on Multiple Regression Aiken amp West 1991 Cohen amp Cohen 1983 Pedhazurl996 Pedhazur amp Schmelkin 1991 for additional information Interaction effects represent the combined effects of variables on the criterion or dependent measure When an interaction effect is present the impact of one variable depends on the level of the other variable Part of the power of MR is the ability to estimate and test interaction effects when the predictor variables are either categorical or continuous As Pedhazur and Schmelkin 1991 note the idea that multiple effects should be studied in research rather than the isolated effects of single variables is one of the important contributions of Sir Ronald Fisher When interaction effects are present it means that interpretation of the individual variables may be incomplete or misleading Kinds of Interactions For example imagine a study that tests the effects of a treatment on an outcome measure The treatment variable is composed of two groups treatment and control The results are that the mean for the treatment group is higher than the mean for the control group But what if the researcher is also interested in whether the treatment is equally effective for females and males 4 5 L That is is there a difference in treatment 4 0 depending on gender group This is a question of interaction Inspect the n g 3 5 results below Interaction results whose g 3 0 x lines do not cross as in the figure at left g x are called ordinal interactions Ifthe E 2 5 slope of lines is not parallel in an ordinal 2 0 GENDER interaction the interaction effect will be n significant given enough statistical l5 Lemme power Ifthe lines are exactly parallel n We then there is no interaction effect In this Treatment Control case a difference in level between the two lines would indicate a main effect of gender a difference in level for both lines between treatment and control would indicate a main effect of treatment However when an interaction is significant and disordinal interpretation of main effects is likely to be misleading To determine exactly which parts of the interaction are significant the omnibus F test must be followed by more focused tests or comparisons TREATMENT FOCUSED TESTS OF INTERACTIONS Whenever interactions are significant the next question that arises is exactly where are the signi cant differences This situation is similar to the issues in post hoc testing of main effects but it is also more complex in that interactions represent the combined effects of two forces or dimensions in the data not just one The following section describes common approaches to obtaining more focused specific information on where differences are in the interaction effect in tow common situations 1 the interaction of one continuous predictor and one categorical predictor and 2 the interaction of two continuous predictors Method 1 Johnson Nevman Regions of Significance This is a common approach to the interpretation of the interaction between one continuous predictor one categorical predictor and the criterion The topic is covered in detail in Pedhazur 1997 chapter 14 The example used in Pedhazur in Table 143 p 588 is presented here The data are available on the web site in the data subdirectory for EDPSY 604 named pedl473sav A syntax file named pedl473sps is also available on the web site Ifyou run the syntax file you will get the results of the regression analysis of the two predictors as well as the interaction of the two predictors The syntax also produces a plot of the data Editing the plot and requesting chartoptionsfit linesubgroups will produce the figure below and the two regression lines fitted for each value of the categorical predictor variable Plot of Y with X by D Satisfaction Rating Teacher Directivenes A0 00 39 1 Tolerance of Ambuguity One of the issues of greatest interest in interpreting the interaction displayed in the figure above is to determine at what points the two subgroup regression lines differ significantly from each other 2 One approach to this question is the J ohnsonNeyman procedure for determining regions of signi cance This approach is described in Pedhazur A syntax le that will calculate the regions of signi cance is also available on the web site and is named regionssps Method 2 Simple Slopes Tests This method is designed for the interpretation of the interaction effect of two continuous predictor variables The method is not discussed in Pedhazur Refer to Aiken and West 1991 or Cohen and Cohen 1983 for further information To illustrate some of the calculations example 21 p 11 from Aiken and West is used The data in matrix format is available on the web site named aiken271sav The regression analysis can be run using aiken271sps The regression analysis results in the following equation Y 2506 843X 3696Z 2621XZ 1 To illustrate and test the signi cant interaction effect separate regression lines are computed plotted and tested for individuals one standard deviation below the mean on predictor Z at the mean of predictor Z and one standard deviation below the mean of predictor Z First the overall regression equation is rearranged so it can be expressed only in terms of values of X Y 843 262ZX 3696Z 2506 2 To calculate an equation for Z one standard deviation above the mean the standard deviation of Z 22 is substituted for Z in equation 2 This results in Y 6607X 106372 for all those 1 SD above the mean on Z 3 For those at the mean of Z a value of 0 is substituted for Z in equation 2 This results in Y 843X 2506 4 To calculate an equation for Z one standard deviation below the mean the standard deviation of Z 22 is substituted for Z and subtracted in equation 2 This results in Y 4921X 56252 for all those 1 SD below the mean on Z 5 Actual values of Y can now be calculated by substituting values of predictor X Commonly values are computed for X at the mean one standard deviation above the mean and one standard deviation below the mean Given that the standard deviation of X is 095 this results in Equation 1 SD on X Mean X 1 SD on X Y for Z 1 SD Above 43606 106372 169139 Y for Mean Z 17052 25060 33069 Y for Z 1 SD Below 09503 56252 103002 3 These values can be entered into SPSS for plotting An example of this is in the le aikenilinessav Using SPSS menus or the syntax below the three regression lines can be plotted resulting in the gure below GRAPH LNEMULTPLEMEANabove MEANaverage MEANbelow BYtime IMISSINGLSTWISE REPORT 200 a I 100 3 s N H a 2 0 0 a Managerial Ability 100 ABOVE AVERAGE 200 BELOW 1 SD Mean 1 SD Time in Position The last step in this process is to test each line to determine whether there is a signi cant relationship to the criterion for each subset of the interaction effect These tests are called simple slopes tests and are directly analogous to simple effects tests of interactions in the analysis of variance The following syntax serves as an example compute zabove z 22 compute zbelow z 22 compute xzabove x zabove compute xzbelow x zbelow execute A regression is then run using predictors X zbelow and xzbelow to test whether there is a signi cant relationship for those below the mean A second regression analysis is run using predictors X zabove and xzabove to test whether there is a signi cant relationship for those above the mean The ttest for xzbelow and xzabove are the simple slope tests 4 References Aiken L S amp West S G 1991 Multiple regression Testing and interpreting interactions Newbury Park Sage Cohen J amp Cohen P 1983 Applied multiple regressiorcorrelation analysis for the behavioral sciences 2quotd Ed Hillsdale NJ Erlbaum Pedhazur E J 1997 Multiple regression in behavioral research 3rd Ed Fort Worth TX Harcourt Brace Pedhazur E J amp Schmelkin L P 1991 Measurement design and analysis An integrated approach Hillsdale NJ Erlbaum Stevens 2000 5