Interpreting interaction effects This web page contains various Excel
worksheets which help interpret twoway and
threeway interaction effects. They use procedures
by Aiken and West (1991), Dawson (2013)
and Dawson and Richter (2006) to plot the
interaction effects, and in the
case of three way interactions test for significant
differences between the
slopes. You can either use the Excel worksheets
directly from this page, or
download them to your computer by rightclicking on
the relevant links. A note about standardisation of
variables. Standardised variables are those
that are both centred around zero and are scaled so
that they have a standard deviation of 1.
Personally, I prefer to use these when testing
interactions because the intepretation of
coefficients can be slightly simpler. Some authors,
such as Aiken and West (1991), recommend that
variables are centred (but not standardised). The
results obtained should be identical whichever
method you use. If you prefer to analyse centred
(but not standardised) variables, you can use the
"unstandardised" versions of the Excel worksheets,
and enter the mean of the variables as zero. Twoway interactions To test for twoway interactions (often thought of as a relationship between an independent variable (IV) and dependent variable (DV), moderated by a third variable), first run a regression analysis, including both independent variables (referred to hence as the IV and moderator) and their interaction (product) term. It is recommended that the independent variable and moderator are standardised before calculation of the product term, although this is not essential. The product term should be significant in the regression equation in order for the interaction to be interpretable. If you have two unstandardised variables, you can plot your interaction effect by entering the unstandardised regression coefficients (including intercept/constant) and means & standard deviations of the IV and moderator in the following worksheet. If you have control variables in your regression, the values of the dependent variable displayed on the plot will be inaccurate unless you standardise (or centre) all control variables first (although the pattern, and therefore the interpretation, will be correct). 2way_unstandardised.xls If you have two standardised variables, you can plot your interaction effect by entering the just unstandardised regression coefficients (including intercept/constant) in the following worksheet. If you have control variables in your regression, the values of the dependent variable displayed on the plot will be inaccurate unless you also standardise (or centre) all control variables first (although the pattern, and therefore the interpretation, will be correct). Note that the interaction term should not be standardised after calculation, but should be based on the standardised values of the IV & moderator. 2way_standardised.xls If you have a binary moderator, you can plot your interaction more usefully by entering the unstandardised regression coefficients (including intercept/constant) and mean & standard deviation of your IV in the following worksheet. Again, if you have control variables in your regression, the values of the dependent variable displayed on the plot will be inaccurate unless you also standardise (or centre) all control variables first (although the pattern, and therefore the interpretation, will be correct). The binary variable should have possible values of 0 and 1, and should not be standardised. 2way_with_binary_moderator.xls If you want to test simple slopes, you can use the following worksheet. Again, control variables should be centered or standardised before the analysis. However, note that simple slope tests are only useful for testing significance at specific values of the moderator. Where possible, meaningful values should be chosen, rather than just one standard deviation above and below the mean. You will also need to request the coefficient covariance matrix as part of the regression output. If you are using SPSS, this can be done by selecting "Covariance matrix" in the "Regression Coefficients" section of the "Statistics" dialog box. Note that the variance of a coefficient is the covariance of that coefficient with itself  i.e. can be found on the diagonal of the coefficient covariance matrix. 2way_unstandardised_with_simple_slopes.xls Other forms of
twoway interaction plots that may be helpful
for experienced users:
To test for threeway interactions (often thought of as a relationship between a variable X and dependent variable Y, moderated by variables Z and W), run a regression analysis, including all three independent variables, all three pairs of twoway interaction terms, and the threeway interaction term. It is recommended that all the independent variable are standardised before calculation of the product terms, although this is not essential. As with twoway interactions, the interaction terms themselves should not be standardised after calculation. The threeway interaction term should be significant in the regression equation in order for the interaction to be interpretable. If you wish to use the Dawson & Richter
(2006) test for differences
between slopes, you should request the coefficient
covariance matrix as part of
the regression output. If you are using SPSS, this
can be done by selecting
"Covariance matrix" in the "Regression Coefficients"
section
of the "Statistics" dialog box. Note that the
variance of a coefficient is the covariance of that
coefficient with itself  i.e. can be found on the
diagonal of the coefficient covariance matrix. If you have used unstandardised variables, you can plot your interaction effect by entering the unstandardised regression coefficients (including intercept/constant) and means & standard deviations of the three independent variables (X, Z and W) in the following worksheet. If you have control variables in your regression, the values of the dependent variable displayed on the plot will be inaccurate unless you standardise all control variables first (although the pattern, and therefore the interpretation, will be correct). To use the test of slope differences, you should also enter the covariances of the XZ, XW and XZW coefficients from the coefficient covariance matrix, and the total number of cases and number of control variables in your regression. 3way_unstandardised.xls If you have used standardised variables, you can plot your interaction effect by entering the just unstandardised regression coefficients (including intercept/constant) in the following worksheet. If you have control variables in your regression, the values of the dependent variable displayed on the plot will be inaccurate unless you also standardise all control variables first (although the pattern, and therefore the interpretation, will be correct). To use the test of slope differences, you should also enter the covariances of the XZ, XW and XZW coefficients from the coefficient covariance matrix, and the total number of cases and number of control variables in your regression. 3way_standardised.xls Other forms of
threeway interaction plots that may be
helpful for experienced users:
Please note: a previous version of the "3 way with all options" sheet included an error in the slope difference test: apologies for any inconvenience caused. This has now been corrected. Quadratic Effects If you wish to plot a quadratic
(curvilinear) effect, you can use one of the
following Excel worksheets. In each case, you test
the quadratic effect by including the main effect
(the IV) along with its squared term (i.e. the
IV*IV) in the regression. In the case of a simple
(unmoderated) relationship, the significance of the
squared term determines whether there is a quadratic
effect. If you are testing a moderated quadratic
relationship, it is the significance of the
interaction between the squared term and the
moderator(s) that determines whether there is a
moderated effect. Note that despite this, all lower
order terms need to be included in the regression:
so, if you have an independent variable A and
moderators B and C, then to test whether there is a
threeway interaction you would need to enter all
the following terms: A, A*A, B, C, A*B, A*C, A*A*B,
A*A*C, B*C, A*B*C, A*A*B*C. It is only the last,
however, that determines the significance of the
threeway quadratic interaction.
Troubleshooting There are a number of common problems encountered when trying to plot these effects. If you are having problems, consider the following:
If you think there are any errors in these sheets, please contact me, Jeremy Dawson.
References Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park, London, Sage. Dawson, J. F. (2014). Moderation in management research: What, why, when and how. Journal of Business and Psychology, 29, 119. Dawson, J. F., & Richter, A. W. (2006). Probing threeway interactions in moderated multiple regression: Development and application of a slope difference test. Journal of Applied Psychology, 91, 917926.
Other online resources Kristopher Preacher's web site contains templates for testing simple slopes, and findings regions of significance, for both 2way and 3way interactions. It also includes options for hierarchical linear modelling (HLM) and latent curve analysis. Yungjui Yang's web site contains SAS macros to plot interaction effects and run the slope difference tests for threeway interactions Cameron Brick's web site contains instructions on how to plot a threeway interaction and test for differences between slopes in Stata
