Interpreting interaction effects
This web page contains various Excel
templates which help interpret two-way and
three-way interaction effects. They use procedures
by Aiken and West (1991), Dawson (2014)
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 templates
directly from this page, or
download them to your computer by right-clicking on
the relevant links.
To test for two-way 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 centred 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.
You can then plot the interaction effect using the following Excel template. You will need to enter the unstandardised regression coefficients (including intercept/constant) and means & standard deviations of the IV and moderator in the cells indicated. If you have control variables in your regression, the values of the dependent variable displayed on the plot will be inaccurate unless you centre (or standardise) all control variables first (although even if you don’t the pattern, and therefore the interpretation, will be correct). If your moderator is binary, ensure you enter the actual values of this variable in cells B31 and B32. 2-way_linear_interactions.xls
Simple slope tests. This template also allows you to perform simple slope tests – these are conditional hypothesis tests of whether the relationship between IV and DV is significant at a particular value of the moderator. If your moderator is a numerical variable, these are not necessary, but may be useful at specific, theoretically interesting values of the moderator. Therefore where possible, meaningful values should be chosen, rather than just one standard deviation above and below the mean (which is where they will be tested by default if you leave cells B31 & B32 blank); otherwise the tests are arbitrary in nature and quite possibly meaningless. To run simple slope tests, 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.
For non-linear two-way interactions (including generalised linear models), you might want to use one of the following templates:
A note about centred and standardised variables. Centred variables are the same as the original version, with the variable mean subtracted so that the new mean is zero. Standardised variables are those
that are both centred around zero and are scaled so
that they have a standard deviation of 1. Following Aiken and West (1991), I recommend that for analysis
variables are centred. Some researchers may prefer to use standardised variables. Each gives some
advantages in interpreting the coefficients - see Dawson (2014)
for more about this (reference below). However, the
results obtained should be identical whichever
method you use. If you choose to analyse centred
(or standardised) variables, you should use the
regular versions of the Excel templates,
and enter the mean of the variables as zero (and standard deviation as 1 if using standardised versions). And for those readers who use the US version of English, for "centred" and "standardised", read "centered" and "standardized"!
To test for three-way 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 two-way interaction terms, and the three-way interaction term. It is recommended that all the independent variable are centred (or standardised) before calculation of the product terms, although this is not essential. As with two-way interactions, the interaction terms themselves should not be centred (or standardised) after calculation. The three-way 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"
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.
You can then plot the interaction effect using the following Excel template. You will need to enter the unstandardised regression coefficients (including intercept/constant) and means & standard deviations of the three independent variables (X, Z and W) in the cells indicated. If you have control variables in your regression, the values of the dependent variable displayed on the plot will be inaccurate unless you centre (or standardise) all control variables first (although even if you don’t 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. If any of your variables is binary, ensure you enter the actual values of this variable where stated. 3-way_linear_interactions.xls
Simple slope tests. As with the 2-way interactions above, this template also allows you to perform simple slope tests, as well as the slope difference tests. See my warnings above about the use of simple slope tests, however.
For non-linear three-way interactions (including generalised linear models), you might want to use one of the following templates:
If you wish to plot a quadratic
(curvilinear) effect, you can use one of the
following Excel templates. 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
three-way 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
three-way quadratic interaction.
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.
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, 1-19.
Dawson, J. F., & Richter, A. W. (2006). Probing three-way interactions in moderated multiple regression: Development and application of a slope difference test. Journal of Applied Psychology, 91, 917-926.
Other online resources
Kristopher Preacher's web site contains templates for testing simple slopes, and findings regions of significance, for both 2-way and 3-way interactions. It also includes options for hierarchical linear modelling (HLM) and latent curve analysis.
Yung-jui Yang's web site contains SAS macros to plot interaction effects and run the slope difference tests for three-way interactions
Cameron Brick's web site contains instructions on how to plot a three-way interaction and test for differences between slopes in Stata
Legacy versions of Excel templates
Previously, some different versions of the linear interaction template were available on this page. I strongly recommend using the above versions, but if you want to see one of the older (legacy) versions, you can click on the appropriate file: 2-way_unstandardised.xls 2-way_standardised.xls 2-way_with_binary_moderator.xls 2-way_with_all_options.xls 3-way_unstandardised.xls 3-way_standardised.xls 3-way_with_all_options.xls