Continuous by continuous interactions in OLS regression can be tricky. Continuous by continuous
interactions in logistic regression can be downright nasty. However, with the assistance of the
**margins** command (introduced in Stata 11) and the **margins** command (introduced
in Stata 12), we will be able to tame those continuous by
continuous logistic interactions.

Most researchers are not comfortable interpreting logistic regression results in terms of the raw coefficients which are scaled in terms of log odds. Interpreting logistic interaction in terms of odds ratios is not much easier. Many researchers prefer to interpret logistic interaction results in terms of probabilities. The shift from log odds to probabilities is a nonlinear transformation which means that the interactions are no longer a simple linear function of the predictors. This FAQ page will try to help you to understand continuous by continuous interactions in logistic regression models both with and without covariates.

We will use an example dataset, **logitconcon**, that has two continuous predictors, **r**
and **m** and a binary response variable **y**. It also has a continuous covariate, **cv1**,
which we will use in a later model.
We will begin by loading the data and running a logistic regression model with an interaction
term.

use http://www.ats.ucla.edu/stat/data/logitconcon, clear logit y c.r##c.m, nologLogistic regression Number of obs = 200 LR chi2(3) = 65.47 Prob > chi2 = 0.0000 Log likelihood = -78.621746 Pseudo R2 = 0.2940 ------------------------------------------------------------------------------ y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- r | .4407548 .1934232 2.28 0.023 .0616522 .8198573 m | .5069182 .1984649 2.55 0.011 .1179343 .8959022 | c.r#c.m | -.0066735 .0032877 -2.03 0.042 -.0131173 -.0002298 | _cons | -32.9762 11.49797 -2.87 0.004 -55.51182 -10.44059 ------------------------------------------------------------------------------

As you can see all of the variables in the above model including the interaction term are
statistically significant. What we will want to do is to see what a one unit change in **r**
has on the probability when **m** is held constant at different values. We can do this easily
using the **margins** command. Here is what the command looks like holding **m** constant
for every five values between 30 and 70. We will use the **post** option so that we can use
**parmest** (**search parmest**) to save the estimates to memory as data.

margins, dydx(r) at(m=(30(5)70)) vsquishAverage marginal effects Number of obs = 200 Model VCE : OIM Expression : Pr(y), predict() dy/dx w.r.t. : r 1._at : m = 30 2._at : m = 35 3._at : m = 40 4._at : m = 45 5._at : m = 50 6._at : m = 55 7._at : m = 60 8._at : m = 65 9._at : m = 70 ------------------------------------------------------------------------------ | Delta-method | dy/dx Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- r | _at | 1 | .0081656 .0069121 1.18 0.237 -.0053818 .0217131 2 | .0089867 .0063677 1.41 0.158 -.0034937 .0214672 3 | .0099785 .0056917 1.75 0.080 -.0011771 .0211341 4 | .0111306 .0049301 2.26 0.024 .0014678 .0207933 5 | .0122375 .0042148 2.90 0.004 .0039767 .0204983 6 | .0123806 .0038803 3.19 0.001 .0047753 .0199858 7 | .0092451 .0051852 1.78 0.075 -.0009176 .0194079 8 | .0016928 .0082169 0.21 0.837 -.014412 .0177977 9 | -.0048499 .0073021 -0.66 0.507 -.0191616 .0094619 ------------------------------------------------------------------------------

We will graph these results using the **marginsplot command** introduced in Stata 12.

marginsplot, ylin(0)

We can make the graph more visually attractive by recasting the confidence intervals as a shaded area.

marginsplot, recast(line) recastci(rarea) ylin(0)

From inspection of the **margins** results and the graph shown above we can see that the
marginal effect is statistically significant between **m** values of 45 to 55 inclusive.
The marginal effects tells the change in probability for a one unit change in the predictor,
in this case, **r**.

**Continuous by continuous interaction with covariate**

Now, let’s add a covariate, **cv1** to the model. The interesting thing about logistic regression
is that the marginal effects for the interaction depend on the values of the covariate even if the
covariate is not part of the interaction itself. Below we show the logistic regression model
with the covariate **cv1** added. Because we used the **parmest** program previously,
we will need to reload the data.

use http://www.ats.ucla.edu/stat/data/logitconcon, clear logit y c.r##c.m cv1, nologLogistic regression Number of obs = 200 LR chi2(4) = 66.80 Prob > chi2 = 0.0000 Log likelihood = -77.953857 Pseudo R2 = 0.3000 ------------------------------------------------------------------------------ y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- r | .4342063 .1961642 2.21 0.027 .0497316 .8186809 m | .5104617 .2011856 2.54 0.011 .1161452 .9047782 | c.r#c.m | -.0068144 .0033337 -2.04 0.041 -.0133483 -.0002805 | cv1 | .0309685 .0271748 1.14 0.254 -.0222931 .08423 _cons | -34.09122 11.73402 -2.91 0.004 -57.08947 -11.09297 ------------------------------------------------------------------------------

This time, everything except for the covariate is statistically significant. As it turns out, it doesn’t matter whether the covariate is significant or not; we still have to take the value of the covariate into account when interpreting the interaction.

Before obtaining the marginal effects we will collect some information on the covariate, namely the values one standard deviation below the mean, the mean, and one standard deviation above the mean.

summarize cv1Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- cv1 | 200 52.405 10.73579 26 71display r(mean)-r(sd) " " r(mean) " " r(mean)+r(sd)41.669207 52.405 63.140793

Now, we can go ahead and run the **margins** command including each of the three values of
**cv1**.

/* holding cv1 at mean minus 1 sd */ margins, dydx(r) at(m=(30(5)70) cv1=(41.669207 52.405 63.140793)) vsquish noatlegendAverage marginal effects Number of obs = 200 Model VCE : OIM Expression : Pr(y), predict() dy/dx w.r.t. : r ------------------------------------------------------------------------------ | Delta-method | dy/dx Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- r | _at | 1 | .0061133 .0065712 0.93 0.352 -.006766 .0189926 2 | .0074917 .0069416 1.08 0.280 -.0061135 .0210969 3 | .0090189 .0073769 1.22 0.221 -.0054396 .0234774 4 | .006587 .0061377 1.07 0.283 -.0054427 .0186167 5 | .0081075 .0063953 1.27 0.205 -.004427 .0206421 6 | .0097902 .0067546 1.45 0.147 -.0034485 .0230289 7 | .0071815 .0056839 1.26 0.206 -.0039586 .0183217 8 | .0088605 .0057648 1.54 0.124 -.0024384 .0201593 9 | .0107094 .0060155 1.78 0.075 -.0010807 .0224994 10 | .0078851 .0052656 1.50 0.134 -.0024354 .0182055 11 | .009721 .0051157 1.90 0.057 -.0003056 .0197476 12 | .0117184 .0052384 2.24 0.025 .0014513 .0219854 13 | .0085235 .004981 1.71 0.087 -.0012391 .0182861 14 | .0104242 .0046175 2.26 0.024 .0013739 .0194744 15 | .0124196 .0046088 2.69 0.007 .0033864 .0214527 16 | .0083341 .0049614 1.68 0.093 -.0013901 .0180583 17 | .00992 .0046688 2.12 0.034 .0007692 .0190708 18 | .0114027 .004686 2.43 0.015 .0022182 .0205871 19 | .0052692 .0059747 0.88 0.378 -.0064411 .0169795 20 | .0058498 .006339 0.92 0.356 -.0065745 .0182741 21 | .006181 .0067253 0.92 0.358 -.0070003 .0193622 22 | -.002175 .0090427 -0.24 0.810 -.0198984 .0155484 23 | -.0021432 .0088189 -0.24 0.808 -.019428 .0151416 24 | -.0020011 .0080879 -0.25 0.805 -.0178531 .0138509 25 | -.0091967 .0089699 -1.03 0.305 -.0267774 .0083839 26 | -.0081533 .0075364 -1.08 0.279 -.0229243 .0066177 27 | -.0069432 .0060361 -1.15 0.250 -.0187739 .0048874 ------------------------------------------------------------------------------

It is difficult to make sense out of all of the numbers in the above table. Plotting the results
will aid us in interpreting the **margins** results. Using the **by** option with
**marginsplot** will get us three plot, one for each of the three values of **cv1**.

marginsplot, by(cv1) yline(0)

Looking at the three plots of **margins** results we see that when the covariate is one
standard deviation below the mean there are no significant marginal effects. When the covariate
is held at it mean value then the marginal effects for **m** at 50 and 55 are

significant.
And, finally when the covariate is held at the mean plus one standard deviation then the
marginal effect for **r** is statistically significant when **m** is between 45 and 55.

It might be useful to look at a single graph combining all three plots. In fact, we’ll go all out and include shaded confidence intervals.

Nice graph, but I don’t know if it really makes the results easier to interpret.