The techniques and methods for this FAQ page were inspired Long (2006) and Hu & Long (2005).

This FAQ page uses features available in Stata 11 and will not work for earlier versions. If you are using an earlier version of Stata go to FAQ page.

Interactions in logistic regression models can be trickier than interactions in comparable OLS regression.

Many researchers are not comfortable interpreting the results in terms of the raw coefficients which are scaled in terms of log odds. The interpretation of interactions in log odds is done basically the same way as in OLS regression. However, many researchers prefer to interpret 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 categorical by continuous interactions in logistic regression models both with and without covariates.

We will use an example dataset, **logitcatcon**, that has one binary predictor, **f**, which
stands for female and one continuous predictor **s**.
In addition, the model will include **fs** which is the **f** by **s** interaction.
We will begin by loading the data and then running the logit model.

use http://www.ats.ucla.edu/stat/data/logitcatcon, clear logit y i.f##c.s, nologLogistic regression Number of obs = 200 LR chi2(3) = 71.01 Prob > chi2 = 0.0000 Log likelihood = -96.28586 Pseudo R2 = 0.2694 ------------------------------------------------------------------------------ y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 1.f | 5.786811 2.302518 2.51 0.012 1.273959 10.29966 s | .1773383 .0364362 4.87 0.000 .1059248 .2487519 | f#c.s | 1 | -.0895522 .0439158 -2.04 0.041 -.1756255 -.0034789 | _cons | -9.253801 1.94189 -4.77 0.000 -13.05983 -5.447767 ------------------------------------------------------------------------------

As you can see all of the variables in the above model including the interaction term are statistically significant. If this were an OLS regression model we could do a very good job of understanding the interaction using just the coefficients in the model. The situation in logistic regression is more complicated because the value of the interaction effect changes depending upon the value of the continuous predictor variable. To begin to understand what is going on consider the Table 1 below.

Table 1: Predicted probabilities when s=40 f=0 f=1 change LB UB .1034 .5111 .4077 .2182 .5972

Table 1 contains predicted probabilities, differences in predicted probabilities and the confidence
interval of the difference in predicted probabilities while holding the continuous predictor at 40.
The first value, .1034, is the predicted probability when **f**=0 (males), the .5111 when
**f**=1 (females).
The third value, .4077, is the difference in probabilities for males and females.
The next two values are the 95% confidence interval on the difference in
probabilities. If the confidence interval contains zero the difference would not be considered
statistically significant. In our example, the confidence interval does not contain zero, thus,
the difference in probabilities is statistically significant.

To get the values for Table 1 we will run **margins** with the **post** option followed by the
**lincom** command.

margins f, at(s=40) postAdjusted predictions Number of obs = 200 Model VCE : OIM Expression : Pr(y), predict() at : s = 40 ------------------------------------------------------------------------------ | Delta-method | Margin Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- f | 0 | .1033756 .0500784 2.06 0.039 .0052238 .2015275 1 | .5111116 .0827069 6.18 0.000 .349009 .6732142 ------------------------------------------------------------------------------lincom 1.f-0.f( 1) - 0bn.f + 1.f = 0 ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- (1) | .407736 .0966865 4.22 0.000 .2182339 .597238 ------------------------------------------------------------------------------

Now that we know how to compute the difference in probabilities including the confidence interval,
we need to do this for a whole rang of values of **s**. Because we used **margins** with
the **post** option we will need to rerun the **logit** command again. We will save the
results from the **lincom** command into a matrix called **t** along with the values of the
continuous variable. Finally, we save the matrix back into the dataset and graph the results.

logit y f##c.s, nolog margins f, at(s=(20(2)70)) post mat t = e(at) mat t = t[1...,"s"] mat t = t,J(26,3,.) forvalue i=1/26 { quietly lincom _b[`i'._at#1.f]-_b[`i'._at#0.f] mat t[`i',2] = r(estimate) mat t[`i',3] = r(estimate) - 1.96*r(se) mat t[`i',4] = r(estimate) + 1.96*r(se) } mat colnames t = at dif ll ul svmat t, names(col) twoway (line dif at)(line ll at)(line ul at), /// yline(0) legend(off) /// xtitle(continuous independent variable) ytitle(probability difference) /// title(male-female difference) scheme(lean1)

Now, we will run the above code fragment and check out the graph.

The above graph shows how the male-female probability difference varies with changes in the value of **s**.
It appears that the difference in probabilities for male and females is statistically
significant between values of **s** of approximately 28 to 55 and is nonsignificant elsewhere.

There is another way that we could compute the male/female difference in probability more
directly. We will use the **margins** command again along with the user written command **parmest**
by Roger Newson (**search parmest**). Here is what the steps look like.

quietly logit y i.f##c.s margins, dydx(f) at(s=(20(2)70)) vsquish postConditional marginal effects Number of obs = 200 Model VCE : OIM Expression : Pr(y), predict() dy/dx w.r.t. : 1.f 1._at : s = 20 2._at : s = 22 3._at : s = 24 4._at : s = 26 5._at : s = 28 6._at : s = 30 7._at : s = 32 8._at : s = 34 9._at : s = 36 10._at : s = 38 11._at : s = 40 12._at : s = 42 13._at : s = 44 14._at : s = 46 15._at : s = 48 16._at : s = 50 17._at : s = 52 18._at : s = 54 19._at : s = 56 20._at : s = 58 21._at : s = 60 22._at : s = 62 23._at : s = 64 24._at : s = 66 25._at : s = 68 26._at : s = 70 ------------------------------------------------------------------------------ | Delta-method | dy/dx Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 1.f | _at | 1 | .1496878 .0987508 1.52 0.130 -.0438602 .3432358 2 | .1724472 .1043526 1.65 0.098 -.0320801 .3769745 3 | .1975119 .1089189 1.81 0.070 -.0159653 .4109891 4 | .2246979 .1121624 2.00 0.045 .0048636 .4445323 5 | .2536377 .1138457 2.23 0.026 .0305042 .4767711 6 | .2837288 .1138312 2.49 0.013 .0606236 .5068339 7 | .3140793 .1121391 2.80 0.005 .0942908 .5338679 8 | .3434564 .1090032 3.15 0.002 .1298141 .5570987 9 | .3702468 .104906 3.53 0.000 .1646348 .5758589 10 | .3924501 .100548 3.90 0.000 .1953797 .5895206 11 | .407736 .0966865 4.22 0.000 .2182339 .5972381 12 | .4136138 .0938187 4.41 0.000 .2297325 .5974952 13 | .4077687 .0918544 4.44 0.000 .2277375 .5878 14 | .3885877 .0901228 4.31 0.000 .2119502 .5652252 15 | .3558056 .0879135 4.05 0.000 .1834983 .5281129 16 | .311046 .0851843 3.65 0.000 .1440879 .4780041 17 | .2579152 .0826761 3.12 0.002 .0958731 .4199574 18 | .2014085 .0810274 2.49 0.013 .0425978 .3602192 19 | .146748 .0798151 1.84 0.066 -.0096868 .3031828 20 | .09816 .0778372 1.26 0.207 -.0543981 .2507182 21 | .0581376 .0741941 0.78 0.433 -.0872801 .2035553 22 | .0273908 .0688343 0.40 0.691 -.1075218 .1623035 23 | .0052927 .0623354 0.08 0.932 -.1168824 .1274678 24 | -.0095202 .0554486 -0.17 0.864 -.1181975 .0991571 25 | -.0186421 .0487849 -0.38 0.702 -.1142588 .0769746 26 | -.0235782 .042708 -0.55 0.581 -.1072843 .0601279 ------------------------------------------------------------------------------ Note: dy/dx for factor levels is the discrete change from the base level.mat at=e(at) mat at=at[1...,"s"]

Notice that we used **dydx(f)** in the **margins** command which computes a discrete
difference between males and females for each value of **c**. Next, we will use
the **parmest** command which will replace the data in memory with vales of the
estimate and confidence intervals.

parmest, fast drop if z==. drop eq parm svmat at twoway (line estimate at1)(line min95 at1)(line max95 at1), /// legend(off) yline(0) /// xtitle(continuous independent variable) ytitle(probability difference) /// title(male-female difference) scheme(lean1)

The graph produced by the **twoway** command is exactly the same as the one shown above.

So, that went fairly well but what if there was a covariate in the model? The model below
includes the covariate **cv1**.

use http://www.ats.ucla.edu/stat/data/logitcatcon, clear logit y f##c.s cv1, nologLogistic regression Number of obs = 200 LR chi2(4) = 114.41 Prob > chi2 = 0.0000 Log likelihood = -74.587842 Pseudo R2 = 0.4340 ------------------------------------------------------------------------------ y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 1.f | 9.983662 3.05269 3.27 0.001 4.0005 15.96682 s | .1750686 .0470033 3.72 0.000 .0829438 .2671933 | f#c.s | 1 | -.1595233 .0570352 -2.80 0.005 -.2713103 -.0477363 | cv1 | .1877164 .0347888 5.40 0.000 .1195316 .2559013 _cons | -19.00557 3.371064 -5.64 0.000 -25.61273 -12.39841 ------------------------------------------------------------------------------

As before, all of the coefficients are statistically significant.

We will run the analysis pretty much as before except that we will do it three times holding the covariate at a different value each time. We begin holding the covariate at a low value of 40, then at a medium value of 50 and finally at a high value of 60. The code fragment below computes the predicted differences in probability for each of the three values of the covariate and produces a separate graph for each one.

/* hold cv1 at 40 */ drop at-ul logit y f##c.s cv1, nolog margins f, at(s=(20(2)70) cv1=40) post mat t = e(at) mat t = t[1...,"s"] mat t = t,J(26,3,.) forvalue i=1/26 { quietly lincom _b[`i'._at#1.f]-_b[`i'._at#0.f] mat t[`i',2] = r(estimate) mat t[`i',3] = r(estimate) - 1.96*r(se) mat t[`i',4] = r(estimate) + 1.96*r(se) } mat colnames t = at dif ll ul svmat t, names(col) twoway (line dif at)(line ll at)(line ul at), /// yline(0) legend(off) ylabel(-1(.5)1) /// xtitle(continuous independent variable) ytitle(probability difference) /// title(male-female difference with cv1 at 40) scheme(lean1) /* hold cv1 at 50 */ drop at-ul logit y f##c.s cv1, nolog margins f, at(s=(20(2)70) cv1=50) post forvalue i=1/26 { quietly lincom _b[`i'._at#1.f]-_b[`i'._at#0.f] mat t[`i',2] = r(estimate) mat t[`i',3] = r(estimate) - 1.96*r(se) mat t[`i',4] = r(estimate) + 1.96*r(se) } svmat t, names(col) twoway (line dif at)(line ll at)(line ul at), /// yline(0) legend(off) ylabel(-1(.5)1) /// xtitle(continuous independent variable) ytitle(probability difference) /// title(male-female difference with cv1 at 50) scheme(lean1) /* hold cv1 at 60 */ drop at-ul logit y f##c.s cv1, nolog margins f, at(s=(20(2)70) cv1=60) post forvalue i=1/26 { quietly lincom _b[`i'._at#1.f]-_b[`i'._at#0.f] mat t[`i',2] = r(estimate) mat t[`i',3] = r(estimate) - 1.96*r(se) mat t[`i',4] = r(estimate) + 1.96*r(se) } svmat t, names(col) twoway (line dif at)(line ll at)(line ul at), /// yline(0) legend(off) ylabel(-1(.5)1) /// xtitle(continuous independent variable) ytitle(probability difference) /// title(male-female difference with cv1 at 60) scheme(lean1)

It seems clear from looking at the three graphs that the male-female difference in probability
increases as **cv1** increases except for high values of **s**. And, yes, I know the upper
limit of the confidence interval exceeds one in some places but that is just an artifact of
how the confidence intervals were created. We aren’t really trying to imply that the probability
can ever exceed one.

## References

Long, J. S. 2006. Group comparisons and other issues in interpreting models for categorical
outcomes using Stata. Presentation at 5th North American Users Group Meeting. Boston,
Massachusetts.

Xu, J. and J.S. Long, 2005. Confidence intervals for predicted outcomes in regression models for
categorical outcomes. The Stata Journal 5: 537-559.