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Stata FAQ
How can I understand a categorical by continuous interaction in logistic regression? (Stata 12)

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 Stata 12 and will not work for earlier versions. In particular, marginsplot and contrast were introduced in Stata 12.

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, nolog

Logistic 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 twice. The second time we will use the dydx option to get the differences in probabilities. with the post option followed by the lincom command. 

margins f, at(s=40) 

Adjusted 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
------------------------------------------------------------------------------

margins, dydx(f) at(s=40)

Conditional marginal effects                      Number of obs   =        200
Model VCE    : OIM

Expression   : Pr(y), predict()
dy/dx w.r.t. : 1.f
at           : s               =          40

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         1.f |    .407736   .0966865     4.22   0.000     .2182339     .597238
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.
Now that we know how to compute the difference in probabilities including the confidence intervals, we need to do this for a whole rang of values of s. The vsquish option just omits extra blank lines in the header. 
margins, dydx(f) at(s=(20(2)70)) vsquish

Conditional 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     .597238
         12  |   .4136138   .0938187     4.41   0.000     .2297325    .5974952
         13  |   .4077687   .0918544     4.44   0.000     .2277375       .5878
         14  |   .3885877   .0901232     4.31   0.000     .2119494     .565226
         15  |   .3558056   .0879135     4.05   0.000     .1834983     .528113
         16  |    .311046   .0851843     3.65   0.000     .1440878    .4780042
         17  |   .2579152   .0826761     3.12   0.002      .095873    .4199574
         18  |   .2014085   .0810274     2.49   0.013     .0425978    .3602193
         19  |    .146748   .0798151     1.84   0.066    -.0096868    .3031828
         20  |     .09816   .0778372     1.26   0.207    -.0543982    .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.
Now, we will graph the differences from the above table using the marginsplot command. 

marginsplot, yline(0)
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.

We can make the above graph a little more visually attractive by shading the confidence intervals. 

marginsplot, recasr(line) recastci(rarea) yline(0)

Logit model with continuous covariate

So, that went fairly well but what if there was a covariate in the model? Adding covariates to a logit model can change the pattern of predicted probabilities even though the covariate does not interact with any of the primary research variables.

Our next model, shown below, includes the covariate cv1.

use http://www.ats.ucla.edu/stat/data/logitcatcon, clear

logit y f##c.s cv1, nolog 

Logistic 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 commands compute the predicted differences in probability for each of the three values of the covariate and produces a separate graph for each one. 

margins, dydx(f) at(s=(20(2)70) cv1=40) noatlegend


Conditional marginal effects                      Number of obs   =        200
Model VCE    : OIM

Expression   : Pr(y), predict()
dy/dx w.r.t. : 1.f

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
1.f          |
         _at |
          1  |   .2307214    .150045     1.54   0.124    -.0633615    .5248042
          2  |   .2361502   .1429905     1.65   0.099     -.044106    .5164064
          3  |   .2416119   .1357653     1.78   0.075    -.0244832    .5077069
          4  |   .2470812   .1284098     1.92   0.054    -.0045973    .4987597
          5  |   .2525225   .1209752     2.09   0.037     .0154154    .4896296
          6  |   .2578855   .1135271     2.27   0.023     .0353765    .4803946
          7  |   .2630995   .1061484     2.48   0.013     .0550525    .4711464
          8  |   .2680645   .0989437     2.71   0.007     .0741384    .4619905
          9  |   .2726404   .0920437     2.96   0.003     .0922382    .4530427
         10  |   .2766309   .0856081     3.23   0.001     .1088422    .4444197
         11  |   .2797622   .0798258     3.50   0.000     .1233066    .4362179
         12  |   .2816551   .0749086     3.76   0.000      .134837    .4284732
         13  |   .2817895   .0710764     3.96   0.000     .1424822    .4210967
         14  |   .2794619   .0685404     4.08   0.000     .1451252    .4137986
         15  |   .2737404   .0675055     4.06   0.000      .141432    .4060488
         16  |   .2634265   .0682395     3.86   0.000     .1296795    .3971735
         17  |    .247046   .0712461     3.47   0.001     .1074062    .3866859
         18  |   .2229048   .0774966     2.88   0.004     .0710142    .3747954
         19  |   .1892608   .0884614     2.14   0.032     .0158797    .3626418
         20  |   .1446609   .1055661     1.37   0.171    -.0622448    .3515666
         21  |     .08845   .1291224     0.69   0.493    -.1646253    .3415252
         22  |   .0213483   .1574555     0.14   0.892    -.2872588    .3299553
         23  |  -.0541486   .1869312    -0.29   0.772     -.420527    .3122298
         24  |  -.1338606   .2130623    -0.63   0.530    -.5514551    .2837339
         25  |   -.212654   .2323119    -0.92   0.360    -.6679769     .242669
         26  |  -.2855934   .2436296    -1.17   0.241    -.7630986    .1919118
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

marginsplot, yline(0)



margins, dydx(f) at(s=(20(2)70) cv1=50) noatlegend


Conditional marginal effects                      Number of obs   =        200
Model VCE    : OIM

Expression   : Pr(y), predict()
dy/dx w.r.t. : 1.f

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
1.f          |
         _at |
          1  |   .6603841   .2089015     3.16   0.002     .2509446    1.069824
          2  |   .6663811   .1945798     3.42   0.001     .2850117    1.047751
          3  |   .6719235   .1806131     3.72   0.000     .3179283    1.025919
          4  |   .6768518    .167067     4.05   0.000     .3494065    1.004297
          5  |   .6809438   .1540312     4.42   0.000     .3790483    .9828394
          6  |   .6838906   .1416328     4.83   0.000     .4062954    .9614859
          7  |   .6852664   .1300559     5.27   0.000     .4303616    .9401712
          8  |   .6844899   .1195626     5.72   0.000     .4501515    .9188282
          9  |   .6807804   .1105069     6.16   0.000     .4641908      .89737
         10  |   .6731136   .1033086     6.52   0.000     .4706324    .8755948
         11  |   .6601906   .0983425     6.71   0.000     .4674428    .8529385
         12  |   .6404474   .0957213     6.69   0.000     .4528371    .8280577
         13  |    .612146   .0950685     6.44   0.000     .4258151    .7984768
         14  |   .5736015   .0955153     6.01   0.000      .386395     .760808
         15  |   .5235847   .0961089     5.45   0.000     .3352148    .7119546
         16  |   .4618751   .0965359     4.78   0.000     .2726681     .651082
         17  |   .3898094   .0976254     3.99   0.000     .1984672    .5811516
         18  |    .310538   .1007595     3.08   0.002      .113053    .5080229
         19  |   .2287029   .1061749     2.15   0.031     .0206039     .436802
         20  |   .1495094   .1121609     1.33   0.183    -.0703219    .3693407
         21  |   .0775412   .1164177     0.67   0.505    -.1506333    .3057156
         22  |   .0158576   .1177948     0.13   0.893     -.215016    .2467312
         23  |  -.0342943   .1167353    -0.29   0.769    -.2630913    .1945027
         24  |  -.0732007   .1146282    -0.64   0.523    -.2978678    .1514664
         25  |   -.102126   .1129167    -0.90   0.366    -.3234388    .1191867
         26  |  -.1227649   .1125169    -1.09   0.275    -.3432941    .0977642
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

marginsplot, yline(0)



margins, dydx(f) at(s=(20(2)70) cv1=60) noatlegend

Conditional marginal effects                      Number of obs   =        200
Model VCE    : OIM

Expression   : Pr(y), predict()
dy/dx w.r.t. : 1.f

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
1.f          |
         _at |
          1  |   .9135206   .0789176    11.58   0.000     .7588449    1.068196
          2  |   .9097497   .0759899    11.97   0.000     .7608121    1.058687
          3  |    .903598   .0750691    12.04   0.000     .7564654    1.050731
          4  |   .8942046   .0769878    11.61   0.000     .7433112    1.045098
          5  |   .8804504   .0825006    10.67   0.000     .7187522    1.042149
          6  |    .860939   .0918979     9.37   0.000     .6808225    1.041056
          7  |   .8340294   .1046826     7.97   0.000     .6288553    1.039203
          8  |     .79797   .1194328     6.68   0.000     .5638859    1.032054
          9  |   .7511866   .1338225     5.61   0.000     .4888992    1.013474
         10  |   .6927478   .1448523     4.78   0.000     .4088424    .9766531
         11  |   .6229422   .1494681     4.17   0.000     .3299901    .9158944
         12  |   .5437595   .1455949     3.73   0.000     .2583987    .8291202
         13  |   .4589623   .1331889     3.45   0.001     .1979168    .7200079
         14  |   .3735294   .1145842     3.26   0.001     .1489486    .5981102
         15  |   .2925756   .0936809     3.12   0.002     .1089644    .4761868
         16  |   .2202105   .0743402     2.96   0.003     .0745063    .3659147
         17  |   .1588388   .0589333     2.70   0.007     .0433317     .274346
         18  |   .1091008   .0478486     2.28   0.023     .0153193    .2028823
         19  |   .0702892   .0401238     1.75   0.080     -.008352    .1489304
         20  |   .0409283   .0345573     1.18   0.236    -.0268028    .1086594
         21  |   .0192749   .0303776     0.63   0.526    -.0402642    .0788139
         22  |   .0036462   .0272628     0.13   0.894    -.0497878    .0570803
         23  |  -.0074154   .0251015    -0.30   0.768    -.0566134    .0417827
         24  |  -.0150913   .0238003    -0.63   0.526     -.061739    .0315564
         25  |  -.0202982   .0232115    -0.87   0.382    -.0657919    .0251955
         26  |  -.0237264     .02315    -1.02   0.305    -.0690996    .0216469
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

marginsplot, yline(0)
That went well, so let's try to combine all three graphs into one. We will rerun the margins command using the quietly option to suppress the output which we have seen above. 
quietly margins, dydx(f) at(s=(20(2)70) cv1=(40 50 60))

marginsplot, yline(0)
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.

One final graph with shaded confidence intervals and we're done. 

marginsplot, recast(line) recastci(rarea) yline(0)
Awesome graph!

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.

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