Question

1. Consider a logistic regression model with a linear predictor that includes an interaction term, say x’β = β₀ + β₁x₁ +β₂x₂+ β₁₂x₁x₂. Does the odds ratio associated with the regressor x₂ have the same interpretation as in the case where the linear predictor does not have the interaction term? Provide mathematical development to support your answer.

Answer 1

In all other model the regression coefficient of a variable corresponds to the change in log odds and its exponentiated form corresponds to the odds ratio. This is only true when our model does not have any interaction terms. When a model has interaction term(s) of two predictor variables, it attempts to describe how the effect of a predictor variable depends on the level/value of another predictor variable. The interpretation of the regression coefficients become more involved.

Let’s take a simple example. (Don't get involved too much into the data like from where it is coming and all. Just go for the logic since I just put these to make you understand clear.)  

logit(p) = log(p/(1-p))= β₀ + β₁*female + β₂*math + β₃*female*math

Logistic regression                               Number of obs   =        200

                                                  LR chi2(3)      =      62.94

                                                  Prob > chi2     =     0.0000

Log likelihood = -79.883301                       Pseudo R2       =     0.2826

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         hon |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]

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      female | -2.899863   3.094186    -0.94   0.349    -8.964357    3.164631

        math |   .1293781   .0358834     3.61   0.000     .0590479    .1997082

femalexmath |   .0669951     .05346     1.25   0.210    -.0377846    .1717749

   intercept | -8.745841    2.12913    -4.11   0.000    -12.91886   -4.572823

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In the presence of interaction term of female by math, we can no longer talk about the effect of female, holding all other variables at certain value, since it does not make sense to fix math and femalexmath at certain value and still allow female change from 0 to 1!

In this simple example where we examine the interaction of a binary variable and a continuous variable, we can think that we actually have two equations: one for males and one for females. For males (female=0), the equation is simply

logit(p) = log(p/(1-p))= β₀ + β₂*math.

For females, the equation is

logit(p) = log(p/(1-p))= (β₀ + β₁) + (β₂ + β₃ )*math.

Now we can map the logistic regression output to these two equations. So we can say that the coefficient for math is the effect of math when female = 0. More explicitly, we can say that for male students, a one-unit increase in math score yields a change in log odds of 0.13. On the other hand, for the female students, a one-unit increase in math score yields a change in log odds of (.13 + .067) = 0.197. In terms of odds ratios, we can say that for male students, the odds ratio is exp(.13) = 1.14 for a one-unit increase in math score and the odds ratio for female students is exp(.197) = 1.22 for a one-unit increase in math score. The ratio of these two odds ratios (female over male) turns out to be the exponentiated coefficient for the interaction term of female by math: 1.22/1.14 = exp(.067) = 1.07.