# Solved – How to interpret logistic regression coefficients with interactions between binary and continuous variables

I know this has been already asked, but I am quite confused about the interpretation of logit regression estimates if I have interacted variables (continuous and binary ones).

I run the following regression:

``model <- glm(elected ~ treat + factor(School) + factor(Race) +                 treat*Treat.City, data = subset(df, Year == 2016),              family = binomial(link = 'logit')) ``

My dependent variable `elected` is equal to 1 if a political candidate got elected, 0 otherwise. `treat` equals 1 if the candidate belongs to a treatment group, 0 if belongs to the control group.

After controlling for schooling and race dummy variables, I have put the interaction `treat*Treat.City`, in which `Treat.City` is a continuous variables indicating the percentage of treatment candidates in relation to the total number of challengers inside candidate's i city.

Running the regression in `R`, I have the following results:

``Call: glm(formula = elected ~ treat + factor(School) + factor(Race) +      treat * Treat.City, family = binomial(link = "logit"), data = subset(df,      Year == 2016))  Deviance Residuals:     Min      1Q  Median      3Q     Max   -1.875  -1.321   1.000   1.039   1.262    Coefficients:                                   Estimate Std. Error z value Pr(>|z|)     (Intercept)                        0.42387    0.15196   2.789 0.005281 **  treat                             -0.22397    0.03879  -5.775 7.71e-09 *** factor(School)MÉDIO_INCOMPLETO     0.04055    0.03452   1.174 0.240227     factor(School)SUPERIOR_COMPLETO    0.11976    0.03221   3.718 0.000201 *** factor(School)SUPERIOR_INCOMPLETO  0.11576    0.02947   3.929 8.55e-05 *** factor(Race)BRANCA                -0.12757    0.15054  -0.847 0.396742     factor(Race)INDÍGENA              -0.57795    0.26393  -2.190 0.028542 *   factor(Race)Preta_Parda           -0.20933    0.15073  -1.389 0.164897     Treat.City                         1.50083    0.61352   2.446 0.014435 *   treat:Treat.City                   2.80625    0.95484   2.939 0.003293 **  --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for binomial family taken to be 1)      Null deviance: 54123  on 39893  degrees of freedom Residual deviance: 54033  on 39884  degrees of freedom AIC: 54053  Number of Fisher Scoring iterations: 4 ``

How can I interpret such coefficients? More specifically, how can I numerically make a statement about the effect of the treatment on the probability in getting elected?

Can I make any clear interpretation about this 'Intensity of treatment' variable that `Treat.City` is?

Contents

The odds of being elected when you are not treated increases by a factor \$exp(1.50083)approx 4.49\$ or \$(4.49-1)times100%=349%\$ if you move from a city with no one treated to a city where everyone is treated.

This effect of `Treat.City` increases by a factor \$exp(2.80625approx16.55)\$ or \$(16.55-1)times100%=1555%\$ if one is treated. For more see: http://maartenbuis.nl/publications/interactions.html

Given the large size of the effect I will assume that `Treat.City` is not a percentage but a proportion. The effects will be more realistic and easier to interpret when you turn `Treat.City` into percentages.

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