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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?

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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 $\exp(2.80625\approx16.55)$ 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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  • $\begingroup$ Thank you! Is there a way I could evaluate, everything else constant, the difference in the predicted probability of being elected of two treated candidates? The only difference between them would be the Treat.City variable - one being on the 25th percentile, the other on the 75th? $\endgroup$ – Arthur Carvalho Brito Oct 18 '17 at 0:28
  • $\begingroup$ the odds are the natural metric for logistic regression. This means that the odds ratio(for the main effects) and ratio of odds ratios (for interaction terms) are a single number. Marginal effects and discrete differences are dependent on the values at which you fix all other variables. Especially for interaction terms you can quickly end up with conclusions like "for some respondents the interaction terms is significantly positive, for others significantly negative, and for other not significant at all". Good luck explaining that to your audience... $\endgroup$ – Maarten Buis Oct 18 '17 at 8:09
  • $\begingroup$ assuming I want to do something simpler: I calculate predicted values for my dependent variable evaluating each covariate at its mean for two hypothetical units. The only difference between them would by the value of Treat.City. If the difference of my predicted elected variable between these two is 0.49, does this means one is 49 percentage points more likely to get elected than the other? $\endgroup$ – Arthur Carvalho Brito Oct 19 '17 at 16:07

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