# Interprete estimates of model with two categorical independent variables in binomial regression (GLM)

I use R for a binomial regression (GLM) to test for a difference in whether people voted (coded as 0/1) in an election based on race (6 categories) and religion (3 categories), see structure of dataset and output below.

Structure of dataset:

GLM Binomial regression:

ANOVA - Test predictors relative to the full model:

1. How can I understand the estimate of the intercept of this model since the model contains two categorical independent variables? Does the estimate for the intercept somehow represent both the first categories for both variable (Religion: catholic; Race: Asian)? How can I understand the P-value of the intercept?

2. As I run an ANOVA for the model both categorical variables render statistically significant P-values but the summary of the regression does only show statistically significant P-values for Religion. How can I understand the result of the ANOVA? Does it provide anything in terms of understanding the relationship between voting behavior and the two categorical variables?

How can I understand the estimate of the intercept of this model since the model contains two categorical independent variables? Does the estimate for the intercept somehow represent both the first categories for both variable (Religion: catholic; Race: white)? How can I understand the P-value of the intercept?

The intercept includes the reference level of both of the variables, so that will be the level of each variable that does not have it's own estimate. The reference level for Religion appears to be catholic - but there isn't enough info provided to see what the reference level for Race is.

As I run an ANOVA for the model both categorical variables render statistically significant P-values but the summary of the regression does only show statistically significant P-values for Religion. How can I understand the result of the ANOVA? Does it provide anything in terms of understanding the relationship between voting behavior and the two categorical variables?

The ANOVA tests are for the whole variables, whereas the tests in the lm() summary output are for individual levels of each variable compared to the reference level for that variable. So you can't really comapare them.

• Sorry, the reference level of the intercept must then "Asian" for Race. Since the ANOVA shows a statistically significant P-value for RACE but there are no statistically significant differences for individual levels, how would you analyze and present this result? Is it correct to state that there are no individual differences between various races in terms of voting behavior, but overall the result of the ANOVA shows that race affects voting behavior overall? Can and should I elaborate further on this?
– JanC
Commented May 20, 2021 at 18:57
• Yes it would make sense for Asian to be the reference for Race. Tha ANOVA is saying that the overall effect of Race is significant, but there is insufficent evidence for any difference between the individual races and Asian. This may be due to an insuficient number of observations for each Race. Commented May 20, 2021 at 19:13
• There could also be another issue here, that there could be a causal ralation between race and religion ? You could look at a cross tablulation which may indicate that, but even if it doesn't there could still be a problem if the sampling was done to keep the group sizes equal. If religion is a mediator then it shouldn't be in the model if you are interested in the total causal effect of Race, but if you want the total causal effect of Religion then you should include Race because Race would then be a confounder. Commented May 20, 2021 at 19:13
• Take a look at my answer here for a detailed look into biases that can arise when you don't consider causal relationships properly Commented May 20, 2021 at 19:15
• Something else I noticed is that you seem to have country data too. Depending on how many countries you should consider either including Country in your existing GLM model as a fixed effect, or if you have a lot of them, perhaps as random intercepts in GLMM model (the glmer function from the lme4 package is usually a good starting point). The point is that observations within the same country are likely to be similar to each other, than to observations in other countries and if you don't account for that, then it can lead to invalid inferences. Commented May 20, 2021 at 19:54