Understanding the output of a linear model with interaction between two binary predictors

Question

I have two categorical variables (A and B) in a linear regression. Each is binary.

Both are statistically significant in a model without an interaction term.

When I include the interaction between them, neither the interaction term nor either of the main effects are significant.

However, if I change the reference category of A, B is significant in the model.

As I have read that the main effects in a model that includes an interaction term represent the effect of one variable when the other is 0, and my explanatory variables are binary, I am confused how this does not constitute an interaction. It seems clear that the effect of B depends on A.

Would anyone mind explaining this to me please? I am unsure how to interpret my results.

Answer 1

• In the model without the interaction, the intercept estimates the response when both A and B are zero. The coefficient for A represents the difference between the response when A is 1, and when A is 0. Likewise, the coefficient for B represents the difference between the response when B is 1, and when B is 0.

• In the model with the interaction, the intercept again estimates the response when both A and B are zero. The coefficient for A estimates the difference between the response when A=1 and A=0, when B is zero. Likewise, the coefficient for B estimates the difference between the response when B=0 and B=1, when A is zero. The interaction itself is literally the product of A and B, $A \times B$, so this will be zero unless A and B are both 1. This means that the interaction estimates the additional difference for the response between A=0 and A=1, when B changes from zero to 1, or vice versa.

Since the coefficients are estimating different things, the t tests for the coefficients are different tests which obviously will have different results.

Regarding what happens when you change the reference level of A, that is simply a re-parameterisation of the model. The global statistics, such as R squared and the F test will be identical, but the effect on the individual estimates will be different in the two models.

• In the model with no interaction, the sign of the coefficient of the "new" A will flip, it's standard error will be unchanged (hence the t test will be the same) but the estimate for the intercept will change by exactly the size of the estimate for A (and it's standard error will change, and hence the t test result will change) because it estimates the difference in the response when A is zero which has now changed by exactly what the original estimate of A was.

• In the model with the interaction, the estimate for the new A and the intercept will change in exactly the same way as for the model with no interaction. The estimate for the interaction should flip signs (with same t test result) but the estimate for B should change by exactly the size of the estimate for the interaction, because it estimates the difference in the response when A is zero, and now A has been reparameterised so of course the standard error and t test result will be different for B

A simple simulation demonstrates all of the above:

> set.seed(15)
> N <- 100
> A <- rbinom(N, 1, 0.4)
> B <- rbinom(N, 1, 0.6)
> Y <- 10 + A + B + 3*A*B + rnorm(N)
> summary(m0 <- lm(Y ~ A + B))

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   8.5643     0.2721   31.48   <2e-16 ***
A             2.9629     0.2666   11.12   <2e-16 ***
B             2.8035     0.2690   10.42   <2e-16 ***

> summary(m1 <- lm(Y ~ A * B))

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   9.7563     0.3091  31.566  < 2e-16 ***
A             1.2174     0.3740   3.255  0.00157 ** 
B             1.1116     0.3682   3.019  0.00325 ** 
A:B           2.7988     0.4736   5.910 5.21e-08 ***

# switch the levels of A
> AA <- (A - 1)^2

> summary(m0.1 <- lm(Y ~ AA + B))

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  11.5271     0.2192   52.60   <2e-16 ***
AA           -2.9629     0.2666  -11.12   <2e-16 ***
B             2.8035     0.2690   10.42   <2e-16 ***

> summary(m1.1 <- lm(Y ~ AA * B))

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  10.9737     0.2106  52.106  < 2e-16 ***
AA           -1.2174     0.3740  -3.255  0.00157 ** 
B             3.9104     0.2978  13.129  < 2e-16 ***
AA:B         -2.7988     0.4736  -5.910 5.21e-08 ***

> coef(m0.1)[1] - coef(m0)[1] - coef(m0)[2]
(Intercept) 
          0 
> coef(m1.1)[1] - coef(m1)[1] - coef(m1)[2]
(Intercept) 
          0 
> coef(m1.1)[3] - coef(m1)[3] - coef(m1)[4]
          B 
          0