# Highly significant positive binary IV: Interaction always positive too

I'm working on a project where the main independent variable is binary. We then interact this variable with different continuous variables. The binary variable is by far the most significant variable in the model (t stats ~20) and the variable is always positive. Now, when I introduce an interaction term between this highly significant binary variable and a continuous variable, the interaction is virtually always significant and positive.

I'm afraid that the highly significant binary var drives my results. Would this be a valid concern? Perhaps you have some suggestions in how to address this issue.

I am not sure what you are asking but it is certainly possible to have a highly significant relationship that isn't involved in interactions.

set.seed(102105) #Random seed
xbinary <- c(rep(1,50), rep(2,50)) #Binary var
xcont <- rnorm(100) #continuous var
y1 <- rnorm(100) + xbinary #invented relationship
m1 <- lm(y1~xbinary + xcont + xbinary*xcont) #regression
summary(m1) #No interaction or effect of xcont

#If xcont is related
y2 <- rnorm(100) + xbinary + xcont #invented relationship
m2 <- lm(y2~xbinary + xcont + xbinary*xcont) #regression
summary(m2) #No interaction


Your first analysis is an example of ANCOVA (Analysis of Covariance) without interaction and the second one is ANCOVA with interaction.

The first analysis can be think of two (because you have a binary factor) regressions combined in one analysis. A common name for this is model with Fixed Slope and Varying Intercept. The combined analysis makes sense because intercept is common for both the levels of binary factor.

The second model have different slopes as well as different intercept and that is the reason you can not combine the two analysis (at different levels of binary factor) in one.

In other word, when interaction is significant the first analysis does not make sense and one has to perform two separate analysis.

The following book have a very nice description with example (I guess, Chapter 18):

Biostatistical Analysis (4th Edition) By Jerrold H. Zar.