How to interpret the results of multiple regression when two dummy coded predictor variables are depended?

I am would like to get the estimated influence of two conditions (variable A and variable B).

When I dummy-coded variable A and B and carried out a multiple regression analysis, both of them were significant predictors.

However, things are little complicated. Variable B is independent, but Variable A depends on B; Variable A can be present (i.e., 1) only when Variable B is present (1).

I am wondering how I can interpret the estimated coefficient for variable B. I thought I could run a simple regression analysis, but I would also like to get an estimated effect of variable A while B is zero (which is what the coefficient for variable A represents, if I am understanding correctly).

Hence, I started wondering this if I can interpret in the way that the estimated influence of B is the coefficient for B plus coefficient for A. However, I am not 100% confident if I am right.

I would really appreciate it if someone could confirm or correct my understanding. Thank you for your help!

• Is the wording in the paragraph starting with "I am wondering..." correct? In the previous paragraph you say that Var A can be present only when Var B is present, but then why would it not make sense that B would be present while A is 0? And are you in fact looking for the estimated effect of A while B is 0? Because, again, that does not go with "Variable A can be present only when Variable B is present". May 31 '19 at 5:38
• Thank you for commenting and answering my question, @AlexK! And sorry for my writing causing confusion. That's right; Var A can be 1 only when Var B is 1, also Var B can be either 0 or 1 regard less of Var A. May 31 '19 at 12:45

The way you are interpreting the coefficients is not quite right. The general interpretation of the coefficient on a dummy variable in a multiple regression is "the expected (or average) difference in the dependent variable between those with $$1$$ and those with $$0$$ values of that dummy variable, holding other independent variables constant.

If you, for example, only have these two $$A$$ and $$B$$ variables as predictors, then the interpretation of the coefficient on variable $$B$$ is "the expected difference in the dependent variable between someone with the value of $$1$$ and someone with the value of $$0$$ of variable $$B$$, if they both had the same value of variable $$A$$." That same value of variable $$A$$ can be anything, not just $$0$$.

Variable A can be present (i.e., 1) only when Variable B is present (1). I am wondering how I can interpret the estimated coefficient for variable B, because the coefficient for B represent the presence of B while A is 0, which logically does not make sense.

I think in these two sentences you may have meant to say that you cannot interpret the effect of $$A$$ while $$B$$ is $$0$$, because $$A$$ should be non-existent when $$B$$ is $$0$$. But the regression model does not know that and the $$\beta$$'s (marginal effects) it provides apply regardless of what the value of the second variable is. So you cannot just take interpret those coefficients as effects of the first variable for a specific value of the second variable.

It is a pretty simple exercise to figure out what each coefficient represents in this kind of regression:

$$Y_i = \beta_0 + \beta_1A_i + \beta_2B_i + \epsilon_i$$

where both $$A$$ and $$B$$ are binary dummy variables. You can just plug in the possible values of those variables and see what you get, like this:

If $$A = 0$$ and $$B = 0$$: The expected value of $$Y$$ will be $$\beta_0$$.

If $$A = 0$$ and $$B = 1$$: The expected value of $$Y$$ will be $$\beta_0 + \beta_2$$.

If $$A = 1$$ and $$B = 0$$: The expected value of $$Y$$ will be $$\beta_0 + \beta_1$$.

If $$A = 1$$ and $$B = 1$$: The expected value of $$Y$$ will be $$\beta_0 + \beta_1 + \beta_2$$.

So, we see that $$\beta_2$$ is the average difference in $$Y$$ between individuals with $$B = 0$$ and $$B = 1$$, regardless of whether $$A = 0$$ or $$A = 1$$.

If you actually want to incorporate the knowledge that $$A$$ can be present only when $$B$$ is present into your model, to allow yourself to interpret results the way you want, you can modify the regression to something like this:

$$Y_i = \beta_0 + \beta_1A_i + \beta_2B_i + \beta_3A_iB_i + \epsilon_i$$

If you perform the same exercise as above with this equation, you'll see that different combinations of coefficients will represent the average difference in $$Y$$ between someone with the value of $$1$$ and someone with the value of $$0$$ of variable $$B$$ if they both had the value of $$0$$ for variable $$A$$ and the average difference in $$Y$$ between someone with the value of $$1$$ and someone with the value of $$0$$ of variable $$B$$ if they both had the value of $$1$$ for variable $$A$$.

• Thank you for your clear and detailed explanation, @AlexK! Now I have a better understanding of coefficients of multiple regression! May 31 '19 at 16:29