# Regression interaction w/ dichotomous predictors: Two levels, two coefficients?

this question is about a way of presenting the results of an interaction I thought of, and I'm looking to see if it makes sense. If there is a better way to do what I'm trying to do (or if what I'm trying to do is not advisable), please let me know.

To simplify, let's say I've got a multiple linear regression equation with two dichotomous predictors (dummies) and an interaction between the two--let's say the DV is test score, predictor 1 is gender (M/F) and predictor 2 is course type (experimental vs. traditional). I understand that in such equations, the coefficient for each non-product term assumes the value of the other non-product term is zero. For the special case of dichotomous predictors only, this means that each coefficient assumes one of the two values of the other: so for example, the coefficient for gender applies to either the experimental or the traditional course, but not both as it would in an equation without an interaction. Which value of the other term is assumed depends on how the data are coded.

So, my question is: if gender is the focal predictor and course type is the moderator, and assuming the interaction is significant, would it be valid to run the model once with course type coded as (0=experimental, 1=traditional) and again with these values switched to (0=traditional, 1=experimental)? Then I could report two coefficients for gender, each of which corresponds to one value of course type. This would be helpful because it would demonstrate the distinct relationships of gender to test score within each course, whereas a single coefficient would describe this relationship in only one course.

From what I understand about interactions, this seems to make sense. But I have no idea whether this is something that's ever done, or if there's a better way to show the coefficients of a single dummy variable at the two levels of another. Thanks in advance!

Your linear regression model appears to be:

$E[Y|Gender,Treatment] = \beta_0 + \beta_1Male +\beta_2Treatment +\beta_3Male*Treatment$

For simplicity, Male = 1 means a male participant, Male = 0 is a female; Treatment = 1 is experimental, and Treatment = 0 is traditional.

In that case, you already have everything you need to know.

If you are interested in absolute values, you can get these as follows: The expected value of your outcome among women who receive the traditional treatment is equal to $\beta_0$. The expected value of your outcome among men who receive the traditional treatment is equal to $\beta_0 + \beta_1$. The expected value of your outcome among women who receive the experimental treatment is equal to $\beta_0 + \beta_2$. The expected value of your outcome among men who receive the experimental treatment is equal to $\beta_0 + \beta_1 + \beta_2 + \beta_3$.

If you are interested in relative values, you can get those by comparing any two of the above four absolute values. So, the ratio of the expected values of the outcome for men versus women, for all those who receive the traditional treatment, is $\frac{\beta_0 + \beta_1}{\beta_0}$. By comparison, the ratio of the expected values of the outcome for women versus men, for all those who receive the traditional treatment, is $\frac{\beta_0}{\beta_0+\beta_1}$.

EDIT: Based on your comments below, you seem to be interested in estimating the difference in test scores between the groups. To get the expected difference in test scores between any two groups, subtract the estimated average test score for those two groups: Δ=E(Y|X=x,Z=z)−E(Y|X=x′,Z=z′). So, for comparing men and women, when both receive traditional treatment, you use $(\beta_0+\beta_1) - (\beta_0) = \beta_1$. Therefore, $\beta_1$ can be interpreted as the expected difference in test scores between men and women, among those receiving the traditional treatment. Similarly, for experimental versus traditional, among women, you compare $(\beta_0 + \beta_2) - (\beta_0) = \beta_2$. So, again, $\beta_2$ can be interpreted as the expected difference in test scores between experimental and traditional treatment, among women.

If you use the same process, you can also get the expected difference in test scores between experimental and traditional treatment, among men: $(\beta_0 + \beta_1 + \beta_2 + \beta_3) - (\beta_0 + \beta_1) = \beta_2 + \beta_3$. Similarly, the expected difference between men and women, among those with the experimental treatment, is $(\beta_0 + \beta_1 + \beta_2 + \beta_3) - (\beta_0 + \beta_2) = \beta_1 + \beta_3$.

You could get these last two by recoding and re-running your regression equation, but there's no need to since you already have all the information you need in your current regression model.

• Hi, thanks for your answer. In Jaccard & Turrisi's book Interaction Effects in Multiple Regression, they state that in interaction models (using a 2-term + interaction model as an example), "The coefficient for X estimates the effect of X on Y when Z is at a specific value, namely, when Z = 0. The coefficient for Z estimates the effect of Z on Y when X is at a specific value, namely, when X = 0." So then wouldn't β1 be the coefficient for gender when treatment = traditional, and β2 the coef for treatment when gender = female? Or is that not what they mean? – intermediary Jun 8 '13 at 2:00
• Yes, the coefficients can be interpreted in that way. For the outcome test score, the expected difference in test scores between men and women, when everyone receives the traditional treatment is: $(\beta_0 + \beta_1) - (\beta_0) = \beta_1$. A similar calculation can be made for the expected difference in test scores between any of the four groups by subtracting the appropriate terms. – Ellie Jun 8 '13 at 23:09
• In your equation above, the β0s just cancel out. Is that what you meant to write? – intermediary Jun 9 '13 at 15:00
• Yes. I wrote it that way to show you how to apply this to a contrast of any two groups. To get the expected difference in test scores between any two groups, subtract the estimated average test score for those two groups: $\Delta = E(Y|X=x,Z=z) - E(Y|X=x',Z=z')$. For comparing men and women, when both receive traditional treatment, this ends up being $\beta_1$; similarly, for experimental versus traditional, among women, this ends up being $\beta_2$. But you can get any comparison of groups using the information you get from the regression equation you have, without needing to recode it. – Ellie Jun 9 '13 at 16:06