Coefficient interpretation- Dependent and independent variable both binary

[Nicole Kashani]

Hi all,

I have run a regression which has passing as an outcome (=1 if passes the exam), and being in a small class as the independent variable (=1 if the student was in the small class).

I get the coefficient 0.0567 with the standard error (0.0219). Knowing that both the outcome and the explanatory variable are binary, how do I interpret this coefficient?
Is it a percentage point? Percentage? Should I multiply it by 100? Is it that being in the small class increases (let's say this relationship is causal) the probability of passing the exam by 0.0567 percentage points? What does it even mean?

[Clyde Schechter]

You don't provide enough information to answer the question. Was this a linear regresion, or did you use a logistic, or probit, or Poisson regression? (Or maybe even some other more exotic possibilities.)

[Nicole Kashani]

You don't provide enough information to answer the question. Was this a linear regresion, or did you use a logistic, or probit, or Poisson regression? (Or maybe even some other more exotic possibilities.)

Sorry, you are right. It is a linear specification.

[Clyde Schechter]

Thank you. So, in a linear regression, the coefficient of one of the explanatory (right hand side) variables represents the expected difference in the outcome (left hand side) variable associated with a 1 unit change in the explanatory variable. In your specific case, this means that, all else equal (i.e. adjusting for the other variables in the regression, if any), the expected probability of passing is 0.0567 higher for a student from a small class than for a student from a non-small class. 0.0567 is a difference in probabilities. It can be converted, if you wish, to a difference in percentages. A difference in percentages is denominated in percentage points, not percents. So your result could be rephrased to say that the chance of passing the test is 5.67 percentage points higher among those from small classes than among those from non-small classes.

When presenting this, whether orally or in writing, it is usually a good idea to also provide some measure of the uncertainty around this estimate. One simple way to do that is to simply also report the standard error, as you did in #1. However, I think that most people feel more comfortable in their intuitions about confidence intervals than in their intuitions about standard errors. So I generally recommend reporting a confidence interval as the clearest way to do that.