Welcome to the CSDE Statistics Core's FAQ page. It is a collection of informal teachings developed from workshops and consulting questions. We welcome comments and contributions from readers, please send them to csde_help@u.washington.edu.
How do I interpret logistic regression coefficients?
If a coefficient is negative (the odds ratio is less than 1), then more of that covariate makes the outcome event less likely (holding the values of the other covariates constant). If a coefficient is positive (the odds ratio is more than 1), then more of that covariate makes the outcome event more likely (holding the values of the other covariates constant).
Odds ratios are used in logistic regression because they show the effect of a variable
that is independent of the values of the other covariates. They make sense in the
context of predicting disease or other rare events. However, researchers often interpret
odds ratios as if they are relative risks, that is the relative probability of the outcome
given a one unit change in the covariate of interest. This is misleading unless the event is rare
(probability of the event about .01 or smaller). You cannot determine the relative risk from the odds ratio.
You only know that the relative risk is between the odds ratio and 1 (see plot).
Probabilities (and relative risks) are a more meaningful way to present results than beta coefficients or odds ratios. This is how the probability would be calculated if there were 5 covariates.
What is the interpretation of coefficients in a linear regression equation where the dependent variable is (natural) log wages?
The regression coefficient may be interpreted as approximately equal to the proportional change in wages due to a unit change in that coefficient's variable holding all other variables constant when the regression coefficient falls in the range of ?0.1, that is, a 10% change in wages.
Why is this the case?
This is true because of properties of the exponential function.
Let:
Then suppose increases by one unit, holding all other x's constant.
Before the change in :
After the change in
The change in wages that occurs when increases by one unit, holding all other x's constant is then
Notice that is the proportional change in wages due to a one unit change in (holding all other covariates constant).
This is where the property of the exponent to the base e comes in. It can be written as an infinite series:
Therefore:
In other words for small enough, the higher order terms are negligible relative to the first term and
Graphically, this looks like this: The larger plot shows the shape of an exponential function. The smaller plot shows more detail of the range inside the black rectangle -- the range where the regression coefficient may be interpreted as approximately equal to the proportional change in wages (due to a unit change in that coefficient's variable holding all other variables constant).
: 
is the proportional change in wages due to a one unit change in (holding all other covariates constant). 
small enough, the higher order terms are negligible relative to the first term and 
