So I thought I had a handle on interpreting exponentiated coefficients from logistic regression in terms of odds and odds ratios, but I'm having more difficulty than I expected.
I'm playing with some example data I generated in R, in about the simplest possible situation. I made 12 observations of two binary predictors and a binary response:
df <- data.frame( x1 = factor(c("High","High","Low","Low","Low","Low","High","High","High","Low", "Low", "High"), levels = c("Low", "High")), x2 = factor(c("No","No","No","No","No","Yes","Yes","Yes","Yes","Yes", "Yes", "No")), y = c(0,1,1,0,1,0,1,0,1,1,1,0))
If I use just one predictor, things work like I expect.
xtabs(~ y + x1, df)
mod1 <- glm(data = df, y ~ x1, "binomial") exp(mod1$coefficients)
I can interpret the exponentiated intercept of 2 as the odds of
y = 1 in the reference level of
x1 (4/2 = 2), and the exponentiated coefficient 0.5 of
x1High as the odds ratio (3/3)/(4/2) = 0.5. So far so good, and I can get the same sort of intelligible results if I only use
x2 as a predictor, though I omit those results in the interest of space.
But when I put both
x2 in the model it all goes to hell, and I realize that I must not know what's going on:
xtabs(~ y + x1 + x2, df)
|x2 = No|
|x2 = Yes|
mod3 <- glm(data = df, y ~ x1 + x2, "binomial") exp(mod3$coefficients)
Now I'm clearly off on the wrong foot, because I naively suppose that the exponentiated intercept should correspond to the odds of
y=1 when the predictors are both at their reference levels -- when
x1 = Low and
x2 = No. But looking at the relevant cells for that condition in the crosstabs, it would appear the odds in that condition are 2/1 = 2, not 1.4294. Changing data the response values of data points that aren't in the reference level of either category also changes the fitted intercept, so that whole interpretation seems straight out the window. And I also can't figure out any way of working with odds ratios that yield either of the coefficients on
I'd appreciate any pointers about how I can relate any odds or odds ratios that could be manually computed from the above cross-tab to any of these exponentiated logistic regression coefficients. I started from a more complex applied problem and reduced it as far as I could to what's shown here, and I'm temporarily stuck at this point, so I thought I'd ask for some help getting unstuck.
P.S. I did check that my
contrasts option for unordered factors is set to the default
contr.treatment, like I expected.