I was trying to get an intuition for the interpretation of the coefficients in a logistic regression that was intended to reproduce to some extent that presented in a youtube video (http://youtu.be/vq-_4kWmzTo). So I created a fictitious data set reflecting the chances of getting accepted (Accepted: int: 0 0 ... 1 0) into a college as related to SAT scores (int 1136 1347 1504) and family/ethnic background (categories = "red" vs "blue").
fit <- glm(Accepted ~ Background - 1,data=dat, family="binomial") exp(cbind(OR = coef(fit),confint(fit)))
OR 2.5 % 97.5 % Backgroundblue 0.7088608 0.5553459 0.9017961 Backgroundred 1.7352941 1.3632702 2.2206569
The interpretation seemed easy: Red applicants have 1.7 times more chances of getting in over the rest; blue applicants were at a disadvantage, and had 7 over 10 odds of getting in.
However, the more complete model,
fit <- glm(Accepted ~ SAT.scores + Background - 1,data=dat, family="binomial") exp(cbind(OR = coef(fit),confint(fit)))
yielded coefficients for background that are difficult to reconcile or interpret:
OR 2.5 % 97.5 % SAT.scores 1.008558e+00 1.006940e+00 1.010297e+00 Backgroundblue 8.730056e-06 8.459031e-07 7.634723e-05 Backgroundred 2.329513e-05 2.426748e-06 1.929259e-04
Can you help point out what I am missing? Thank you.
Thanks to the enlightening answer from Maarten below, I was able to make some progress, and obtain the correct Odds Ratios without and with the SAT confounder:
Here is just regressing to Background ("Red" versus "Blue"):
fit <- glm(Accepted ~ Background, data = dat, family = "binomial") exp(cbind(Odds_Ratio_RedvBlue = coef(fit), confint(fit))) Odds_Ratio_RedvBlue 2.5 % 97.5 % (Intercept) 0.7088608 0.5553459 0.9017961 Backgroundred 2.4480042 1.7397640 3.4595454
Which brought up a couple of additional questions (probably very basic): 1. Why is the Odds Ratio of the (Intercept) - 0.7088608 - the same for this model as for the Odds - as opposed to Odds Ratio - of the Background Blue in the model without the intercept above? And 2. Shouldn't the OddsRatio of Blue and Red be the reciprocal of each other $OddsRatioBlue = 1 / OddsRatioRed$?