Prediction in logistic regression with prediction criteria ranges

Question

I’m not sure how to best explain my problem but I’ll try. I can’t be too specific because this is a homework assignment; I just would like some guidance from the experts on how to approach it.

So I’ve fit a logistic regression model. Now I am trying to predict some response based on some values for a bunch of predictors. Two of those predictors are Income and Rank. Income (in \$) is coded by {1 = less than 5000, 2 = 5000 to 9999, 3 = 10000 to 14999, 4 = 15000 to 19999, etc. up to 15} and Rank is coded by {5 = top 10%, 4 = next 10%, 3 = next 20%, 2 = next 20%, bottom = last 40%}. How do I go about prediction if I want to predict the probability of the response based on (some concrete values for other predictors), an income of less than $10000 and a rank in the top 15%?

I’m not sure how to really predict when I have to combine multiple categories of a predictor and also split between categories if that makes sense? For Rank, I’m almost thinking about using Rank = 4.5 for prediction but that doesn’t seem right.

Thanks for all the help!

Answer 1

If Rank and Income are independent, you can do as follows:

Top 15% observations have a 1 in 3 chance of being in the top 20% but not in the top 10% and a 2 in 3 chance of actually being in the top 10, so you can compute both predictions and calculate expectation taking that into account.

The "less than $10000" is though harder to treat. I would estimate the probability of it being in any of the possible ranges (1 or 2) according to the observed proportion in the sample you used to estimate the model. Now multiply the probabilities together to get the chance of each observation being in each of the four goups.

If the variables are not independent, use the sample to calculate the chance of the four different (Rank-Income) combinations of interest.