Using a Logit model to predict unknown outcome

Usually when fitting a logit or probit model to data, one has a dependent binary outcome variable (say, university attendance = [0, 1]) and the usual set of independent variables (for the sake of it, say, parent income, gender and some measure of ability).

In my case, things are a bit different. To stay within the example, for my main data set I do not know whether people attended university. The only aspects I have information about are parent income, gender and other independent variables. However, I was able to obtain data on university attendance for a random subset of the data. Using this subset I fitted a logistic model, obtaining coefficients for all independent variables.

How can I use these coefficients to predict the probability of university attendance for everyone else?

[Please note that the education example is only made to simplify things.]

• i don't believe that you can say somethng like "given that your family income is a, the probability to attend at university is b". I believe coeeficients can be interpreted as "given the income a, the probability a student attend at a university is b times higher/lower than someone else with an income c" Jan 16 '17 at 20:08
• I think that there is a group here (cross validated) where there are myuch more xperienced members who can help you with statistical inference Jan 16 '17 at 20:11
• @Yorgos: if it is a logistic model then it is multiplying the odds rather than the probability. But given the data and a decent model you can make predictions about the odds and thus the probabilities for other individuals for whom you have the independent data though not the outcome Jan 16 '17 at 21:23
• What do you mean exactly by: "I was able to obtain data on university attendance for a random subset of the data." - do you have specific attendance data for individuals, or not? How do you know that the subset is "random", and what do you mean by "random" in this case? Feb 17 '17 at 2:02

If I understood you correctly, $\Lambda(x_i'\hat\beta)$, where $\Lambda(x) = e^x / (1+e^x)$. You can use predict in both R and Stata. Try in Stata:

clear all
*** Generate data
set obs 10
set seed 1
gen x = rnormal()
gen y = 1+x+rnormal() > 0
replace y = . in 7/10
list
*** Estimate
logit y x
*** Predict
predict phat, p
list

For clarification, I presumed that your outcome variable is in $\{0,1 \}$ (not in $[0,1]$). You observe $y_i$ for some $i$ but not for all, you fitted a logit regression, and then you want to predict the probability for every $i$. Please let me know if I misinterpreted your question.