Question 14 of our Applied Regression final exam (and solution to question 13)

Here’s question 14 of our exam:

14. You are predicting whether a student passes a class given pre-test score. The fitted model is, Pr(Pass) = logit^−1(a_j + 0.1x),
for a student in classroom j whose pre-test score is x. The pre-test scores range from 0 to 50. The a_j’s are estimated to have a normal distribution with mean 1 and standard deviation 2.

(a) Draw the fitted curve Pr(Pass) given x, for students in an average classroom.

(b) Draw the fitted curve for students in a classroom at the 25th and the 75th percentile of classrooms.

And the solution to question 13:

13. You fit a model of the form: y ∼ x + u full + (1 | group). The estimated coefficients are 2.5, 0.7, and 0.5 respectively for the intercept, x, and u full, with group and individual residual standard deviations estimated as 2.0 and 3.0 respectively. Write the above model as
y_i = a_j[i] + bx + ε_i
a_j = A + Bu_j + η_j.

(a) Give the estimates of b, A, and B together with the estimated distributions of the error terms.

(b) Ignoring uncertainty in the parameter estimates, give the predictive standard deviation for a new observation in an existing group and for a new observation in a new group.

(a) The estimates of b, A, and B are 0.7, 2.5, and 0.5, respectively, and the estimated distributions are ε ~ normal(0, 3.0) and η ~ normal(0, 2.0).

(b) 3.0 and sqrt(3.0^2 + 2.0^2) = 3.6.

Common mistakes

Almost everyone got part (a) correct, and most people got (b) also, but there was some confusion about the uncertainty for a new observation in a new group.