Here’s question 13 of our exam:
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.
And the solution to question 12:
12. In the regression above, suppose you replaced height in inches by height in centimeters. What would then be the intercept and slope of the regression? (One inch is 2.54 centimeters.)
The intercept remains the same at -21.51, and the slope is divided by 2.54, so it becomes 0.28/2.54 = 0.11: a change of one centimeter corresponds to 1/2.54 inches.
Common mistakes
About half the students go the right answer here; the others guessed in various ways: some changed the intercept as well as the slope, some multiplied by 2.54 instead of dividing. The problem’s trickier than it looks, and to make sure my answer was correct I had to think twice and make sure I wasn’t getting it backward.
I’m not quite sure how I could teach the class better so that students would get this question right. Except of course by drilling them, giving them lots of homeworks covering this material.
Reading this I thought the definition of “heavy” changed as well.
If I’m understanding question 12 correctly, one option for teaching the concept might be to do what managed to get me (barely) through physics–focus on dimensional analysis.
Every term in a regression has units. The intercept is always already in terms of [target]. Every coefficient is in terms of [target]/[feature]. For categorical targets, the units are probability of category.
When applying a unit conversion, you can only apply it to terms that already have that unit in them.
This also solves the divide vs multiply confusion. You know you have to end up with probability of heavy/cm from probability of heavy/in.
+1 to Dimensional analysis ! ! !
It’s surprising students have trouble with this! Start with height_in_cm=height_in_inches*2.54. Divide both sides by 2.54, substitute into the regression equation.
Kaiser:
I think the problem is that students don’t have good general problem-solving skills. If they can’t guess the answer, and there’s no formula, they don’t know where to start or how to check their answer.
😣