From: Brian Junker <bj20@andrew.cmu.edu>
Date: Fri, 4 Feb 2022 13:43:23 -0500
Subject: Re: late hw01's
To: Siddhaarth Sarkar <siddhaas@andrew.cmu.edu>

I haven't got solutions for hw02 for you yet, but here are some general notes:

Problem #1: see my response on Piazza
(https://piazza.com/class/kykc45ull1j7dh?cid=19) for a rather complete
example of the sort of thing I am expecting (most students' answers
will not be that complete, but as long as it is clear they know what
they are doing, that's OK)

Problem #2: The four models are

y ~ 1 + x + x + u + (1 + x | County)
y ~ 1 + x + x * u + (1 + x | County)
y ~ 1 + x + x + nj + (1 + x | County)
y ~ 1 + x + x * nj + (1 + x | County)

Problem #3: literally this is just calculating -2*(logLik(M1) -
logLik(M2)) and using pchisq() with 1 df in R to calculate a p-value
and draw an inference.

Problem #4: Part (a) is a ggplot with facets similar to one of the
problems in hw01.  For part (b) the model should be something like

y_i = a_{0j[i]} + a_{1j[i]} t_i  + b_2 u_{j[i]} + \epsilon_i

a_{0j} = b_0 + \eta_{0j}

a_{1j} = b_1 + \eta_{1j}

where:

* i is the observation index number (row in the data frame)
* j[i] is the child j that observation i belongs to
* t_i is the time of a visit for child i
* y_i is sqrt(CD4PCT)
* u_{j[i]} is the treatment status (0 or 1) for the child that
  observation i belongs to

Note that the term b_2 u_{j[i]} could appear in the level 1 equation
as it does above, or it could occur in the level 2 equation for
a_{0j}.

For part (c) the model with the cross-level interaction would be the
same, but with this level 1 equation (adding in the cross-level
interaction):

y_i = a_{0j[i]} + a_{1j[i]} t_i  + b_2 u_{j[i]} + b_3 t_i u_{j[i]} + \epsilon_i

of course, the notation the students use may be slightly different...

hope this helps,

-BJ