VPC in three and four levels binary response models

[IngarB]

Dear all,

I'm trying to estimat three and four levels binary response models, and would like to calculate the VPC for the different levels.
I have searched through the learning materials, manuals and some books to find out how this is done. But have only found how
I may do it for a continous response variable. For a two level binary response (logit) model we normally use a threshold model setting the level
one residual variance = 3,29. I guess this also apply for the three and four level logit models? And if so do I have to sum level 2 and 3 (and 4) residual variance when calculating the VPC for the highest level (3 or 4) ? So it this the right way to do it?:

Three level model:
VPC (for highest level =3) = level 3 residual variance / level 1 residual variance (=3.29) + level 2 residual variance + level 3 residual variance

VPC (for level =2) = lev 2 residual var + lev 3 residual var / lev 1 residual var (=3.29) + lev 2 residual var ( + lev 3 residual var ? )

Four level model:
VPC (for highest level =4) = lev 4 residual var / lev 1 residual var (=3.29) + lev 2 residual var + lev 3 residual var + lev 4 residual var

VPC (for level 3) = lev 3 residual var + lev 4 residual var / lev 1 residual var (=3.29) + lev 2 residual var + lev 3 residual var ( + lev 4 residual 4 var ? )

VPC (for level 2) = lev 2 residual var + lev 3 residual var + lev 4 residual var / lev 1 residual var (=3.29) + lev 2 residual var (+ lev 3 residual var + lev 4 residual var ? )

I hope that someone may help me with this or have any suggestions on where I may find the answer:)
Thanks!!

Best wishes,
Ingar

[Lydia]

VPCs are beautifully simple for 2 level models, but become a very complicated topic when you have more levels than that. There are many different versions depending on what you actually want from the VPC.

One approach I've seen is just to take

variance at a particular level / total variance

so for a three level model you'd get 3 VPCs:

level 1 VPC = sigma^2_e / (sigma^2_v + sigma^2_u + sigma^2_e)
level 2 VPC = sigma^2_u / (sigma^2_v + sigma^2_u + sigma^2_e)
level 3 VPC = sigma^2_v / (sigma^2_v + sigma^2_u + sigma^2_e)

You might interpret these as giving 'the percentage of variance assigned to each level', and, with a pupils within schools within areas example, the level 3 VPC is the percentage of variance due to differences between areas, or the correlation between pupils in the same area; and the level 2 VPC is the percentage of variance due to differences between schools in the same area

Another approach is to go more for the interpretations and calculate these 2 VPCs (the level 3 VPC being the same as above):

level 3 VPC = sigma^2_v / (sigma^2_v + sigma^2_u + sigma^2_e) = percentage of variance due to differences between areas, or correlation between pupils in the same area
level 2 VPC = (sigma^2_v + sigma^2_u) / (sigma^2_v + sigma^2_u + sigma^2_e) = percentage of variance due to differences between schools in different areas, or correlation between pupils in the same school

Additionally, you might be interested in:

sigma^2_u / (sigma^2_v + sigma^2_u) = correlation between schools in the same area

This can all be extended when you have 4 levels, or 5, or ...

So, for a many level model, when you talk about the VPC there isn't only one thing that you could mean, and if you want to calculate it then you need to think about which of these it is that you want.

The good news is that it barely gets any more complicated if you have a binary instead of a continuous response. As you mention below,

For a two level binary response (logit) model we normally use a threshold model setting the level
one residual variance = 3,29.

And yes indeed, if we have more levels we still set the level 1 variance to be 3.29. So your instinct was correct: once you've worked out which of the above formulae for the VPC you're interested in, simply substitute 3.29 for sigma^2_e.