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Re: st: Cumulative probabilities


From   Evans Jadotte <evans.jadotte@uab.es>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Cumulative probabilities
Date   Wed, 21 Oct 2009 15:08:42 +0200

Austin Nichols wrote:
Evans Jadotte <evans.jadotte@uab.es> :
What's z in (z-xb-...) below? If you are calculating an estimate of e
in the numerator, and dividing by the estimate of the SD of e, then
you are calculating the Z score of the idiosyncratic error, and
Phi(Z).  What is this for?  Can you provide refs for what "some books
suggest" ?

On Tue, Oct 20, 2009 at 11:16 AM, Evans Jadotte <evans.jadotte@uab.es> wrote:
Hello listers,

Sorry for sending this message again but I realized some characters did not
appear too well.

I am estimating cumulative probabilities of the following function:

Yijk = b0 +b1Xijk + eijk + u.jk + u..k



where u.jk  and u..k  are two random intercepts with variance Sigma^2 (u.jk)
and Sigma^2 (u..k). The variance of my raw residuals is Sigma^2 (eijk).  The
cumulative probabilities I want to calculate are of the form:

Phi((z-xb-uhat.jk - uhat../k/)/sqrt(?))

where Phi denotes the standard normal cumulative density. My question is:
should the square root, sqrt, in the denominator contain just the variance
of the raw residuals, i.e. Sigma^2 (eijk), as some books suggest? Or should
it bear, according to my logic, the total variance of the model, which would
be the sum Sigma^2 (e ijk) + Sigma^2 (u.jk) + Sigma^2 (u..k)? And finally,
what would be the statistics rationale for using the former  instead of the
latter formula?

Thanks in advance,

Evans

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Hi Austin,

z is a threshold (e.g. a deprivation line) and xb are the fitted values (yhat) of the fixed part of the estimation. The Phi is to calculate the cumulative probabilities of the function:

Pr(Yijk < z) = Phi((z-xb-uhat.jk - uhat..k)/sqrt(?))

For instance, in their book "Multilevel and Longitudinal Modelling Using Stata", Rabe-Hesketh and Skrondal (2005: 167), section 5.11, use only the SD of e in the denominator, other papers adopt the same stance (e.g. "Estimating Vulnerability to Idiosyncratic and Covariate Shocks": Gunther and Harttgen (2009)). I am trying to understand the statistics rationale for not accounting for the variances of the random intercepts Sigma^2 (u.jk) and Sigma^2 (u..k) in the denominator.

Thanks!

Evans
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