# Re: st: Cumulative probabilities

 From Evans Jadotte To statalist@hsphsun2.harvard.edu Subject Re: st: Cumulative probabilities Date Wed, 21 Oct 2009 16:10:48 +0200

```Austin Nichols wrote:
```
```Evans Jadotte <evans.jadotte@uab.es> :
Rabe-Hesketh and Skrondal (2005: 167) are not predicting in that
equation--they are describing a model of unit-specific variance in e
with common cut points in an ordered probit framework.  Presumably,
this is not related to your desideratum.  I can't see Gunther and
Harttgen (July 2009. "Estimating Vulnerability to Idiosyncratic and
Covariate Shocks." World Development 37(7):1222-1234) so maybe you can
describe what they are actually doing.

On Wed, Oct 21, 2009 at 9:08 AM, Evans Jadotte <evans.jadotte@uab.es> wrote:
```
```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" ?

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?

Evans

```
```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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```
Yes indeed,  Rabe-Hesketh and Skrondal (2005: 167)is not directly related to my desideratum. However, the method should be standard I suppose. And what Gunther and Harttgen (2009)are doing is estimating a vulnerability index with the Pr(Yijk < z). My model is an extension of theirs though since they do not have the two random intercepts I incorporated in the numerator (in fact I believe they should have one random intercept in the numerator Phi (.) for consistency of their model. I would send the Gunther and Harttgen (2009)but I was suggested not to attach files in this thread.

Thanks,

Evans

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