# Re: st: Cumulative probabilities

 From Austin Nichols To statalist@hsphsun2.harvard.edu Subject Re: st: Cumulative probabilities Date Wed, 21 Oct 2009 09:44:17 -0400

```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:
>>
>> 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?
>>>
>>>
>>> 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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