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Re: st: Query regarding formula for calculating the conventional variance for the xtgee command


From   clindsey@stata.com (Charles Lindsey, StataCorp)
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Query regarding formula for calculating the conventional variance for the xtgee command
Date   Tue, 20 Mar 2012 14:38:43 -0500

Rachael Hughes (rachael.hughes@bristol.ac.uk) asked about the formula used to
compute the conventional variance-covariance estimator in -xtgee-:

> I have a query regarding the formula for the calculation of the
> conventional variance when the model is the generalized estimating
> equations logistic regression model with an exchange working correlation
> matrix, i.e. Stata command
>
> xtgee outcome covariates, family(binomial) link(logit) corr(exchangeable) 
>   vce(conventional) nmp
>
> In keeping with Liang and Zeger's 1986 Biometrika paper I am using Stata's
> vce option nmp.
>
> Dropping the cluster subscripts, the Stata documentation for the xtgee
> command states that the conventional variance is calculated as
> {sum(D'V^{-1}D)}^-1, where V = A^{1/2}R(alpha)A^{1/2}. Using this formula I
> obtain the same result as Stata.
>
> My query is that in Liang and Zeger's 1986 paper
> V = A^{1/2}R(alpha)A^{1/2}/phi.
> 
> Note that the estimate of phi (the scale parameter) is close to 1 but not
> exactly 1. I have confirmed that my estimate for phi is the same as Stata's
> by confirming I get the same answer for the parameter estimate alpha, which
> is dependent upon phi. (alpha is the parameter of the exchangeable working 
> correlation matrix).
> 
> Could you please confirm if I am correct that Stata's xtgee command 
> calculates V different to the formula stated in the Liang and Zeger 1986 
> paper?
> 
> References
> Liang and Zeger, Longitudinal data analysis using generalized linear 
> models, Biometrika, 1986, 73,13-22.

In the Methods and Formulas section of the -xtgee- manual entry, there is an
unfortunate mix of notation from McCullagh and Nelder (1989) and Liang and
Zeger (1986) which led to the typo that Rachael discovered in the definition
of V on p. 136.  The definition is missing a scale parameter phi.  We will
update the documentation to fix this.

The -xtgee- command does properly account for the scale parameter when
computing the conventional variance-covariance estimates.

References:

Liang and Zeger, Longitudinal data analysis using generalized linear
models, Biometrika, 1986, 73,13-22.

McCullagh, P., and J. A. Nelder. 1989. Generalized Linear Models.
2nd ed. London: Chapman & Hall/CRC.


-- Charles Lindsey
clindsey@stata.com
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