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Re: st: Why t- rather than z-statistics in svylogit?


From   jpitblado@stata.com (Jeff Pitblado, StataCorp LP)
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Why t- rather than z-statistics in svylogit?
Date   Mon, 24 Nov 2003 17:14:59 -0600

Eric Neumayer <E.Neumayer@lse.ac.uk> asks why -svy- estimation commands report
-t- statistics instead of -z- statistics:

> I was surprised to find that svylogit produces t-statistics rather than
> z-statistics. Logit produces z-statistics as one would expect. Anybody knows
> the answer?

The following assumptions rely on asymptotic results similar to the central
limit theorem for samples taken from finite populations.

1. Let's assume that the distribution of the coefficient estimates (bhat) is
essentially that of the normal distribution.

2. Using the same asymptotic results, we can typically assume that the
associated variance estimates (vhat) are essentially distributed as some
constant times a chi-square random variable.

This is the basic setup for tests and confidence intervals for the mean;
however, in this case, our point estimates come from estimating equations that
sum to zero and the variance estimates are derived from the same estimation
equations.

Well, the usual test statistic is formed by:

	statistic = (bhat - b_0)/sqrt(vhat)

where b_0 is the hypothesized value (the table of results always tests
H_0: b_0 = 0, thus we will drop b_0 from the discussion).  So our assumptions
are basically,

	bhat ~ Normal(0, V)

	df*vhat/V ~ chi-square(df)

The student t distribution with df=n-1 (degrees of freedom) is derived from 

	t = Z/sqrt(X) ~ t(df)

where Z ~ Normal(0, 1), df*X ~ chi-square(df), Z and X are indenpendent.

Thus the usual test statistic is

	bhat          bhat/sqrt(V)   Z
	---------- =  ------------ ~ ------- ~ t(df)
	sqrt(vhat)    sqrt(vhat/V)   sqrt(X)

It is true that the t distribution converges to the standard normal
distribution as the number of degees of freedom increases, and they are
basically indistinguishable beyond 50 degrees of freedom.

All -svy- estimation commands work this way, the individual test statistics
are based on these principles, the details are in the functional form of the
estimating equations.

--Jeff
jpitblado@stata.com
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