Re: st: Ttest and Welch's degrees of freedom

[Roger Newson <r.newson@imperial.ac.uk>]

As far as I can see, there is no reason that the Welch degrees of 
freedom (or even the Satterthwaite degrees of freedom) shouldn't be 
greater than the homoskedastic (equal-variance) degrees of freedom, 
which is (as Garry says) n1 + n2 - 2. Of course, this is not the case 
most of the time, but (as Garry has shown) it is the case some of the time.

In statistical confidence interval formulas, the term "degrees of 
freedom" is a shorthand for "twice the inverse-squared coefficient of 
variation of the variance estimator itself", where the "variance 
estimator" is the estimated sampling variance of the estimated 
parameter. In the case of an unequal-variance t-test, the parameter of 
interest is the difference between 2 sub-population means, and its 
sampling-variance variance estimator is the square root of the sum of 
the 2 squared standard errors of the 2 sample means. This 
sampling-variance estimator is itself subject to sampling variation, 
which is why we use the t-distribution instead of the Normal 
distribution to calculate confidence limits. IF the 2 sub-population 
variances are equal, THEN you can use the equal-variance standard error 
for the difference between 2 means, which works by using the sample 
variance of the larger sample to estimate the sub-population variance of 
the smaller sample. And, IF the 2 sub-population variances are equal, 
THEN, by definition, this is a reasonable thing to do. And, IF the 2 
subpopulation variances are equal, THEN the equal-variance standard 
error for the difference between the 2 means will be subject (at least 
asymptotically) to less sampling-variation than the unequal-variance 
standard error for the difference between the 2 means, and therefore 
will be allowed to have more degrees of freedom. HOWEVER, IF the 2 
sub-population variances are unequal, THEN the equal-variance standard 
error of the difference between the 2 means will be biassed anyway, and 
may or may not be subject to less sampling variability than the 
unequal-variance standard error of the difference between the 2 means.

I hope this helps.

Best wishes

Roger


Roger B Newson BSc MSc DPhil
Lecturer in Medical Statistics
Respiratory Epidemiology and Public Health Group
National Heart and Lung Institute
Imperial College London
Royal Brompton Campus
Room 33, Emmanuel Kaye Building
1B Manresa Road
London SW3 6LR
UNITED KINGDOM
Tel: +44 (0)20 7352 8121 ext 3381
Fax: +44 (0)20 7351 8322
Email: r.newson@imperial.ac.uk
Web page: http://www.imperial.ac.uk/nhli/r.newson/
Departmental Web page:
http://www1.imperial.ac.uk/medicine/about/divisions/nhli/respiration/popgenetics/reph/

Opinions expressed are those of the author, not of the institution.

On 18/04/2011 09:37, Garry Anderson wrote:
> Dear Statalist,
>
> I was reading the -ttest- entry in the manual on page 1998 (example 3)
> and noticed that use of Welch's degrees of freedom can increase the
> degrees of freedom compared with the usual degrees of freedom obtained
> from an unpaired t-test.
>
> Should Welch's degrees of freedom be larger than n1 + n2 - 2 ?
>
> The commands and output are shown below.
>
> . use http://www.stata-press.com/data/r11/fuel3
>
> . ttest mpg, by(treated)
>
> Two-sample t test with equal variances
> ------------------------------------------------------------------------
> ------
>     Group |     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf.
> Interval]
> ---------+--------------------------------------------------------------
> ------
>         0 |      12          21    .7881701    2.730301    19.26525
> 22.73475
>         1 |      12       22.75    .9384465    3.250874    20.68449
> 24.81551
> ---------+--------------------------------------------------------------
> ------
> combined |      24      21.875    .6264476    3.068954    20.57909
> 23.17091
> ---------+--------------------------------------------------------------
> ------
>      diff |               -1.75    1.225518               -4.291568
> .7915684
> ------------------------------------------------------------------------
> ------
>      diff = mean(0) - mean(1)                                      t =
> -1.4280
> Ho: diff = 0                                     degrees of freedom =
> 22
>
>      Ha: diff<  0                 Ha: diff != 0                 Ha: diff
> > 0
>   Pr(T<  t) = 0.0837         Pr(|T|>  |t|) = 0.1673          Pr(T>  t) =
> 0.9163
>
> . ttest mpg, by(treated) welch
>
> Two-sample t test with unequal variances
> ------------------------------------------------------------------------
> ------
>     Group |     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf.
> Interval]
> ---------+--------------------------------------------------------------
> ------
>         0 |      12          21    .7881701    2.730301    19.26525
> 22.73475
>         1 |      12       22.75    .9384465    3.250874    20.68449
> 24.81551
> ---------+--------------------------------------------------------------
> ------
> combined |      24      21.875    .6264476    3.068954    20.57909
> 23.17091
> ---------+--------------------------------------------------------------
> ------
>      diff |               -1.75    1.225518                -4.28369
> .7836902
> ------------------------------------------------------------------------
> ------
>      diff = mean(0) - mean(1)                                      t =
> -1.4280
> Ho: diff = 0                             Welch's degrees of freedom =
> 23.2465
>
>      Ha: diff<  0                 Ha: diff != 0                 Ha: diff
> > 0
>   Pr(T<  t) = 0.0833         Pr(|T|>  |t|) = 0.1666          Pr(T>  t) =
> 0.9167
>
> .
>
> Kind regards, Garry
>
>
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