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# RE: st: Ttest and Welch's degrees of freedom

 From Garry Anderson <[email protected]> To <[email protected]> Subject RE: st: Ttest and Welch's degrees of freedom Date Tue, 19 Apr 2011 18:29:43 +1000

```Thank you Roger for your explanation.

You mention you can see no reason why even the Satterthwaite degrees of
freedom should not be greater than (n1 + n2 - 2). However, according to
the Satterthwaite formula on page 2002 of the reference manual (3rd
equation on the page) any combination of variance and sample size for
the two samples does not seem to provide a degrees of freedom that is
greater than n1 + n2 - 2.

If I use Welch's formula on page 2002 with data
SD1 = 10
SD2 = 10
n1 = 11
n2 = 11

Welch's df = 22
This is larger than the equal-variance df of 20
This seems to provide a bonus of 2 degrees of freedom.
Satterthwaite's df = 20

. ttesti 11 10 10 11 10 10,welch

Two-sample t test with unequal variances
------------------------------------------------------------------------
------
|     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf.
Interval]
---------+--------------------------------------------------------------
------
x |      11          10    3.015113          10    3.281909
16.71809
y |      11          10    3.015113          10    3.281909
16.71809
---------+--------------------------------------------------------------
------
combined |      22          10    2.080626    9.759001    5.673101
14.3269
---------+--------------------------------------------------------------
------
diff |                   0    4.264014               -8.843024
8.843024
------------------------------------------------------------------------
------
diff = mean(x) - mean(y)                                      t =
0.0000
Ho: diff = 0                             Welch's degrees of freedom =
22

Ha: diff < 0                 Ha: diff != 0                 Ha: diff
> 0
Pr(T < t) = 0.5000         Pr(|T| > |t|) = 1.0000          Pr(T > t) =
0.5000

Kind regards, Garry

-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of Roger Newson
Sent: Monday, 18 April 2011 10:49 PM
To: [email protected]
Subject: Re: st: Ttest and Welch's degrees of freedom

Sorry, the sentence:

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.

should have been:

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 estimator is the sum of  > the 2 squared standard
errors of the 2 sample means.

Sorry for any inconvenience caused.

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: [email protected]
Web page: http://www.imperial.ac.uk/nhli/r.newson/
Departmental Web page:
http://www1.imperial.ac.uk/medicine/about/divisions/nhli/respiration/pop
genetics/reph/

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

On 18/04/2011 12:49, Roger Newson wrote:
> 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: [email protected]
> Web page: http://www.imperial.ac.uk/nhli/r.newson/
> Departmental Web page:
> http://www1.imperial.ac.uk/medicine/about/divisions/nhli/respiration/p
> opgenetics/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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