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| From | Roger Newson <r.newson@imperial.ac.uk> |
| To | "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu> |
| Subject | Re: st: Ttest and Welch's degrees of freedom |
| Date | Mon, 18 Apr 2011 13:49:22 +0100 |
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: 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 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: 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: diff0Pr(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: diff0Pr(T< t) = 0.0833 Pr(|T|> |t|) = 0.1666 Pr(T> t) = 0.9167 . 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