Re: [R] sample size for t-tests
Look into the code of power.t.test in the stats package. For example,
the sample size for two-sample t-test, two-tail testing and strict
interpretation of tail probability can be found by solving the following
equation iteratively :
\begin{equation}
1 - \beta = \Pr ( t_{v,ncp}^{*} t_{v, \alpha/2 } )
+ \Pr ( t_{v,ncp}^{*} t_{v, 1 - \alpha/2 } )
\end{equation}
where :
1) $t_{v, \alpha/2}$ is the $\alpha/2$ quantile of a central
t-distribution with $v$ degrees of freedom and $v = n1 + n2 - 2$
2) $t_{v,ncp}^{*}$ follows a non-central t-distribution $v$ degrees of
freedom and non-centrality parameter of $ncp$
3) non-centralitity parameter in 2) estimated by
$ncp = \delta / ( \sigma \sqrt{ \frac{1}{n_1} + \frac{1}{n_2} } )$
As usual, $alpha$ and $beta$ represent type I and type II error and
\delta, \sigma represents the desired difference in group means and
variance.
This would be explained in a textbook but none comes to my mind at the
moment. There is an approximate analytical solution based on normality
assumption but the results are very close for large sample sizes. It is
better to use the exact equation as the computations (via uniroot) is
fast anyway.
Regards, Adai
On Tue, 2004-08-31 at 18:49, Caimiao Wei wrote:
Dear all,
Could any one please tell me the exact formula R uses to calculate the sample size
for one-sample and two-sample t-tests? Thanks,
Caimiao
[[alternative HTML version deleted]]
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Re: [R] sample size for t-tests
Adaikalavan Ramasamy [EMAIL PROTECTED] writes:
Look into the code of power.t.test in the stats package. For example,
the sample size for two-sample t-test, two-tail testing and strict
interpretation of tail probability can be found by solving the following
equation iteratively :
\begin{equation}
1 - \beta = \Pr ( t_{v,ncp}^{*} t_{v, \alpha/2 } )
+ \Pr ( t_{v,ncp}^{*} t_{v, 1 - \alpha/2 } )
\end{equation}
where :
1) $t_{v, \alpha/2}$ is the $\alpha/2$ quantile of a central
t-distribution with $v$ degrees of freedom and $v = n1 + n2 - 2$
2) $t_{v,ncp}^{*}$ follows a non-central t-distribution $v$ degrees of
freedom and non-centrality parameter of $ncp$
3) non-centralitity parameter in 2) estimated by
$ncp = \delta / ( \sigma \sqrt{ \frac{1}{n_1} + \frac{1}{n_2} } )$
As usual, $alpha$ and $beta$ represent type I and type II error and
\delta, \sigma represents the desired difference in group means and
variance.
This would be explained in a textbook but none comes to my mind at the
moment. There is an approximate analytical solution based on normality
assumption but the results are very close for large sample sizes. It is
better to use the exact equation as the computations (via uniroot) is
fast anyway.
Yep, just a few nits/qualifications:
- sigma is the standard deviation, not variance
- n1 and n2 are generally set equal in sample size calculations since
that is known to be optimal (largest power for given n1+n2)
- all the solutions are based on normality assumptions, the analytical
formulas come out of assuming the variance known in advance (so that
you can use the normal distribution instead of the t distribution).
These ignore the uncertainty from having to estimate the variance
and so give somewhat too small sample sizes.
--
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c/ /'_ --- Dept. of Biostatistics 2200 Cph. N
(*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918
~~ - ([EMAIL PROTECTED]) FAX: (+45) 35327907
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[R] sample size for t-tests
Dear all, Could any one please tell me the exact formula R uses to calculate the sample size for one-sample and two-sample t-tests? Thanks, Caimiao [[alternative HTML version deleted]] __ [EMAIL PROTECTED] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
Re: [R] sample size for t-tests
On Tue, 31 Aug 2004, Caimiao Wei wrote: Dear all, Could any one please tell me the exact formula R uses to calculate the sample size for one-sample and two-sample t-tests? Thanks, There isn't a formula in closed form. The exact procedure is in the code. In words: the critical value for the t-test comes from a t-distribution. The distribution of the t-statistic under the alternative is a non-central t-distribution, and the sample size is adjusted to get the requested power. -thomas __ [EMAIL PROTECTED] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
