R help - Aug 2004 - 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]]

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

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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