On May 24, 2011, at 8:44 AM, stocd...@mail.tu-berlin.de wrote:

Dear R-User,

I'm trying to visualize the results of the power calculation with the function power.t.test(). Therefore I want to plot the related t- distributions and shade the surfaces indicatingt the type I error, the type II error and the power. For sample sizes greater 30 I got results which are very satisfying. For small sample sizes I got stuck and did'nt find a mistake.
To show you the problem I wrote some lines in R:

par(mfrow = c(4,2))
for(n in c(2,6,10,14,18,22,26,30))
{
temp = power.t.test(n = n, sd = 1, power = 0.5, sig.level = 0.05, # power calculation --> power is specified with 50%
                   type = "one.sample", alternative = "one.sided")
s = temp $ sd # get standard deviation out of test distribution n = temp $ n # get sample size out of test distribution delta = temp $ delta # get delta (distance between centrality points) out of test ditribution

plot(1:10, xlim = c(-5,10), ylim = c(0, 0.5), type = "n") # create plot window

x = seq(-5, 10, length = 400 ) # x -values y1 = dt(x, df = n -1 ) # y -values calculated with related t-distribution (df=n-1)

lines(x, y1, col = 2 ) # plot related t-distribution lines(x + delta/(s/sqrt(n)), y1 ) # plot related t-distribution shifted with normalized delta

abline(v = qt(0.95, df = n -1 )) # draws a vertical line at the
abline(v=delta/(s/sqrt(n)),lty=2)

legend("topright",legend=c(paste("n=",n),
paste("bias=",round(qt(0.95, df = n-1)-delta/(s/sqrt(n)), 2)))) # creates legend
}

This code creates some plots with different sample size n. I would expect the solid vertical line and the dotted vertical line one above the other. But indeed the space between both of them is increasing with a decreasing sample size. Whe re is my mistake? Is it a error in reasoning or is it "just" not possible to visualize this problem for small sample sizes?

You do realize that you should be using the non-central t distribution when considering the alternate hypothesis, right? By the time you get down to sample sizes of 6 there should be a visible skew to the distribution of "observed" differences.



I look foward to any suggestions and hints. So much thanks in advance.

Étienne


--
David Winsemius, MD
West Hartford, CT

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