R help - May 2011 - power.t.test visualization problem

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?

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

?tienne

On May 24, 2011, at 8:44 AM, stocdarf at 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.
>
> ?tienne

-- 
David Winsemius, MD
West Hartford, CT