R help - Aug 2010 - one (small) sample wilcox.test confidence intervals

Dear R people,

I notice that the confidence intervals of a very  small sample (e.g. n=6)
derived from the one-sample wilcox.test are  just the maximum and minimum values
of the sample. This only occurs  when the required confidence level is higher
than 0.93. Example:

> sample <- c(1.22, 0.89, 1.14, 0.98, 1.37, 1.06)
> summary(sample)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
   0.89    1.00    1.10    1.11    1.20    1.37
>  wilcox.test(sample,conf.int=TRUE,conf.lev=0.95)
        Wilcoxon signed rank test

data:  sample
V = 21, p-value = 0.03125
alternative hypothesis: true location is not equal to 0
95 percent confidence interval:
 0.89 1.37
sample estimates:
(pseudo)median
           1.1

According  to "help", since my sample contains less than 50 values, an
exact  p-value is calculated that should enable the confidence interval to be 
obtained from Bauer (1972) (I have not read it):

<< By default (if ‘exact’ is not specified), an exact p-value is
     computed if the samples contain less than 50 finite values and
     there are no ties.  Otherwise, a normal approximation is used.
     ...........
     If exact p-values are available, an exact
     confidence interval is obtained by the algorithm described in
     Bauer (1972), and the Hodges-Lehmann estimator is employed.
     Otherwise, the returned confidence interval and point estimate are
     based on normal approximations.  These are continuity-corrected
     for the interval but _not_ the estimate (as the correction depends
     on the ‘alternative’). With small samples it may not be possible to achieve
very high
     confidence interval coverages. If this happens a warning will be
     given and an interval with lower coverage will be substituted. >>

The latter indeed happens if I ask for confidence level of 0.99:
> wilcox.test(sample,mu=0,conf.int=TRUE,conf.lev=0.99)
        Wilcoxon signed rank test

data:  sample
V = 21, p-value = 0.03125
alternative hypothesis: true location is not equal to 0
96.9 percent confidence interval:
 0.89 1.37
sample estimates:
(pseudo)median
           1.1

Warning message:
In wilcox.test.default(sample, mu = 0, conf.int = TRUE, conf.lev = 0.99) :
  Requested conf.level not achievable

My questions (finally!) are:

1.  Why the above warning for conf.lev = 0.99 does not appear for 0.93 < 
conf.lev < 0.98 although it produces the same summary?

2. For  conf.lev = 0.95, is there anything else I can do in order to obtain 
confidence intervals other than the max. and min. values of my sample  or I am
limited from my sample's size ?

Thanks for your patience in reading this,

Panos

 -------------------------------------------------------
  Dr Panos Hadjinicolaou
  
  Energy Environment&  Water Research Center (EEWRC)
  The Cyprus Institute
-------------------------------------------------------
	[[alternative HTML version deleted]]

On Aug 10, 2010, at 1:30 PM, Panos Hadjinicolaou wrote:
> My questions (finally!) are:
> 
> 1.  Why the above warning for conf.lev = 0.99 does not appear for 0.93 <
conf.lev < 0.98 although it produces the same summary?
> 
> 2. For  conf.lev = 0.95, is there anything else I can do in order to obtain
confidence intervals other than the max. and min. values of my sample  or I am
limited from my sample's size ?
> 
> Thanks for your patience in reading this,
> 
> Panos

It is as it should be. 

I think it is instructive to look at explicitly shifted samples, e.g.
> wilcox.test(sample-1.369)
	Wilcoxon signed rank test

data:  sample - 1.369 
V = 1, p-value = 0.0625
alternative hypothesis: true location is not equal to 0 
> wilcox.test(sample-1.371)
	Wilcoxon signed rank test

data:  sample - 1.371 
V = 0, p-value = 0.03125
alternative hypothesis: true location is not equal to 0 

Notice how the p-value jumps as the shift crosses 1.37. You can shift the
distribution by 1.369999 to the left and have nonsignificant test that the
center is at zero. However if you shift by more than 1.37, then you do get
significance. This is true for all significance levels between 0.03125 and
0.0625 (and 0.03125 == 1/32, the probability that all ranks have the same sign).

The above explains almost everything if you think a little about the
definitions. The only slightly puzzling thing is why confidence levels larger
than 1-0.03125 are considered achievable. The actual code has

                if (achieved.alpha - alpha > alpha/2) {
                  warning("Requested conf.level not achievable")
                  conf.level <- 1 - signif(achieved.alpha, 2)
                }

so I have to assume that the author has considered this with some care.


-- 
Peter Dalgaard
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com