[R] Wilcoxon signed rank test and its requirements

[Frank E Harrell Jr]

On 06/24/2010 12:40 PM, David Winsemius wrote:
>
> On Jun 23, 2010, at 9:58 PM, Atte Tenkanen wrote:
>
>> Thanks. What I have had to ask is that
>>
>> how do you test that the data is symmetric enough?
>> If it is not, is it ok to use some data transformation?
>>
>> when it is said:
>>
>> "The Wilcoxon signed rank test does not assume that the data are
>> sampled from a Gaussian distribution. However it does assume that the
>> data are distributed symmetrically around the median. If the
>> distribution is asymmetrical, the P value will not tell you much about
>> whether the median is different than the hypothetical value."
>
> You are being misled. Simply finding a statement on a statistics
> software website, even one as reputable as Graphpad (???), does not mean
> that it is necessarily true. My understanding (confirmed reviewing
> "Nonparametric statistical methods for complete and censored data" by M.
> M. Desu, Damaraju Raghavarao, is that the Wilcoxon signed-rank test does
> not require that the underlying distributions be symmetric. The above
> quotation is highly inaccurate.
>

To add to what David and others have said, look at the kernel that the 
U-statistic associated with the WSR test uses: the indicator (0/1) of xi 
+ xj > 0.  So WSR tests H0:p=0.5 where p = the probability that the 
average of a randomly chosen pair of values is positive.  [If there are 
ties this probably needs to be worded as P[xi + xj > 0] = P[xi + xj < 
0], i neq j.

Frank

-- 
Frank E Harrell Jr   Professor and Chairman        School of Medicine
                      Department of Biostatistics   Vanderbilt University