Assignment 9 

1. Here we have before and after ("paired") data.  Under the null hypothesis,
   the before and after values within each subject are exchangeable ==> the 
   sign of the difference is equally likely to be positive or negative.  

   (a) The permutation idea corresponds to randomly assigning signs to the 
       observed differences.  You can use as a test statistic any function 
       of the differences that is relevant; in this context, presumably some
       measure of location of the differences is most appropriate.  Could be 
       the average, the median, a trimmed mean, an m-estimate, the paired-t 
       statistic, etc.  The permutation p-value you get will depend upon 
       which statistic you use -- and to a minor extent (provided you sample
       enough permutations) on the permutations you sample.  For this data,
       using the paired-t statistic as the test statistic, the value for 
       the observed data is 2.0244. My 5,000 random assignments of 
       signs led to 129 assignments for which the paired-t statistic 
       was greater than +2.0244, so the permutation p-value is 0.026.

   (b) Comparing the the paired-t statistic of 2.0244 to Student's t with 
       df = 19 leads to a p-value = 0.029.  Using the Wilcoxon signed-rank
       test statistic leads to a p-value = 0.033.  

   (c) The true permutation p-value is not much different from
       the p-value obtained from Student's t with df = 19.  As generally 
       expected, the Wilcoxon leads to a slightly larger p-value (that is,
       the assessment is not quite as sensitive) -- though not much larger.