Statistical Methods in Medical Research 4ed

284 Analysing non-normal data
High protein
Low protein 83 97 104 107 113 119 123 124 129 134 146 161
70 13 27 34 37 43 49 53 54 59 64 76 91
85 2 12 19 22 28 34 38 39 44 49 61 76
94 11 3 10 13 19 25 29 30 35 40 52 67
101 18 4 3 6 12 18 22 23 28 33 45 60
107 24 10 3 0 6 12 16 17 22 27 39 54
118 35 21 14 11 5 1 5 6 11 16 28 43
132 49 35 28 25 19 13 9 8 3 2 14 29
For values outside the range of Table A7 an approximation to the number of
differences to be excluded from each end of the ordered set is given by the integer
part of
1
2 n1n2 z
r
n1n2…n1 ‡ n2 ‡ 1†
12
The method of estimation described above could be applied to the data of
Example 10.3. The 95% confidence limits are calculated as 66 and 2, in the
units of percentage change used in Table 10.2. However, the method is inappropriate
here, since the hypothesis of a constant displacement in the distributions
is quite unrealistic in view of the bunching of observations at the lower
bound of 100%. Other parameters need to be used to describe the difference
between the groups. For instance, one might report the difference in the proportions
of observations at the lower bound, or less than 90%. An alternative
approach, which is available for any application of the Wilcoxon/Mann±Whitney
test, is to note that the statistic UXY =n1n2 is clearly an estimate of the
probability that a randomly chosen value of x is less than a randomly chosen
value of y. However, the variance of this statistic is difficult to evaluate since it
depends on the precise way in which the two distributions differ; it is important
to realize that the usual binomial variance for a proportion with n1n2 observations
is wholly inappropriate here since the n1n2 differences are not independent.
Normal scores
An alternative approach to the two-sample distribution-free problem is provided
by the Fisher±Yates normal scores. Instead of using ranks, the observations are
transformed to a different set of scores, which depend purely on the ranks in the
combined sample of size n. The score for the observation of rank number r is, in
fact, numerically equal to the mean value of the rth smallest observation in a
sample of n from a standardized normal distribution, N(0, 1). The scores are
tabulated for various sample sizes by Fisher and Yates (1963, Table XX).