STT 430/530, Nonparametric Statistics

1

• pairing data values (before-after, method1 vs.
  method2 on same subjects, subject is own control,
  etc.) often reduces variability and the analysis
  reduces to single sample methods.
• but be careful about giving the same subject
  both treatments - if the treatments are drugs for
  example, what about residual effects of the first
  drug? how long does it take to completely leave
  the subject's system?
• see the example data in Table 4.1.1 on page 110
  - both methods of estimating calorie intake are
  given to 5 subjects (though not mentioned, I
  would randomize the order of the method - i.e.,
  flip a coin to see which of the 2 methods is
  asked first.)
• if there were no differences between the two
  methods, then the 2 calorie values could have
  come from either method, so the difference
  between the two could be either or - see
  Table 4.1.2 on page 111 - this lists all the
  possible /- differences in calories as estimated
  by the two methods (note there are 25 32
  (15101051) such permutations).
• now use this table to perform a permutation
  test compare our observed mean difference with
  the distribution of all possible mean differences

2

• When there are more than a few subjects, 2n
  becomes too large, so we'll do random sampling of
  signs of differences by randomly picking the sign
  (1 or -1) and multiplying it by the absolute
  value of the differences, then get the mean
  difference. Do this 5000 times and compare our
  observed mean difference to this permutation
  distribution.
• HW (due next Tuesday) Do a permutation test
  (randomly selected permutations, not all
  217131,072 possible differences in signs!) for
  the data in Table 4.1.3 on page 112. Use R as
  we've been doing. HINT See section 4.1.2 to
  guide you through the process. Look at the
  function sample(c(-1,1),1,probc(.5,.5))
• Instead of using the mean difference as the test
  statistic, we may use either S or S- , defined
  as the sum of the positive and negative
  differences, respectively.
• Since
• we see that either one of the sums may be
  used as the test statistic in place of the mean
  of the differences - you may also use one of the
  sums in your permutation test if you'd like to
• Go over the formulas in section 4.1.3

3

• Assuming no difference in treatments in this
  paired-comparison situation, lets find the means
  and variances of the permutation distributions of
  the important statistics in this context (they
  will tend to normality)
• Let Ui /- 1 with probability .5 First show
  that E(Ui ) 0 and Var(Ui ) 1
• Now
• so
• Now do similar computations of mean and variance
  for
• Thus for large samples we may use the normal
  approximations to the permutation distributions
  as in Example 4.1.2
• HW Complete reading section 4.1 well begin
  section 4.2 next time...

4

• One last note about this paired data procedure
• We may use it to test hypotheses about the median
  of a symmetric population of measurements as
  follows... If Xi the ith observation in the
  sample from this population and if M is the
  hypothesized median, then if the null hypothesis
  is true, Di Xi M is equally likely to be
  positive or negative and so the Di s are
  symmetrically distributed around 0 so our paired
  data procedure may be used instead of the
  parametric one-sample t-test for example...