KolmogorovSmirnov Test

1
Kolmogorov-Smirnov Test

• Assume Xi from treatment 1 and Yj from treatment
  2 and that they have empirical cdf's
• given by
• Suppose we don't know how the differences in
  treatments might show up - shift? change in
  variance? ? . We would like an omnibus test as
  the textbook calls it, that is "designed to pick
  up differences among treatments regardless of the
  nature of the differences". We'll use the
  Kolmogorov-Smirnov test which essentially finds
  the maximum absolute difference between the two
  empirical cdf's
• Large values of D indicate the samples are
  coming from different populations - there are
  tables of D or we could do permutation tests, or
  large sample approximations as in SAS
• PROC NPAR1WAY KS CLASS class_variable
• VAR response_variable
• the EXACT KS statement may also be used see
  SAS help...
• See Table 2.8.2 on page 57 for a simple example

2

• The complete permutation distribution for the D
  statistic, assuming there is no difference in the
  population cdfs, is given in Table 2.8.3note
  that the p-value is found by computing the
  probability of getting our value of D (.4) or
  greater (13070182)/252 220/252 .873
• There is also a function in R called ks.testsee
  the help.start() menu and then try
  ks.test(before,after) to confirm the value of D
  and the p-value given in Example 2.8.4.
• below is some code for plotting the two ecdfs
  and graphically checking for the maximum
  difference...
• help(ecdf)
• help(plot.stepfun)
• plot(ecdf(trt1),col.h"red")
• plot(ecdf(trt2),addT,col.h"blue")