A Note concerning Matching by Rank
Kenneth J. Levy, Subhash C. Narula · International Statistical Review · 1977
parameters yxl, #x2, 21 , a22, and Pxxx2, all unknown. To test the null hypothesis that x, = #x2, one performs a paired t-test on the N pairs of differences (X1 X-2), i = 1, ..., N; and, when the null hypothesis is true, the appropriate test statistic is distributed as a central t with N1 degrees of freedom. For situation (c), as described by McNemar (1969), we can think of having drawn N individuals at random for one group, then forming the second group by selecting individuals who can be paired with the members of the first group on the basis of variables which need to be controlled. Hypothetically, an experimenter might be interested in comparing the effects on reading comprehension of two methods of teaching reading. If the experimenter had reason to suspect that reading comprehension scores (X) are highly correlated with IQ scores (Y), then, as described by McNemar (1969), the experimenter might draw N individuals at random from the target population under study and measure each subject's IQ. The experimenter would then form a second group by selecting subjects who are paired with members of the first group on the basis of IQ. One method of teaching reading would then be administered to the first group of subjects; the second method, to the second group of subjects. The resulting pairs of reading comprehension scores (Xi1 and X,2), i = 1, ..., N would then be analyzed by means of a paired t-test. McNemar's procedure would be very time consuming at the very best in that great costs might be incurred in attempting to select a group of subjects whose IQ scores match exactly those in the first group. An alternative procedure for forming matched pairs is as follows: The experimenter simply draws 2N individuals at random from the target population. The 2N subjects are then ranked with respect to their scores on the matching variable (e.g. IQ) and the highest ranked pair are