Hand's paradox
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In
Paradox
Comparisons of two treatments often involve comparing the responses of a
This has been called Hand's
Examples
Example 1
Label the two treatments A and B and suppose that:
Patient 1 would have responded values 2 and 3 to A and B respectively. Patient 2 would have responded values 4 and 5 to A and B respectively. Patient 3 would have responded values 6 and 1 to A and B respectively.
Then the probability that the response to A of a randomly chosen patient is greater than the response to B of a randomly chosen patient is 6/9 = 2/3. But the probability that a randomly chosen patient will have a greater response to A than B is 1/3. Thus a simple comparison of two independent groups may suggest that patients have a higher probability of doing better under A, whereas in fact patients have a higher probability of doing better under B.
Example 2
Suppose we have two random variables, and , corresponding to the effects of two treatments. If we assume that and are independent, then , suggesting that A is more likely to benefit a patient than B. In contrast, the
