Exercise 10.5.1 presented the following problem: Patients with painful knee osteoarthritis were randomly assigned in a clinical trial to one of five treatments: glucosamine, chondroitin, both, placebo, or Celebrex, the standard therapy. One outcome recorded was whether or not each patient experienced substantial improvement in pain or in ability to function. The data are given in the following table.

(a) Suppose we wished to compare only the success rates of each of the treatments to the control (placebo) using four separate 2 × 2 chi-square tests. The P-values for these comparisons follow. Using a Bonferroni adjustment with aew = 0.05, which treatments perform significantly different from the placebo? Indicate the value of acw used.
Treatments compared to placebo P-value
Glucosamine.................................... 0.346
Chondroitin..................................... 0.088
Both.............................................. 0.024
Celebrex.......................................... 0.007
(b) The P-value of the chi-square test that considers the entire 5 × 2 table is 0.054, which provides insufficient evidence to demonstrate any difference among the success rates of the five treatments using α = 0.05. Explain why this result does not contradict the results of part (a). [How many comparisons are being considered by this chi-square test as compared to the number of comparisons in part (a)? To achieve aew = 0.05 using a Bonferroni adjustment, how large would acw need to be? How large was it in part (a)? How does conducting many tests with a Bonferroni adjustment affect the power of each test?)

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Successful outcome Treatment Sample size Number Percentage Glucosamine Chondroitin Both Placebo Celebrex 317 318 317 313 318 192 202 208 178 214 60.6 63.5 65.6 56.9 67.3