Clinical Trials

Clinical Trials: A Practical Guide ■❚❙❘Table 3. Calculation of chi-squared statistics using the 2 × 2 contingency table in Table 1.Death Treatment TotalActive drugPlaceboObserved numbersYes 110 165 275No 1935 1857 3792Total 2045 2022 4067Expected numbersYes 138.3 136.7 275No 1906.7 1885.3 3792Total 2045 2022 4067Hence, the further the observed values are from the expected, the larger χ 2 will be.Using the data in Table 3 and formula (4), we have:χ 2 = (110 – 138.3)2 + (1935 – 1906.7)2 + (165 – 136.7)2 (1857 – 1885.3)2+138.3 1906.7 136.7 1885.3= 12.48Under the null hypothesis of no difference in the proportion of deaths betweenthe two groups, χ 2 should follow a chi-squared distribution with 1 degree offreedom (df) (see reference [3] for more about degrees of freedom). Like thet-distribution, the shape of the chi-squared distribution depends on the number ofdegrees of freedom.Table 4 shows some selected critical values of the chi-squared distribution (χ 2 ) α,dfwith different degrees of freedom and significance levels (α). For example, whendf = 1 and α = 0.05, χ 2 = 3.84. To determine the probability that the observed0.05,1result or more extreme results would be observed if the null hypothesis were true,we need to compare χ 2 with χ 2 . For an observed α,df χ2 , P ≤ α if, and only if, χ 2 ≥ χ 2 . α,dfTherefore, using the principles described in Chapter 18, with the chi-squared test,we would have evidence against the null hypothesis in equation (3) and state thatthe treatment difference is significant at the α level if χ 2 ≥ χ 2 . α,dfFor the MI trial data, since χ 2 (12.48) > χ 2 (10.83), P < 0.001. We can0.001,1therefore say that the treatment difference in the proportion of deaths betweenthe two treatment groups is highly significant at the α = 0.1% level.223