Clinical Trials

Clinical Trials: A Practical Guide ■❚❙❘Step 2:Determine the number of individuals at risk and the number of events ateach time in each group, as well as in the two groups combined. At eachtime t j(t 1≤ t j≤ t r), we assume there are n 1j, n 2j, and n jindividuals at riskof the event and d 1j, d 2j, and d jindividuals who have had an event ingroups 1, 2, and the two groups combined, respectively.Step 3: Calculate the expected number of events and the variance of v 1jin group 1at each time t j. The expected number of events is given by:e 1j= n 1j– d jn jand the variance of v 1jis given by:v 1j= n 1jn 2jd j(n j– d j)n 2 j(n j– 1)Step 4:Calculate the log-rank statistic using the following formula:rU L= Σ (d 1j– e 1j)j=1rV L= Σ (v 1j)j=1χ 2 = U2 LV Lwhere χ 2 is a chi-squared statistic that follows chi-squared distribution with onedegree of freedom (see Chapter 20 for more about the chi-squared [χ 2 ] test).The detailed calculation for the pancreatic cancer trial data is displayed in Table 4:U L= –4.4776, V L= 3.6967, so:χ 2 = (–4.4776)2 = 5.423.6967This χ 2 value is converted to a P-value of 0.020. As P < 0.05, the log-rank test hasshown a significant survival difference between the new treatment A and standardtreatment B. This test readily generalizes to three or more groups, with the nullhypothesis that all groups have the same survival function. If the null hypothesisis true, the test statistic has a chi-squared distribution with the degrees of freedomequal to the number of groups minus 1.243