{"id":2439,"date":"2013-04-05T18:19:04","date_gmt":"2013-04-05T09:19:04","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=2439"},"modified":"2014-07-25T14:49:37","modified_gmt":"2014-07-25T05:49:37","slug":"how-to-calculate-appropriate-sample-size-in-cox-proportional-hazard-analysis-with-cross-tabulation","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=2439","title":{"rendered":"How to calculate appropriate sample size in Cox proportional hazard analysis with cross tabulation?"},"content":{"rendered":"

In this article, I’d like to describe how to calculate appropriate sample size in Cox proportional analysis with cross tabulation, a<\/span> error and b<\/span> error. a<\/span> error is called as statistical significance or type 1 error and b<\/span> error is called as type 2 error, respectively. 1 – b<\/span> is called as statistical power. a<\/span> is usually configured at 0.05 (two-tailed) and b<\/span> is configured at 0.2 (one-sided), respectively. As a result, Za<\/span>\/2<\/sub> is 1.96 and Zb<\/span><\/sub> is 0.84, respectively.<\/p>\n

I’d like to assume that S1<\/sub> is survival rate of risk group or intervention group and S0<\/sub> is survival rate of control group, without risk or intervention. q<\/span> is ratio of logarithm of them.<\/p>\n

\\displaystyle LN(S_1) = Exp(B)LN(S_0)\\vspace{0.1in}\\\\\\theta = Exp(B) = \\frac{LN(S_1)}{LN(S_0)}<\/p>\n<\/p>\n

I’d like to use cross tabulation here. You can replace endpoint with death or failure.<\/p>\n\n\n\n\n\n\n
<\/td>\nENDPOINT<\/span><\/strong><\/td>\nCENSOR<\/span><\/strong><\/td>\nMarginal total<\/span><\/td>\n<\/tr>\n
POSITIVE<\/span><\/strong><\/td>\na<\/span><\/td>\nb<\/span><\/span><\/td>\na + b<\/span><\/td>\n<\/tr>\n
NEGATIVE<\/span><\/strong><\/td>\nc<\/span><\/td>\nd<\/span><\/td>\nc + d<\/span><\/td>\n<\/tr>\n
Marginal total<\/span><\/td>\na + c<\/span><\/td>\nb + d<\/span><\/td>\nN<\/span> <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\\displaystyle S_1 = \\frac{b}{a+b}\\vspace{0.1in}\\\\S_0 = \\frac{d}{c+d}<\/p>\n<\/p>\n

You can calculate estimated number of death (e) in both group as following formula by Freedman’s approximate calculation.<\/p>\n

\\displaystyle e = \\left(\\frac{\\theta+1}{\\theta-1}\\right)^2(Z_{\\alpha\/2}+Z_\\beta)^2<\/p>\n<\/p>\n

You can calculate entry size (n) in each group, as following formula.<\/p>\n

\\displaystyle e = n(1-S_0)+n(1-S_1)\\vspace{0.1in}\\\\n = \\frac{e}{2 - S_0 - S_1}<\/p>\n<\/p>\n

You have to correct entry size with drop-out rate (w) as following formula. Throughout trial, two times of n is needed.<\/p>\n

\\displaystyle n = \\frac{e}{(2 - S_0 - S_1)(1-w)}<\/p>\n<\/p>\n