Methods to evaluate the relationship between survival times

For two groups of patients selected by the value of a prognostic factor from a larger set, the common practice is to evaluate the difference by the logrank test with some variants. Now if we have a set of indexed rules that link the survival times of the two groups it is natural to choose the rules which minimize the logrank test. To find this minimum is a difficult task in the general case because the functions are not analytical ones. Our strategy is to transform the observations of one group by a set of predefined indexed rules to identify at least one rule that minimize the log rank test. If T is survival time for one group, let say basic group, we solved the problem for the sets of rules as {aT|a real value}, {a + T|a real value}and {a + bT|a, b real value}. Mathematical foundations for an algorithm and a generalization for {ag(T)|a real value}, {a + g(T)|a real value} and {a + bg(T)|a, b real value} with g(.) an increasing function are presented. For breast cancer, some examples solved by Mathematica programs are presented.