Logrank test

The question: H0: F0(.) = F1(.) or S0(.) = S1(.) or h0(.) = h1(.)

Types of tests

Directional test: oriented to a specific type of difference

Omnibus test:not specific to one type of difference, tends to has some power against some or all types of difference.

How to choose from these two tests: a. prior expectations of the likely difference; b. properties of various tests for a variety of settings; c. practical consequences of a false negative result.

Basic logrank test

Construction: order distinct event times --> construct 2 X 2 table for each distinct event time (event number for one group follows hypergeometric distribution given the margins) --> calculate the expected value and variance of number of events for each event time table --> calculate the normal (or chi-squre) test statistic

The test statistic is only affected by ranks of the observed times (both censored or failure)

In constructing the test statistic, we assume the contributions from each time are independent

The construction of the test statistic is very similar to that of the chi-square test for contingency table

For 2 X 2 table, as long as the number of at risk of one group become 0, the construction of tables should stop since the observed and expected values become same.

Some extensions of the logrank test

Weighted logrank test

Stratified logrank test

Logrank test for > 2 groups

Stratify the sample by another categorical variable besides the original survival group. Compute O, E, V within each level of the stratification variable then sum up over the levels.

This method assumes values in each stratum are uncorrelated

If there are too many strata, the test could be low in power

This test also arises as a score test from Cox's model

To impose a way that emphasizes certain times more than others: impose a weight wj > 0 for each distinct event time in the calculation of test statistic

Choose larger weight for times where larger differences are anticipated. (how to define difference?)

Generalized Wilcoxon test: If choose number at risk for each time point . It places greater emphasis on early differences since number at risk decrease with time

Ordinary logrank test is a special case of weighted test when all the weights are same

similar as 2-group test, but can expressed using matrix representation

Logrank trend test

A modification from the test for > 2 groups, supposing there is an order in the survival groups, e.g. dose.

Assign a weight to each level (NOT time) and construct the TS with the weight

Using the relation of the (weighted logrank test to the Cox partial likelihood approach, one can test whether the trend is the only cause of variation between the p groups)

How to choose the weight c depends part on the setting