# Comparison of weights of log-rank test in survival analysis

This page explains the difference in the types of weight in log rank test in survival analysis.

We use, as a test statistic,

$$Z_W = \frac{\sum_{k=1}^D W(t_k) \left( dN_1 (t_k) - Y_1(t_k) dN(t_k)/Y(t_k) \right)}{\left[ \sum_{k=1}^D W(t_k)^2 Y_1(t_k) \frac{dN(t_k)}{Y(t_k)} \frac{Y(t_k) - dN(t_k)}{Y(t_k)} \frac{Y_0(t_k)}{Y(t_k) - 1} \right]^{1/2}}$$

where $$t_k$$ : is the times of the event, $$Y_i(\cdot)$$ : number at risk at specific time for group $$i=0,1$$,$$D$$ : number of events, $$dN_i(\cdot)$$ : number of events at specific time of group $$i$$, and $$W(\cdot)$$ : any predictable process which serves as a "weight".

The above page explains that Gehan's weight, where $$W(t) = Y(t)$$ is used if one is interested in early differences in survival times. It's understandable, since early times have larger $$Y(t)$$. But I don't understand why Peto-Peto is usefule for situations where many observations are censored.