# The logrank test statistic is equivalent to the score of a Cox regression. Is there an advantage of using a logrank test over a Cox regression?

I have understood the logrank test as a "safe" or "conservative" way to check for a difference between two survival curves. It is "safe" in the sense that it is a nonparametric test of $$h_1(t) = h_2(t)$$, where $$h_i( \cdot)$$ is the hazard function of the $$i^{th}$$ group. I can't find sources that support this claim of "safety", so I've either invented this interpretation myself or observed it from others over the years.

I have seen sources that say that the logrank test statistic is equivalent to the score test (also called the Legrange multiplier test). I am pulling definitions from these course notes (pg. 14) for anyone who wants to look. The score of a Cox model is $$\sum_{i=1}^{n} \delta_{i}\left\{Z_{i}-E\left(Z ; T_{i}\right)\right\},$$ where $$\delta_i$$ is an indicator of whether the subject $$i$$ has an event, $$T_i$$ is the possibly-censored survival time, $$Z_i$$ is a covariate like treatment assignment, and $$E\left(Z ; T_{i}\right)$$ is the expectation of $$Z_i$$. This is equivalent (or analogous??) to the "Observed - Expected" form of the logrank test.

I have two related questions:

1. Since the logrank test and Cox regression have this equivalence, is my perception incorrect that the logrank test is "safer" than Cox regression?
2. A cox model assumes that the hazard functions are proportional with the same baseline hazard: $$h(t | Z_i) = h_0(t) * \exp(Z_i \times \beta)$$. I am not aware of any assumptions for the logrank test. My perception has always been that a logrank test statistics must be better than Cox regression in terms of power or asymptotic efficiency or something since it is not making an assumption of proportional hazards. Is this actually true?
• A good discussion, but not a complete answer to your questions, here: discourse.datamethods.org/t/… Sep 9 '20 at 18:46
• Could you please let me know where I can find the references for the presented notes? I've searched on books and articles but I haven't been able to find the sources for pages 26, 27 and 28. Thank you so in advance! Aug 11 '21 at 10:08
• @Raquel course notes like these often don't include references. The content on pages 26-28 is based on standard log-partial-likelihood formulas for a Cox model, to be found starting on page 13 of those notes and in most survival-analysis texts, and its score test. What's missing is direct comparison to the equivalent form of the usual logrank test, in the notes for lecture 3 in this course.
– EdM
Aug 11 '21 at 16:54