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Depending on which statistics text you read you will get 2 different formulations of the log-rank test statistic. In some texts you will see it specified as:

$$ \frac{(O-E)^2}{E} $$ Example 1

Whilst in others you will see it as $$ \frac{(O-E)^2}{V} $$

Example 1
Example 2

My questions are:

  • Do these variants have specific names?
  • Do they have any differences in properties?
  • When should one be used over the other?
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1 Answer 1

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The first one is related to the chi-squared test. The second one is the log-rank test that I am aware of. The problem with running a chi-squared test in survival statistics is that the tests at various time points are not independent (i.e. to be considered for a later time point someone must survive an earlier time point), so I would not recommend using it. In general with survival analysis use the second variant (truly I think it is the primary methodology).

It can be shown that the second variant approaches a z-distribution (N(0,1)) and a chi-squared distribution with one degree of freedom when squaring the statistic as shown here.

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  • $\begingroup$ Is there any comparative study you are aware of which demonstrates that second one is better, e.g, in terms of power and type I error? $\endgroup$
    – Statisfun
    Nov 11, 2023 at 19:27

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