Definitions

# Logrank test

In statistics, the logrank test (sometimes called the Mantel-Cox test) is a hypothesis test to compare the survival distributions of two samples. It is a nonparametric test and appropriate to use when the data are right censored (technically, the censoring must be non-informative). It is widely used in clinical trials to establish the efficacy of new drugs compared to a control group (often a placebo) when the measurement is the time to event (such as a heart attack).

The test was first proposed by Nathan Mantel and was named the logrank test by Richard and Julian Peto.

## Definition

The logrank test statistic compares estimates of the hazard functions of the two groups at each observed event time. It is constructed by computing the observed and expected number of events in one of the groups at each observed event time and then adding these to obtain an overall summary across all time points where there is an event.

Let j = 1, ..., J be the distinct times of observed events in either group. For each time $j$, let $N_\left\{1j\right\}$ and $N_\left\{2j\right\}$ be the number of subjects "at risk" (have not yet had an event or been censored) at the start of period $j$ in the groups respectively. Let $N_j = N_\left\{1j\right\} + N_\left\{2j\right\}$. Let $O_\left\{1j\right\}$ and $O_\left\{2j\right\}$ be the observed number of events in the groups respectively at time $j$, and define $O_j = O_\left\{1j\right\} + O_\left\{2j\right\}$.

Given that $O_j$ events happened across both groups at time $j$, under the null hypothesis $O_\left\{1j\right\}$ has the hypergeometric distribution with parameters $N_j$, $N_\left\{1j\right\}$, and $O_j$. This distribution has expected value $E_j = O_jfrac\left\{N_\left\{1j\right\}\right\}\left\{N_j\right\}$ and variance $V_j = frac\left\{O_j \left(N_\left\{1j\right\}/N_j\right) \left(1 - N_\left\{1j\right\}/N_j\right) \left(N_j - O_j\right)\right\}\left\{N_j - 1\right\}$.

The logrank statistic compares each $O_\left\{1j\right\}$ to its expectation $E_j$ under the null hypothesis and is defined as

$Z = frac \left\{sum_\left\{j=1\right\}^J \left(O_\left\{1j\right\} - E_j\right)\right\} \left\{sqrt \left\{sum_\left\{j=1\right\}^J V_j\right\}\right\}.$

## Asymptotic distribution

If the two groups have the same survival function, the logrank statistic is approximately standard normal. A one-sided level $alpha$ test will reject the null hypothesis if $Z>z_alpha$ where $z_alpha$ is the upper $alpha$ quantile of the standard normal distribution. If the hazard ratio is $lambda$, there are $n$ total subjects, $d$ is the probability a subject in either group will eventually have an event, and the proportion of subjects randomized to each group is 50%, then the logrank statistic is approximately normal with mean $\left(log\left\{lambda\right\}\right) , sqrt \left\{frac \left\{n , d\right\} \left\{4\right\}\right\}$ and variance 1. For a one-sided level $alpha$ test with power $1-beta$, the sample size required is $n = frac \left\{4 , \left(z_alpha + z_beta\right)^2 \right\} \left\{dlog^2\left\{lambda\right\}\right\}$ where $z_alpha$ and $z_beta$ are the quantiles of the standard normal distribution.

## Joint distribution

Suppose $Z_1$ and $Z_2$ are the logrank statistics at two different time points in the same study ($Z_1$ earlier). Again, assume the hazard functions in the two groups are proportional with hazard ratio $lambda$ and $d_1$ and $d_2$ are the probabilities that a subject will have an event at the two time points. $Z_1$ and $Z_2$ are approximately bivariate normal with means $log\left\{lambda\right\} , sqrt \left\{frac \left\{n , d_1\right\} \left\{4\right\}\right\}$ and $log\left\{lambda\right\} , sqrt \left\{frac \left\{n , d_2\right\} \left\{4\right\}\right\}$ and correlation $sqrt \left\{frac \left\{d_1\right\} \left\{d_2\right\}\right\}$. Calculations involving the joint distribution are needed to correctly maintain the error rate when the data are examined multiple times within a study by a Data Monitoring Committee.

## Relationship to other statistics

• The logrank statistic can be derived as the score test for the Cox proportional hazards model comparing two groups. It is therefore asymptotically equivalent to the likelihood ratio test statistic based from that model.
• The logrank statistic is asymptotically equivalent to the likelihood ratio test statistic for any family of distributions with proportional hazard alternative. For example, if the data from the two samples have exponential distributions.
• If $Z$ is the logrank statistic, $D$ is the number of events observed, and $hat \left\{lambda\right\}$ is the estimate of the hazard ratio, then $log\left\{hat \left\{lambda\right\}\right\} approx Z , sqrt\left\{4/D\right\}$. This relationship is useful when two of the quantities are known (e.g. from a published article), but the third one is needed.

## References

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