Definitions

alpha-naphthol test

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_{1j} and N_{2j} 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_{1j} + N_{2j}. Let O_{1j} and O_{2j} be the observed number of events in the groups respectively at time j, and define O_j = O_{1j} + O_{2j}.

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

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

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

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 (log{lambda}) , sqrt {frac {n , d} {4}} and variance 1. For a one-sided level alpha test with power 1-beta, the sample size required is n = frac {4 , (z_alpha + z_beta)^2 } {dlog^2{lambda}} 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{lambda} , sqrt {frac {n , d_1} {4}} and log{lambda} , sqrt {frac {n , d_2} {4}} and correlation sqrt {frac {d_1} {d_2}} . 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 {lambda} is the estimate of the hazard ratio, then log{hat {lambda}} approx Z , sqrt{4/D} . This relationship is useful when two of the quantities are known (e.g. from a published article), but the third one is needed.

See also

References

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