Logrank test

Logrank test

In statistics, the logrank 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 skewed and censored (technically, the censoring must be non-informative). It is widely used in clinical trials to establish the efficacy of a new treatment compared to a control treatment when the measurement is the time to event (such as the time from initial treatment to a heart attack). The test is sometimes called the Mantel–Cox test, named after Nathan Mantel and David Cox. The logrank test can also be viewed as a time stratified Cochran–Mantel–Haenszel test.

The test was first proposed by Nathan Mantel and was named the logrank test by Richard and Julian Peto.[1][2][3]



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 N1j and N2j 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 Nj = N1j + N2j. Let O1j and O2j be the observed number of events in the groups respectively at time j, and define Oj = O1j + O2j.

Given that Oj events happened across both groups at time j, under the null hypothesis (of the two groups having identical survival and hazard functions) O1j has the hypergeometric distribution with parameters Nj, N1j, and Oj. This distribution has expected value E_{1j} = O_j\frac{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 O1j to its expectation E1j under the null hypothesis and is defined as

Z = \frac {\sum_{j=1}^J (O_{1j} - E_{1j})} {\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 α test will reject the null hypothesis if Z > zα where zα is the upper α quantile of the standard normal distribution. If the hazard ratio is λ, there are n total subjects, d is the probability a subject in either group will eventually have an event (so that nd is the expected number of events at the time of the analysis), 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.[4] For a one-sided level α test with power 1 − β, the sample size required is  n = \frac {4 \, (z_\alpha + z_\beta)^2 } {d\log^2{\lambda}} where zα and zβ are the quantiles of the standard normal distribution.

Joint distribution

Suppose Z1 and Z2 are the logrank statistics at two different time points in the same study (Z1 earlier). Again, assume the hazard functions in the two groups are proportional with hazard ratio λ and d1 and d2 are the probabilities that a subject will have an event at the two time points where d1d2. Z1 and Z2 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.
  • The logrank statistic can be used when observations are censored. If censored observations are not present in the data then the Wilcoxon rank sum test is appropriate.
  • The logrank statistic gives all calculations the same weight, regardless of the time at which an event occurs. The Peto logrank statistic gives more weight to earlier events when there are a large number of observations.

See also


  1. ^ Mantel, Nathan (1966). "Evaluation of survival data and two new rank order statistics arising in its consideration.". Cancer Chemotherapy Reports 50 (3): 163–70. PMID 5910392. 
  2. ^ Peto, Richard; Peto, Julian (1972). "Asymptotically Efficient Rank Invariant Test Procedures". Journal of the Royal Statistical Society. Series A (General) (Blackwell Publishing) 135 (2): 185–207. doi:10.2307/2344317. JSTOR 2344317. 
  3. ^ Harrington, David (2005). "Linear Rank Tests in Survival Analysis". Encyclopedia of Biostatistics. Wiley Interscience. doi:10.1002/0470011815.b2a11047. 
  4. ^ Schoenfeld, D (1981). "The asymptotic properties of nonparametric tests for comparing survival distributions". Biometrika 68: 316–319. JSTOR 2335833. 

External links

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Student's t-test — A t test is any statistical hypothesis test in which the test statistic follows a Student s t distribution if the null hypothesis is supported. It is most commonly applied when the test statistic would follow a normal distribution if the value of …   Wikipedia

  • Pearson's chi-squared test — (χ2) is the best known of several chi squared tests – statistical procedures whose results are evaluated by reference to the chi squared distribution. Its properties were first investigated by Karl Pearson in 1900.[1] In contexts where it is… …   Wikipedia

  • Gold standard (test) — For other uses, see Gold standard (disambiguation). In medicine and statistics, gold standard test refers to a diagnostic test or benchmark that is the best available under reasonable conditions. It does not have to be necessarily the best… …   Wikipedia

  • Chi-squared test — Chi square test is often shorthand for Pearson s chi square test. A chi square test, also referred to as chi squared test or χ2 test, is any statistical hypothesis test in which the sampling distribution of the test statistic is a chi square… …   Wikipedia

  • Cochran's C test — In statistics, Cochran s C test [1], named after William G. Cochran, is a one sided upper limit variance outlier test. The C test is used to decide if a single estimate of a variance (or a standard deviation) is significantly larger than a group… …   Wikipedia

  • Mauchly's sphericity test — is a statistical test used to validate repeated measures factor ANOVAs. The test was introduced by ENIAC co inventor John Mauchly in 1940.[1] Contents 1 What is sphericity? 2 Formulation of the test …   Wikipedia

  • Survival analysis — is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, and duration analysis or duration modeling in economics or …   Wikipedia

  • Statistical hypothesis testing — This article is about frequentist hypothesis testing which is taught in introductory statistics. For Bayesian hypothesis testing, see Bayesian inference. A statistical hypothesis test is a method of making decisions using data, whether from a… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • Mann–Whitney U — In statistics, the Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW) or Wilcoxon rank sum test) is a non parametric statistical hypothesis test for assessing whether one of two samples of independent observations tends to have… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”