logrank test

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.

Consider two groups of patients, e.g., treatment vs. control. Let $1,\; \backslash ldots,\; J$ be the distinct times of observed events in either group. Let $N\_\{1,j\}$ and $N\_\{2,j\}$ be the number of subjects "at risk" (who have not yet had an event or been censored) at the start of period $j$ in the groups, respectively. Let $O\_\{1,j\}$ and $O\_\{2,j\}$ be the observed number of events in the groups at time $j$. Finally, define $N\_j\; =\; N\_\{1,j\}\; +\; N\_\{2,j\}$ and $O\_j\; =\; O\_\{1,j\}\; +\; O\_\{2,j\}$.

The null hypothesis is that the two groups have identical hazard functions, $H\_0\; :\; h\_1(t)\; =\; h\_2(t)$. Hence, under $H\_0$, for each group $i\; =\; 1,\; 2$, $O\_\{i,j\}$ follows a hypergeometric distribution with parameters $N\_j$, $N\_\{i,j\}$, $O\_j$. This distribution has expected value $E\_\{i,j\}\; =\; O\_j\; \backslash frac\{N\_\{i,j\}\}\{N\_j\}$ and variance $V\_\{i,j\}\; =\; E\_\{i,j\}\; \backslash left(\; \backslash frac\{N\_j\; -\; O\_j\}\{N\_j\}\; \backslash right)\; \backslash left(\; \backslash frac\{N\_j\; -\; N\_\{i,j\}\}\{N\_j\; -\; 1\}\; \backslash right)$.

For all $j\; =\; 1,\; \backslash ldots,\; J$, the logrank statistic compares $O\_\{i,j\}$ to its expectation $E\_\{i,j\}$ under $H\_0$. It is defined as

:$Z\_i\; =\; \backslash frac\; \{\backslash sum\_\{j=1\}^J\; (O\_\{i,j\}\; -\; E\_\{i,j\})\}\; \{\backslash sqrt\; \{\backslash sum\_\{j=1\}^J\; V\_\{i,j\}\}\}\backslash \; \backslash xrightarrow\{d\}\backslash \; \backslash mathcal\; N(0,1)$      (for $i=1$ or $2$)

It is easy to see that for all $j$, $O\_\{2,j\}\; -\; E\_\{2,j\}\; =\; -(O\_\{1,j\}\; -\; E\_\{1,j\})$ and $V\_\{2,j\}\; =\; V\_\{1,j\}$,

so $Z\_2\; =\; -Z\_1$.

By the central limit theorem, the distribution of each $Z\_i$ converges to that of a standard normal distribution as $J$ approaches infinity and therefore can be approximated by the standard normal distribution for a sufficiently large $J$. An improved approximation can be obtained by equating this quantity to Pearson type I or II (beta) distributions with matching first four moments, as described in Appendix B of the Peto and Peto paper.

If the two groups have the same survival function, the logrank statistic is approximately standard normal. A one-sided level $\backslash alpha$ test will reject the null hypothesis if $Z>z\_\backslash alpha$ where $z\_\backslash alpha$ is the upper $\backslash alpha$ quantile of the standard normal distribution. If the hazard ratio is $\backslash lambda$, 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 $(\backslash log\{\backslash lambda\})\; \backslash ,\; \backslash sqrt\; \{\backslash frac\; \{n\; \backslash ,\; d\}\; \{4\}\}$ and variance 1.{{cite journal | last=Schoenfeld | first=D | year=1981 | title=The asymptotic properties of nonparametric tests for comparing survival distributions | journal=Biometrika | volume=68 | issue=1 | pages=316–319 | jstor=2335833 | doi=10.1093/biomet/68.1.316}} For a one-sided level $\backslash alpha$ test with power $1-\backslash beta$, the sample size required is

$n\; =\; \backslash frac\; \{4\; \backslash ,\; (z\_\backslash alpha\; +\; z\_\backslash beta)^2\; \}\; \{d\backslash log^2\{\backslash lambda\}\}$

where $z\_\backslash alpha$ and $z\_\backslash beta$ are the quantiles of the standard normal 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 $\backslash lambda$ and $d\_1$ and $d\_2$ are the probabilities that a subject will have an event at the two time points where $d\_1\; \backslash leq\; d\_2$. $Z\_1$ and $Z\_2$ are approximately bivariate normal with means $\backslash log\{\backslash lambda\}\; \backslash ,\; \backslash sqrt\; \{\backslash frac\; \{n\; \backslash ,\; d\_1\}\; \{4\}\}$ and $\backslash log\{\backslash lambda\}\; \backslash ,\; \backslash sqrt\; \{\backslash frac\; \{n\; \backslash ,\; d\_2\}\; \{4\}\}$ and correlation $\backslash sqrt\; \{\backslash frac\; \{d\_1\}\; \{d\_2\}\}$. 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.

• 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 $\backslash hat\; \{\backslash lambda\}$ is the estimate of the hazard ratio, then $\backslash log\{\backslash hat\; \{\backslash lambda\}\}\; \backslash approx\; Z\; \backslash ,\; \backslash sqrt\{4/D\}$. 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 test statistic gives more weight to earlier events when there are a large number of observations.

The logrank test is based on the same assumptions as the Kaplan-Meier survival curve—namely, that censoring is unrelated to prognosis, the survival probabilities are the same for subjects recruited early and late in the study, and the events happened at the times specified. Deviations from these assumptions matter most if they are satisfied differently in the groups being compared, for example if censoring is more likely in one group than another.{{Cite journal | year = 2004 | pages = 1073 | pmid = 15117797 | pmc = 403858 | doi = 10.1136/bmj.328.7447.1073 | issue = 7447 | volume = 328 | last2 = Altman | first1 = J. M. | first2 = D. G. | author-link1=Martin Bland| title = The logrank test | journal = BMJ | last1 = Bland| author-link2=Doug Altman}}

{{Portal|Mathematics}}

• Kaplan–Meier estimator
• Hazard ratio

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{{Statistics|analysis}}

{{DEFAULTSORT:Logrank Test}}

Category:Survival analysis

Category:Time series statistical tests