en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T12:23:42Z

WP:CHECKWIKI error fixes using AWB (10093)

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	~~In [[statistics]], the '''log-rank test''' is a [[hypothesis test]] to compare the [[survival analysis\|survival]] distributions of two samples~~. It is a [[nonparametric]] test and appropriate to use when the data are right skewed and [[Censoring (statistics)\|censored]] (technically, the censoring must be non-informative). It is widely used in [[clinical trials]] to establish the efficacy of a new treatment in comparison with 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 [[Sir David Cox (statistician)\|David Cox]]. The log-rank test can also be viewed as a time-stratified [[Cochran–Mantel–Haenszel statistics\|Cochran–Mantel–Haenszel test]].		Hi there. Let me begin by introducing the author, her name is Sophia. As a woman what she truly likes is fashion and she's been doing it for fairly a whilst. Mississippi is exactly where her home is but her husband desires them to transfer. For years she's been operating as a travel agent.<br><br>Feel free to surf to my weblog ... online psychic ([http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 findyourflirt.net])

	~~The test was first proposed~~ by ~~[[Nathan Mantel]] and was named~~ the ~~''log-rank test'' by [[Richard Peto\|Richard]] and [[Julian Peto]].<ref name=Mantel1966>{{cite journal~~
	\| author ~~= Mantel~~, ~~Nathan \|authorlink=Nathan Mantel~~
	~~\| year = 1966~~
	~~\| title = Evaluation of survival data and two new rank order statistics arising in its consideration.~~
	~~\| journal = Cancer Chemotherapy Reports~~
	~~\| volume = 50~~
	~~\| issue = 3~~
	~~\| pages = 163–70~~
	~~\| pmid = 5910392~~
	~~}}</ref><ref~~ name~~=Peto1972>{{cite journal~~
	~~\| author = Peto, Richard \| authorlink=Richard Peto~~
	~~\| coauthors = Peto, Julian~~
	~~\| year = 1972~~
	~~\| title = Asymptotically Efficient Rank Invariant Test Procedures~~
	~~\| journal = Journal of the Royal Statistical Society, Series A~~
	~~\| volume = 135~~
	~~\| issue = 2~~
	~~\| pages = 185–207~~
	~~\| doi = 10~~.~~2307/2344317~~
	~~\| jstor = 2344317~~
	~~\| publisher = Blackwell Publishing~~
	~~}}</ref><ref>~~
	~~{{Cite book\| title=Encyclopedia of Biostatistics \| chapter=Linear Rank Tests in Survival Analysis~~
	~~\| first=David \|last=Harrington\|doi=10.1002/0470011815.b2a11047 \|year=2005\| publisher=Wiley Interscience}}</ref>~~

	~~==Definition==~~
	~~The log-rank test statistic compares estimates of the [[Failure rate#Failure rate in the continuous sense\|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 <math>j</math>, let <math>N_{1j}</math> and <math>N_{2j}</math> be the number of subjects "at risk" (have not yet had an event or been censored) at the start of period <math>j</math> in the two groups (often treatment vs. control), respectively. Let <math>N_j = N_{1j} + N_{2j}</math>. Let <math>O_{1j}</math> and <math>O_{2j}</math> be the observed number of events in the groups respectively at time <math>j</math>, and define <math>O_j = O_{1j} + O_{2j}</math>.

	Given that <math>O_j</math> events happened across both groups at time <math>j</math>, under the null hypothesis (of the two groups having identical survival and hazard functions) <math>O_{1j}</math> has the [[hypergeometric distribution]] with parameters <math>N_j</math>, <math>N_{1j}</math>, and <math>O_j</math>. ~~This distribution has expected value <math>E_{1j} = \frac{O_j}{N_j}N_{1j}</math> and variance <math>V_j = \frac{O_j (N_{1j}/N_j) (1 - N_{1j}/N_j) (N_j - O_j)}{N_j - 1}</math>.~~

	~~The log-rank statistic compares each <math>O_{1j}</math> to its expectation <math>E_{1j}</math> under the null hypothesis and~~ is ~~defined as~~

	~~:<math>Z = \frac {\sum_{j=1}^J (O_{1j} - E_{1j})} {\sqrt {\sum_{j=1}^J V_j}}.</math>~~

	~~==Asymptotic distribution==~~
	If the two groups have the same survival function, the log-rank statistic is approximately standard normal. A one-sided level <math>\alpha</math> test will reject the null hypothesis if <math>Z>z_\alpha</math> where ~~<math>z_\alpha</math>~~ is the upper <math>\alpha</math> quantile of the standard normal distribution. If the hazard ratio is <math>\lambda</math>, there are <math>n</math> total subjects, <math>d</math> is the probability a subject in either group will eventually have an event (so that <math>nd</math> 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 log-rank statistic is approximately normal with mean <math> (\log{\lambda}) \, \sqrt {\frac {n \, d} {4}} </math> and variance 1~~.<ref>{{cite journal \| last=Schoenfeld \| first=D \| year=1981 \| title=The asymptotic properties of nonparametric tests for comparing survival distributions \| journal=Biometrika \| volume=68 \| pages=316–319 \| jstor=2335833}}</ref> For a ~~one-sided level <math>\alpha</math> test with power <math>1-\beta</math>, the sample size required is~~
	~~<math> n = \frac {4 \, (z_\alpha + z_\beta)^2 } {d\log^2{\lambda}}</math>~~
	~~where <math>z_\alpha</math> and <math>z_\beta</math> are the quantiles of the standard normal distribution.~~

	~~==Joint distribution==~~

	Suppose <math> Z_1 </math> and <math> Z_2 </math> are the log-rank statistics at two different time points in the same study (<math> Z_1 </math> earlier). Again, assume the hazard functions in the two groups are proportional with hazard ratio <math>\lambda</math> and <math> d_1 </math> and <math> d_2 </math> are the probabilities that a subject will have an event at the two time points where <math> d_1 \leq d_2 </math>. ~~<math> Z_1 </math> and <math> Z_2 </math> are approximately bivariate normal with means <math> \log{\lambda} \, \sqrt {\frac {n \, d_1} {4}} </math> and~~ <~~math~~> ~~\log{\lambda} \, \sqrt {\frac {n \, d_2} {4}}~~ <~~/math~~> and correlation <math>\sqrt {\frac {d_1} {d_2}} </math>. 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 log-rank test has been shown to be too permissive a test, allowing significant results for survivorship prediction models that have low accuracy. The F test was developed in response to ~~these observations and has been shown~~ to ~~be more critical and to track accuracy of the prediction models with higher fidelity~~.~~<ref>{{cite journal \|last=Berty \|first=H~~. P. ~~\|last2=Shi \|first2=H. \|last3=Lyons-Weiler \|first3=J. \|title=Determining the statistical significance of survivorship prediction models \|year=2010 \|journal=~~[~~[Journal of Evaluation in Clinical Practice\|J Eval Clin Pract]] \|volume=16 \|issue=1 \|pages=155–165 \|pmid=20367827 \|doi=10.1111~~/~~j.1365-2753.2009.01199.x }}<~~/~~ref>~~

	*The log-rank statistic can be derived as the [[score test]] for the [[Proportional hazards models\|Cox proportional hazards model]] comparing two groups. It is therefore asymptotically equivalent to the [[likelihood ratio test]] statistic based from that model.

	*The log-rank 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 distribution]]s.~~

	*If <math> Z </math> is the log-rank statistic, <math> D </math> is the number of events observed, and <math>\hat {\lambda} </math> is the estimate of the hazard ratio, then <math> \log{\hat {\lambda}} \approx Z \, \sqrt{4/D} </math>. This relationship is useful when two of the quantities are known (e.g. from a published article), but the third one is needed.

	*The log-rank 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 log-rank 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~~==
	~~{{Portal\|Statistics}}~~
	*[[Kaplan–Meier estimator]]
	*[[Hazard ratio]]

	~~==References==~~
	~~{{Reflist}}~~

	~~==External links==~~
	*{{cite doi\|10.1136/bmj.328.7447.~~1073}}~~

	~~{{Statistics\|analysis}}~~

	~~{{DEFAULTSORT:Logrank Test}}~~
	~~[[Category:Survival analysis]]~~
	~~[[Category:Statistical tests]~~]

en>Binksternet: /* Waveform of Mathieu wavelets */ no need for em dash here

2012-10-13T11:59:34Z

Waveform of Mathieu wavelets: no need for em dash here

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en>Glenn: /* Multiresolution analysis filters and Mathieu's equation */ +links

2012-08-31T17:31:51Z

Multiresolution analysis filters and Mathieu's equation: +links

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Mathieu wavelet - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

en>Binksternet: /* Waveform of Mathieu wavelets */ no need for em dash here

en>Glenn: /* Multiresolution analysis filters and Mathieu's equation */ +links