en>EmilJ: /* Definitions and basic properties */ fix notation

2012-11-30T17:22:37Z

Definitions and basic properties: fix notation

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	~~Pleased to meet you! That name~~ is ~~Eusebio Ledbetter~~. It's ~~not~~ a ~~common stuff but what I like doing~~ is ~~bottle blouses collecting~~ and ~~now I~~ have ~~time~~ to ~~take on new things~~. ~~Software developing~~ is ~~how As well as support my family~~. ~~My house~~ is at the ~~present~~ in ~~Vermont~~. I'~~ve been jogging on my website~~ with ~~respect to some time now~~. ~~Inspection~~ it ~~out here~~: ~~http~~://[http://www.~~Twitpic~~.~~com~~/~~tag/circuspartypanama circuspartypanama~~].~~com<br><br>Have a look at my web blog -~~ [http://~~circuspartypanama~~.~~com clash~~ of ~~clans hack tool password~~]		{{Expert-subject\|Statistics\|date=August 2011}}

			In [[statistics]], the '''Page test''' for multiple comparisons between ordered correlated variables is the counterpart of [[Spearman's rank correlation coefficient]] which summarizes the association of continuous variables. It is also known as '''Page's trend test''' or '''Page's ''L'' test'''. It is a repeated measure trend test.

			The Page test is useful where:
			*there are three or more conditions,
			*a number of subjects (or other randomly sampled entities) are all observed in each of them, and
			*we predict that the observations will have a particular order.

			For example, a number of subjects might each be given three trials at the same task, and we predict that performance will improve from trial to trial. A test of the significance of the trend between conditions in this situation was developed by Page (1963). More formally, the test considers the [[null hypothesis]] that, for ''n'' conditions, where ''m''<sub>''i''</sub> is a measure of the [[central tendency]] of the ''i''th condition,

			:<math>m_1 = m_2 = m_3 = \cdots = m_n\, </math>

			against the [[alternative hypothesis]] that

			:<math> m_1 > m_2 > m_3 > \cdots > m_n.\,</math>

			It has more [[statistical power]] than the [[Friedman test]] against the alternative that there is a difference in trend. Friedman's test considers the alternative hypothesis that the central tendencies of the observations under the ''n'' conditions are different without specifying their order.

			Procedure for the Page test, with ''k'' subjects each exposed to ''n'' conditions:

			*Arrange the ''n'' conditions in the order implied by the alternative hypothesis, and assign each of them a rank ''Y''<sub>''i''.</sub>
			*For each of the ''k'' subjects separately, rank the ''n'' observations from 1 to ''n''.
			*Add the ranks for each condition to give a total ''X''<sub>''i''</sub>.
			*Multiply ''X''<sub>''i''</sub> by ''Y''<sub>''i''</sub> and add all the products together; this sum is called ''L''.
			*To test whether there is a significant trend, values of ''L'' can be compared with those tabulated by Page (1963).

			:Alternatively, the quantity

			::<math> { (12L - 3kn(n+1)^2)^2 \over kn^2(n^2 - 1)(n + 1) } </math>

			:may be compared with values of the [[chi-squared distribution]] with one [[degrees of freedom (statistics)\|degree of freedom]]. This gives a [[two-tailed test]]. The approximation is reliable for more than 20 subjects with any number of conditions, for more than 12 subjects when there are 4 or more conditions, and for any number of subjects when there are 9 or more conditions.
			*If a measure of the overall correlation between the conditions and the data is required, it can be calculated as

			::ρ = 12''L''/''k''(''n''<sup>3</sup> − ''n'') − 3(''n'' + 1)/(''n'' − 1)

			:if ''k'' = 1, this reduces to the familiar Spearman coefficient.

			The Page test is most often used with fairly small numbers of conditions and subjects. The minimum values of ''L'' for significance at the 0.05 level, one-tailed, with three conditions, are 56 for 4 subjects (the lowest number that is capable of giving a significant result at this level), 54 for 5 subjects, 91 for 7 subjects, 128 for 10 subjects, 190 for 15 subjects and 251 for 20 subjects.

			A corresponding extension of [[Kendall's tau]] was developed by [[Jonckheere's trend test\| Jonckheere]] (1954).

			==References==
			*{{cite journal \|author=Jonckheere, A. R. \|title=A test of significance for the relation between ''m'' rankings and ''k'' ranked categories \|journal=British Journal of Statistical Psychology \|volume=7 \|pages=93–100 \|year=1954 \|doi=10.1111/j.2044-8317.1954.tb00148.x \|issue=2}}
			*[http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=468904#B4 Jonckheere-Terpstra test]
			*{{cite journal \|author=Page, E. B. \|title=Ordered hypotheses for multiple treatments: A significance test for linear ranks \|journal=Journal of the American Statistical Association \|volume=58 \|pages=216–30 \|year=1963 \|doi=10.2307/2282965 \|issue=301 \|jstor=2282965}}
			*[http://www.psychology.nottingham.ac.uk/staff/pal/stats/C81MSA/Overheads/Lecture%208.pdf Nonparametric Tests of Trend]

			[[Category:Statistical tests]]

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Pleased to meet you! That name is Eusebio Ledbetter. It's not a common stuff but what I like doing is bottle blouses collecting and now I have time to take on new things. Software developing is how As well as support my family. My house is at the present in Vermont. I've been jogging on my website with respect to some time now. Inspection it out here: http://[http://www.Twitpic.com/tag/circuspartypanama circuspartypanama].com<br><br>Have a look at my web blog - [http://circuspartypanama.com clash of clans hack tool password]

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