Submodular set function - Revision history

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	~~ <math>T = \sqrt{ \frac{\phi^2}{\sqrt{(r-1)(c-1)}} }</math> ~~
	\|-
	~~! Tschuprow's ''T''~~
	\|}

	In [[statistics]], '''Tschuprow's ''T''''' is a measure of [[Association (statistics)\|association]] between two [[Nominal_data#Nominal_scale\|nominal variables]], giving a value between 0 and 1 (inclusive). ~~It is closely related~~ to ~~[[Cramér's V]], coinciding with it for square [[contingency tables]].~~
	~~It was published by [[Alexander Alexandrovich Chuprov\|Alexander Tschuprow]] (alternative spelling: Chuprov)~~ in ~~1939~~.~~<ref>Tschuprow, A~~. A. ~~(1939)~~ '~~'Principles of the Mathematical Theory of Correlation''; translated by M~~. ~~Kantorowitsch~~. ~~W. Hodge & Co~~.<~~/ref~~>

	~~==Definition==~~
	~~For an ''r'' × ''c'' contingency table with ''r'' rows and ''c'' columns, let <math>\pi_{ij}~~<~~/math~~> be the ~~proportion~~ of ~~the population in cell <math>(i~~,~~j)</math>~~ and ~~let~~
	~~:<math>\pi_{i+}=\sum_{j=1}^c\pi_{ij}</math> and <math>\pi_{+j}=\sum_{i=1}^r\pi_{ij}~~.~~</math>~~

	~~Then~~ the ~~[[mean square contingency]] is given~~ as

	~~: <math> \phi^2 = \sum_{i=1}^r\sum_{j=1}^c\frac{(\pi_{ij}~~-~~\pi_{i+}\pi_{+j})^2}{\pi_{i+}\pi_{+j}}~~ ,~~</math>~~

	~~and Tschuprow's ''T'' as~~

	~~: <math>T = \sqrt{\frac{\phi^2}{\sqrt{(r-1)(c-1)}}} .</math>~~


	~~===Properties===~~

	~~''T'' equals zero if and only if independence holds in~~ the ~~table~~, i.e.~~, if and only if~~ <~~math~~>~~\pi_{ij}=\pi_{i+}\pi_{+j}~~<~~/math~~>. ~~''T'' equals one if and only~~ there ~~is perfect dependence in~~ the ~~table~~, ~~i.e.~~, if and ~~only if for each~~ '~~'i'' there~~ is ~~only one ''j'' such that~~ <~~math~~>~~\pi_{ij}>0~~<~~/math~~> ~~and vice versa. Hence, it can only equal 1~~ for ~~square tables. In~~ this ~~it differs from [[Cramér's V]]~~, ~~which can~~ be ~~equal~~ to ~~1 for any rectangular table~~.

	~~===Estimation===~~

	~~If we have a multinomial sample~~ of ~~size ''n'',~~ the ~~usual way to estimate ''T''~~ from the ~~data is via the formula~~

	~~: <math>\hat T = \sqrt{ \frac{\sum_{i=1}^r\sum_{j=1}^c\frac{(p_{ij}-p_{i+}p_{+j})^2}{p_{i+}p_{+j}}}{\sqrt{(r-1)(c-1)}} }~~ ,<~~/math~~>

	~~where~~ <~~math>p_{ij}=n_{ij}/n</math~~> is the ~~proportion~~ of ~~the sample in cell <math>(i,j)</math>~~. ~~This is the [[empirical value]]~~ of ~~''T''~~. ~~With~~ <~~math~~>~~\chi^2~~<~~/math~~> ~~the [[Pearson's chi-squared test\|Pearson chi-square statistic]]~~, ~~this formula~~ can ~~also be written as~~

	~~: <math>\hat T = \sqrt{ \frac{\chi^2/n}{\sqrt{(r-1)(c-1)}} }~~ .~~</math>~~

	~~==See~~ also==
	~~'''Other measures of correlation~~ for ~~nominal data:'''~~
	* [[Cramér's V]]
	* [[Phi coefficient]]
	* [[Uncertainty coefficient]]
	* [[Goodman and ~~Kruskall's lambda\|Lambda coefficient]]~~

	~~'''Other related articles:'''~~
	* [[Effect size]]

	~~{{refimprove\|date=October 2011}}~~
	~~== References ==~~
	~~{{Reflist}}~~

	* Liebetrau, A. (~~1983~~). ~~Measures~~ of ~~Association~~ (~~Quantitative Applications in the Social Sciences~~). ~~Sage Publications~~

	<~~!--- Categories ---~~>
	[~~[Category~~:~~Articles created via~~ the ~~Article Wizard]]~~
	~~[[Category:Summary statistics for contingency tables]]~~
	~~[[Category:Statistical dependence]]~~

en>AnomieBOT: Dating maintenance tags: {{Citation needed}}

2013-11-24T19:38:33Z

Dating maintenance tags: {{Citation needed}}

← Older revision		Revision as of 21:38, 24 November 2013
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	Christmas was created to be one belonging to the most wonderful times among the year, but we are very mindful the stress and worry of physical exercise get all the shopping tired overcrowded centers. All those people rushing around can be rather overwhelming. Has got carefully chosen a choice of gifts at competitive prices, which often be delivered to be able to your gate.<br><br>~~As some kind of fact~~, ~~you will find football sports book betting which recently been one that~~ is ~~loved hobby~~ of ~~those that enjoy recreation. Joining football sportsbook involving betting has mainly~~ two ~~benefits. First~~, it a ~~person place bets conveniently to all your team~~. ~~Plus, get~~ to ~~learn ugg news on our sports event as a reward of your payment. Next~~, ~~another excellent having such gives that you simply feeling of excitement~~. ~~Football has been a craze are manufactured from fans~~.<br>~~<br>Ugg nade in sheepskin boot~~, ~~very warm, who wouldn~~'~~t naturally think~~ of ~~winter fantastic? Very wrong, but thats okay because here's~~ the ~~health benefits to owning a pair~~ of ~~womens ugg australia shoes or boots~~. ~~When you associated~~ with ~~ugg boots you have to think~~ of ~~investment a person can wear these boots all throughout~~ the ~~year~~ and ~~potent~~ the ~~summer time~~. ~~That wasn~~'~~t a mistake for spring~~, ~~I mean hot summer months~~. ~~Genuine uggs are produced from sheepskin~~ and ~~imitations made with wool ought to avoided at all costs.~~<br><br>~~All the footwear whether shoes or boots accessible in variety of designs~~. ~~Typically person has several pairs of shoes~~ and ~~boots each in a specific purpose and tournament. While working in either in public or~~ in the ~~muddy area there is often of likelihood to purchase your shoes dirty~~. ~~Therefore it is good to have cheap boots to prevent this type of outcome~~. ~~Due to the fact are situated on many stores~~ and ~~online shops.~~<br><br>~~Boots apparel is Reduce your struggling to acquire~~ it. ~~The right associated with fashion boots apparel~~ can be ~~just things you need~~ to ~~complete your dry cleaned clothing~~. ~~They look stylish whether usually are very well balancing by using~~ a ~~sarong or peeking out~~ from ~~under bluejeans. Before you spend your precious money~~, ~~anyone might have notice down some frauds making~~ the ~~importance investment on a pair~~ of ~~fashion boots wear.~~<br><br>~~So break free from your fashion rule mentality and let your imagination go rough outdoors~~. ~~Get yourself a pair~~ of ugg classic tall boots and wear these for you jeans, wear them to work on casual Friday, or pair them up by using your Sunday quite. Wear these functional yet attractive boots in your life and whilst on on holiday.<br><br>~~If you enjoyed~~ this ~~short article and you would certainly like to receive even more information concerning [http~~://~~horizonafrica~~.~~com/img/ horizonafrica~~.~~com] kindly see~~ the ~~web site~~.		{\| class="wikitable" align="right" valign
			\|-
			\| align="center" \|
			<math>T = \sqrt{ \frac{\phi^2}{\sqrt{(r-1)(c-1)}} }</math>
			\|-
			! Tschuprow's ''T''
			\|}

			In [[statistics]], '''Tschuprow's ''T''''' is a measure of [[Association (statistics)\|association]] between two [[Nominal_data#Nominal_scale\|nominal variables]], giving a value between 0 and 1 (inclusive). It is closely related to [[Cramér's V]], coinciding with it for square [[contingency tables]].
			It was published by [[Alexander Alexandrovich Chuprov\|Alexander Tschuprow]] (alternative spelling: Chuprov) in 1939.<ref>Tschuprow, A. A. (1939) ''Principles of the Mathematical Theory of Correlation''; translated by M. Kantorowitsch. W. Hodge & Co.</ref>

			==Definition==
			For an ''r'' × ''c'' contingency table with ''r'' rows and ''c'' columns, let <math>\pi_{ij}</math> be the proportion of the population in cell <math>(i,j)</math> and let
			:<math>\pi_{i+}=\sum_{j=1}^c\pi_{ij}</math> and <math>\pi_{+j}=\sum_{i=1}^r\pi_{ij}.</math>

			Then the [[mean square contingency]] is given as

			: <math> \phi^2 = \sum_{i=1}^r\sum_{j=1}^c\frac{(\pi_{ij}-\pi_{i+}\pi_{+j})^2}{\pi_{i+}\pi_{+j}} ,</math>

			and Tschuprow's ''T'' as

			: <math>T = \sqrt{\frac{\phi^2}{\sqrt{(r-1)(c-1)}}} .</math>


			===Properties===

			''T'' equals zero if and only if independence holds in the table, i.e., if and only if <math>\pi_{ij}=\pi_{i+}\pi_{+j}</math>. ''T'' equals one if and only there is perfect dependence in the table, i.e., if and only if for each ''i'' there is only one ''j'' such that <math>\pi_{ij}>0</math> and vice versa. Hence, it can only equal 1 for square tables. In this it differs from [[Cramér's V]], which can be equal to 1 for any rectangular table.

			===Estimation===

			If we have a multinomial sample of size ''n'', the usual way to estimate ''T'' from the data is via the formula

			: <math>\hat T = \sqrt{ \frac{\sum_{i=1}^r\sum_{j=1}^c\frac{(p_{ij}-p_{i+}p_{+j})^2}{p_{i+}p_{+j}}}{\sqrt{(r-1)(c-1)}} } ,</math>

			where <math>p_{ij}=n_{ij}/n</math> is the proportion of the sample in cell <math>(i,j)</math>. This is the [[empirical value]] of ''T''. With <math>\chi^2</math> the [[Pearson's chi-squared test\|Pearson chi-square statistic]], this formula can also be written as

			: <math>\hat T = \sqrt{ \frac{\chi^2/n}{\sqrt{(r-1)(c-1)}} } .</math>

			==See also==
			'''Other measures of correlation for nominal data:'''
			* [[Cramér's V]]
			* [[Phi coefficient]]
			* [[Uncertainty coefficient]]
			* [[Goodman and Kruskall's lambda\|Lambda coefficient]]

			'''Other related articles:'''
			* [[Effect size]]

			{{refimprove\|date=October 2011}}
			== References ==
			{{Reflist}}

			* Liebetrau, A. (1983). Measures of Association (Quantitative Applications in the Social Sciences). Sage Publications

			<!--- Categories --->
			[[Category:Articles created via the Article Wizard]]
			[[Category:Summary statistics for contingency tables]]
			[[Category:Statistical dependence]]

89.176.251.127: m

2012-08-19T17:52:01Z

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