en>Kku: /* Simulation */ Stochastic simulation

2013-08-05T10:21:11Z

Simulation: Stochastic simulation

← Older revision		Revision as of 12:21, 5 August 2013
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	~~She~~ is ~~known by~~ the ~~name~~ of ~~Myrtle Shryock~~. ~~My working day occupation~~ is a ~~meter reader~~. It's ~~not~~ a ~~common factor but what she likes performing~~ is ~~base jumping and now she~~ is ~~attempting~~ to ~~earn cash with it~~. ~~Puerto Rico~~ is where he and ~~his wife live.~~<br><br>~~Here~~ is ~~my page ::~~ [http://www.~~aseandate~~.com/~~index.php~~?m=~~member_profile&p=profile&id=13345213~~ www.~~aseandate~~.~~com~~]		In [[statistics]], '''Levene's test'''<ref name=Levene1960>{{cite book \|last=Levene \|first=Howard \|chapter=Robust tests for equality of variances \|year=1960 \| booktitle=Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling \| editor=[[Ingram Olkin]], [[Harold Hotelling]], et alia \| publisher=Stanford University Press \| pages=278–292}}</ref> is an inferential statistic used to assess the equality of [[variance]]s for a variable calculated for two or more groups. Some common statistical procedures assume that variances of the populations from which different samples are drawn are equal. Levene's test assesses this assumption. It tests the [[null hypothesis]] that the population variances are equal (called ''homogeneity of variance'' or ''[[homoscedasticity]]''). If the resulting ''P''-value of Levene's test is less than some critical value (typically 0.05), the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances. Thus, the null hypothesis of equal variances is rejected and it is concluded that there is a difference between the variances in the population.

			Some of the procedures typically assuming homoscedasticity, for which one can use Levene's tests, include [[analysis of variance]] and [[Student's t-test\|t-tests]].

			Levene's test is often used before a comparison of means. When Levene's test shows significance, one should switch to generalized tests, free from homoscedasticity assumptions.

			Levene's test may also be used as a main test, for answering a stand-alone question whether two sub-samples in a given population have equal or different variances.


			==Definition==
			The test statistic, ''W'', is defined as follows:

			: <math>W = \frac{(N-k)}{(k-1)} \frac{\sum_{i=1}^k N_i (Z_{i\cdot}-Z_{\cdot\cdot})^2} {\sum_{i=1}^k \sum_{j=1}^{N_i} (Z_{ij}-Z_{i\cdot})^2},</math>

			where

			* <math>W</math> is the result of the test,
			* <math>k</math> is the number of different groups to which the sampled cases belong,
			* <math>N</math> is the total number of cases in all groups,
			* <math>N_i</math> is the number of cases in the <math>i</math>th group,
			* <math>Y_{ij}</math> is the value of the measured variable for the<math>j</math>th case from the <math>i</math>th group,
			* <math>Z_{ij} = \left\{\begin{matrix}
			\|Y_{ij} - \bar{Y}_{i\cdot}\|, & \bar{Y}_{i\cdot} \mbox{ is a mean of i-th group } \\
			\|Y_{ij} - \tilde{Y}_{i\cdot}\|, & \tilde{Y}_{i\cdot} \mbox{ is a median of i-th group } \end{matrix}\right.</math>
			(Both definitions are in use though the second one is, strictly speaking, the [[Brown–Forsythe test]] – see below for comparison)
			* <math>Z_{\cdot\cdot} = \frac{1}{N} \sum_{i=1}^{k} \sum_{j=1}^{N_i} Z_{ij}</math> is the mean of all <math>Z_{ij}</math>,
			* <math>Z_{i\cdot} = \frac{1}{N_i} \sum_{j=1}^{N_i} Z_{ij}</math> is the mean of the <math>Z_{ij}</math> for group <math>i</math>.
			The significance of <math>W</math> is tested against <math>F(\alpha,k-1,N-k)</math> where <math>F</math> is a quantile of the [[F-test]] distribution, with <math>k-1</math> and <math>N-k</math> its degrees of freedom, and <math>\alpha</math> is the chosen level of significance (usually 0.05 or 0.01).

			==Comparison with the Brown–Forsythe test==
			The [[Brown–Forsythe test]] uses the median instead of the mean. Although the optimal choice depends on the underlying distribution, the definition based on the median is recommended as the choice that provides good [[Robust statistics\|robustness]] against many types of non-normal data while retaining good [[statistical power]]. If one has knowledge of the underlying distribution of the data, this may indicate using one of the other choices. Brown and Forsythe performed [[Monte Carlo]] studies that indicated that using the [[trimmed mean]] performed best when the underlying data followed a [[Cauchy distribution]] (a [[heavy-tailed]] distribution) and the median performed best when the underlying data followed a [[Chi-squared distribution]] with four degrees of freedom (a heavily [[skewness\|skewed distribution]]). Using the mean provided the best power for symmetric, moderate-tailed, distributions.

			==See also==
			*[[Bartlett's test]]

			==References==
			{{Reflist}}

			==External links==
			* [http://www.youtube.com/watch?v=O6taUlWejB0 Parametric and nonparametric Levene's test in SPSS]
			* http://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm

			[[Category:Analysis of variance]]
			[[Category:Statistical tests]]

129.215.90.221: Stationary distribution is Chi-squared, not gamma ~~~~

2012-07-17T15:17:38Z

Stationary distribution is Chi-squared, not gamma ~~~~

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She is known by the name of Myrtle Shryock. My working day occupation is a meter reader. It's not a common factor but what she likes performing is base jumping and now she is attempting to earn cash with it. Puerto Rico is where he and his wife live.<br><br>Here is my page :: [http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13345213 www.aseandate.com]

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en>Kku: /* Simulation */ Stochastic simulation

129.215.90.221: Stationary distribution is Chi-squared, not gamma ~~~~