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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク multinomial test ： ウィキペディア英語版 multinomial test In statistics, the multinomial test is the test of the null hypothesis that the parameters of a multinomial distribution equal specified values. It is used for categorical data; see Read and Cressie.〔Read, T. R. C. and Cressie, N. A. C. (1988). ''Goodness-of-fit statistics for discrete multivariate data.'' New York: Springer-Verlag. ISBN 0-387-96682-X.〕 We begin with a sample of $N$ items each of which has been observed to fall into one of $k$ categories. We can define $\backslash mathbf\; =\; (x\_1,\; x\_2,\; \backslash dots,\; x\_k)$ as the observed numbers of items in each cell. Hence $\backslash textstyle\; \backslash sum\_^k\; x\_\; =\; N$. Next, we define a vector of parameters $H\_0:\; \backslash mathbf\; =\; (\backslash pi\_,\; \backslash pi\_,\; \backslash dots,\; \backslash pi\_)$, where :$\backslash textstyle\; \backslash sum\_^k\; \backslash pi\_\; =\; 1$. These are the parameter values under the null hypothesis. The exact probability of the observed configuration $\backslash mathbf$ under the null hypothesis is given by :$\backslash Pr(\backslash mathbf\; =\; N!\; \backslash prod\_^k\; \backslash frac\}.$ The significance probability for the test is the probability of occurrence of the data set observed, or of a data set less likely than that observed, if the null hypothesis is true. Using an exact test, this is calculated as :$\backslash Pr(\backslash mathbf)=\backslash sum\_\}\; \backslash Pr(\backslash mathbf)$ where the sum ranges over all outcomes as likely as, or less likely than, that observed. In practice this becomes computationally onerous as $k$ and $N$ increase so it is probably only worth using exact tests for small samples. For larger samples, asymptotic approximations are accurate enough and easier to calculate. One of these approximations is the likelihood ratio. We set up an alternative hypothesis under which each value $\backslash pi\_$ is replaced by its maximum likelihood estimate $p\_=x\_/N$. The exact probability of the observed configuration $\backslash mathbf$ under the alternative hypothesis is given by : $\backslash Pr(\backslash mathbf\; =\; N!\; \backslash prod\_^k\; \backslash frac\}.$ The natural logarithm of the ratio between these two probabilities multiplied by $-2$ is then the statistic for the likelihood ratio test : $-2\backslash ln(L/R)\; =\; \backslash textstyle\; -2\backslash sum\_^k\; x\_\backslash ln(\backslash pi\_/p\_)\; .$ If the null hypothesis is true, then as $N$ increases, the distribution of $-2\backslash ln(LR)$ converges to that of chi-squared with $k-1$ degrees of freedom. However it has long been known (e.g. Lawley 1956) that for finite sample sizes, the moments of $-2\backslash ln(LR)$ are greater than those of chi-squared, thus inflating the probability of type I errors (false positives). The difference between the moments of chi-squared and those of the test statistic are a function of $N^$. Williams (1976) showed that the first moment can be matched as far as $N^$ if the test statistic is divided by a factor given by : $q\_1\; =\; 1+\backslash frac^-1\}.$ In the special case where the null hypothesis is that all the values $\backslash pi\_$ are equal to $1/k$ (i.e. it stipulates a uniform distribution), this simplifies to : $q\_1\; =\; 1+\backslash frac.$ Subsequently, Smith et al. (1981) derived a dividing factor which matches the first moment as far as $N^$. For the case of equal values of $\backslash pi\_$, this factor is : $q\_2\; =\; 1+\backslash frac+\backslash frac.$ The null hypothesis can also be tested by using Pearson's chi-squared test :$\backslash chi^2\; =\; \backslash sum\_^$ where $E\_i=N\backslash pi\_i$ is the expected number of cases in category $i$ under the null hypothesis. This statistic also converges to a chi-squared distribution with $k-1$ degrees of freedom when the null hypothesis is true but does so from below, as it were, rather than from above as $-2\backslash ln(LR)$ does, so may be preferable to the uncorrected version of $-2\backslash ln(LR)$ for small samples. == References == 〔 * * * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「multinomial test」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.