Boxscher M test

The Boxsche M-test is a method from the mathematical statistics . It was developed by GEP Box in 1949 and is an extension of the Bartlett test for equality of variances for the multivariate case. It is used in multivariate methods , for example in discriminant analysis to test for equality of scatter in the groups.

It is assumed that the -dimensional data in the groups have a multivariate normal distribution: distributed with expected value vectors and covariance matrices ( ). ${\displaystyle m}$ ${\displaystyle p}$ ${\displaystyle {\mathcal {N}}(\mu _{k};\Sigma _{k})}$ ${\displaystyle \mu _{k}=(\mu _{k1},...,\mu _{km})}$ ${\displaystyle \Sigma _{k}=(\sigma _{k;rs})_{r,s=1,...,m}}$ ${\displaystyle k=1,...,p}$

The hypothesis to be tested that all covariance matrices are the same, so

${\displaystyle H_{0}:\Sigma _{1}=\Sigma _{2}=...=\Sigma _{p}}$vs. there are min. a couple and with . ${\displaystyle H_{1}:}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle \Sigma _{i}\neq \Sigma _{j}}$

The test variable for the test is the so-called M from Box,

${\displaystyle M=\gamma \sum _{k=1}^{p}(n_{k}-1)\cdot \log |S_{k}^{-1}\cdot S|}$

in which

${\displaystyle \gamma =1-{\frac {2m^{2}+3m-1}{6\cdot (m+1)\cdot (p-1)}}\left(\sum _{k=1}^{p}{\frac {1}{n_{k}-1}}-{\frac {1}{n-p}}\right)}$

serves as a correction. The covariance matrix is from the observations to the group , estimated include ${\displaystyle \Sigma _{k}}$ ${\displaystyle k}$

${\displaystyle s_{k;rs}={\frac {1}{n_{k}-1}}\sum _{i=1}^{n_{k}}(x_{ir}-{\bar {x}}_{r})(x_{is}-{\bar {x}}_{s})}$

and the pooled, i.e. mean, covariance matrix

${\displaystyle S={\frac {1}{n-p}}\sum _{k=1}^{p}(n_{k}-1)\cdot S_{k}.}$

If the test variable is sufficiently large, it is approximately distributed in chi-squares with degrees of freedom . If they are very different from overall , the test quantity value becomes high. is therefore rejected at the level of significance if M is greater than the - quantile of the chi-square distribution with degrees of freedom. ${\displaystyle n_{k}}$ ${\displaystyle m(m+1)(p-1)/2}$ ${\displaystyle S_{k}}$ ${\displaystyle S}$ ${\displaystyle H_{0}}$ ${\displaystyle \alpha }$ ${\displaystyle 1-\alpha }$ ${\displaystyle m(m+1)(p-1)/2}$

The test reacts sensitively to violations of the precondition of a multidimensional normal distribution .

Individual evidence

1. ^ Box, GEP (1949). A general distribution theory for a class of likelihood criteria. Biometrika, 36, 317-346, doi : 10.1093 / biomet / 36.3-4.317 , JSTOR 2332671 .