Multivariate analysis of variance: Difference between revisions

Revision as of 18:58, 11 December 2023

Revision as of 06:47, 20 December 2023

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== Model ==

Assume <math display="inline">n</math> <math display="inline">q</math>-dimensional observations, where the <math display="inline">i</math>’th observation <math display="inline">y_i</math> is assigned to the group <math display="inline">g(i)\in \{1,\dots,m\}</math> and is distributed around the group center <math display="inline">\mu^{(g(i))}\in \mathbb R^q</math> with [[Multivariate normal distribution|Multivariatemultivariate Gaussian]] noise: <math display="block">

y_i = \mu^{(g(i))} + \varepsilon_i\quad \varepsilon_i \overset{\text{i.i.d.}}{\sim} \mathcal N_q (0, \Sigma) \quad \text{ for } i=1,\dots, n

</math> where <math display="inline">\Sigma</math> is the [[covariance matrix]]. Then we formulate our [[null hypothesis]] as

<math display="block">H_0\!:\;\mu^{(1)}=\mu^{(2)}=\dots =\mu^{(m)}</math>

==Relationship with ANOVA==