Multivariate analysis of variance: Difference between revisions

Revision as of 11:54, 4 August 2023

Revision as of 01:55, 8 August 2023

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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|Multivariate 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|covariance matrix]]. Then we formulate our [[Null hypothesis|null hypothesis]] as

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

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* <math display="inline">\bar Y</math>: where the <math display="inline">i</math>-th row is the best prediction given no information. That is the [[Sample mean and covariance|empirical mean]] over all <math display="inline">n</math> observations <math display="inline">\frac{1}{n}\sum_{k=1}^n y_k</math>

Then the matrix <math display="inline">S_{\text{model}} := (\hat Y - \bar Y)^T(\hat Y - \bar Y)</math> is a generalization of the sum of squares explained by the group, and <math display="inline">S_{\text{res}} := (Y - \hat Y)^T(Y - \hat Y)</math> is a generalization of the [[Residual sum of squares|residual sum of squares]].<ref name="Anderson1994">{{cite book |last=Anderson |first=T. W. |title=An Introduction to Multivariate Statistical Analysis |year=1994 |publisher=Wiley}}</ref> <ref name="Krzanowski1988">{{cite book |last=Krzanowski |first=W. J. |title=Principles of Multivariate Analysis. A User’s Perspective |year=1988 |publisher=Oxford University Press}}</ref>

Note that alternatively one could also speak about covariances when the abovementioned matrices are scaled by 1/(n-1) since the subsequent test statistics do not change by multiplying <math display="inline">S_{\text{model}}</math> and <math display="inline">S_{\text{res}}</math> by the same non-zero constant.

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{{main|Multivariate analysis of covariance}}

One can also test if there is a group effect after adjusting for covariates. For this, follow the procedure above but substitute <math display="inline">\hat Y</math> with the predictions of the [https://en.wikipedia.org/wiki/General_linear_model [general linear model]], containing the group and the covariates, and substitute <math display="inline">\bar Y</math> with the predictions of the general linear model containing only the covariates (and an intercept). Then <math display="inline">S_{\text{model}}</math> are the additional sum of squares explained by adding the grouping information and <math display="inline">S_{\text{res}}</math> is the residual sum of squares of the model containing the grouping and the covariates.<ref name="Krzanowski1988" />

Note that in case of unbalanced data, the order of adding the covariates matter.