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Agenda of Week VII Review of Week VI Multiple regression Canonical correlation

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Review of Week VI Inference on MV Mean Vector Two populations Inference on MV Mean Vector Multi-populations: MANOVA

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Multiple Regression Basic assumptions Linearity X is an n by K matrix with rank K Regression Homoscedasticity

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Multiple Regression Hypotheses By ANOVA When the null hypothesis is rejected, For all i.

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Multiple Regression OLS model OLS solution

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Multiple Regression ANOVA table

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Multiple Regression t-statistics (d.f. = n-k) Where S kk is the k th diagonal element of (XX) -1 Standard error of the regression

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Multiple Regression Standard error of coefficient

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Canonical Correlation Correlation structure between a group of independent variables and a group of dependent variables A kind of multiple regression with more than two dependent variables Example: Physical size group vs. Exercise group How much explained the variation of a group of variables by other group of variables?

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Canonical Correlation Basic model Group of independent variables Group of dependent variables Covariance matrixes

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Canonical Correlation Basic model LC of two groups of variables Correlation between V and W

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Canonical Correlation Correlation testing Hypotheses Correlation between V and W

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Canonical Correlation Objective: To find α and β maximizing ρ or r.

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Canonical Correlation 1 st canonical variable k th canonical variable

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Canonical Correlation 1 st canonical correlation coefficient k th canonical coefficient coefficient

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Canonical Correlation

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Canonical loading matrix Canonical cross loading matrix

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