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Multivariate statistical analysis Regression analysis

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Regression vs. correlation 分析性解釋變數與反應變量之間的 ( 先驗 ) 因果關係 衡量變數之間的關聯 (association) 強度

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Regression model (Y 1, Y 2, … Y j )= f (X 1, X 2, … X k ) k ≧ 2, multiple regression( 複迴歸 ) j ≧ 2, multivariate regression( 多元迴歸 ) The assumed model, y n =β 0 +β 1 x 1 +β 2 x 2 + … β n x n +e n, e n is the random error term based on some prerequisite assumptions Normal i.i.d. ~N(0, σ2) Normality Independence Variance equality

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Modeling the regression line Ref.

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ANOVA table for regression analysis — total model testing

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Sum of errors Sum of squares for error (SSE) Sum of squares for model (SSM) Sum of squares for total (SST) MSE=SSE/d.f. of error=SSE/K MSM=SSM/d.f. of model=SSM/(N-K-1) d.f. of total=N-1 F=MSM/MSE

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Determination Coefficient of determination R2=SSM/SST=1-SSE/SST, 0≦R2≦1 Adjusted coefficient of determination Adjusted by means of dividing by degree of freedom Adj. R2=1-[SSE/(N-K-1)]/[SST/(N-1)]=1-(1-R2)[(N- 1)/(N-K-1)] N>K+1, 必須比解釋變數之個數加一還多 Determining the goodness of fit of a sampled regression line

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t-test for the coefficients of explaining variables — Marginal testing

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Conflicts between total testing and marginal testing Confidence interval vs. confidence region (a region composed with several more narrower interval confidence intervals respectively)

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Determine the predictors Checking the contribution of additional variables Stepwise regression Forward regression Backward regression

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Testing the assumptions Normality testing Wilk-shapiro statistics Q-Q/ P-P plotting (expected distribution vs. real distribution) Variance equality testing Scatter the error term along x n Verify the randomized pattern Durbin-Watson test for testing the first autocorrelation of residuals Mean=2, if >2, “ - ” relation, if <2, “ + ” relation Independence testing Assumed the random & independent sampling process for the cross-sectional data Time-series analysis for the longitudinal data

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Colinearity A pair of predictor variables that are strongly correlated Tolerance, 1-Rj2, if there exists strong correlation, the Tolerance will be smaller and near to zero VIF (variance inflation factor) The inverse of tolerance, if tolerance is small, VIF will inflate very large

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Outliers Leverage h jj, (<1) h jj =1/n+[square(obj j - obj mean)]/Σ[ square(obj j - obj mean)] If h jj is comparatively too large, remove this observation.

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Weighted regression The different impact of sample data Outliers set the influence weight near to 0

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Data transformation Transformation for normality, variance equality Transformation by log, or inverse, square

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