CH 10 Notes

# CH 10 Notes - Multiple Linear Regression Model Y = 0 1 x1 2...

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Multiple Linear Regression Model Y = ß 0 + ß 1 x 1 + ß 2 x 2 + … + ß k x k + ε deterministic component random component Note: Linear w.r.t. Parameters ß 0, ß 1 … ß k and not predictor variables. Assumptions: 1. The specified regression model has the correct form . Therefore for given values of x 1 , x 2 , … x k , E(Y) = ß 0 + ß 1 x 1 + … + ß k x k and E(ε) = 0 This implies that when the least squares estimates are determined, the least squares equation Ŷ = b 0 + b 1 x 1 + … + b k x k estimates the average value of Y, given a set of values for the predictor variables x; and, the model correctly represents the form of the association between the response variable and the predictor variables. 2. The error variance is constant . The σ 2 ε is constant over all values of the predictor variables. Thus, the range of deviations of the Y- values from the regression model is the same regardless of the values of x 1 , x 2 , …,x k . 3. Random errors, ε’s, are independent and normally distributed . Hence, the random errors associated with the Y-values are statistically independent of one another and normally distributed. The residual variance S 2 e is defined by, S 2 e = SSE / (n-k-1) where SSE = ∑ (Y i – Ŷ i ) 2 S 2 e remains an absolute measure of how well the least squares equation fits the sample Y – values. If the fit were perfect, all residuals would equal 0 and S 2 e = 0.

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Partitioning the Total Variation SST = SSR + SSE where _ _ SST = ∑(Y i –Y) 2 SSR = ∑(Ŷ – Y) 2 SSE = ∑( - Ŷ) Ү 2 The coefficient of determination has the same interpretation as in simple linear regression, that is, the fraction of the total variation in the sample Y-values that has been explained by the predictor variables in the least
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CH 10 Notes - Multiple Linear Regression Model Y = 0 1 x1 2...

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