sta108_handout13

# sta108_handout13 - pmax-1 under consideration[For the...

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Handout 13 Polynomial regression: determination of the degree . If a polynomial of degree p - 1 has been ﬁtted, then the number of beta parameters estimated is p and we will denote the residual sum of squares of this ﬁtted mode by SSE p . Note that df ( SSE p ) = n - p , MSE p = SSE p / ( n - p ), R 2 p = SSR p /SSTO = 1 - SSE p /SSTO , R 2 adj,p = 1 - MSE p /MSTO = 1 - n - 1 n - p (1 - R 2 p ˙ ) . It is worthwhile noting that (see the table below), that SSE p will invariably decrease (or the R 2 p will increase) as the degree of the polynomial increases. So one cannot use SSE p (or R 2 p ) as a criterion for model selection. One may decide to choose the model for which MSE p is the smallest (or equivalently R 2 adj,p is the largest). However, this is not a very good criterion for choosing models. Here are some well-known criteria functions. For any of these methods, model selection is acheived by selecting the model at which the criterion function is the smallest. Akaike’s FPE(ﬁnal prediction error): FPE p = n + p n - p SSE p Mallows’ C p : C p = SSE p MSE pmax - ( n - 2 p ) where pmax is the mean square error of the highest degree polynomial model (i.e., of order
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Unformatted text preview: pmax-1) under consideration. [For the growth data, p max is taken to be equal to 6.] AIC (Akaike information criterion): AIC p = n ln( SSE p )-n ln( n ) + 2 p SBC (Schwartz’s Bayesian criterion): SBC p = n ln( SSE p )-n ln( n ) + [ln( n )] p Degree of df ( SSE p ) SSE p R 2 p MSE p R 2 adj,p FPE p C p AIC p SBC p polynomial p-1 n-p 11 1096.00 99.64 1295.32 186.42 56.17 56.66 1 10 891.73 .186 89.17 .105 1248.38 151.81 55.70 56.67 2 9 49.06 .955 5.45 .945 81.75 2.79 22.90 24.35 3 8 44.31 .960 5.54 .944 88.64 3.94 23.68 25.62 4 7 37.12 .966 5.30 .947 90.10 4.65 23.55 25.98 5 6 33.50 .969 5.58 .944 100.44 6.00 24.32 27.23 According to FPE, C p , AIC and SBC, the quadratic model seems to be the most appropriate one. Fitted quadratic model is ˆ Y = 89 . 8479-. 4005 x-. 1290 x 2 , where x = X-¯ X = X-22 . 0833 with R 2 = . 955, R 2 adj = . 945 and SSE = 49 . 0643 . 1...
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