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Deviance Plots: Assess the overall t of the model. log (1;ii) = 0 + 1X1i + + rXri 2 3 66 Y.1 77 Y=4 . 5 Yn binary responses D(^ ; Y) = n X i=1 = ;2 d(^ i; Yi) n X [Yi log(^i) i=1 +(1 ; Yi) log(1 ; ^i)] 2 3 6 ^1 7 ^ = 64 .. 75 estimates ^n Global deviance: of i = P r(Yi = 1j(X1i; : : : ; Xpi)) from the tted model, i.e. = G2 for testing H0 : proposed model vs. HA : general alternative 0 i 1 exp(b0+b1X1i++brXpi) ^i = 1+exp( b0+b1X1i++br Xpi) 1144 1145 Deviance plot: When the model is appropriate, the global deviance tends to be small. When the model is inappropriate, the data will exhibit systematic deviations from predicted values of i (a lack-of-t component) that tend to inate the global deviance. G2 does not have a 2 distribution when there is only one observed response for each Xi, even if H0 is true. 1146 1. Partition the n observational units into K non-overlapping clusters, with nk units in the k-th cluster. (a) Ward's method (b) Complete linkage (c) Centroid method 2. Create a model matrix Z where the k8-th column has < 1 if in k;th cluster : 0 otherwise 1147 Let ^i denote the estimate of i from the model in Step 3. For the `-th unit in the k-th cluster, dene 3. Fit the model: 2 1 ) 3 log( 1.;1 77 66 64 . 75 n log( 1;n ) = X + Z = dk` 2 3 2 3 64X Z 75 4 5 4. Use the model t in Step 3. to compute \local" deviance contributions by summing deviances within each cluster. = d ^hk`; Yk` i = ;2 Yk` log ^k` + (1 ; Yk`) log 1 ; ^k` 5. Order the K clusters so that S1 S2 Sk where Sk is a measure of \within" cluster inhomogeneity, e.g., nk 1 X Sk = (Xik ; X k)0(Xik ; X k) nk i=1 1148 6. Compute running means of local deviances: 0 nk 1, t t X X t = @ dk`A X (nK ; 1) D k=1 `=1 k=1 t against the \d.f." 7. Plot D t X (nt ; 1) for t = 1; 2; : : : ; k. k=1 Superimpose a horizontal line representing the global mean deviance = D(^; Y)=(n ; r ; 1) D If the proposed model is appropriate the values of D t will approach D . 1150 1149 Smoothed partial residual plots: These are used to determine how the response is related to each particular covariate after adjusting for the eects of (other) covariates included in the proposed model. Partial residuals: With respect to covariate rpar;i = 2 3 6 V1 7 V = 64 .. 75 Vn (Yi ; ^^i) + ^^ V i ^^i(1 ; ^^i) 1151 1. Plot rpar;i against Vi. where ^^ and ^^ are obtained from tting the model: 8 2 > X + V 1 ) 3 if V is not a log( (1.;1) 77 >>< 66 column of X 64 . 75 = > > log( (1;nn) > : X;v ;v + V otherwise 2. Draw a \smooth" curve on the partial residual plot by using Cleveland's (1979, JASA, pp 829{836) method for tting a robust locally weighted regression of rpar;i on Vi. (loess) (i) \Straight-line" curves suggest the Vi should enter the model as BiVi. (ii) horizontal line: suggest Vi is not important (i.e., i = 0). (iii) a \curved" curve suggests that Vi should enter the model as coecient * function(Vi) + 1152 1153 Based on this analysis: 1. Add new variables or functions of variables to the variable list in the model search. 2. Do a new selection of variabales. 3. Look at diagnostics for the new model. (i) leverage, inuence, residuals, etc. (ii) deviance plots True Model: log 1; = ;1 + X5i + X6i ; 2X62i Fitted Model: log 1; = 0 + 1X5i + 2X6i i i (iii) partial residual plots i i Iterate on these steps until you identify a reasonable model. 1154 from Landwher, et al. (1984) JASA 1155