Model Selection in GLMs • Last class: estimability/identifiability, analysis of deviance, stan- dard errors & confidence intervals (should be able to implement frequentist GLM analyses!) • Today: standard frequentist methods for model selection 1 Model Uncertainty In most data analyses, there is uncertainty about the model & you need to do some form of model comparison. 1. Select q out of p predictors to form a parsimonius model 2. Select the link function (e.g., logit or probit) 3. Select the distributional form (normal, t-distributed) We focus first on problem 1 - Variable Selection 2 Frequentist Strategies for Variable Selection It is standard practice to sequentially add or drop variables from a model one at a time and examine the change in model fit If model fit does not improve much (or significantly at some arbitrary level) when adding a predictor, then that predictor is left out of the model (forward selection) Similarly, if model fit does not decrease much when removing a pre- dictor, then that predictor is removed (backwards elimination) Implementation: step() function in S-PLUS. 3 Illustration • Suppose we have 30 candidate predictors, X1,...,X30. • The true model is 0 logit Pr(yi = 1 | xi) = xiβ, 0 where xi = (xi1, . , xi,30) (i = 1, . , n = 100) • Note that predictors with β = 0 are effectively not in the model • Let γj = 1(βj =6 0) be a 0/1 indicator that the jth predictor is included 4 • Interest focuses on selection of the subset of q ≤ p important predictors out of the p candidate predictors: Xγ = {Xj : γj = 1, βj =6 0} ∼ (q × 1). • There is a list of 2p possible models corresponding to the different subsets Xγ ⊂ X. • The size of the model list, 2p, is calculated by noting that we can either include or exclude each of the p candidate predictors. 5 • Letting M denote the model list, we can introduce a model index M ∈ M • The trick is how to decide between competing models M ∈ M and M 0 ∈ M. • An important issue is also efficient searching of the (potentially huge!) model space M for good models • Often not possible to visit and perform calculations for all models in M (230 = 1073741824) 6 • Returning to the logistic regression with 30 candidate predictors example, we simulate data & run stepwise selection • Simulation conditions: n = 100, q = 5, p = 30, xi ∼ Np(0, Ip), X1 − X5 have coefficients βγ = (1, 2, 3, 4, 5)/2, X6 − X30 have coefficients 0. • S-PLUS implementation: n<- 1000 # sample size p<- 30 # number of candidate predictors X<- matrix(0,n,p) # simulate predictor values for(j in 1:p) X[,j]<- rnorm(n,0,1) Xl<- data.frame(X) beta<- c((1:5)/2,rep(0,25)) # true values of parameters py<- 1/(1+exp(0.5-X%*%beta)) # probability of response # note that intercept is assumed to be -0.5 7 y<- rbinom(n,1,py) # simulate response from true model fit.true<- glm(y ~ X.1 + X.2 + X.3 + X.4 + X.5, family=binomial, data=Xl) # fit true model X2<- Xl[,ind] # scramble order of predictors fit.full<- glm(y ~ ., family=binomial, data=X2) # implement stepwise selection using AIC criteria fit.step<- step(fit.full, scope=list(upper = ~., lower=~1), trace=F) # intercept in all fit.step$anova # summarize results 8 • Comments: start with full model & performs backwards elimi- nation - removes predictors sequentially which reduce AIC • Definition of Akaike Information Criterion (AIC): AIC = −2 maximized log likelihood + 2# parameters • Why put in the number of parameters? 9 • Results from stepwise analysis of simulated data: Stepwise Model Path Analysis of Deviance Table Initial Model: y ~ X.4 + X.15 + X.26 + X.13 + X.16 + X.25 + X.9 + X.3 + X.18 + X.29 + X.1 + X.30 + X.24 + X.11 + X.12 + X.7 + X.21 + X.10 + X.17 + X.8 + X.5 + X.28 + X.22 + X.27 + X.2 + X.23 + X.6 + X.14 + X.19 + X.20 Final Model: y ~ X.4 + X.15 + X.26 + X.25 + X.3 + X.1 + X.12 + X.17 + X.5 + X.28 + X.22 + X.2 10 Step Df Deviance Resid. Df Resid. Dev AIC 1 969 528.2075 590.2075 2 - X.9 1 0.018536 970 528.2260 588.2260 3 - X.20 1 0.020528 971 528.2466 586.2466 4 - X.11 1 0.024500 972 528.2711 584.2711 5 - X.18 1 0.045945 973 528.3170 582.3170 6 - X.13 1 0.057255 974 528.3743 580.3743 7 - X.16 1 0.080202 975 528.4545 578.4545 8 - X.24 1 0.100360 976 528.5548 576.5548 9 - X.21 1 0.278245 977 528.8331 574.8331 10 - X.30 1 0.296931 978 529.1300 573.1300 11 - X.27 1 0.440982 979 529.5710 571.5710 12 - X.10 1 0.445923 980 530.0169 570.0169 13 - X.7 1 0.447855 981 530.4648 568.4648 11 14 - X.23 1 0.625145 982 531.0899 567.0899 15 - X.8 1 1.000717 983 532.0906 566.0906 16 - X.6 1 1.025047 984 533.1157 565.1157 17 - X.29 1 1.366004 985 534.4817 564.4817 18 - X.19 1 1.445814 986 535.9275 563.9275 19 - X.14 1 1.623275 987 537.5508 563.5508 12 • Selected model contains predictors X1 − X5 belonging to true model (we got lucky & coefficients large) • Also contains predictors X12,X15,X17,X22,X25,X26,X28, which should have been excluded (not so good) • Results of maximum likelihood estimation for true model: Value Std. Error t value (Intercept) -0.4494802 0.1106592 -4.061844 X.1 0.5525122 0.1115489 4.953095 X.2 0.9541896 0.1176940 8.107376 X.3 1.7647030 0.1489397 11.848442 X.4 2.0024375 0.1638457 12.221483 X.5 2.8375060 0.2018184 14.059703 13 • Results of maximum likelihood estimation for full model (focus- ing on important predictors): Value Std. Error t value (Intercept) -0.47183494 0.1181678 -3.9929235 X.1 0.58449537 0.1214352 4.8132281 X.2 1.06182854 0.1289719 8.2330242 X.3 1.91001906 0.1626665 11.7419354 X.4 2.13608265 0.1772401 12.0519146 X.5 3.04082332 0.2230194 13.6347913 • Full model MLEs differ somewhat from MLEs under true model and standard errors are higher 14 • Now, consider the estimates from the final selected model: Value Std. Error t value (Intercept) -0.4661101 0.1148807 -4.057341 X.1 0.5675410 0.1157651 4.902522 X.2 1.0170453 0.1233606 8.244489 X.3 1.8653303 0.1574395 11.847921 X.4 2.1040769 0.1731561 12.151332 X.5 2.9939133 0.2168262 13.807893 X.12 0.1725280 0.1124232 1.534630 X.15 0.1593174 0.1071552 1.486790 X.17 -0.2245846 0.1059882 -2.118958 X.22 -0.2398270 0.1128883 -2.124464 X.25 -0.1953539 0.1166990 -1.673999 X.26 -0.2685661 0.1112931 -2.413143 X.28 -0.1726031 0.1110436 -1.554373 15 • Comparing the parameters for the predictors that should have been included to the earlier results, we find results in between those for true model and full model. • Interesting result: many of the predictors known to be unimpor- tant have coefficients “significantly” different from zero • This is a very common occurrence when using stepwise selection! • If we had chosen smaller values for the coefficients for the im- portant predictors, we would have missed one or more of them (based on repeating simulation) 16 Some Issues with Stepwise Procedures Normal Linear & Orthogonal Case • In normal models with orthogonal X, forward and backwards selection will yield the same model (i.e., the selection process is not order-dependent). • However, the selection of the significance level for inclusion in the model is arbitrary and can have a large impact on the final model selected. Potentially, one can use some goodness of fit criterion (e.g., AIC, BIC). • In addition, if interest focus on inference on the β’s, stepwise procedures can result in biased estimates and invalid hypothesis tests (i.e., if one naively uses the final model selected without correction for the selection process) 17 Issues with Stepwise Procedures (General Case) For GLMs other than orthogonal, linear Gaussian models, the order in which parameters are added or dropped from the model can have a large impact on the final model selected. This is not good, since choices of the order of selection are typically (if not always) arbitrary. Model selection is a challenging area, but there are alternatives 18 Goodness of Fit Criteria There are a number of criteria that have been proposed for comparing models based on a measure of goodness of fit penalized by model complexity 1. Akaike’s Information Criterion (AIC): AICM = DM + 2pM , where DM is the deviance for model M and pM is the number of predictors 2. Bayesian Information Criterion (BIC): BICM = DM + pM log(n). 19 Note that deviance decreases as variables are added and the likelihood increases. The AIC and BIC differ in the penalty for model complexity, with the AIC using twice the number of parameters and the BIC using the number of parameters multiplied by the logarithm of the sample size. The BIC tends to place a larger penalty on the number of predictors and hence more parsimonius models are selected. 20 Uncertainty in the link function & distribution • Note that we can similarly use the AIC or BIC criteria to select the link function or distribution • For example, suppose that we have binary outcome data • There are many possible link functions - any smooth, monotone function mapping from < → [0, 1] (i.e., cumulative distribution functions for continuous densities) 21 • We simulated data from a logistic regression model with xi = (1, dosei), where dosei ∼ Uniform(0, 1), i = 1,..., 100.