Model Selection in GLMs. (should be able to implement frequentist GLM analyses!) Today: standard frequentist methods for model selection

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1 Model Selection in GLMs Last class: estimability/identifiability, analysis of deviance, standard errors & confidence intervals (should be able to implement frequentist GLM analyses!) Today: standard frequentist methods for model selection 1

2 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

3 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 predictor, then that predictor is removed (backwards elimination) Implementation: step() function in S-PLUS. 3

4 Illustration Suppose we have 30 candidate predictors, X 1,..., X 30. The true model is logit Pr(y i = 1 x i ) = x iβ, where x i = (x i1,..., x i,30 ) (i = 1,..., n = 100) Note that predictors with β = 0 are effectively not in the model Let γ j = 1(β j 0) be a 0/1 indicator that the jth predictor is included 4

5 Interest focuses on selection of the subset of q p important predictors out of the p candidate predictors: Xγ = {X j : γ j = 1, β j 0} (q 1). There is a list of 2 p possible models corresponding to the different subsets Xγ X. The size of the model list, 2 p, is calculated by noting that we can either include or exclude each of the p candidate predictors. 5

6 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 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 (2 30 = ) 6

7 Returning to the logistic regression with 30 candidate predictors example, we simulate data & run stepwise selection Simulation conditions: n = 100, q = 5, p = 30, x i N p (0, I p ), X 1 X 5 have coefficients βγ = (1, 2, 3, 4, 5)/2, X 6 X 30 have coefficients 0. S-PLUS implementation: n< p<- 30 # sample size # 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

8 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

9 Comments: start with full model & performs backwards elimination - 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

10 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

11 Step Df Deviance Resid. Df Resid. Dev AIC X X X X X X X X X X X X

12 14 - X X X X X X

13 Selected model contains predictors X 1 X 5 belonging to true model (we got lucky & coefficients large) Also contains predictors X 12, X 15, X 17, X 22, X 25, X 26, X 28, which should have been excluded (not so good) Results of maximum likelihood estimation for true model: Value Std. Error t value (Intercept) X X X X X

14 Results of maximum likelihood estimation for full model (focusing on important predictors): Value Std. Error t value (Intercept) X X X X X Full model MLEs differ somewhat from MLEs under true model and standard errors are higher 14

15 Now, consider the estimates from the final selected model: Value Std. Error t value (Intercept) X X X X X X X X X X X X

16 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 unimportant 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 important predictors, we would have missed one or more of them (based on repeating simulation) 16

17 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

18 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

19 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): AIC M = D M + 2p M, where D M is the deviance for model M and p M is the number of predictors 2. Bayesian Information Criterion (BIC): BIC M = D M + p M log(n). 19

20 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

21 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 R [0, 1] (i.e., cumulative distribution functions for continuous densities) 21

22 We simulated data from a logistic regression model with x i = (1, dose i ), where dose i Uniform(0, 1), i = 1,..., 100. The parameters were chosen to be β = ( 3, 5) As an alternative to the logistic model, we considered the probit: Pr(y i = 1 x i ) = Φ(x iβ), where Φ(z) = z (2π) 1/2 exp( z 2 /2) is the standard normal cdf We also considered the complementary log-log model: Pr(y i = 1 x i ) = 1 exp { exp(x iβ)}. 22

23 Pr(response) dose 23

24 Summary of results Logistic regression results: β = ( 3.67, 6.14), AIC = , BIC = Probit regression results: β = ( 2.14, 3.60), AIC = , BIC = Complementary log-log results: β = ( 3.14, 4.39), AIC = BIC = Low values of AIC & BIC are preferred, so we select the probit model (which happens to be wrong) 24

25 Some things to think about It is very often the case that many models are consistent with the data This is particularly true when there is a large number of models in the list of plausible models Ideally, substantive information can be brought to bear to reduce the size of the list For example, certain models may be more consistent with biology 25

26 However, one is typically still faced with many possible models & concerned about sensitivity of inferences to the model chosen or selected by some algorithm Given the selection bias that occurs, it may be better to focus a priori on a single model rather than run stepwise selection However, better yet would be to formally account for model uncertainty (e.g., through Bayesian model averaging) 26

27 DDE and Pre-Term Birth Example (Longnecker et al., 2001, Lancet) Scientific Interest: Association between DDE exposure (continuous predictor) and preterm birth (binary response). Previous Data: DDT is a reproductive toxin in birds, rabbits and possibly sea lions. In humans, DDT has been associated with preterm birth in small studies. 27

28 Current Data: Data drawn from the US Collaborative Perinatal Project (CPP) - complete data for n = 2380 children out of which 361 were born preterm. Possibly Confounding Factors: Study center, serum triglycerides and cholesterol, infant ethnicity and gender, mother s age, height, body mass index before pregnancy, rate of weight gain during pregnancy, parity, socioeconomic status, and smoking during pregnancy. 28

29 Notation: y i = 1 if preterm birth and y i = 0 if full-term birth d i = dose of dde for ith woman (i = 1,..., n) z i = (z i1,..., z ip ) = vector of confounding factors x i = vector of predictors (i.e., covariates) What model should we use to investigate the association between dose of dde and the probability of preterm birth? 29

30 Logistic Regression Analysis log [ Pr(y i = 1 x i )] = βdi + α 1 z i α p z ip = βd i + z Pr(y i = 0 x i ) iα where β is the increase in the log-odds of a response associated with a unit increase in dose of dde This log-odds ratio is adjusted for possible confounding variables z i 30

31 Typically, investigators are interested in the exposure odds ratio: Pr(y i = 1 x i = c, z i )/Pr(y i = 0 x i = c, z i ) Pr(y i = 1 x i = c 1, z i )/Pr(y i = 0 x i = c 1, z i ) associated with a unit increase in exposure (e.g., dde dose) Exponentiating the logit model, the exposure odds ratio is equal to exp{βc + z iα} exp{β(c 1) + z iα} = exp(β), which is simply the exponentiated regression coefficient for exposure 31

32 Rare Disease Assumption Often, investigators are interested in the Relative Risk, RR = Pr(y i = 1 x i = c, z i ) Pr(y i = 1 x i = c 1, z i ), which (unfortunately) depends on the value of c However, under the assumption that the disease (i.e., the response) is rare, Pr(y i = 1 x i = c, z i ) Pr(y i = 1 x i = c, z i ) Pr(y i = 0 x i = c, z i ), and it follows that the Relative Risk is approximately equal to the exposure odds ratio 32

33 Returning to the Longnecker et al. example, what assumption about the dose-response are we making by using the model: log [ Pr(y i = 1 x i )] = βdi + z Pr(y i = 0 x i ) iα Do they use this model in the published analysis? 33