Statistics 262: Intermediate Biostatistics Model selection
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1 Statistics 262: Intermediate Biostatistics Model selection Jonathan Taylor & Kristin Cobb Statistics 262: Intermediate Biostatistics p.1/??
2 Today s class Model selection. Strategies for model selection. Model selection in survival analysis. Statistics 262: Intermediate Biostatistics p.2/??
3 Model selection Up to now, we had a fixed model which we presumed was good, in both regression and survival models. In applied settings, we may be faced with MANY covariates, some of which may not be a priori related to the outcome of interest. Model selection is the process of building a model from possible many covariates. Statistics 262: Intermediate Biostatistics p.3/??
4 Model selection: goals Which main effects do we include? Which interactions do we include? The previous two steps define a collection of models: we need an algorithm to choose a model from this collection. Statistics 262: Intermediate Biostatistics p.4/??
5 Model selection: general This is an unsolved problem in statistics: there are no magic procedures to get you the best model. In some sense, model selection is data mining. Data miners / machine learners often work with very many predictors. Statistics 262: Intermediate Biostatistics p.5/??
6 Model selection: strategies Model selection can be done according to a fixed set of rules (as in Hosmer & Lemeshow): purposeful selection of variables. Alternatively, can be done algorithmically. To automate the procedure, we first need a criterion or benchmark to compare two models. We also need a search strategy. Statistics 262: Intermediate Biostatistics p.6/??
7 Purposeful selections 1. First, fit all models with only one predictor variable. 2. Fit a model with all variables that had p-value less than 0.2 in previous step. 3. Add variables not added in previous step, one at a time to the multivariable model to see if we have missed any important confounders. Statistics 262: Intermediate Biostatistics p.7/??
8 Continuous covariates 1. Plot #1: (X i, M i ) where M i are the martingale residuals, add a lowess smooth to get an idea of the functional relationship. 2. Plot #2: (X i, log(δ i,sm /Ĥi,sm)). 3. Above, δ i,sm is a smoothed version of the plot (X i, δ i ) and Ĥi,sm is a smoothed version of the plot (X i, Ĥi). Statistics 262: Intermediate Biostatistics p.8/??
9 WHAS: length of stay Martingale residual Length of stay Statistics 262: Intermediate Biostatistics p.9/??
10 WHAS: length of stay Smoothed log(c/h) Length of stay Statistics 262: Intermediate Biostatistics p.10/??
11 WHAS: age Martingale residual Length of stay Statistics 262: Intermediate Biostatistics p.11/??
12 WHAS: age Smoothed log(c/h) Age Statistics 262: Intermediate Biostatistics p.12/??
13 Interactions 1. Interactions should be added after determining scales of continuous covariates. 2. In the purposeful search, H&L suggest only including plausible interactions, and include all interactions significant at a level of 0.05 or lower. 3. Suggest following rule of including both terms of an interaction, even if one of the main effects is not significant. Statistics 262: Intermediate Biostatistics p.13/??
14 Stepwise search (likelihood tests) Includes / deletes variables based on partial likelihood tests. Choose p E < p R (E for entry, R for reject). Works for many different regression model selection not just survival analysis. Statistics 262: Intermediate Biostatistics p.14/??
15 Stepwise search (likelihood tests) Start with a model with no coefficients, just an intercept. For each covariate {X 1,..., X p } fit model with just one covariate. If any covariate has p-value less than p E choose covariate with smallest p-value. Statistics 262: Intermediate Biostatistics p.15/??
16 Stepwise search (likelihood tests) Assume we have chosen a variable, X i For each covariate X j {X 1,..., X p }\{X i } fit a model with two covariates: X i, X j. If any partial likelihood tests have p-values less than p E choose covariate with the smallest p-value. Statistics 262: Intermediate Biostatistics p.16/??
17 Stepwise search (likelihood tests) Assume we now have two variables X i1, X i2. Test whether any variable can be dropped, by looking at partial likelihood tests: drop variable with largest p-value > p R. After dropping (or not), check to see if any variable can be added with p-value less than p E. Repeat until no variables can be added or dropped. Statistics 262: Intermediate Biostatistics p.17/??
18 Variations Begin at an initially well-chosen model, possibly including treatment + some confounders. Search only forward or only backward. Statistics 262: Intermediate Biostatistics p.18/??
19 Best subsets regression If the number of predictors (including interactions) is not too large, then it is possible to fit all models of a certain size. Models are compared (within a fixed size) on the basis of the score test χ 2 statistic. This identifies best model of a given size, but does not allow you to compare models of different sizes. Statistics 262: Intermediate Biostatistics p.19/??
20 SAS example: WHAS stepwise PROC PHREG DATA=WHASMOD; MODEL LENFOL*CENSOR(0) = AGE SEXNUM CPK SHONUM CHFNUM LENSTAY / SELECTION=STEPWISE DETAILS; RUN; PROC PHREG DATA=WHASMOD; MODEL LENFOL*CENSOR(0) = AGE SEXNUM CPK SHONUM CHFNUM LENSTAY / SELECTION=BACKWARD DETAILS; RUN; PROC PHREG DATA=WHASMOD; MODEL LENFOL*CENSOR(0) = AGE SEXNUM CPK SHONUM CHFNUM LENSTAY / SELECTION=FORWARD DETAILS; RUN; Statistics 262: Intermediate Biostatistics p.20/??
21 SAS example: WHAS best subsets PROC PHREG DATA=WHASMOD; MODEL LENFOL*CENSOR(0) = AGE SEXNUM CPK SHONUM CHFNUM LENSTAY / SELECTION=SCORE BEST=3; RUN; PROC PRINT DATA=SELOUT; RUN; Statistics 262: Intermediate Biostatistics p.21/??
22 Akaike Information Criterion An alternative stepwise search uses the AIC (Akaike Information Criterion). AIC(M) = 2 log L(M) + 2 #M where 2 log L(M) is partial likelihood test statistic comparing to null model and #M is the number of parameters in M. AIC is a criterion that penalized larger models, with equal fit. It can be used to compare models of different size. Statistics 262: Intermediate Biostatistics p.22/??
23 AIC continued AIC also defined for many other models: method is not specific to survival analysis data. If we replace 2 df M with log n df M then we get Schwarz s Bayesian Information Criterion (BIC). Other penalties have been proposed in the literature, but AIC and BIC are the most popular. Statistics 262: Intermediate Biostatistics p.23/??
24 Stepwise AIC search Start with a model with no coefficients, just an intercept. For each covariate {X 1,..., X p } fit model with just one covariate. If any models have lower AIC, choose the one with the lowest AIC. Statistics 262: Intermediate Biostatistics p.24/??
25 Stepwise AIC search Next, try to add one of the remaining variables, also try to drop each of the variables in the model. If any models have lower AIC, choose the one with the lowest AIC. Repeat until no further additions / deletions will decrease the AIC converges to a local minimum of AIC. Statistics 262: Intermediate Biostatistics p.25/??
26 Variations Begin at an initially well-chosen model, possibly including treatment + some confounders. Search only forward or only backward. Statistics 262: Intermediate Biostatistics p.26/??
27 Caveats In choosing a model automatically, even if the full model is correct (unbiased) our resulting model may be biased a fact we have ignored so far. Inference (F, χ 2 tests, etc) is not quite correct for biased models. Diagnostics are still necessary! Just because an algorithm tells us it is good doesn t mean it is! Statistics 262: Intermediate Biostatistics p.27/??
28 Alternatives: penalized models Alternatively: we can choose biased models by imposing constraints. Here, large β is interpreted as complex model. Goal is really to penalize complex models, i.e. Occam s razor. Idea is to accept a little bias to get a better model : bias-variance tradeoff. Statistics 262: Intermediate Biostatistics p.28/??
29 Ridge regression Assume that columns (X j ) 1 j p have zero mean, and length 1. Minimize L p,λ (β) = 2 log L p (β) + λ p j=1 where L p is the Cox partial likelihood. Corresponds (through Lagrange multiplier) to a quadratic constraint on β s. LASSO, another penalized regression uses p j=1 β j. β 2 j Statistics 262: Intermediate Biostatistics p.29/??
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