Fractional Polynomials and Model Averaging

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Fractional Polynomials and Model Averaging Paul C Lambert Center for Biostatistics and Genetic Epidemiology University of Leicester UK paul.lambert@le.ac.uk Nordic and Baltic Stata Users Group Meeting, Stockholm 7th September 2007 Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 /28

Fractional Polynomials Fractional Polynomials are used in regression models to fit non-linear functions. Often preferable to cut-points. Functions from fractional polynomials more flexible than from standard polynomials. See (Royston and Altman, 994) or (Sauerbrei and Royston, 999) for more details. Implemented in Stata with fracpoly and mfp commands. Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 2/28

Powers The linear predictor for a fractional polynomial of order M for covariate x can be defined as, β 0 + M β m x pm m= where each power p m is chosen from a restricted set. The usual set of powers is x 0 is taken as ln(x) { 2,, 0.5, 0, 0.5,, 2, 3} Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 3/28

Selecting the Best Fitting Model All combinations of powers are fitted and the best fitting model obtained. Using the default set of powers for an FP2 model there are 8 FP Models 36 FP2 Models (including 8 repeated powers) The best fitting model for fractional polynomials of the same degree can be obtain by minimising the deviance. When comparing models of a different degree, e.g. FP2 and FP models, the model can be selected using a formal significance test or the Akaike Information Criterion (AIC). Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 4/28

Selecting the Best Fitting Model All combinations of powers are fitted and the best fitting model obtained. Using the default set of powers for an FP2 model there are 8 FP Models 36 FP2 Models (including 8 repeated powers) The best fitting model for fractional polynomials of the same degree can be obtain by minimising the deviance. When comparing models of a different degree, e.g. FP2 and FP models, the model can be selected using a formal significance test or the Akaike Information Criterion (AIC). Model selection uncertainty is ignored. Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 4/28

German Breast Cancer Study Group Data 686 women with primary node positive breast cancer (Sauerbrei and Royston, 999). Time to recurrence or death (299 events). Covariates include, Age (years) Menopausal staus Tumour Size (mm) Tumour Grade Number of positive lymph nodes Progesterone Receptor (fmol) Oestrogen Receptor (fmol) Hormonal Therapy Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 5/28

German Breast Cancer Study Group Data 686 women with primary node positive breast cancer (Sauerbrei and Royston, 999). Time to recurrence or death (299 events). Covariates include, Age (years) Menopausal staus Tumour Size (mm) Tumour Grade Number of positive lymph nodes Progesterone Receptor (fmol) Oestrogen Receptor (fmol) Hormonal Therapy 5 covariates were selected using mfp command. Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 5/28

Breast Cancer - Best Fitting Model for Age 5 Best Fitting Model ( 2.5), AIC = 3562.73 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 6/28

Breast Cancer - Best Fitting Model for Age 5 Best Fitting Model ( 2.5), AIC = 3562.73 4 3 2 0 20 40 60 80 age, years FP2(-2-0.5): ln(h(t)) = ln(h 0 (t)) + β Age 2 + β 2 Age 0.5 Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 6/28

Breast Cancer - The 5 Best Fitting Model for Age Powers AIC (-2,-0.5) 3562.73 (-,-) 3562.77 (-2,-) 3562.78 (-2,0) 3562.83 (-2,0.5) 3563.05 Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 7/28

Breast Cancer - Age 5 Best Fitting Model ( 2.5), AIC = 3562.73 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 8/28

Breast Cancer - Age 5 Powers ( ), AIC = 3562.77 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 8/28

Breast Cancer - Age 5 Powers ( 2 ), AIC = 3562.78 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 8/28

Breast Cancer - Age 5 Powers ( 2 0), AIC = 3562.83 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 8/28

Breast Cancer - Age 5 Powers ( 2.5), AIC = 3563.05 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 8/28

Breast Cancer - No. Positive Lymph Nodes Best Fitting Model ( 2), AIC = 3498.99 3 3 5 7 0 5 0 5 20 25 30 35 40 45 50 55 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 9/28

Breast Cancer - No. Positive Lymph Nodes Powers ( ), AIC = 3499.05 3 3 5 7 0 5 0 5 20 25 30 35 40 45 50 55 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 9/28

Breast Cancer - No. Positive Lymph Nodes Powers ( 2.5), AIC = 3499.2 3 3 5 7 0 5 0 5 20 25 30 35 40 45 50 55 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 9/28

Breast Cancer - No. Positive Lymph Nodes Powers ( 2 ), AIC = 3499.44 3 3 5 7 0 5 0 5 20 25 30 35 40 45 50 55 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 9/28

Breast Cancer - No. Positive Lymph Nodes Powers ( 3), AIC = 3499.64 3 3 5 7 0 5 0 5 20 25 30 35 40 45 50 55 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 9/28

Model Averaging In FP models the model selection process is usually ignored when calculating fitted values and their associated confidence intervals. Model Averaging is popular Bayesian research area (Hoeting et al., 999), (Congdon, 2007). Increasing interest from frequentist perspective (Burnham and Anderson, 2004) (Buckland et al., 2007) (Congdon, 2007) (Faes et al., 2007) Usually interest lies in model averaging for a parameter. Here we are interested in averaging over the functional form obtained from different models. Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 0/28

Model Averaging 2 If there are K contending models, M k, k =,..., K with weights, w k, which are scaled so that w k =, then the estimate of a parameter or quantity, θ (assumed to be common to all models) is taken to be, θ a = K w k θk k= The variance of θ a is, ( ) var θa = K k= w 2 k ( ) var ( θk M k + ( θk θ ) ) 2 a Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 /28

Obtaining the Weights, w k In a Bayesian context we want, w k = P(M k Data) These probabilities are not trivial to calculate and various approximations are available. One such approximation is to use the Bayesian Information Criterion (BIC) BIC k = ln(l k ) 2 p ln(n) The AIC can also be used to derive the model weights (Buckland et al., 2007) AIC k = ln(l k ) 2p Recently Faes used the AIC to derive model weights for fractional polynomial models (Faes et al., 2007). Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 2/28

Obtaining the Weights, w k Let k = BIC k BIC min or k = AIC k AIC min The weights, w k, are then defined as, w k = exp ( ) 2 k K j= exp ( ) 2 j Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 3/28

Using Bootstrapping to Obtain the Weights, w k An alternative to using the AIC or BIC for the model weights, w k, is to use bootstrapping (Holländer et al., 2006). For each bootstrap sample the best fitting fractional polynomial model is selected. The weights w k, are simply obtained using the frequencies of the models selected over the B bootstrap samples. If comparing fractional polynomial models of different degrees then some selection process is needed. This is usually done by setting a value for α. Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 4/28

Using fpma Using fpma.. fpma x, ic(aic) xpredict: stcox x Models Included (in order of weight) Powers AIC deltaaic weight cum. weight -2 -.5 3562.73 0.00 0.0802 0.0802 2 - - 3562.77 0.03 0.0789 0.59 3-2 - 3562.78 0.04 0.0785 0.2376 4-2 0 3562.83 0.09 0.0766 0.342 5-2.5 3563.05 0.3 0.0686 0.3827 6 - -.5 3563.05 0.32 0.0685 0.452 7-2 -2 3563.26 0.53 0.066 0.528 8-2 3563.38 0.65 0.0580 0.5709 9-0 3563.50 0.77 0.0546 0.6255 0 -.5 -.5 3563.52 0.79 0.0540 0.6795 (output omitted ) 43 2 3578.8 5.44 0.0000.0000 44 3 3578.32 5.58 0.0000.0000 New variables created xb ma xb ma se xb ma lci xb ma uci Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 5/28

Using fpma - Bootstrapping (α = 0.05) Using fpma with bootstrapping.. fpma x, ic(bootstrap) xpredict xpredname(x_ma_boot) reps(000): stcox x Running 000 bootstrap samples to determine model weights (bootstrap: maboot) Powers Freq. weight cum. weight -2-2 252 0.2520 0.2520 2-2 - 67 0.670 0.490 3 - - 63 0.630 0.5820 4 - -.5 88 0.0880 0.6700 5 7 0.070 0.740 6-2 67 0.0670 0.8080 7 -.5 -.5 63 0.0630 0.870 8-2 -.5 5 0.050 0.9220 9 -.5 0 30 0.0300 0.9520 0 0 0 9 0.090 0.970 0.5 6 0.0060 0.9770 (output omitted ) 22-0 0.000 0.9990 23 -.5.5 0.000.0000 Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 6/28

Breast Cancer - Age 5 Model Average BIC 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 7/28

Breast Cancer - Age 5 Model Average AIC 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 7/28

Breast Cancer - Age 5 Model Average Bootstrap (alpha = 0.05) 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 7/28

Breast Cancer - Age 5 Model Average Bootstrap (alpha = ) 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 7/28

Breast Cancer - No. of Positive Lymph Nodes Model Average BIC 3 3 5 7 0 5 0 5 20 25 30 35 40 45 50 55 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 8/28

Breast Cancer - No. of Positive Lymph Nodes Model Average AIC 3 3 5 7 0 5 0 5 20 25 30 35 40 45 50 55 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 8/28

Breast Cancer - No. of Positive Lymph Nodes Model Average Bootstrap (alpha = 0.05) 3 3 5 7 0 5 0 5 20 25 30 35 40 45 50 55 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 8/28

Breast Cancer - No. of Positive Lymph Nodes Model Average Bootstrap (alpha = ) 3 3 5 7 0 5 0 5 20 25 30 35 40 45 50 55 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 8/28

Breast Cancer - No. of Positive Lymph Nodes 20 2.5 Model Average BIC.5 0.5.5 2 0 5 0 5 20 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 9/28

Breast Cancer - No. of Positive Lymph Nodes 20 2.5 Model Average AIC.5 0.5.5 2 0 5 0 5 20 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 9/28

Breast Cancer - No. of Positive Lymph Nodes 20 2.5 Model Average Bootstrap (alpha = 0.05).5 0.5.5 2 0 5 0 5 20 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 9/28

Breast Cancer - No. of Positive Lymph Nodes 20 2.5 Model Average Bootstrap (alpha = ).5 0.5.5 2 0 5 0 5 20 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 9/28

Multivariable Fractional Polynomials The above only really applies when using fractional polynomials for only one of the covariates in the model. However, is is common to use models with fractional polynomials for more than one covariate. A simple approach is to model average over various fractional polynomial models for the covariate of interest, while keeping the functional form of the remaining covariates constant. The usemfp option will do this for you. Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 20/28

Using mfp with Model Averaging mfp.. mfp stcox x x2 x3 x4a x4b x5 x6 x7 hormon, nohr alpha(.05) select(0.05) (output omitted ) Final multivariable fractional polynomial model for _t Variable Initial Final df Select Alpha Status df Powers x 4 0.0500 0.0500 in 4-2 -.5 x2 0.0500 0.0500 out 0 x3 4 0.0500 0.0500 out 0 x4a 0.0500 0.0500 in x4b 0.0500 0.0500 out 0 x5 4 0.0500 0.0500 in 4-2 - x6 4 0.0500 0.0500 in 2.5 x7 4 0.0500 0.0500 out 0 hormon 0.0500 0.0500 in Cox regression -- Breslow method for ties (output omitted ) Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 2/28

Using fpma after mfp - Age Using fpma after mfp.. fpma x, ic(aic) xpredict: usemfp Models Included (in order of weight) Powers AIC deltaaic weight cum. weight -2 -.5 3434.72 0.00 0.0986 0.0986 2-2 - 3434.76 0.03 0.0969 0.956 3 - - 3434.85 0.3 0.0925 0.288 4-2 0 3434.89 0.7 0.0907 0.3788 5-2.5 3435.24 0.52 0.0760 0.4548 6 - -.5 3435.30 0.58 0.0740 0.5288 7-2 -2 3435.43 0.70 0.0695 0.5982 8-2 3435.75.03 0.0590 0.6572 9-0 3435.96.24 0.053 0.703 0 -.5 -.5 3436.00.27 0.0522 0.7624 (output omitted ) 43 3452.04 7.3 0.0000.0000 44.5 3452.05 7.32 0.0000.0000 Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 22/28

Model Averaging after mfp - Age 5 AIC weights 4 3 2 0 20 40 60 80 age, years Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 23/28

Model Averaging after mfp - No. of Positive Lymph Nodes AIC weights 3 3 5 7 0 5 0 5 20 25 30 35 40 45 50 55 number of positive nodes Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 24/28

Discussion Fractional Polynomials very useful for modelling non-linear functions. Model selection uncertainty is usually ignored after final model is obtained. Model averaging is easy to implement and incorporates FP model selection uncertainty. Still further work needed. For example, Statistical properties (coverage etc). Comparison with fully Bayesian model averaging. Paul C Lambert Fractional Polynomials and Model Averaging Stockholm, 7th September 2007 25/28

References I Buckland, S., Burnham, K., and Augustin, N. (2007). Model selection: An intergral part of inference. Biometrics, 53(2):603 68. Burnham, K. P. and Anderson, D. R. (2004). Multimodal inference: Understanding AIC and BIC in model selection. Sociological Methods and Research, 33(2):26 304. Congdon, P. (2007). Model weights for model choice and averaging. Statistical Methodology, 4:43 57. Faes, C., Aerts, M., H., G., and Molenberghs, G. (2007). Model averaging using fractional polynomials to estimate a safe level of exposure. Risk Analysis, 27(): 23. Hoeting, J. A., Madigan, D., E., R. A., and Volinsky, C. T. (999). Bayesian model averaging: A tutorial. Statistical Science, 4(4):382 47. Holländer, N., Augustin, N., and Sauerbrei, W. (2006). Investigation on the improvement of prediction by bootstrap model averaging. Methods of Information in Medicine, 45:44 50. Royston, P. and Altman, D. (994). Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling. JRSSA, 43(3):429 467. Sauerbrei, W. and Royston, P. (999). Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials. JRSSA, 62():7 94.

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