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Multivariable Model - Building: A Pragmatic Approach to Regression Anaylsis based on Fractional Polynomials for Modelling Continuous Variables

Multivariable Model - Building: A Pragmatic Approach to Regression Anaylsis based on Fractional Polynomials for Modelling Continuous Variables

Patrick Royston, Willi Sauerbrei

ISBN: 978-0-470-02842-1 May 2008 322 Pages


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Multivariable regression models are of fundamental importance in all areas of science in which empirical data must be analyzed. This book proposes a systematic approach to building such models based on standard principles of statistical modeling. The main emphasis is on the fractional polynomial method for modeling the influence of continuous variables in a multivariable context, a topic for which there is no standard approach. Existing options range from very simple step functions to highly complex adaptive methods such as multivariate splines with many knots and penalisation. This new approach, developed in part by the authors over the last decade, is a compromise which promotes interpretable, comprehensible and transportable models.

1. Introduction.

1.1 Real-Life Problems as Motivation for Model Building.

1.2 Issues in Modelling Continuous Predictors.

1.3 Types of Regression Model Considered.

1.4 Role of Residuals.

1.5 Role of Subject-Matter Knowledge in Model Development.

1.6 Scope of Model Building in our Book.

1.7 Modelling Preferences.

1.8 General Notation.

2. Selection of Variables.

2.1 Introduction.

2.2 Background.

2.3 Preliminaries for a Multivariable Analysis.

2.4 Aims of Multivariable Models.

2.5 Prediction: Summary Statistics and Comparisons.

2.6 Procedures for Selecting Variables.

2.7 Comparison of Selection Strategies in Examples.

2.8 Selection and Shrinkage.

2.9 Discussion.

3. Handling Categorical and Continuous Predictors.

3.1 Introduction.

3.2 Types of Predictor.

3.3 Handling Ordinal Predictors.

3.4 Handling Counting and Continuous Predictors: Categorization.

3.5 Example: Issues in Model Building with Categorized Variables.

3.6 Handling Counting and Continuous Predictors: Functional Form.

3.7 Empirical Curve Fitting.

3.8 Discussion.

4. Fractional Polynomials for One Variable.

4.1 Introduction.

4.2 Background.

4.3 Definition and Notation.

4.4 Characteristics.

4.5 Examples of Curve Shapes with FP1 and FP2 Functions.

4.6 Choice of Powers.

4.7 Choice of Origin.

4.8 Model Fitting and Estimation.

4.9 Inference.

4.10 Function Selection Procedure.

4.11 Scaling and Centering.

4.12 FP Powers as Approximations to Continuous Powers.

4.13 Presentation of Fractional Polynomial Functions.

4.14 Worked Example.

4.15 Modelling Covariates with a Spike at Zero.

4.16 Power of Fractional Polynomial Analysis.

4.17 Discussion.

5. Some Issues with Univariate Fractional Polynomial Models.

5.1 Introduction.

5.2 Susceptibility to Influential Covariate Observations.

5.3 A Diagnostic Plot for Influential Points in FP Models.

5.4 Dependence on Choice of Origin.

5.5 Improving Robustness by Preliminary Transformation.

5.6 Improving Fit by Preliminary Transformation.

5.7 Higher Order Fractional Polynomials.

5.8 When Fractional Polynomial Models are Unsuitable.

5.9 Discussion.

6. MFP: Multivariable Model-Building with Fractional Polynomials.

6.1 Introduction.

6.2 Motivation.

6.3 The MFP Algorithm.

6.4 Presenting the Model.

6.5 Model Criticism.

6.6 Further Topics.

6.7 Further Examples.

6.8 Simple Versus Complex Fractional Polynomial Models.

6.9 Discussion.

7. Interactions.

7.1 Introduction.

7.2 Background.

7.3 General Considerations.

7.4 The MFPI Procedure.

7.5 Example 1: Advanced Prostate Cancer.

7.6 Example 2: GBSG Breast Cancer Study.

7.7 Categorization.

7.8 STEPP.

7.9 Example 3: Comparison of STEPP with MFPI.

7.10 Comment on Type I Error of MFPI.

7.11 Continuous-by-Continuous Interactions.

7.12 Multi-Category Variables.

7.13 Discussion.

8. Model Stability.

8.1 Introduction.

8.2 Background.

8.3 Using the Bootstrap to Explore Model Stability.

8.4 Example 1: Glioma Data.

8.5 Example 2: Educational Body-Fat Data.

8.6 Example 3: Breast Cancer Diagnosis.

8.7 Model Stability for Functions.

8.8 Example 4: GBSG Breast Cancer Data.

8.9 Discussion.

9. Some Comparisons of MFP with Splines.

9.1 Introduction.

9.2 Background.

9.3 MVRS: A Procedure for Model Building with Regression Splines.

9.4 MVSS: A Procedure for Model Building with Cubic Smoothing Splines.

9.5 Example 1: Boston Housing Data.

9.6 Example 2: GBSG Breast Cancer Study.

9.7 Example 3: Pima Indians.

9.8 Example 4: PBC.

9.9 Discussion.

10. How To Work with MFP.

10.1 Introduction.

10.2 The Dataset.

10.3 Univariate Analyses.

10.4 MFP Analysis.

10.5 Model Criticism.

10.6 Stability Analysis.

10.7 Final Model.

10.8 Issues to be Aware of .

10.9 Discussion.

11. Special Topics Involving Fractional Polynomials.

11.1 Time-Varying Hazard Ratios in the Cox Model.

11.2 Age-specific Reference Intervals.

11.3 Other Topics.

12. Epilogue.

12.1 Introduction.

12.2 Towards Recommendations for Practice.

12.3 Omitted Topics and Future Directions.

12.4 Conclusion.

Appendix A: Data and Software Resources.

A.1 Summaries of Datasets.

A.2 Datasets used more than once.

A.2.1 Research Body Fat.

A.2.2 GBSG Breast Cancer.

A.2.3 Educational Body Fat.

A.2.4 Glioma.

A.2.5 Prostate Cancer.

A.2.6 Whitehall I.

A.2.7 PBC.

A.2.8 Oral Cancer.

A.2.9 Kidney Cancer.

A.3 Software.

Appendix B: Glossary of Abbreviations.



“This new approach, developed in part by the authors over the last decade, is a compromise which promotes interpretable, comprehensible and transportable models.”  (Zentralblatt Math, 1 October 2013)

“The book is very useful for practicing statisticians and can also be recommended for teaching purposes.” (Biometrical Journal, July 2009)

“It is an excellent book on multivariable model-building, presenting the material in an easy-to-understand and informal style.” (Mathematical Reviews, 2009)

"This excellent book fills a gap in the current literature on statistical modelling. It is the first time that a book is devoted to the whole breadth of application of fractional polynomials. The authors are the experts on this useful methodology." (Statistics in Medicine, Feb 2009)