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From the preface:

“[…] Our general objective is to provide a readable text giving the rationale of, and practical advice on, a unified approach to multivariable modelling which aims to make such models simpler and more effective. […] No multivariable model-building strategy has rigorous theoretical underpinnings. Even those approaches most used in practice have not had their properties studied adequately by simulation. In particular, handling continuous variables in a multivariable context has largely been ignored. Since there is no consensus among researchers on the ‘best’ strategy, a pragmatic approach is required. Our book reflects our views derived from wide experience. The text assumes a basic understanding of multiple regression modelling, but it can be read without detailed mathematical knowledge. […] As expressed in a very readable paper by Chatfield (2002), we aim to ‘encourage and guide practitioners, and also to counterbalance a literature that can be overly concerned with theoretical matters far removed from the day-to-day concerns of many working statisticians’. […]“

 

Table of Contents

1	Introduction			1
	1.1	Real-Life Problems as Motivation for Model Building		1
		1.1.1	Many Candidate Models	1
		1.1.2	Functional Form for Continuous Predictors	2
		1.1.3	Example 1: Continuous Response	2
		1.1.4	Example 2: Multivariable Model for Survival Data	5
	1.2	Issues in Modelling Continuous Predictors		8
		1.2.1	Effects of Assumptions	8
		1.2.2	Global versus Local Influence Models	9
		1.2.3	Disadvantages of Fractional Polynomial Modelling	9
		1.2.4	Controlling Model Complexity	10
	1.3	Types of Regression Model Considered		10
		1.3.1	Normal-Errors Regression	10
		1.3.2	Logistic Regression	12
		1.3.3	Cox Regression	12
		1.3.4	Generalized Linear Models	14
		1.3.5	Linear and Additive Predictors	14
	1.4	Role of Residuals		15
		1.4.1	Uses of Residuals	15
		1.4.2	Graphical Analysis of Residuals	15
	1.5	Role of Subject-Matter Knowledge in Model Development		16
	1.6	Scope of Model Building in our Book		17
	1.7	Modelling Preferences		18
		1.7.1	General Issues	18
		1.7.2	Criteria for a Good Model	18
		1.7.3	Personal Preferences	19
	1.8	General Notation		20

2	Variable Selection			23
	2.1	Introduction		23
	2.2	Background		24
	2.3	Preliminaries for a Multivariable Analysis		25
	2.4	Aims of Multivariable Models		26
	2.5	Prediction: Summary Statistics and Comparisons		29
	2.6	Procedures for Selecting Variables		29
		2.6.1	Strength of Predictors	30
		2.6.2	Stepwise Procedures	31
		2.6.3	All-Subsets Model Selection Using Information Criteria	32
		2.6.4	Further Considerations	33
	2.7	Comparison of Selection Strategies in Examples		35
		2.7.1	Myeloma Study	35
		2.7.2	Educational Body-Fat Data	36
		2.7.3	Glioma Study	38
	2.8	Selection and Shrinkage		40
		2.8.1	Selection Bias	40
		2.8.2	Simulation Study	40
		2.8.3	Shrinkage to Correct for Selection Bias	42
		2.8.4	Post-estimation Shrinkage	44
		2.8.5	Reducing Selection Bias	45
		2.8.6	Example	46
	2.9	Discussion		47
		2.9.1	Model Building in Small Datasets	47
		2.9.2	Full, Pre-specified or Selected Model?	47
		2.9.3	Comparison of Selection Procedures	49
		2.9.4	Complexity, Stability and Interpretability	49
		2.9.5	Conclusions and Outlook	50

3	Handling Categorical and Continuous Predictors			53
	3.1	Introduction		53
	3.2	Types of Predictor		54
		3.2.1	Binary	54
		3.2.2	Nominal	54
		3.2.3	Ordinal, Counting, Continuous	55
		3.2.4	Derived	55
	3.3	Handling Ordinal Predictors		55
		3.3.1	Coding Schemes	55
		3.3.2	Effect of Coding Schemes on Variable Selection	56
	3.4	Handling Counting and Continuous Predictors: Categorization		58
		3.4.1	‘Optimal’ Cutpoints: A Dangerous Analysis	58
		3.4.2	Other Ways of Choosing a Cutpoint	59
	3.5	Example: Issues in Model Building with Categorized Variables		60
		3.5.1	One Ordinal Variable	61
		3.5.2	Several Ordinal Variables	62
	3.6	Handling Counting and Continuous Predictors: Functional Form		64
		3.6.1	Beyond Linearity	64
		3.6.2	Does Nonlinearity Matter?	65
		3.6.3	Simple versus Complex Functions	66
		3.6.4	Interpretability and Transportability	66
	3.7	Empirical Curve Fitting		67
		3.7.1	General Approaches to Smoothing	68
		3.7.2	Critique of Local and Global Influence Models	68
	3.8	Discussion		69
		3.8.1	Sparse Categories	69
		3.8.2	Choice of Coding Scheme	69
		3.8.3	Categorizing Continuous Variables	70
		3.8.4	Handling Continuous Variables	70

4	Fractional Polynomials for One Variable			71
	4.1	Introduction		72
	4.2	Background		72
		4.2.1	Genesis	72
		4.2.2	Types of Model	73
		4.2.3	Relation to Box–Tidwell and Exponential Functions	73
	4.3	Definition and Notation		74
		4.3.1	Fractional Polynomials	74
		4.3.2	First Derivative	74
	4.4	Characteristics		75
		4.4.1	FP1 and FP2 Functions	75
		4.4.2	Maximum or Minimum of a FP2 Function	75
	4.5	Examples of Curve Shapes with FP1 and FP2 Functions		76
	4.6	Choice of Powers		78
	4.7	Choice of Origin		79
	4.8	Model Fitting and Estimation		79
	4.9	Inference		79
		4.9.1	Hypothesis Testing	79
		4.9.2	Interval Estimation	80
	4.10	Function Selection Procedure		82
		4.10.1	Choice of Default Function	82
		4.10.2	Closed Test Procedure for Function Selection	82
		4.10.3	Example	83
		4.10.4	Sequential Procedure	83
		4.10.5	Type I Error and Power of the Function Selection Procedure	84
	4.11	Scaling and Centering		84
		4.11.1	Computational Aspects	84
		4.11.2	Examples	85
	4.12	FP Powers as Approximations to Continuous Powers		85
		4.12.1	Box–Tidwell and Fractional Polynomial Models	85
		4.12.2	Example	85
	4.13	Presentation of Fractional Polynomial Functions		86
		4.13.1	Graphical	86
		4.13.2	Tabular	87
	4.14	Worked Example		89
		4.14.1	Details of all Fractional Polynomial Models	89
		4.14.2	Function Selection	90
		4.14.3	Details of the Fitted Model	90
		4.14.4	Standard Error of a Fitted Value	91
		4.14.5	Fitted Odds Ratio and its Confidence Interval	91
	4.15	Modelling Covariates with a Spike at Zero		92
	4.16	Power of Fractional Polynomial Analysis		94
		4.16.1	Underlying Function Linear	95
		4.16.2	Underlying Function FP1 or FP2	95
		4.16.3	Comment	96
	4.17	Discussion		97

5	Some Issues with Univariate Fractional Polynomial Models			71
	5.1	Introduction		99
	5.2	Susceptibility to Influential Covariate Observations		100
	5.3	A Diagnostic Plot for Influential Points in FP Models		100
		5.3.1	Example 1: Educational Body-Fat Data	101
		5.3.2	Example 2: Primary Biliary Cirrhosis Data	101
	5.4	Dependence on Choice of Origin		103
	5.5	Improving Robustness by Preliminary Transformation		105
		5.5.1	Example 1: Educational Body-Fat Data	106
		5.5.2	Example 2: PBC Data	107
		5.5.3	Practical Use of the Pre-transformation gδ(x)	107
	5.6	Improving Fit by Preliminary Transformation		108
		5.6.1	Lack of Fit of Fractional Polynomial Models	108
		5.6.2	Negative Exponential Pre-transformation	108
	5.7	Higher Order Fractional Polynomials		109
		5.7.1	Example 1: Nerve Conduction Data	109
		5.7.2	Example 2: Triceps Skinfold Thickness	110
	5.8	When Fractional Polynomial Models are Unsuitable		111
		5.8.1	Not all Curves are Fractional Polynomials	111
		5.8.2	Example: Kidney Cancer	112
	5.9	Discussion		113

6	MFP: Multivariable Model-Building with Fractional Polynomials			115
	6.1	Introduction		115
	6.2	Motivation		116
	6.3	The MFP Algorithm		117
		6.3.1	Remarks	118
		6.3.2	Example	118
	6.4	Presenting the Model		120
		6.4.1	Parameter Estimates	120
		6.4.2	Function Plots	121
		6.4.3	Effect Estimates	121
	6.5	Model Criticism		123
		6.5.1	Function Plots	123
		6.5.2	Graphical Analysis of Residuals	124
		6.5.3	Assessing Fit by Adding More Complex Functions	125
		6.5.4	Consistency with Subject-Matter Knowledge	129
	6.6	Further Topics		129
		6.6.1	Interval Estimation	129
		6.6.2	Importance of the Nominal Significance Level	130
		6.6.3	The Full MFP Model	131
		6.6.4	A Single Predictor of Interest	132
		6.6.5	Contribution of Individual Variables to the Model Fit	134
		6.6.6	Predictive Value of Additional Variables	136
	6.7	Further Examples		138
		6.7.1	Example 1: Oral Cancer	138
		6.7.2	Example 2: Diabetes	139
		6.7.3	Example 3: Whitehall I	140
	6.8	Simple Versus Complex Fractional Polynomial Models		144
		6.8.1	Complexity and Modelling Aims	144
		6.8.2	Example: GBSG Breast Cancer Data	144
	6.9	Discussion		146
		6.9.1	Philosophy of MFP	147
		6.9.2	Function Complexity, Sample Size and Subject-Matter Knowledge	148
		6.9.3	Improving Robustness by Preliminary Covariate Transformation	148
		6.9.4	Conclusion and Future	149

7	Interactions			151
	7.1	Introduction		151
	7.2	Background		152
	7.3	General Considerations		152
		7.3.1	Effect of Type of Predictor	152
		7.3.2	Power	153
		7.3.3	Randomized Trials and Observational Studies	153
		7.3.4	Predefined Hypothesis or Hypothesis Generation	153
		7.3.5	Interactions Caused by Mismodelling Main Effects	154
		7.3.6	The ‘Treatment–Effect’ Plot	154
		7.3.7	Graphical Checks, Sensitivity and Stability Analyses	154
		7.3.8	Cautious Interpretation is Essential	155
	7.4	The MFPI Procedure		155
		7.4.1	Model Simplification	156
		7.4.2	Check of the Results and Sensitivity Analysis	156
	7.5	Example 1: Advanced Prostate Cancer		157
		7.5.1	The Fitted Model	158
		7.5.2	Check of the Interactions	160
		7.5.3	Final Model	161
		7.5.4	Further Comments and Interpretation	162
		7.5.5	FP Model Simplification	163
	7.6	Example 2: GBSG Breast Cancer Study		163
		7.6.1	Oestrogen Receptor Positivity as a Predictive Factor	163
		7.6.2	A Predefined Hypothesis: Tamoxifen–Oestrogen Receptor Interaction	163
	7.7	Categorization		165
		7.7.1	Interaction with Categorized Variables	165
		7.7.2	Example: GBSG Study	166
	7.8	STEPP		167
	7.9	Example 3: Comparison of STEPP with MFPI		168
		7.9.1	Interaction in the Kidney Cancer Data	168
		7.9.2	Stability Investigation	168
	7.10	Comment on Type I Error of MFPI		171
	7.11	Continuous-by-Continuous Interactions		172
		7.11.1	Mismodelling May Induce Interaction	173
		7.11.2	MFPIgen: An FP Procedure to Investigate Interactions	174
		7.11.3	Examples of MFPIgen	175
		7.11.4	Graphical Presentation of Continuous-by-Continuous Interactions	179
		7.11.5	Summary	180
	7.12	Multi-Category Variables		181
	7.13	Discussion		181

8	Model Stability			183
	8.1	Introduction		183
	8.2	Background		184
	8.3	Using the Bootstrap to Explore Model Stability		185
		8.3.1	Selection of Variables within a Bootstrap Sample	185
		8.3.2	The Bootstrap Inclusion Frequency and the Importance of a Variable	186
	8.4	Example 1: Glioma Data		186
	8.5	Example 2: Educational Body-Fat Data		188
		8.5.1	Effect of Influential Observations on Model Selection	189
	8.6	Example 3: Breast Cancer Diagnosis		190
	8.7	Model Stability for Functions		191
		8.7.1	Summarizing Variation between Curves	191
		8.7.2	Measures of Curve Instability	192
	8.8	Example 4: GBSG Breast Cancer Data		193
		8.8.1	Interdependencies among Selected Variables and Functions in Subsets	193
		8.8.2	Plots of Functions	193
		8.8.3	Instability Measures	195
		8.8.4	Stability of Functions Depending on Other Variables Included	196
	8.9	Discussion		197
		8.9.1	Relationship between Inclusion Fractions	198
		8.9.2	Stability of Functions	198

9	Some Comparisons of MFP with Splines			201
	9.1	Introduction		201
	9.2	Background		202
	9.3	MVRS: A Procedure for Model Building with Regression Splines		203
		9.3.1	Restricted Cubic Spline Functions	203
		9.3.2	Function Selection Procedure for Restricted Cubic Splines	205
		9.3.3	The MVRS Algorithm	205
	9.4	MVSS: A Procedure for Model Building with Cubic Smoothing Splines		205
		9.4.1	Cubic Smoothing Splines	205
		9.4.2	Function Selection Procedure for Cubic Smoothing Splines	206
		9.4.3	The MVSS Algorithm	206
	9.5	Example 1: Boston Housing Data		207
		9.5.1	Effect of Reducing the Sample Size	208
		9.5.2	Comparing Predictors	212
	9.6	Example 2: GBSG Breast Cancer Study		214
	9.7	Example 3: Pima Indians		215
	9.8	Example 4: PBC		217
	9.9	Discussion		219
		9.9.1	Splines in General	220
		9.9.2	Complexity of Functions	221
		9.9.3	Optimal Fit or Transferability?	221
		9.9.4	Reporting of Selected Models	221
		9.9.5	Conclusion	222

10	How ToWork with MFP			223
	10.1	Introduction		223
	10.2	The Dataset		223
	10.3	Univariate Analyses		226
	10.4	MFP Analysis		227
	10.5	Model Criticism		228
		10.5.1	Function Plots	228
		10.5.2	Residuals and Lack of Fit	228
		10.5.3	Robustness Transformation and Subject-Matter Knowledge	229
		10.5.4	Diagnostic Plot for Influential Observations	230
		10.5.5	Refined Model	231
		10.5.6	Interactions	231
	10.6	Stability Analysis		232
	10.7	Final Model		235
	10.8	Issues to be Aware of		235
		10.8.1	Selecting the Main-Effects Model	235
		10.8.2	Further Comments on Stability	236
		10.8.3	Searching for Interactions	238
	10.9	Discussion		238

11	Special Topics Involving Fractional Polynomials			241
	11.1	Time-Varying Hazard Ratios in the Cox Model		241
		11.1.1	The Fractional Polynomial Time Procedure	242
		11.1.2	The MFP Time Procedure	243
		11.1.3	Prognostic Model with Time-Varying Effects for Patients with Breast Cancer	243
		11.1.4	Categorization of Survival Time	245
		11.1.5	Discussion	246
	11.2	Age-specific Reference Intervals		247
		11.2.1	Example: Fetal growth	247
		11.2.2	Using FP Functions as Smoothers	248
		11.2.3	More Sophisticated Distributional Assumptions	249
		11.2.4	Discussion	249
	11.3	Other Topics		250
		11.3.1	Quantitative Risk Assessment in Developmental Toxicity Studies	250
		11.3.2	Model Uncertainty for Functions	251
		11.3.3	Relative Survival	252
		11.3.4	Approximating Smooth Functions	253
		11.3.5	Miscellaneous Applications	254

12	Epilogue			255
	12.1	Introduction		255
	12.2	Towards Recommendations for Practice		255
		12.2.1	Variable Selection Procedure	255
		12.2.2	Functional Form for Continuous Covariates	257
		12.2.3	Extreme Values or Influential Points	257
		12.2.4	Sensitivity Analysis	257
		12.2.5	Check for Model Stability	258
		12.2.6	Complexity of a Predictor	258
		12.2.7	Check for Interactions	258
	12.3	Omitted Topics and Future Directions		258
		12.3.1	Measurement Error in Covariates	258
		12.3.2	Meta-analysis	258
		12.3.3	Multi-level (Hierarchical) Models	259
		12.3.4	Missing Covariate Data	259
		12.3.5	Other Types of Model	259
	12.4	Conclusion		259

Appendix A: Data and Software Resources				261
	A.1	Summaries of Datasets		261
	A.2	Datasets used more than once		262
	A.3	Software		267

Appendix B: Glossary of Abbreviations				269
References				271
Index				285

 

Datasets and some information are available for download here

Datasets in available formats – Stata – SAS – Excel – ASCII
For more details about the data see the Appendix A of the book.