Regression Modeling Strategies

Author

Affiliation

Department of Biostatistics
School of Medicine
Vanderbilt University

Published

October 13, 2024

flowchart LR
rms[Multivariable Model Development] --> est[Estimation] --> pred[Prediction] --> val[Validation]

Preface

All standard regression models have assumptions that must be verified for the model to have power to test hypotheses and for it to be able to predict accurately. Of the principal assumptions (linearity, additivity, distributional), this course will emphasize methods for assessing and satisfying the first two. Practical but powerful tools are presented for validating model assumptions and presenting model results. This course provides methods for estimating the shape of the relationship between predictors and response using the widely applicable method of augmenting the design matrix using restricted cubic splines. Even when assumptions are satisfied, overfitting can ruin a model’s predictive ability for future observations. Methods for data reduction will be introduced to deal with the common case where the number of potential predictors is large in comparison with the number of observations. Methods of model validation (bootstrap and cross-validation) will be covered, as will auxiliary topics such as modeling interaction surfaces, efficiently utilizing partial covariable data by using multiple imputation, variable selection, overly influential observations, collinearity, and shrinkage, and a brief introduction to the R rms package for handling these problems. The methods covered will apply to almost any regression model, including ordinary least squares, longitudinal models, logistic regression models, ordinal regression, quantile regression, longitudinal data analysis, and survival models. Statistical models will be contrasted with machine learning so that the student can make an informed choice of predictive tools.

Target Audience

Statisticians and persons from other quantitative disciplines who are interested in multivariable regression analysis of univariate responses, in developing, validating, and graphically describing multivariable predictive models and in covariable adjustment in clinical trials. The course will be of particular interest to applied statisticians and developers of applied statistics methodology, graduate students, clinical and pre-clinical biostatisticians, health services and outcomes researchers, econometricians, psychometricians, and quantitative epidemiologists. A good command of ordinary multiple regression is a prerequisite.

Learning Goals

Students will

be able to fit multivariable regression models:
- accurately
- in a way the sample size will allow, without overfitting
- uncovering complex non–linear or non–additive relationships
- testing for and quantifying the association between one or more predictors and the response, with possible adjustment for other factors
- making maximum use of partial data rather than deleting observations containing missing variables
be able to validate models for predictive accuracy and to detect overfitting and understand problems caused by overfitting.
learn techniques of “safe data mining” in which significance levels, confidence limits, and measures such as $R^2$ have the claimed properties.
learn how to interpret fitted models using both parameter estimates and graphics
learn about the advantages of semiparametric ordinal models for continuous $Y$
learn about some of the differences between frequentist and Bayesian approaches to statistical modeling
learn differences between machine learning and statistical models, and how to determine the better approach depending on the nature of the problem

Course Philosophy

The audio narration starts with the third bullet point listed here.

Modeling is the endeavor to transform data into information and information into either prediction or evidence about the data generating mechanism¹
Models are usually the best descriptive statistics
- adjust for one variable while displaying the association with $Y$ and another variable
- descriptive statistics do not work in higher dimensions
Satisfaction of model assumptions improves precision and increases statistical power
- Be aware of assumptions, especially those mattering the most
It is more productive to make a model fit step by step (e.g., transformation estimation) than to postulate a simple model and find out what went wrong
- Model diagnostics are often not actionable
- Changing the model in reaction to observed patterns $\uparrow$ uncertainty but is reflected by an apparent $\downarrow$ in uncertainty
Graphical methods should be married to formal inference
Overfitting occurs frequently, so data reduction and model validation are important
Software without multiple facilities for assessing and fixing model fit may only seem to be user-friendly
Carefully fitting an improper model is better than badly fitting (and overfitting) a well-chosen one
- E.g. small $N$ and overfitting vs. carefully formulated right hand side of model
Methods which work for all types of regression models are the most valuable.
In most research projects the cost of data collection far outweighs the cost of data analysis, so it is important to use the most efficient and accurate modeling techniques, to avoid categorizing continuous variables, and to not remove data from the estimation sample just to be able to validate the model.
- A $100 analysis can make a $1,000,000 study worthless.
The bootstrap is a breakthrough for statistical modeling and model validation.
Bayesian modeling is ready for prime time.
- Can incorporate non-data knowledge
- Provides full exact inferential tools even when penalizing $\beta$
- Rational way to account for model uncertainty
- Direct inference: evidence for all possible values of $\beta$
- More accurate way of dealing with missing data
Using the data to guide the data analysis is almost as dangerous as not doing so.
A good overall strategy is to decide how many degrees of freedom (i.e., number of regression parameters) can be "spent", where they should be spent, to spend them with no regrets. See the excellent text Clinical Prediction Models (Steyerberg, 2009)

¹ Thanks to Drew Levy for ideas that greatly improved this section.

Key Messages of the Course

Consider RMS as practicing a craft, and the RMS text book is the foundational treatise for the craft. Get, read–and periodically re-read–the textbook. There is a lot in the book not covered in the short course. Reading the text will help by providing students a comprehensive tool set, and reinforce the overall coherence in RMS. And there is also a lot of wisdom throughout the textbook.
A fundamental RMS principle of “methods that enable an analyst to develop models that will make accurate predictions of responses for future observations.” The RMS program (philosophy, methodology, tools) is predicated on this.
Phantom degrees of freedom : Information about the relationship between the dependent variable and candidate predictors introduced into model specification or model selection that is not later properly accounted for in the variance estimates and inference that goes into the ultimate report of analysis results. Put another way: assessments made in model development and then later elided that compromise the fidelity of standard errors, alpha, Type-I assertion probability $\alpha$, etc. It is ‘phantom’ not only because it disappears, but also because this information continues to haunt the analysis, typically with overconfidence.
$\chi^{2}$ - df : The potential magnitude of signal, penalized for possibility of accident. Or— the potential magnitude of signal, adjusted for risk of coincidence.
Chunk tests : Simultaneous or aggregate assessment of the importance (e.g., proportion of total variance explained) of collinear or dependent (e.g. related polynomial or interaction) terms or a group of non-collinear candidate predictors. Important for making coherent judgments.
Spending df’s : Given a fixed amount of information in the data available for the analysis there is an ‘information budget’ that should be used judiciously: more important predictors should represented in a richer way (e.g. make the number of knots in splines proportional to the overall importance and complexity in the variable) than predictors that are less important. “Spending df’s with no regrets”: to preserve the operating characteristics of formal inference, once assessed in the modeling process even the less important candidate predictors should remain in view and not elided. All candidate predictors get a portion of the ‘df budget’, but to a varying extent based on their predictive potential.
Partial plots: Pay special attention to partial plots. Partial plots reveal important information about where the information is in predicting the dependent variable. Partial plots elucidate the form and strength of individual predictors (independent of covariate effects) and can reveal the structure in the data and where the information is situated in the model.
Be aware that cognitive biases — and an analysts’ interests, incentives and ethics — are all also a latent (often insidious) part of the overall analysis and reporting process. Be honest with others and yourself.

Written by Drew Levy

Annotating the Notes

For information about adding annotations, comments, and questions inside the text click here: Comments

Symbols Used in the Right Margin of the Text

in the right margin is a hyperlink to a YouTube video related to the subject.
is a hyperlink to the discussion topic in datamethods.org devoted to the specific topic. You can go directly to the discussion about chapter n by going to datamethods.org/rmsn.
An audio player symbol indicates that narration elaborating on the notes is available for the section. Red letters and numbers in the right margin are cues referred to within the audio recordings.
blog in the right margin is a link to a blog entry that further discusses the topic.

Other Information

Discussion board for the overall course and hints for registering
How to annotate and pose questions on web pages
RMS main web site
Twitter: #rmscourse
Publicly available videos of the short course
- Four hour short course from useR!2022
BBR course
Go directly to a YouTube video for RMS Session n by going to bit.ly/yt-rmsn
Study questions
Datamethods discussion board
Statistical papers written for clinical researchers
RMS overview and template for statistical analysis plan
Statistical Thinking blog
Glossary of statistical terms
R Workflow

R Packages

To be able to run all the examples in the book, install current versions of the following CRAN packages:

Hmisc, rms, data.table, nlme, rmsb, ggplot2
kableExtra, pcaPP, VGAM, MASS, leaps, rpart

License

Regression Modeling Strategies Course is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at https://hbiostat.org/rmsc.

Update History

Date	Sections	Changes
2024-10-13	2.9 Bayesian Modeling	Expanded to include collapsible section on uncertainty in model performance metrix, plus problems with CLT and $\delta$-method
2024-08-11	Preface	Added Drew Levy’s key messages
2024-08-02	5.3.8 Validation of Bayesian Models	New subsection on validation issues in Bayesian modeling
2024-04-21	10.11 Bayesian Logistic Model Example	Updated `blrm` from using `keepsep` to `pcontrast`
2024-03-03	5.3.7 Validation of Data Reduction	New section of validatiion of data reduction
2024-02-18	Contrasts	New section on contrasts
2023-09-17	5.1.3 Relative Explained Variation	New subsection on relative explained variation
2023-08-02	3.7.1 Predictive Mean Matching With Constraints	New subsection on constraints for imputed values
2023-08-01	24 Bacteremia: Case Study in Nonlinear Data Reduction with Imputation	New chapter for bacteremia case study
2023-07-28	1.1 Hypothesis Testing, Estimation, and Prediction	Added paired tests
2023-07-21	23 Body Fat: Case Study in Linear Modeling	New chapter: linear model case study
2023-07-14	9.7 AIC & BIC	New material and links for AIC/BIC
2023-05-30	4.12 Summary: Possible Modeling Strategies	Added consideration of confounding
2023-05-24	4.4 Overfitting and Limits on Number of Predictors	Better effective sample size for binary $Y$
2023-05-20	11.4 Regression on Original Variables, Principal Components and Pretransformations, 8.6 Data Reduction Using Principal Components	Added graphical display of PC loadings
2023-04-30	10 Binary Logistic Regression	Many improvements in graphics, and code using data.table
2023-04-29	2.8 Complex Curve Fitting Example	Add likelihood ratio tests
2023-04-22	9.4 The Hauck-Donner Effect	New section on Hauck-Donner effect ruining Wald statistics
2023-04-22	10 Binary Logistic Regression, 12.3 Binary Logistic Model with Casewise Deletion of Missing Values	Added new `anova(..., test='LR')`
2023-03-06	15 Regression Models for Continuous Y and Case Study in Ordinal Regression	Several changes; replaced lattice graphics with ggplot2 and added validation with simultaneous multiple imputation
2023-03-01	5.3.6 Multiple Imputation and Resampling-Based Model Validation	New section on simultaneous validation and imputation
2023-02-20	8 Case Study in Data Reduction, 11 Binary Logistic Regression Case Study 1	Used new Hmisc 5.0-0 function princmp for principal components
2023-02-12		Used rms 6.5-0 to improve code, removing results=‘asis’ from chunk headers
2023-02-07		Started moving study questions to end of chapters
2022-10-28	1 Introduction	3 new flowcharts
2022-10-28	1.6.1 Model Uncertainty and Model Checking	New subsection of model uncertainty and GOF
2022-09-16	9.5 Confidence Intervals	Link to nice profile likelihood CI example

Review Questions

Consider inference from comparing central tendencies of two groups on a continuous response variable Y. What assumptions are you willing to make when selecting a statistical test? Why are you willing to make those assumptions?
Consider the comparison of 5 groups on a continuous Y. Suppose you observe that two of the groups have a similar mean and the other three also have a similar sample mean. What is wrong with combining the two samples and combining the three samples, then comparing two means? How does this compare to stepwise variable selection?
Name a specific statistical test for which we don’t have a corresponding statistical model
Concerning a multi-group problem or a sequential testing problem what is the frequentist approach to multiplicity correction? The Bayesian approach?

```{r include=FALSE} require(Hmisc) options(qproject='rms') getRs('qbookfun.r') ``` <img src="logo.png" width=40%> ```{mermaid} %%| column: screen-inset-right %%| fig-width: 8 flowchart LR rms[Multivariable Model Development] --> est[Estimation] --> pred[Prediction] --> val[Validation] ``` # Preface {.unnumbered} All standard regression models have assumptions that must be verified for the model to have power to test hypotheses and for it to be able to predict accurately. Of the principal assumptions (linearity, additivity, distributional), this course will emphasize methods for assessing and satisfying the first two. Practical but powerful tools are presented for validating model assumptions and presenting model results. This course provides methods for estimating the shape of the relationship between predictors and response using the widely applicable method of augmenting the design matrix using restricted cubic splines. Even when assumptions are satisfied, overfitting can ruin a model's predictive ability for future observations. Methods for data reduction will be introduced to deal with the common case where the number of potential predictors is large in comparison with the number of observations. Methods of model validation (bootstrap and cross-validation) will be covered, as will auxiliary topics such as modeling interaction surfaces, efficiently utilizing partial covariable data by using multiple imputation, variable selection, overly influential observations, collinearity, and shrinkage, and a brief introduction to the `R` `rms` package for handling these problems. The methods covered will apply to almost any regression model, including ordinary least squares, longitudinal models, logistic regression models, ordinal regression, quantile regression, longitudinal data analysis, and survival models. Statistical models will be contrasted with machine learning so that the student can make an informed choice of predictive tools. ## Target Audience Statisticians and persons from other quantitative disciplines who are interested in multivariable regression analysis of univariate responses, in developing, validating, and graphically describing multivariable predictive models and in covariable adjustment in clinical trials. The course will be of particular interest to applied statisticians and developers of applied statistics methodology, graduate students, clinical and pre-clinical biostatisticians, health services and outcomes researchers, econometricians, psychometricians, and quantitative epidemiologists. A good command of ordinary multiple regression is a prerequisite. ## Learning Goals Students will * be able to fit multivariable regression models: + accurately + in a way the sample size will allow, without overfitting + uncovering complex non--linear or non--additive relationships + testing for and quantifying the association between one or more predictors and the response, with possible adjustment for other factors + making maximum use of partial data rather than deleting observations containing missing variables * be able to validate models for predictive accuracy and to detect overfitting and understand problems caused by overfitting. * learn techniques of "safe data mining" in which significance levels, confidence limits, and measures such as $R^2$ have the claimed properties. * learn how to interpret fitted models using both parameter estimates and graphics * learn about the advantages of semiparametric ordinal models for continuous $Y$ * learn about some of the differences between frequentist and Bayesian approaches to statistical modeling * learn differences between machine learning and statistical models, and how to determine the better approach depending on the nature of the problem ## Course Philosophy {#sec-philosophy} `r mrg(sound('philosophy'))` [The audio narration starts with the third bullet point listed here.]{.aside} * Modeling is the endeavor to transform data into information and information into either prediction or evidence about the data generating mechanism^[Thanks to Drew Levy for ideas that greatly improved this section.] * Models are usually the best descriptive statistics + adjust for one variable while displaying the association with $Y$ and another variable + descriptive statistics do not work in higher dimensions * Satisfaction of model assumptions improves precision and increases statistical power + Be aware of assumptions, especially those mattering the most * It is more productive to make a model fit step by step (e.g., transformation estimation) than to postulate a simple model and find out what went wrong + Model diagnostics are often not actionable + Changing the model in reaction to observed patterns $\uparrow$ uncertainty but is reflected by an apparent $\downarrow$ in uncertainty * Graphical methods should be married to formal inference * Overfitting occurs frequently, so data reduction and model validation are important * Software without multiple facilities for assessing and fixing model fit may only seem to be user-friendly * Carefully fitting an improper model is better than badly fitting (and overfitting) a well-chosen one + E.g. small $N$ and overfitting vs. carefully formulated right hand side of model * Methods which work for all types of regression models are the most valuable. * In most research projects the cost of data collection far outweighs the cost of data analysis, so it is important to use the most efficient and accurate modeling techniques, to avoid categorizing continuous variables, and to not remove data from the estimation sample just to be able to validate the model. + A $100 analysis can make a $1,000,000 study worthless. * The bootstrap is a breakthrough for statistical modeling and model validation. * Bayesian modeling is ready for prime time. + Can incorporate non-data knowledge + Provides full exact inferential tools even when penalizing $\beta$ + Rational way to account for model uncertainty + Direct inference: evidence for all possible values of $\beta$ + More accurate way of dealing with missing data * Using the data to guide the data analysis is almost as dangerous as not doing so. * A good overall strategy is to decide how many degrees of freedom (i.e., number of regression parameters) can be \"spent\", where they should be spent, to spend them with no regrets. See the excellent text _Clinical Prediction Models_ [@cpm] ## Key Messages of the Course 1. Consider RMS as practicing a craft, and the RMS text book is the foundational treatise for the craft. Get, read--and periodically re-read--the textbook. There is a lot in the book not covered in the short course. Reading the text will help by providing students a comprehensive tool set, and reinforce the overall coherence in RMS. And there is also a lot of wisdom throughout the textbook. [Written by Drew Levy]{.aside} 1. A fundamental RMS principle of "methods that enable an analyst to develop models that will make accurate predictions of responses for future observations." The RMS program (philosophy, methodology, tools) is predicated on this. 1. **Phantom degrees of freedom** : Information about the relationship between the dependent variable and candidate predictors introduced into model specification or model selection that is not later properly accounted for in the variance estimates and inference that goes into the ultimate report of analysis results. Put another way: assessments made in model development and then later elided that compromise the fidelity of standard errors, alpha, Type-I assertion probability $\alpha$, etc. It is 'phantom' not only because it disappears, but also because this information continues to haunt the analysis, typically with overconfidence. 1. **$\chi^{2}$ - df** : The potential magnitude of signal, penalized for possibility of accident. Or--- the potential magnitude of signal, adjusted for risk of coincidence. 1. **Chunk tests** : Simultaneous or aggregate assessment of the importance (e.g., proportion of total variance explained) of collinear or dependent (e.g. related polynomial or interaction) terms or a group of non-collinear candidate predictors. Important for making coherent judgments. 1. **Spending df's** : Given a fixed amount of information in the data available for the analysis there is an 'information budget' that should be used judiciously: more important predictors should represented in a richer way (e.g. make the number of knots in splines proportional to the overall importance and complexity in the variable) than predictors that are less important. "Spending df's with no regrets": to preserve the operating characteristics of formal inference, once assessed in the modeling process even the less important candidate predictors should remain in view and not elided. All candidate predictors get a portion of the 'df budget', but to a varying extent based on their predictive potential. 1. **Partial plots**: Pay special attention to partial plots. Partial plots reveal important information about where the information is in predicting the dependent variable. Partial plots elucidate the form and strength of individual predictors (independent of covariate effects) and can reveal the structure in the data and where the information is situated in the model. 1. Be aware that cognitive biases --- and an analysts' interests, incentives and ethics --- are all also a latent (often insidious) part of the overall analysis and reporting process. Be honest with others and yourself. ### Annotating the Notes For information about adding annotations, comments, and questions inside the text click here: `r hypcomment` ### Symbols Used in the Right Margin of the Text * <img src="movie.png" width="15px"> in the right margin is a hyperlink to a YouTube video related to the subject. * <img src="discourse.png" width="15px"> is a hyperlink to the discussion topic in `datamethods.org` devoted to the specific topic. You can go directly to the discussion about chapter `n` by going to `datamethods.org/rmsn`. * An audio player symbol indicates that narration elaborating on the notes is available for the section. Red letters and numbers in the right margin are cues referred to within the audio recordings. * blog in the right margin is a link to a blog entry that further discusses the topic. ## Other Information * [Discussion board for the overall course](http://datamethods.org/rms) and [hints for registering](https://hbiostat.org/datamethods) * How to [annotate and pose questions on web pages](https://hbiostat.org/comment) * [RMS main web site](https://hbiostat.org/rms) * Twitter: `#rmscourse` * Publicly available videos of the short course + [Four hour short course](https://youtu.be/UPcpkIJXlvY) from useR!2022 * [BBR course](https://hbiostat.org/bbrc) * Go directly to a YouTube video for RMS Session `n` by going to `bit.ly/yt-rmsn` * [Study questions](https://hbiostat.org/rms/qstudy.html) * [Datamethods discussion board](http://datamethods.org) * [Statistical papers written for clinical researchers](https://hbiostat.org/bib) * [RMS overview and template for statistical analysis plan](https://fharrell.com/post/modplan) * [Statistical Thinking blog](https://fharrell.com) * [Glossary of statistical terms](https://hbiostat.org/glossary) * [R Workflow](https://hbiostat.org/rflow) ## R Packages To be able to run all the examples in the book, install current versions of the following `CRAN` packages: ``` Hmisc, rms, data.table, nlme, rmsb, ggplot2 kableExtra, pcaPP, VGAM, MASS, leaps, rpart ``` ## License <a rel="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png" /></a> Regression Modeling Strategies Course is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>. Based on a work at <a xmlns:dct="http://purl.org/dc/terms/" href="https://hbiostat.org/rmsc" rel="dct:source">https://hbiostat.org/rmsc</a>. ::: {.callout-note collapse="true"} # Update History | Date | Sections | Changes | |:---|:-|:--------| | 2024-10-13 | [-@sec-genreg-bayes] | Expanded to include collapsible section on uncertainty in model performance metrix, plus problems with CLT and $\delta$-method | | 2024-08-11 | Preface | Added Drew Levy's key messages | | 2024-08-02 | [-@sec-val-bayes] | New subsection on validation issues in Bayesian modeling | | 2024-04-21 | [-@sec-lrm-bayesex] | Updated `blrm` from using `keepsep` to `pcontrast` | | 2024-03-03 | [-@sec-val-dr] | New section of validatiion of data reduction | | 2024-02-18 | [-@sec-val-contrast] | New section on contrasts | | 2023-09-17 | [-@sec-val-rev] | New subsection on relative explained variation | | 2023-08-02 | [-@sec-missing-constraints] | New subsection on constraints for imputed values | | 2023-08-01 | [-@sec-bacteremia] | New chapter for bacteremia case study | | 2023-07-28 | [-@sec-intro-tep] | Added paired tests | | 2023-07-21 | [-@sec-bodyfat] | New chapter: linear model case study | | 2023-07-14 | [-@sec-mle-aic] | New material and links for AIC/BIC | | 2023-05-30 | [-@sec-multivar-strategy]| Added consideration of confounding | | 2023-05-24 | [-@sec-multivar-overfit] | Better effective sample size for binary $Y$ | | 2023-05-20 | [-@sec-lrcase1-pc], [-@sec-impred-pc] | Added graphical display of PC loadings | | 2023-04-30 | [-@sec-lrm] | Many improvements in graphics, and code using data.table | | 2023-04-29 | [-@sec-genreg-gtrans] | Add likelihood ratio tests | | 2023-04-22 | [-@sec-mle-hd] | New section on Hauck-Donner effect ruining Wald statistics | | 2023-04-22 | [-@sec-lrm;-@sec-titanic-cd] | Added new `anova(..., test='LR')` | | 2023-03-06 | [-@sec-cony] | Several changes; replaced lattice graphics with ggplot2 and added validation with simultaneous multiple imputation | | 2023-03-01 | [-@sec-val-mival] | New section on simultaneous validation and imputation | | 2023-02-20 | [-@sec-impred;-@sec-lrcase1] | Used new Hmisc 5.0-0 function princmp for principal components | | 2023-02-12 | | Used rms 6.5-0 to improve code, removing results='asis' from chunk headers | | 2023-02-07 | | Started moving study questions to end of chapters | | 2022-10-28 | [-@sec-intro] | 3 new flowcharts | | 2022-10-28 | [-@sec-intro-gof] | New subsection of model uncertainty and GOF | | 2022-09-16 | [-@sec-mle-ci] | Link to nice profile likelihood CI example | ::: ## Review Questions 1. Consider inference from comparing central tendencies of two groups on a continuous response variable Y. What assumptions are you willing to make when selecting a statistical test? Why are you willing to make those assumptions? 1. Consider the comparison of 5 groups on a continuous Y. Suppose you observe that two of the groups have a similar mean and the other three also have a similar sample mean. What is wrong with combining the two samples and combining the three samples, then comparing two means? How does this compare to stepwise variable selection? 1. Name a specific statistical test for which we don't have a corresponding statistical model 1. Concerning a multi-group problem or a sequential testing problem what is the frequentist approach to multiplicity correction? The Bayesian approach?