StepReg: Stepwise Regression Analysis
29 October 2024
Abstract
The StepReg package, developed for exploratory model building tasks, offers support across diverse scenarios. It facilitates model construction for various response variable types, including continuous (linear regression), binary (logistic regression), and time-to-event (Cox regression), among others. StepReg encompasses all commonly used model selection strategies, including forward selection, backward elimination, bidirectional elimination, and best subsets. Notably, it offers flexibility in selection metrics, accommodating both information criteria (AIC, BIC, etc.) and significance level cutoffs. This vignettes provide numerous examples showcasing the effective utilization of StepReg for model development in diverse contexts. Furthermore, it delves into considerations for selecting appropriate strategies and metrics, empowering users to make informed decisions throughout the modeling process.
Package
StepReg 1.5.5
1 Introduction
Model selection is the process of choosing the most relevant features from a set of candidate variables. This procedure is crucial because it ensures that the final model is both accurate and interpretable while being computationally efficient and avoiding overfitting. Stepwise regression algorithms iteratively add or remove features from the model based on certain criteria (e.g., significance level or P-value, information criteria like AIC or BIC, etc.). The process continues until no further improvements can be made according to the chosen criterion. At the end of the stepwise procedure, you’ll have a final model that includes the selected features and their coefficients.
StepReg simplifies model selection tasks by providing a unified programming interface. It currently supports model buildings for five distinct response variable types (section 3.1), four model selection strategies (section 3.2) including the best subsets algorithm, and a variety of selection metrics (section 3.3). Moreover, StepReg detects and addresses the multicollinearity issues if they exist (section 3.4). The output of StepReg includes multiple tables summarizing the final model and the variable selection procedures. Additionally, StepReg offers a plot function to visualize the selection steps (section 4). For demonstration, the vignettes include four use cases covering distinct regression scenarios (section 5). Non-programmers can access the tool through the iterative Shiny app detailed in section 6.
2 Quick demo
The following example selects an optimal linear regression model with the mtcars dataset.
library(StepReg)
data(mtcars)
formula <- mpg ~ .
res <- stepwise(formula = formula,
data = mtcars,
type = "linear",
include = c("qsec"),
strategy = "bidirection",
metric = c("AIC"))Breakdown of the parameters:
formula: specifies the dependent and independent variablestype: specifies the regression category, depending on your data, choose from “linear”, “logit”, “cox”, etc.include: specifies the variables that must be in the final modelstrategy: specifies the model selection strategy, choose from “forward”, “backward”, “bidirection”, “subset”metric: specifies the model fit evaluation metric, choose one or more from “AIC”, “AICc”, “BIC”, “SL”, etc.
The output consists of multiple tables, which can be viewed with:
resTable 1. Summary of arguments for model selection
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Parameter Value
—————————————————————————————————————————————
included variable qsec
strategy bidirection
metric AIC
tolerance of multicollinearity 1e-07
multicollinearity variable NULL
intercept 1
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Table 2. Summary of variables in dataset
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Variable_type Variable_name Variable_class
——————————————————————————————————————————————
Dependent mpg numeric
Independent cyl numeric
Independent disp numeric
Independent hp numeric
Independent drat numeric
Independent wt numeric
Independent qsec numeric
Independent vs numeric
Independent am numeric
Independent gear numeric
Independent carb numeric
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Table 3. Summary of selection process under bidirection with AIC
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Step EffectEntered EffectRemoved NumberParams AIC
——————————————————————————————————————————————————————————————
1 1 1 149.94345
2 qsec 2 145.776054
3 wt 3 97.90843
4 am 4 95.307305
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Table 4. Summary of coefficients for the selected model with mpg under bidirection and AIC
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Variable Estimate Std. Error t value Pr(>|t|)
—————————————————————————————————————————————————————————
(Intercept) 9.617781 6.959593 1.381946 0.177915
qsec 1.225886 0.28867 4.246676 0.000216
wt -3.916504 0.711202 -5.506882 7e-06
am 2.935837 1.410905 2.080819 0.046716
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗You can also visualize the variable selection procedures with:
plot(res)$bidirection
$bidirection$detail
$bidirection$overview
The (+)1 refers to original model with intercept being added, (+) indicates variables being added to the model while (-) means variables being removed from the model.
Additionally, you can generate reports of various formats with:
report(res, report_name = "path_to/demo_res", format = "html")Replace "path_to/demo_res" with desired output file name, the suffix ".html" will be added automatically. For detailed examples and more usage, refer to section 4 and 5.
3 Key features
3.1 Regression categories
StepReg supports multiple types of regressions, including linear, logit, cox, poisson, and gamma regressions. These methods primarily vary by the type of response variable, which are summarized in the table below. Additional regression techniques can be incorporated upon user requests.
| Regression | Reponse |
|---|---|
| linear | continuous |
| logit | binary |
| cox | time-to-event |
| poisson | count |
| gamma | continuous and positively skewed |
3.2 Model selection strategies
Model selection aims to identify the subset of independent variables that provide the best predictive performance for the response variable. Both stepwise regression and best subsets approaches are implemented in StepReg. For stepwise regression, there are mainly three methods: Forward Selection, Backward Elimination, Bidirectional Elimination.
| Strategy | Description |
|---|---|
| Forward Selection | In forward selection, the algorithm starts with an empty model (no predictors) and adds in variables one by one. Each step tests the addition of every possible predictor by calculating a pre-selected metric. Add the variable (if any) whose inclusion leads to the most statistically significant fit improvement. Repeat this process until more predictors no longer lead to a statistically better fit. |
| Backward Elimination | In backward elimination, the algorithm starts with a full model (all predictors) and deletes variables one by one. Each step test the deletion of every possible predictor by calculating a pre-selected metric. Delete the variable (if any) whose loss leads to the most statistically significant fit improvement. Repeat this process until less predictors no longer lead to a statistically better fit. |
| Bidirectional Elimination | Bidirectional elimination is essentially a forward selection procedure combined with backward elimination at each iteration. Each iteration starts with a forward selection step that adds in predictors, followed by a round of backward elimination that removes predictors. Repeat this process until no more predictors are added or excluded. |
| Best Subsets | Stepwise algorithms add or delete one predictor at a time and output a single model without evaluating all candidates. Therefore, it is a relatively simple procedure that only produces one model. In contrast, the Best Subsets algorithm calculates all possible models and output the best-fitting models with one predictor, two predictors, etc., for users to choose from. |
Given the computational constraints, when dealing with datasets featuring a substantial number of predictor variables greater than the sample size, the Bidirectional Elimination typically emerges as the most advisable approach. Forward Selection and Backward Elimination can be considered in sequence. On the contrary, the Best Subsets approach requires the most substantial processing time, yet it calculates a comprehensive set of models with varying numbers of variables. In practice, users can experiment with various methods and select a final model based on the specific dataset and research objectives at hand.
3.3 Selection metrics
Various selection metrics can be used to guide the process of adding or removing predictors from the model. These metrics help to determine the importance or significance of predictors in improving the model fit. In StepReg, selection metrics include two categories: Information Criteria and Significance Level of the coefficient associated with each predictor. Information Criteria is a means of evaluating a model’s performance, which balances model fit with complexity by penalizing models with a higher number of parameters. Lower Information Criteria values indicate a better trade-off between model fit and complexity. Note that when evaluating different models, it is important to compare them within the same Information Criteria framework rather than across multiple Information Criteria. For example, if you decide to use AIC, you should compare all models using AIC. This ensures consistency and fairness in model comparison, as each Information Criterion has its own scale and penalization factors. In practice, multiple metrics have been proposed, the ones supported by StepReg are summarized below.
Importantly, given the discrepancies in terms of the precise definitions of each metric, StepReg mirrors the formulas adopted by SAS for univariate multiple regression (UMR) except for HQ, IC(1), and IC(3/2). A subset of the UMR can be easily extended to multivariate multiple regression (MMR), which are indicated in the following table.
| Statistic | Meanings |
|---|---|
| \({n}\) | Sample Size |
| \({p}\) | Number of parameters including the intercept |
| \({q}\) | Number of dependent variables |
| \(\sigma^2\) | Estimate of pure error variance from fitting the full model |
| \({SST}\) | Total sum of squares corrected for the mean for the dependent variable, which is a numeric value for UMR and a matrix for multivariate regression |
| \({SSE}\) | Error sum of squares, which is a numeric value for UMR and a matrix for multivariate regression |
| \(\text{LL}\) | The natural logarithm of likelihood |
| \({| |}\) | The determinant function |
| \(\ln()\) | The natural logarithm |
| Abbreviation | Definition | Formula | |
|---|---|---|---|
| linear | logit, cox, poisson and gamma | ||
| AIC | Akaike’s Information Criterion |
\(n\ln\left(\frac{|\text{SSE}|}{n}\right) + 2pq + n + q(q+1)\) (Clifford M. Hurvich 1989; Al-Subaihi 2002)\(^1\) |
\(-2\text{LL} + 2p\) (Darlington 1968; George G. Judge 1985) |
| AICc | Corrected Akaike’s Information Criterion |
\(n\ln\left(\frac{|\text{SSE}|}{n}\right) + \frac{nq(n+p)}{n-p-q-1}\) (Clifford M. Hurvich 1989; Edward J. Bedrick 1994)\(^2\) |
\(-2\text{LL} + \frac{n(n+p)}{n-p-2}\) (Clifford M. Hurvich 1989) |
| BIC | Sawa Bayesian Information Criterion |
\(n\ln\left(\frac{SSE}{n}\right) + 2(p+2)o - 2o^2, o = \frac{n\sigma^2}{SSE}\) (Sawa 1978; George G. Judge 1985) not available for MMR |
not available |
| Cp | Mallows’ Cp statistic |
\(\frac{SSE}{\sigma^2} + 2p - n\) (Mallows 1973; Hocking 1976) not available for MMR |
not available |
| HQ | Hannan and Quinn Information Criterion |
\(n\ln\left(\frac{|\text{SSE}|}{n}\right) + 2pq\ln(\ln(n))\) (E. J. Hannan 1979; Allan D R McQuarrie 1998; Clifford M. Hurvich 1989) |
\(-2\text{LL} + 2p\ln(\ln(n))\) (E. J. Hannan 1979) |
| IC(1) | Information Criterion with Penalty Coefficient Set to 1 |
\(n\ln\left(\frac{|\text{SSE}|}{n}\right) + p\) (J. A. Nelder 1972; A. F. M. Smith 1980) not available for MMR |
\(-2\text{LL} + p\) (J. A. Nelder 1972; A. F. M. Smith 1980) |
| IC(3/2) | Information Criterion with Penalty Coefficient Set to 3/2 |
\(n\ln\left(\frac{|\text{SSE}|}{n}\right) + \frac{3}{2}p\) (A. F. M. Smith 1980) not available for MMR |
\(-2\text{LL} + \frac{3}{2}p\) (A. F. M. Smith 1980) |
| SBC | Schwarz Bayesian Information Criterion |
\(n\ln\left(\frac{|\text{SSE}|}{n}\right) + pq \ln(n)\) (Clifford M. Hurvich 1989; Schwarz 1978; George G. Judge 1985; Al-Subaihi 2002) not available for MMR |
\(-2\text{LL} + p\ln(n)\) (Schwarz 1978; George G. Judge 1985) |
| SL | Significance Level (pvalue) | \(\textit{F test}\) for UMR and \(\textit{Approximate F test}\) for MMR |
Forward: LRT and Rao Chi-square test (logit, poisson, gamma); LRT (cox); Backward: Wald test |
| adjRsq | Adjusted R-square statistic |
\(1 - \frac{(n-1)(1-R^2)}{n-p}\), where \(R^2=1 - \frac{SSE}{SST}\) (Darlington 1968; George G. Judge 1985) not available for MMR |
not available |
|
1 Unsupported AIC formula (which does not affect the selection process as it only differs by constant additive and multiplicative factors): \(AIC=n\ln\left(\frac{SSE}{n}\right) + 2p\) (Darlington 1968; George G. Judge 1985) |
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2 Unsupported AICc formula (which does not affect the selection process as it only differs by constant additive and multiplicative factors): \(AICc=\ln\left(\frac{SSE}{n}\right) + 1 + \frac{2(p+1)}{n-p-2}\) (Allan D R McQuarrie 1998) |
No metric is necessarily optimal for all datasets. The choice of them depends on your data and research goals. We recommend using multiple metrics simultaneously, which allows the selection of the best model based on your specific needs. Below summarizes general guidance.
AIC: AIC works by penalizing the inclusion of additional variables in a model. The lower the AIC, the better performance of the model. AIC does not include sample size in penalty calculation, and it is optimal in minimizing the mean square error of predictions (Mark J. Brewer 2016).
AICc: AICc is a variant of AIC, which works better for small sample size, especially when
numObs / numParam < 40(Kenneth P. Burnham 2002).Cp: Cp is used for linear models. It is equivalent to AIC when dealing with Gaussian linear model selection.
IC(1) and IC(3/2): IC(1) and IC(3/2) have 1 and 3/2 as penalty factors respectively, compared to 2 used by AIC. As such, IC(1) turns to return a complex model with more variables that may suffer from overfitting issues.
BIC and SBC: Both BIC and SBC are variants of Bayesian Information Criterion. The main distinction between BIC/SBC and AIC lies in the magnitude of the penalty imposed: BIC/SBC are more parsimonious when penalizing model complexity, which typically results to a simpler model (SAS Institute Inc 2018; Sawa 1978; Clifford M. Hurvich 1989; Schwarz 1978; George G. Judge 1985; Al-Subaihi 2002).
The precise definitions of these criteria can vary across literature and in the SAS environment. Here, BIC aligns with the definition of the Sawa Bayesion Information Criterion as outlined in SAS documentation, while SBC corresponds to the Schwarz Bayesian Information Criterion. According to Richard’s post, whereas AIC often favors selecting overly complex models, BIC/SBC prioritize a small models. Consequently, when dealing with a limited sample size, AIC may seem preferable, whereas BIC/SBC tend to perform better with larger sample sizes.
HQ: HQ is an alternative to AIC, differing primarily in the method of penalty calculation. However, HQ has remained relatively underutilized in practice (Kenneth P. Burnham 2002).
Rsq: The R-squared (R²) statistic measures the proportion of variations that is explained by the model. It ranges from 0 to 1, with 1 indicating that all of the variability in the response variables is accounted for by the independent variables. As such, R-squared is valuable for communicating the explanatory power of a model. However, R-squared alone is not sufficient for selection because it does not take into account the complexity of the model. Therefore, while R-squared is useful for understanding how well the model fits the data, it should not be the sole criterion for model selection.
adjRsq: The adjusted R-squared (adj-R²) seeks to overcome the limitation of R-squared in model selection by considering the number of predictors. It serves a similar purpose to information criteria, as both methods compare models by weighing their goodness of fit against the number of parameters. However, information criteria are typically regarded as superior in this context (Stevens 2016).
SL: SL stands for Significance Level (P-value), embodying a distinct approach to model selection in contrast to information criteria. The SL method operates by calculating a P-value through specific hypothesis testing. Should this P-value fall below a predefined threshold, such as 0.05, one should favor the alternative hypothesis, indicating that the full model significantly outperforms the reduced model. The effectiveness of this method hinges upon the selection of the P-value threshold, wherein smaller thresholds tend to yield simpler models.
3.4 Multicollinearity
This blog by Jim Frost gives an excellent overview of multicollinearity and when it is necessary to remove it.
Simply put, a dataset contains multicollinearity when input predictors are correlated. When multicollinearity occurs, the interpretability of predictors will be badly affected because changes in one input variable lead to changes in other input variables. Therefore, it is hard to individually estimate the relationship between each input variable and the dependent variable.
Multicollinearity can dramatically reduce the precision of the estimated regression coefficients of correlated input variables, making it hard to find the correct model. However, as Jim pointed out, “Multicollinearity affects the coefficients and p-values, but it does not influence the predictions, precision of the predictions, and the goodness-of-fit statistics. If your primary goal is to make predictions, and you don’t need to understand the role of each independent variable, you don’t need to reduce severe multicollinearity.”
In StepReg, QC Matrix Decomposition is performed ahead of time to detect and remove input variables causing multicollinearity.
4 StepReg output
StepReg provides multiple functions for summarizing the model building results. The function stepwise() generates a list of tables that describe the feature selection steps and the final model. To facilitate collaborations, you can redirect the tables into various formats such as “xlsx”, “html”, “docx”, etc. with the function report(). Furthermore, you can easily compare the variable selection procedures for multiple selection metrics by visualizing the steps with the function plot(). Details see below.
Depending on the number of selected regression strategies and metrics, you can expect to receive at least four tables from stepwise(). Below describes the content of each of 4 tables from Quick Demo 2.
| Table_Name | Table_Description |
|---|---|
| Summary of arguments for model selection | Arguments used in the stepwise function, either default or user-supplied values. |
| Summary of variables in dataset | Variable names, types, and classes in dataset. |
| Summary of selection process under xxx(strategy) with xxx(metric) | Overview of the variable selection process under specified strategy and metric. |
| Summary of coefficients for the selected model with xxx(dependent variable) under xxx(strategy) and xxx(metric) | Coefficients for the selected models under specified strategy with metric. Please note that this table will not be generated for the strategy ‘subset’ when using the metric ‘SL’. |
You can save the output in different format like “xlsx”, “docx”, “html”, “pptx”, and others, facilitating easy sharing. Of note, the suffix will be automatically added to the report_name. For instance, the following example generates both “results.xlsx” and “results.docx” reports.
report(res, report_name = "results", format = c("xlsx", "docx"))5 Use cases
Please choose the regression model that best suits the type of response variable. For detailed guidance, see section 3.1. Below, we present various examples utilizing different models tailored to specific datasets.
5.1 Linear regression with the mtcars dataset
In this section, we’ll demonstrate how to perform linear regression analysis using the mtcars dataset, showcasing different scenarios with varying numbers of predictors and dependent variables. We set type = "linear" to direct the function to perform linear regression.
Description of the mtcars dataset
The mtcars is a classic dataset in statistics and is included in the base R installation. It was sourced from the 1974 Motor Trend US magazine, comprising 32 observations on 11 variables. Here’s a brief description of the variables included:
- mpg: miles per gallon (fuel efficiency)
- cyl: number of cylinders
- disp: displacement (engine size) in cubic inches
- hp: gross horsepower
- drat: rear axle ratio
- wt: weight (in thousands of pounds)
- qsec: 1/4 mile time (in seconds)
- vs: engine type (0 = V-shaped, 1 = straight)
- am: transmission type (0 = automatic, 1 = manual)
- gear: number of forward gears
- carb: number of carburetors
Why choose linear regression
Linear regression is an ideal choice for analyzing the mtcars dataset due to its inclusion of continuous variables like “mpg”, “hp”, or “weight”, which can serve as response variables. Furthermore, the dataset exhibits potential linear relationships between the response variable and other variables.
5.1.1 Example1: single dependent variable (“mpg”)
In this example, we employ “forward” strategy with “AIC” as the selection criteria. Additionally, we specify using the include argument that “disp”, “cyl” must always be included in the model.
library(StepReg)
data(mtcars)
formula <- mpg ~ .
res1 <- stepwise(formula = formula,
data = mtcars,
type = "linear",
include = c("disp", "cyl"),
strategy = "forward",
metric = "AIC")
res1Table 1. Summary of arguments for model selection
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Parameter Value
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included variable disp cyl
strategy forward
metric AIC
tolerance of multicollinearity 1e-07
multicollinearity variable NULL
intercept 1
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Table 2. Summary of variables in dataset
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Variable_type Variable_name Variable_class
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Dependent mpg numeric
Independent cyl numeric
Independent disp numeric
Independent hp numeric
Independent drat numeric
Independent wt numeric
Independent qsec numeric
Independent vs numeric
Independent am numeric
Independent gear numeric
Independent carb numeric
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Table 3. Summary of selection process under forward with AIC
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Step EffectEntered NumberParams AIC
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1 1 1 149.94345
2 disp cyl 3 108.333571
3 wt 4 98.746294
4 hp 5 97.525537
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Table 4. Summary of coefficients for the selected model with mpg under forward and AIC
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Variable Estimate Std. Error t value Pr(>|t|)
—————————————————————————————————————————————————————————
(Intercept) 40.828537 2.757468 14.806532 0
disp 0.011599 0.011727 0.989122 0.331386
cyl -1.29332 0.655877 -1.971894 0.058947
wt -3.853904 1.015474 -3.795178 0.000759
hp -0.020538 0.012147 -1.690851 0.102379
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗To visualize the selection process:
plot_list <- plot(res1)
cowplot::plot_grid(plotlist = plot_list$forward, ncol = 1)
To exclude the intercept from the model, adjust the formula as follows:
formula <- mpg ~ . + 0To limit the model to a specific subset of predictors, adjust the formula as follows, which will only consider “cyp”, “disp”, “hp”, “wt”, “vs”, and “am” as the predictors.
formula <- mpg ~ cyl + disp + hp + wt + vs + am + 0Another way is to use minus symbol("-") to exclude some predictors for variable selection. For example, include all variables except “disp”, “wt”, and intercept.
formula <- mpg ~ . - 1 - disp - wtYou can simultaneously provide multiple selection strategies and metrics. For example, the following code snippet employs both “forward” and “backward” strategies using metrics “AIC”, “BIC”, and “SL”. It’s worth mentioning that when “SL” is specified, you may also want to set the significance level for entry (“sle”) and stay (“sls”), both of which default to 0.15.
formula <- mpg ~ .
res2 <- stepwise(formula = formula,
data = mtcars,
type = "linear",
strategy = c("forward", "backward"),
metric = c("AIC", "BIC", "SL"),
sle = 0.05,
sls = 0.05)
res2Table 1. Summary of arguments for model selection
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Parameter Value
————————————————————————————————————————————————————————
included variable NULL
strategy forward & backward
metric AIC & BIC & SL
significance level for entry (sle) 0.05
significance level for stay (sls) 0.05
test method F
tolerance of multicollinearity 1e-07
multicollinearity variable NULL
intercept 1
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Table 2. Summary of variables in dataset
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable_type Variable_name Variable_class
——————————————————————————————————————————————
Dependent mpg numeric
Independent cyl numeric
Independent disp numeric
Independent hp numeric
Independent drat numeric
Independent wt numeric
Independent qsec numeric
Independent vs numeric
Independent am numeric
Independent gear numeric
Independent carb numeric
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 3. Summary of selection process under forward with AIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectEntered NumberParams AIC
———————————————————————————————————————————————
1 1 1 149.94345
2 wt 2 107.217363
3 cyl 3 97.197999
4 hp 4 96.664562
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 4. Summary of selection process under forward with BIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectEntered NumberParams BIC
———————————————————————————————————————————————
1 1 1 115.061344
2 wt 2 74.373395
3 cyl 3 66.190251
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 5. Summary of selection process under forward with SL
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectEntered NumberParams SL
—————————————————————————————————————————————
1 1 1 1
2 wt 2 0
3 cyl 3 0.001064
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 6. Summary of selection process under backward with AIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectRemoved NumberParams AIC
———————————————————————————————————————————————
1 11 104.897744
2 cyl 10 102.915068
3 vs 9 100.973242
4 carb 8 99.121264
5 gear 7 97.456669
6 drat 6 96.161901
7 disp 5 95.515302
8 hp 4 95.307305
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 7. Summary of selection process under backward with BIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectRemoved NumberParams BIC
——————————————————————————————————————————————
1 11 83.872801
2 cyl 10 80.827738
3 vs 9 77.795923
4 carb 8 74.805755
5 gear 7 71.925757
6 drat 6 69.307272
7 disp 5 67.229924
8 hp 4 65.713833
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 8. Summary of selection process under backward with SL
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectRemoved NumberParams SL
—————————————————————————————————————————————
1 11 1
2 cyl 10 0.916087
3 vs 9 0.843258
4 carb 8 0.746958
5 gear 7 0.619641
6 drat 6 0.462401
7 disp 5 0.298972
8 hp 4 0.223088
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 9. Summary of coefficients for the selected model with mpg under forward and AIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error t value Pr(>|t|)
—————————————————————————————————————————————————————————
(Intercept) 38.751787 1.786864 21.687038 0
wt -3.166973 0.740576 -4.276365 0.000199
cyl -0.941617 0.550916 -1.709183 0.09848
hp -0.018038 0.011876 -1.518838 0.140015
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 10. Summary of coefficients for the selected model with mpg under forward and BIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error t value Pr(>|t|)
—————————————————————————————————————————————————————————
(Intercept) 39.686261 1.714984 23.140893 0
wt -3.190972 0.756906 -4.215808 0.000222
cyl -1.507795 0.414688 -3.635972 0.001064
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 11. Summary of coefficients for the selected model with mpg under forward and SL
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error t value Pr(>|t|)
—————————————————————————————————————————————————————————
(Intercept) 39.686261 1.714984 23.140893 0
wt -3.190972 0.756906 -4.215808 0.000222
cyl -1.507795 0.414688 -3.635972 0.001064
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 12. Summary of coefficients for the selected model with mpg under backward and AIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error t value Pr(>|t|)
—————————————————————————————————————————————————————————
(Intercept) 9.617781 6.959593 1.381946 0.177915
wt -3.916504 0.711202 -5.506882 7e-06
qsec 1.225886 0.28867 4.246676 0.000216
am 2.935837 1.410905 2.080819 0.046716
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 13. Summary of coefficients for the selected model with mpg under backward and BIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error t value Pr(>|t|)
—————————————————————————————————————————————————————————
(Intercept) 9.617781 6.959593 1.381946 0.177915
wt -3.916504 0.711202 -5.506882 7e-06
qsec 1.225886 0.28867 4.246676 0.000216
am 2.935837 1.410905 2.080819 0.046716
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 14. Summary of coefficients for the selected model with mpg under backward and SL
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error t value Pr(>|t|)
—————————————————————————————————————————————————————————
(Intercept) 9.617781 6.959593 1.381946 0.177915
wt -3.916504 0.711202 -5.506882 7e-06
qsec 1.225886 0.28867 4.246676 0.000216
am 2.935837 1.410905 2.080819 0.046716
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗plot_list <- plot(res2)
cowplot::plot_grid(plotlist = plot_list$forward, ncol = 1, rel_heights = c(2, 1))
cowplot::plot_grid(plotlist = plot_list$backward, ncol = 1, rel_heights = c(2, 1))
5.1.2 Example2: multivariate regression (“mpg” and “drat”)
In this scenario, there are two dependent variables, “mpg” and “drat”. The model selection aims to identify the most influential predictors that affect both variables.
formula <- cbind(mpg, drat) ~ . + 0
res3 <- stepwise(formula = formula,
data = mtcars,
type = "linear",
strategy = "bidirection",
metric = c("AIC", "HQ"))
res3Table 1. Summary of arguments for model selection
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Parameter Value
—————————————————————————————————————————————
included variable NULL
strategy bidirection
metric AIC & HQ
tolerance of multicollinearity 1e-07
multicollinearity variable NULL
intercept 0
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 2. Summary of variables in dataset
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable_type Variable_name Variable_class
—————————————————————————————————————————————————
Dependent cbind(mpg, drat) nmatrix.2
Independent cyl numeric
Independent disp numeric
Independent hp numeric
Independent wt numeric
Independent qsec numeric
Independent vs numeric
Independent am numeric
Independent gear numeric
Independent carb numeric
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 3. Summary of selection process under bidirection with AIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectEntered EffectRemoved NumberParams AIC
——————————————————————————————————————————————————————————————
1 0 0 Inf
2 gear 1 204.83021
3 carb 2 176.482276
4 qsec 3 158.756091
5 wt 4 149.906176
6 carb 3 148.903913
7 am 4 147.093775
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 4. Summary of selection process under bidirection with HQ
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectEntered EffectRemoved NumberParams HQ
——————————————————————————————————————————————————————————————
1 0 0 Inf
2 gear 1 167.80191
3 carb 2 140.425676
4 qsec 3 123.67119
5 wt 4 115.792975
6 carb 3 113.819012
7 am 4 112.980575
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 5. Summary of coefficients for the selected model with Response mpg under bidirection and AIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error t value Pr(>|t|)
——————————————————————————————————————————————————————
gear 0.487986 0.883509 0.552328 0.585106
qsec 1.53717 0.153471 10.01606 0
wt -3.295432 0.52766 -6.245372 1e-06
am 3.512686 1.761811 1.993793 0.055997
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 6. Summary of coefficients for the selected model with Response drat under bidirection and AIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error t value Pr(>|t|)
——————————————————————————————————————————————————————
gear 0.381969 0.121585 3.141571 0.003946
qsec 0.118085 0.02112 5.591137 6e-06
wt -0.025826 0.072615 -0.355665 0.724757
am 0.387525 0.242454 1.598344 0.121192
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 7. Summary of coefficients for the selected model with Response mpg under bidirection and HQ
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error t value Pr(>|t|)
——————————————————————————————————————————————————————
gear 0.487986 0.883509 0.552328 0.585106
qsec 1.53717 0.153471 10.01606 0
wt -3.295432 0.52766 -6.245372 1e-06
am 3.512686 1.761811 1.993793 0.055997
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 8. Summary of coefficients for the selected model with Response drat under bidirection and HQ
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error t value Pr(>|t|)
——————————————————————————————————————————————————————
gear 0.381969 0.121585 3.141571 0.003946
qsec 0.118085 0.02112 5.591137 6e-06
wt -0.025826 0.072615 -0.355665 0.724757
am 0.387525 0.242454 1.598344 0.121192
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗plot(res3)$bidirection
$bidirection$detail
$bidirection$overview
5.2 Logistic regression with the remission dataset
In this example, we’ll showcase logistic regression using the remission dataset. By setting type = "logit", we instruct the function to perform logistic regression.
Description of the remission dataset
The remission dataset, obtained from the online course STAT501 at Penn State University, has been integrated into StepReg. It consists of 27 observations across seven variables, including a binary variable named “remiss”:
- remiss: whether leukemia remission occurred, a value of 1 indicates occurrence while 0 means non-occurrence
- cell: cellularity of the marrow clot section
- smear: smear differential percentage of blasts
- infil: percentage of absolute marrow leukemia cell infiltrate
- li: percentage labeling index of the bone marrow leukemia cells
- blast: the absolute number of blasts in the peripheral blood
- temp: the highest temperature before the start of treatment
Why choose logistic regression
Logistic regression effectively captures the relationship between predictors and a categorical response variable, offering insights into the probability of being assigned into specific response categories given a set of predictors. It is suitable for analyzing binary outcomes, such as the remission status (“remiss”) in the remission dataset.
5.2.1 Example1: using “forward” strategy
In this example, we employ a “forward” strategy with “AIC” as the selection criteria, while force ensuring that the “cell” variable is included in the model.
data(remission)
formula <- remiss ~ .
res4 <- stepwise(formula = formula,
data = remission,
type = "logit",
include= "cell",
strategy = "forward",
metric = "AIC")[1] "There was an error message."res4Table 1. Summary of arguments for model selection
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Parameter Value
—————————————————————————————————————————
included variable cell
strategy forward
metric AIC
tolerance of multicollinearity 1e-07
multicollinearity variable NULL
intercept 1
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 2. Summary of variables in dataset
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable_type Variable_name Variable_class
——————————————————————————————————————————————
Dependent remiss numeric
Independent cell numeric
Independent smear numeric
Independent infil numeric
Independent li numeric
Independent blast numeric
Independent temp numeric
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 3. Summary of selection process under forward with AIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectEntered NumberParams AIC
——————————————————————————————————————————————
1 1 1 36.371765
2 cell 2 35.791792
3 li 3 30.340719
4 temp 4 29.953368
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 4. Summary of coefficients for the selected model with remiss under forward and AIC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error z value Pr(>|z|)
—————————————————————————————————————————————————————————
(Intercept) 67.633906 56.887547 1.188905 0.234477
cell 9.652152 7.751076 1.245266 0.213034
li 3.8671 1.778278 2.174632 0.029658
temp -82.073774 61.712382 -1.32994 0.183538
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗plot(res4)$forward
$forward$detail
$forward$overview
5.2.2 Example2: using “subset” strategy
In this example, we employ a “subset” strategy, utilizing “SBC” as the selection criteria while excluding the intercept. Meanwhile, we set best_n = 3 to restrict the output to the top 3 models for each number of variables.
data(remission)
formula <- remiss ~ . + 0
res5 <- stepwise(formula = formula,
data = remission,
type = "logit",
strategy = "subset",
metric = "SBC",
best_n = 3)[1] "There was an error message."res5Table 1. Summary of arguments for model selection
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Parameter Value
————————————————————————————————————————
included variable NULL
strategy subset
metric SBC
tolerance of multicollinearity 1e-07
multicollinearity variable NULL
intercept 0
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 2. Summary of variables in dataset
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable_type Variable_name Variable_class
——————————————————————————————————————————————
Dependent remiss numeric
Independent cell numeric
Independent smear numeric
Independent infil numeric
Independent li numeric
Independent blast numeric
Independent temp numeric
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 3. Summary of selection process under subset with SBC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
NumberOfVariables SBC VariablesInModel
—————————————————————————————————————————————————————————————————
1 37.627761 0 temp
1 38.645147 0 cell
1 38.908893 0 smear
2 32.450079 0 li temp
2 37.241906 0 blast temp
2 37.692052 0 cell li
3 33.57889 0 cell li temp
3 35.101444 0 infil li temp
3 35.582915 0 smear li temp
4 36.711937 0 cell li blast temp
4 36.809659 0 cell smear li temp
4 36.841948 0 cell infil li temp
5 38.980486 0 cell smear infil li temp
5 39.686807 0 cell smear li blast temp
5 39.752516 0 cell infil li blast temp
6 42.25891 0 cell smear infil li blast temp
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 4. Summary of coefficients for the selected model with remiss under subset and SBC
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error z value Pr(>|z|)
——————————————————————————————————————————————————————
li 2.941731 1.195358 2.460962 0.013856
temp -3.855005 1.404155 -2.745426 0.006043
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗plot(res5)$subset
$subset$detail
$subset$overview
Here, the 0 in the above plot means that there is no intercept in the model.
5.3 Cox regression with the lung dataset
In this example, we’ll demonstrate how to perform Cox regression analysis using the lung dataset. By setting type = "cox", we instruct the function to conduct Cox regression.
Description of the lung dataset
The lung dataset, available in the "survival" R package, includes information on survival times for 228 patients with advanced lung cancer. It comprises ten variables, among which the “status” variable codes for censoring status (1 = censored, 2 = dead), and the “time” variable denotes the patient survival time in days. To learn more about the dataset, use ?survival::lung.
Why choose Cox regression
Cox regression, also termed the Cox proportional hazards model, is specifically designed for analyzing survival data, making it well-suited for datasets like lung that include information on the time until an event (e.g., death) occurs. This method accommodates censoring and assumes proportional hazards, enhancing its applicability to medical studies involving time-to-event outcomes.
5.3.1 Example1: using “forward” strategy
In this example, we employ a “forward” strategy with “AICc” as the selection criteria.
library(dplyr)
library(survival)
# Preprocess:
lung <- survival::lung %>%
mutate(sex = factor(sex, levels = c(1, 2))) %>% # make sex as factor
na.omit() # get rid of incomplete records
formula = Surv(time, status) ~ .
res6 <- stepwise(formula = formula,
data = lung,
type = "cox",
strategy = "forward",
metric = "AICc")
res6Table 1. Summary of arguments for model selection
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Parameter Value
—————————————————————————————————————————————
included variable NULL
strategy forward
metric AICc
significance level for entry (sle) 0.15
significance level for stay (sls) 0.15
test method efron
tolerance of multicollinearity 1e-07
multicollinearity variable NULL
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 2. Summary of variables in dataset
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable_type Variable_name Variable_class
———————————————————————————————————————————————————
Dependent Surv(time, status) nmatrix.2
Independent inst numeric
Independent age numeric
Independent sex factor
Independent ph.ecog numeric
Independent ph.karno numeric
Independent pat.karno numeric
Independent meal.cal numeric
Independent wt.loss numeric
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 3. Summary of selection process under forward with AICc
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectEntered NumberParams AICc
————————————————————————————————————————————————
1 ph.ecog 1 1127.927415
2 sex 2 1122.958293
3 inst 3 1121.654289
4 wt.loss 4 1120.224249
5 ph.karno 5 1118.535984
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 4. Summary of coefficients for the selected model with Surv(time, status) under forward and AICc
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable coef exp(coef) se(coef) z Pr(>|z|)
———————————————————————————————————————————————————————————————
ph.ecog 0.993224 2.699926 0.232115 4.279014 1.9e-05
sex2 -0.571959 0.564419 0.198865 -2.876111 0.004026
inst -0.030042 0.970404 0.012931 -2.323316 0.020162
wt.loss -0.0148 0.985309 0.007664 -1.931026 0.05348
ph.karno 0.021492 1.021725 0.011222 1.915171 0.055471
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗plot(res6)$forward
$forward$detail
$forward$overview
5.4 Poisson regression with the creditCard dataset
In this example, we’ll demonstrate how to perform Poisson regression analysis using the creditCard dataset. We set type = "poisson" to direct the function to perform Poisson regression.
Descprition of the creditCard dataset
The creditCard dataset contains credit history information for a sample of applicants for a specific type of credit card, included in the "AER" package. It encompasses 1319 observations across 12 variables, including “reports”, “age”, “income”, among others. The “reports” variable represents the number of major derogatory reports. For detailed information, refer to ?AER::CreditCard.
Why choose Poisson regression
Poisson regression is frequently employed method for analyzing count data, where the response variable represents the occurrences of an event within a defined time or space frame. In the context of the creditCard dataset, Poisson regression can model the count of major derogatory reports (“reports”), enabling assessment of predictors’ impact on this variable.
5.4.1 Example1: using “forward” strategy
In this example, we employ a “forward” strategy with “SL” as the selection criteria. We set the significance level for entry to 0.05 (sle = 0.05).
data(creditCard)
formula = reports ~ .
res7 <- stepwise(formula = formula,
data = creditCard,
type = "poisson",
strategy = "forward",
metric = "SL",
sle = 0.05)[1] "There was an error message."res7Table 1. Summary of arguments for model selection
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Parameter Value
—————————————————————————————————————————————
included variable NULL
strategy forward
metric SL
significance level for entry (sle) 0.05
test method Rao
tolerance of multicollinearity 1e-07
multicollinearity variable NULL
intercept 1
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 2. Summary of variables in dataset
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable_type Variable_name Variable_class
——————————————————————————————————————————————
Dependent reports numeric
Independent card factor
Independent age numeric
Independent income numeric
Independent share numeric
Independent expenditure numeric
Independent owner factor
Independent selfemp factor
Independent dependents numeric
Independent months numeric
Independent majorcards numeric
Independent active numeric
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 3. Summary of selection process under forward with SL
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Step EffectEntered NumberParams SL
—————————————————————————————————————————————
1 1 1 1
2 card 2 0
3 active 3 0
4 expenditure 4 0.000167
5 months 5 0.001497
6 owner 6 0.00029
7 majorcards 7 0.008537
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Table 4. Summary of coefficients for the selected model with reports under forward and SL
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
Variable Estimate Std. Error z value Pr(>|z|)
——————————————————————————————————————————————————————————
(Intercept) -0.298644 0.109685 -2.722729 0.006475
cardyes -2.703522 0.117196 -23.068395 0
active 0.06543 0.003998 16.367447 0
expenditure 0.000672 0.000178 3.785384 0.000153
months 0.002125 0.00053 4.006288 6.2e-05
owneryes -0.34377 0.092648 -3.710493 0.000207
majorcards 0.274039 0.104513 2.622062 0.00874
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗plot(res7)Warning in min(x): no non-missing arguments to min; returning InfWarning in max(x): no non-missing arguments to max; returning -Inf$forward
$forward$detail
$forward$overview
6 Interactive app
We have developed an interactive Shiny application to simplify model selection tasks for non-programmers. You can access the app through the following URL:
https://junhuili1017.shinyapps.io/StepReg/
You can also access the Shiny app directly from your local machine with the following code:
library(StepReg)
StepRegShinyApp()Here is the user interface.
7 Session info
R version 4.4.1 (2024-06-14)
Platform: x86_64-apple-darwin20
Running under: macOS Sonoma 14.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.4-x86_64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-x86_64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.0
locale:
[1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/New_York
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] survival_3.7-0 dplyr_1.1.4 kableExtra_1.4.0 knitr_1.48
[5] StepReg_1.5.5 BiocStyle_2.32.1
loaded via a namespace (and not attached):
[1] tidyselect_1.2.1 viridisLite_0.4.2 farver_2.1.2
[4] fastmap_1.2.0 fontquiver_0.2.1 promises_1.3.0
[7] shinyjs_2.1.0 digest_0.6.37 timechange_0.3.0
[10] mime_0.12 lifecycle_1.0.4 magrittr_2.0.3
[13] compiler_4.4.1 rlang_1.1.4 sass_0.4.9
[16] tools_4.4.1 utf8_1.2.4 yaml_2.3.10
[19] data.table_1.16.2 labeling_0.4.3 askpass_1.2.1
[22] summarytools_1.0.1 htmlwidgets_1.6.4 plyr_1.8.9
[25] xml2_1.3.6 withr_3.0.2 purrr_1.0.2
[28] grid_4.4.1 fansi_1.0.6 gdtools_0.4.0
[31] xtable_1.8-4 colorspace_2.1-1 ggplot2_3.5.1
[34] scales_1.3.0 MASS_7.3-61 tinytex_0.53
[37] cli_3.6.3 rmarkdown_2.28 ragg_1.3.3
[40] generics_0.1.3 rstudioapi_0.17.0 reshape2_1.4.4
[43] ggcorrplot_0.1.4.1 cachem_1.1.0 pander_0.6.5
[46] stringr_1.5.1 shinythemes_1.2.0 splines_4.4.1
[49] BiocManager_1.30.25 matrixStats_1.4.1 base64enc_0.1-3
[52] vctrs_0.6.5 Matrix_1.7-0 jsonlite_1.8.9
[55] fontBitstreamVera_0.1.1 bookdown_0.41 rapportools_1.1
[58] ggrepel_0.9.6 systemfonts_1.1.0 magick_2.8.5
[61] jquerylib_0.1.4 tidyr_1.3.1 glue_1.8.0
[64] codetools_0.2-20 cowplot_1.1.3 DT_0.33
[67] lubridate_1.9.3 stringi_1.8.4 flextable_0.9.7
[70] gtable_0.3.6 later_1.3.2 shinycssloaders_1.1.0
[73] munsell_0.5.1 tibble_3.2.1 pillar_1.9.0
[76] htmltools_0.5.8.1 openssl_2.2.2 R6_2.5.1
[79] tcltk_4.4.1 textshaping_0.4.0 evaluate_1.0.1
[82] shiny_1.9.1 lattice_0.22-6 highr_0.11
[85] backports_1.5.0 fontLiberation_0.1.0 httpuv_1.6.15
[88] pryr_0.1.6 bslib_0.8.0 Rcpp_1.0.13
[91] zip_2.3.1 uuid_1.2-1 svglite_2.1.3
[94] checkmate_2.3.2 officer_0.6.7 xfun_0.48
[97] pkgconfig_2.0.3