3 Fitting glms

BranchGLM() allows fitting of gaussian, binomial, gamma, and poisson glms with a variety of links available.
Parallel computation can also be done to speed up the fitting process, but it is only useful for larger datasets.

3.1 Optimization methods

The optimization method can be specified, the default method is fisher scoring, but BFGS and L-BFGS are also available.
BFGS and L-BFGS typically perform better when there are many predictors in the model (at least 50 predictors), otherwise fisher scoring is typically faster.
The grads argument is for L-BFGS only and it is the number of gradients that are stored at a time and are used to approximate the inverse information. The default value for this is 10, but another common choice is 5.
The tol argument controls how strict the convergence criteria are, higher tolerance will lead to more accurate results, but may also be slower.
The method argument is ignored for linear regression and the OLS solution is used.

3.2 Examples

An offset can be specified using offset, it should be a numeric vector.

### Using mtcars

library(BranchGLM)

cars <- mtcars

### Fitting linear regression model with Fisher scoring

LinearFit <- BranchGLM(mpg ~ ., data = cars, family = "gaussian", link = "identity")

LinearFit
#> Results from gaussian regression with identity link function 
#> Using the formula mpg ~ .
#> 
#>             Estimate      SE       t p.values  
#> (Intercept)  12.3034 49.6061  0.6573   0.5110  
#> cyl          -0.1114  2.7695 -0.1066   0.9151  
#> disp          0.0133  0.0473  0.7468   0.4552  
#> hp           -0.0215  0.0577 -0.9868   0.3237  
#> drat          0.7871  4.3341  0.4813   0.6303  
#> wt           -3.7153  5.0206 -1.9612   0.0499 *
#> qsec          0.8210  1.9369  1.1234   0.2613  
#> vs            0.3178  5.5774  0.1510   0.8800  
#> am            2.5202  5.4505  1.2254   0.2204  
#> gear          0.6554  3.9574  0.4389   0.6607  
#> carb         -0.1994  2.1964 -0.2406   0.8098  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Dispersion parameter taken to be 7.0235
#> 32 observations used to fit model
#> (0 observations removed due to missingness)
#> 
#> Residual Deviance: 147 on 21 degrees of freedom
#> AIC: 166

### Fitting gamma regression with inverse link with L-BFGS

GammaFit <- BranchGLM(mpg ~ ., data = cars, family = "gamma", link = "inverse",
                      method = "LBFGS", grads = 5)

GammaFit
#> Results from gamma regression with inverse link function 
#> Using the formula mpg ~ .
#> 
#>             Estimate      SE       z p.values   
#> (Intercept)  -0.0679  0.0029 -2.2012   0.0277 * 
#> cyl           0.0018  0.0002  0.9049   0.3655   
#> disp          0.0000  0.0000 -0.2273   0.8202   
#> hp           -0.0001  0.0000 -1.6589   0.0971 . 
#> drat         -0.0004  0.0003 -0.1565   0.8756   
#> wt           -0.0092  0.0003 -2.7445   0.0061 **
#> qsec          0.0017  0.0001  1.4919   0.1357   
#> vs           -0.0003  0.0003 -0.0933   0.9256   
#> am            0.0006  0.0003  0.1823   0.8554   
#> gear          0.0049  0.0002  1.8936   0.0583 . 
#> carb         -0.0010  0.0001 -0.6930   0.4883   
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Dispersion parameter taken to be 0.0087
#> 32 observations used to fit model
#> (0 observations removed due to missingness)
#> 
#> Residual Deviance: 0 on 21 degrees of freedom
#> AIC: 152
#> Algorithm converged in 2 iterations using L-BFGS

3.3 Useful functions

BranchGLM also has many utility functions for glms such as
- coef() to extract the coefficients
- logLik() to extract the log likelihood
- AIC() to extract the AIC
- BIC() to extract the BIC
- predict() to obtain predictions from the fitted model
The coefficients, standard errors, wald test statistics, and p-values are stored in the coefficients slot of the fitted model
Unlike glm there is no summary method, all the important information is included in the BranchGLM object.

### Predict method

predict(GammaFit)
#>  [1] 21.18615 20.58497 25.08262 19.73161 16.84319 19.66668 14.40835 22.08084
#>  [9] 23.72765 18.71060 19.08313 15.34283 16.20932 16.27113 12.61543 12.23133
#> [17] 12.09623 28.31611 31.25422 32.12053 22.43103 17.18572 17.63337 13.73915
#> [25] 15.78662 29.52933 26.61283 30.53478 16.86586 19.39727 14.08581 21.53070

### Accessing coefficients matrix

GammaFit$coefficients
#>                  Estimate           SE           z    p.values
#> (Intercept) -6.792915e-02 2.876056e-03 -2.20120066 0.027721822
#> cyl          1.760788e-03 1.813520e-04  0.90486903 0.365534775
#> disp        -7.992537e-06 3.276651e-06 -0.22732922 0.820167753
#> hp          -6.719661e-05 3.774991e-06 -1.65894623 0.097126627
#> drat        -4.270142e-04 2.542503e-04 -0.15652428 0.875619781
#> wt          -9.224571e-03 3.132502e-04 -2.74445162 0.006061209
#> qsec         1.738884e-03 1.086275e-04  1.49187199 0.135732706
#> vs          -3.101872e-04 3.096972e-04 -0.09334419 0.925630130
#> am           6.273623e-04 3.207733e-04  0.18227243 0.855368935
#> gear         4.880930e-03 2.402244e-04  1.89359104 0.058279314
#> carb        -1.026295e-03 1.380129e-04 -0.69303204 0.488289447

4 Performing variable selection

Forward selection, backward elimination, and branch and bound selection can be done using VariableSelection().
VariableSelection() can accept either a BranchGLM object or a formula along with the data and the desired family and link to perform the variable selection.
Available metrics are AIC and BIC, which are used to compare models and to select the best model.
VariableSelection() returns the final model and some other information about the search.
Note that VariableSelection() will not properly handle interaction terms, i.e. it may keep an interaction term while removing the lower-order terms.
keep can also be specified if any set of variables are desired to be kept in every model.

4.1 Stepwise methods

Forward selection and backward elimination are both stepwise variable selection methods.
They are not guaranteed to find the best model or even a good model, but they are very fast.
Forward selection is recommended if the number of variables is greater than the number of observations or if many of the larger models don’t converge.
Parallel computation can be used for the methods, but is generally only necessary for large datasets.

4.1.1 Forward selection example

### Forward selection with mtcars

VariableSelection(GammaFit, type = "forward")
#> Variable Selection Info:
#> -------------------------------------------
#> Variables were selected using forward selection with AIC
#> The best value of AIC obtained was 142
#> Number of models fit: 27
#> 
#> Order the variables were added to the model:
#> 
#> 1). wt
#> 2). hp
#> -------------------------------------------
#> Final Model:
#> -------------------------------------------
#> Results from gamma regression with inverse link function 
#> Using the formula mpg ~ hp + wt
#> 
#>             Estimate      SE       z p.values    
#> (Intercept)  -0.0089  0.0003 -3.1800   0.0015 ** 
#> hp           -0.0001  0.0000 -4.2117   <2e-16 ***
#> wt           -0.0098  0.0001 -7.0999   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Dispersion parameter taken to be 0.0104
#> 32 observations used to fit model
#> (0 observations removed due to missingness)
#> 
#> Residual Deviance: 0 on 29 degrees of freedom
#> AIC: 142
#> Algorithm converged in 2 iterations using Fisher's scoring

4.1.2 Backward elimination example

### Backward elimination with mtcars

VariableSelection(GammaFit, type = "backward")
#> Variable Selection Info:
#> --------------------------------------------
#> Variables were selected using backward elimination with AIC
#> The best value of AIC obtained was 142
#> Number of models fit: 49
#> 
#> Order the variables were removed from the model:
#> 
#> 1). vs
#> 2). drat
#> 3). am
#> 4). disp
#> 5). carb
#> 6). cyl
#> --------------------------------------------
#> Final Model:
#> --------------------------------------------
#> Results from gamma regression with inverse link function 
#> Using the formula mpg ~ hp + wt + qsec + gear
#> 
#>             Estimate      SE       z p.values    
#> (Intercept)  -0.0469  0.0017 -2.6323   0.0085 ** 
#> hp           -0.0001  0.0000 -2.0838   0.0372 *  
#> wt           -0.0095  0.0002 -5.2124   <2e-16 ***
#> qsec          0.0013  0.0001  1.7385   0.0821 .  
#> gear          0.0027  0.0002  1.6114   0.1071    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Dispersion parameter taken to be 0.0091
#> 32 observations used to fit model
#> (0 observations removed due to missingness)
#> 
#> Residual Deviance: 0 on 27 degrees of freedom
#> AIC: 142
#> Algorithm converged in 2 iterations using Fisher's scoring

4.2 Branch and bound

Branch and bound is much slower than the other methods, but it is guaranteed to find the best model.
The branch and bound method can be much faster than an exhaustive search and can also be made many times faster if parallel computation is used.
One way to judge how much faster the branch and bound algorithm is compared to an exhaustive search is to look at the ratio of the number of models fit against the total number of possible models.
- If this ratio is close to 1, then it is performing poorly and has to fit almost every model, if it is close to 0, then it is performing well and ruled out many models without having to fit them.

4.2.1 Branch and bound example

If showprogress is true, then progress of the branch and bound algorithm will be reported occasionally.
Parallel computation can be used with this method and can lead to very large speedups.

### Branch and bound with mtcars

VariableSelection(GammaFit, type = "branch and bound", showprogress = FALSE)
#> Variable Selection Info:
#> --------------------------------------------
#> Variables were selected using branch and bound selection with AIC
#> The best value of AIC obtained was 142
#> Number of models fit: 112
#> 
#> 
#> --------------------------------------------
#> Final Model:
#> --------------------------------------------
#> Results from gamma regression with inverse link function 
#> Using the formula mpg ~ hp + wt + qsec + gear
#> 
#>             Estimate      SE       z p.values    
#> (Intercept)  -0.0469  0.0017 -2.6323   0.0085 ** 
#> hp           -0.0001  0.0000 -2.0838   0.0372 *  
#> wt           -0.0095  0.0002 -5.2124   <2e-16 ***
#> qsec          0.0013  0.0001  1.7385   0.0821 .  
#> gear          0.0027  0.0002  1.6114   0.1071    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Dispersion parameter taken to be 0.0091
#> 32 observations used to fit model
#> (0 observations removed due to missingness)
#> 
#> Residual Deviance: 0 on 27 degrees of freedom
#> AIC: 142
#> Algorithm converged in 2 iterations using Fisher's scoring

### Can also use a formula and data

FormulaVS <- VariableSelection(mpg ~ . ,data = cars, family = "gamma", 
                               link = "inverse", type = "branch and bound",
                               showprogress = FALSE)

### Number of models fit divided by the number of possible models

FormulaVS$numchecked / 2^(length(FormulaVS$variables))
#> [1] 0.109375

### Extracting final model

FormulaVS$finalmodel
#> Results from gamma regression with inverse link function 
#> Using the formula mpg ~ hp + wt + qsec + gear
#> 
#>             Estimate      SE       z p.values    
#> (Intercept)  -0.0469  0.0017 -2.6323   0.0085 ** 
#> hp           -0.0001  0.0000 -2.0838   0.0372 *  
#> wt           -0.0095  0.0002 -5.2124   <2e-16 ***
#> qsec          0.0013  0.0001  1.7385   0.0821 .  
#> gear          0.0027  0.0002  1.6114   0.1071    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Dispersion parameter taken to be 0.0091
#> 32 observations used to fit model
#> (0 observations removed due to missingness)
#> 
#> Residual Deviance: 0 on 27 degrees of freedom
#> AIC: 142
#> Algorithm converged in 2 iterations using Fisher's scoring

4.3 Using keep

Specifying variables via keep will ensure that those variables are kept through the selection process.

### Example of using keep

VariableSelection(mpg ~ . ,data = cars, family = "gamma", 
                               link = "inverse", type = "branch and bound",
                               keep = c("hp", "cyl"), metric = "AIC",
                               showprogress = FALSE)
#> Variable Selection Info:
#> --------------------------------------------
#> Variables were selected using branch and bound selection with AIC
#> The best value of AIC obtained was 143
#> Number of models fit: 49
#> Variables that were kept in each model:  hp, cyl
#> 
#> --------------------------------------------
#> Final Model:
#> --------------------------------------------
#> Results from gamma regression with inverse link function 
#> Using the formula mpg ~ cyl + hp + wt + qsec + gear
#> 
#>             Estimate      SE       z p.values    
#> (Intercept)  -0.0646  0.0026 -2.3934   0.0167 *  
#> cyl           0.0014  0.0002  0.8601   0.3897    
#> hp           -0.0001  0.0000 -2.2650   0.0235 *  
#> wt           -0.0104  0.0002 -4.9820   <2e-16 ***
#> qsec          0.0018  0.0001  1.9186   0.0550 .  
#> gear          0.0039  0.0002  1.7910   0.0733 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Dispersion parameter taken to be 0.0089
#> 32 observations used to fit model
#> (0 observations removed due to missingness)
#> 
#> Residual Deviance: 0 on 26 degrees of freedom
#> AIC: 143
#> Algorithm converged in 2 iterations using Fisher's scoring

4.4 Convergence issues

It is not recommended to use branch and bound if the upper models do not converge since it can make the algorithm very slow.
Sometimes when using backwards selection and all the upper models that are tested do not converge, no final model can be selected.
For these reasons, if there are convergence issues it is recommended to use forward selection.

5 Utility functions for binomial glms

BranchGLM also has some utility functions for binomial glms
- Table() creates a confusion matrix based on the predicted classes and observed classes
- ROC() creates an ROC curve which can be plotted with plot()
- AUC() and Cindex() calculate the area under the ROC curve
- MultipleROCCurves() allows for the plotting of multiple ROC curves on the same plot

5.1 Table

### Predicting if a car gets at least 18 mpg

catData <- ToothGrowth

catFit <- BranchGLM(supp ~ ., data = catData, family = "binomial", link = "logit")

Table(catFit)
#> Confusion matrix:
#> ----------------------
#>             Predicted
#>              OJ   VC
#> 
#>          OJ  17   13
#> Observed
#>          VC  7    23
#> 
#> ----------------------
#> Measures:
#> ----------------------
#> Accuracy:  0.6667 
#> Sensitivity:  0.7667 
#> Specificity:  0.5667 
#> PPV:  0.6389

5.2 ROC


catROC <- ROC(catFit)

plot(catROC, main = "ROC Curve", col = "indianred")

5.3 Cindex/AUC


Cindex(catFit)
#> [1] 0.7127778

AUC(catFit)
#> [1] 0.7127778

5.4 MultipleROCPlots

### Showing ROC plots for logit, probit, and cloglog

probitFit <- BranchGLM(supp ~ . ,data = catData, family = "binomial", 
                       link = "probit")

cloglogFit <- BranchGLM(supp ~ . ,data = catData, family = "binomial", 
                       link = "cloglog")

MultipleROCCurves(catROC, ROC(probitFit), ROC(cloglogFit), 
                  names = c("Logistic ROC", "Probit ROC", "Cloglog ROC"))

5.5 Using predictions

For each of the methods used in this section predicted probabilities and observed classes can also be supplied instead of the BranchGLM object.


preds <- predict(catFit)

Table(preds, catData$supp)
#> Confusion matrix:
#> ----------------------
#>             Predicted
#>              OJ   VC
#> 
#>          OJ  17   13
#> Observed
#>          VC  7    23
#> 
#> ----------------------
#> Measures:
#> ----------------------
#> Accuracy:  0.6667 
#> Sensitivity:  0.7667 
#> Specificity:  0.5667 
#> PPV:  0.6389

AUC(preds, catData$supp)
#> [1] 0.7127778

ROC(preds, catData$supp) |> plot(main = "ROC Curve", col = "deepskyblue")

BranchGLM Vignette

1 Description

2 Installation

3 Fitting glms

3.1 Optimization methods

3.2 Examples

3.3 Useful functions

4 Performing variable selection

4.1 Stepwise methods

4.1.1 Forward selection example

4.1.2 Backward elimination example

4.2 Branch and bound

4.2.1 Branch and bound example

4.3 Using keep

4.4 Convergence issues

5 Utility functions for binomial glms

5.1 Table

5.2 ROC

5.3 Cindex/AUC

5.4 MultipleROCPlots

5.5 Using predictions