# Multiple (Linear) Regression

R provides comprehensive support for multiple linear regression. The topics below are provided in order of increasing complexity.

## Fitting the Model

```
# Multiple Linear Regression Example
```

fit <- lm(y ~ x1 + x2 + x3, data=mydata)

summary(fit) # show results

`# Other useful functions `

coefficients(fit) # model coefficients

confint(fit, level=0.95) # CIs for model parameters

fitted(fit) # predicted values

residuals(fit) # residuals

anova(fit) # anova table

vcov(fit) # covariance matrix for model parameters

influence(fit) # regression diagnostics

## Diagnostic Plots

Diagnostic plots provide checks for heteroscedasticity, normality, and influential observerations.

`# diagnostic plots `

layout(matrix(c(1,2,3,4),2,2)) # optional 4 graphs/page

plot(fit)

For a more comprehensive evaluation of model fit see regression diagnostics or the exercises in this interactive course on regression.

## Comparing Models

You can compare nested models with the anova( ) function. The following code provides a simultaneous test that x3 and x4 add to linear prediction above and beyond x1 and x2.

`# compare models`

fit1 <- lm(y ~ x1 + x2 + x3 + x4, data=mydata)

fit2 <- lm(y ~ x1 + x2)

anova(fit1, fit2)

## Cross Validation

You can do K-Fold cross-validation using the cv.lm( ) function in the DAAG package.

`# K-fold cross-validation`

library(DAAG)

cv.lm(df=mydata, fit, m=3) # 3 fold cross-validation

Sum the MSE for each fold, divide by the number of observations, and take the square root to get the cross-validated standard error of estimate.

You can assess **R2 shrinkage** via K-fold cross-validation. Using the **crossval() **function from the **bootstrap **package, do the following:

```
# Assessing R2 shrinkage using 10-Fold Cross-Validation
```

fit <- lm(y~x1+x2+x3,data=mydata)

library(bootstrap)

# define functions

theta.fit <- function(x,y){lsfit(x,y)}

theta.predict <- function(fit,x){cbind(1,x)%*%fit$coef}

# matrix of predictors

X
<- as.matrix(mydata[c("x1","x2","x3")])

# vector of predicted values

y <- as.matrix(mydata[c("y")])

results <- crossval(X,y,theta.fit,theta.predict,ngroup=10)

cor(y, fit$fitted.values)**2 # raw R2

cor(y,results$cv.fit)**2 # cross-validated R2

## Variable Selection

Selecting a subset of predictor variables from a larger set (e.g., stepwise selection) is a controversial topic. You can perform stepwise selection (forward, backward, both) using the **stepAIC( )** function from the **MASS **package. **stepAIC( ) ** performs stepwise model selection by exact AIC.

`# Stepwise Regression`

library(MASS)

fit <- lm(y~x1+x2+x3,data=mydata)

step <- stepAIC(fit, direction="both")

step$anova # display results

Alternatively, you can perform all-subsets regression using the **leaps( )** function from the leaps package. In the following code nbest indicates the number of subsets of each size to report. Here, the ten best models will be reported for each subset size (1 predictor, 2 predictors, etc.).

`# All Subsets Regression`

library(leaps)

attach(mydata)

leaps<-regsubsets(y~x1+x2+x3+x4,data=mydata,nbest=10)

# view results

summary(leaps)

# plot a table of models showing variables in each model.

#
models are ordered by the selection statistic.

plot(leaps,scale="r2")

# plot statistic by subset size

library(car)

subsets(leaps, statistic="rsq")

Other options for **plot( )** are bic, Cp, and adjr2. Other options for plotting with**
subset( )** are bic, cp, adjr2, and rss.

## Relative Importance

The relaimpo package provides measures of relative importance for each of the predictors in the model. See help(calc.relimp) for details on the four measures of relative importance provided.

`# Calculate Relative Importance for Each Predictor`

library(relaimpo)

calc.relimp(fit,type=c("lmg","last","first","pratt"),

rela=TRUE)

# Bootstrap Measures of Relative Importance (1000 samples)

boot <- boot.relimp(fit, b = 1000, type = c("lmg",

"last", "first", "pratt"), rank = TRUE,

diff = TRUE, rela = TRUE)

booteval.relimp(boot) # print result

plot(booteval.relimp(boot,sort=TRUE)) # plot result

## Graphic Enhancements

The car package offers a wide variety of plots for regression, including added variable plots, and enhanced diagnostic and Scatterplots.

## Going Further

### Nonlinear Regression

The **nls** package provides functions for nonlinear regression. See John Fox's Nonlinear Regression and Nonlinear Least Squares for an overview. Huet and colleagues' Statistical Tools for Nonlinear Regression: A Practical Guide with S-PLUS and R Examples is a valuable reference book.

### Robust Regression

There are many functions in R to aid with robust regression. For example, you can perform robust regression with the **rlm( )** function in the **MASS** package. John Fox's (who else?) Robust Regression provides a good starting overview. The UCLA Statistical Computing website has Robust Regression Examples.

The robust package provides a comprehensive library of robust methods, including regression. The robustbase package also provides basic robust statistics including model selection methods. And David Olive has provided an detailed online review of Applied Robust Statistics with sample R code.

## To Practice

This course in machine learning in R includes excercises in multiple regression and cross validation.