Multiple regression
SCCS R packageWikipedia:Variance_inflation_factor#Interpretation vif > 5 multicollinearity
Multiple regression. Instead of seeing if variables measure the same thing, we may want to see if different variables combine to predict others, possibly through their causal effects. It may be that the combination of the mother's role and the father's role in the nuclear family combine to reduce rates of crime. Here two separate variables may have an outcome for the third. Spss commands /Analyze/Regression ... etc (to be continued: Wikipedia:Regression).
Contents |
Texts and Examples for Robust Regression
New software: Quick R Multiple (Linear) Regression
http://www.statmethods.net/stats/regression.html 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 observations.
- diagnostic plots
layout(matrix(c(1,2,3,4),2,2))
- optional 4 graphs/page
plot(fit)
click to view
For a more comprehensive evaluation of model fit see regression diagnostics.
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")
click to view
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
click to view
GRAPHIC ENHANCEMENTS
The car package offers a wide variety of plots for regression, including added variable plots, and enhanced diagnostic and scatter plots.
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.
New software: Quick R Regression Diagnostics
Regression Diagnostics An excellent review of regression diagnostics is provided in John Fox's aptly named Overview of Regression Diagnostics. Dr. Fox's car package provides advanced utilities for regression modeling.
- Assume that we are fitting a multiple linear regression on the MTCARS data
library(car) fit <- lm(mpg~disp+hp+wt+drat, data=mtcars) This example is for exposition only. We will ignore the fact that this may not be a great way of modeling the this particular set of data! OUTLIERS
- Assessing Outliers
outlierTest(fit)
- Bonferroni p-value for most extreme obs
qqPlot(fit, main="QQ Plot") #qq plot for studentized resid leveragePlots(fit) *leverage plots
click to view leverage - outlier test
Bonferroni: The sequential Bonferroni (Holm 1979, Rice 1989) evaluates the increasing probability discounts in a series of ordered significance values. E.g., for Brown and Eff (2009) the only variables that pass the test are negative PCsize.2 (squared) (p = 0.011) and PCAP (p=0.012 ), two variables created by combining variables through principal components and then taking the first principal component. Food scarcity (p=.016) is a close third but even with a p<.05 threshold the probability of at least one of these three (.95*.95=.90, .95*.95*.95=.86 (.95*.95*.95=.86 occurring at random, even with a p<.05 threshold, the third variable does not pass the group significance test. With a p<.10 threshold and excluding the control variable for Date of observation (logdate) the “hidden variables” model also passes the Bonferroni test, p=.008 for No_rain_Dry and p=.034 for AnimXbridewealth, with Missions a close third (p=.043).
INFLUENTIAL OBSERVATIONS
- Influential Observations
- added variable plots
av.Plots(fit)
- Cook's D plot
- identify D values > 4/(n-k-1)
cutoff <- 4/((nrow(mtcars)-length(fit$coefficients)-2)) plot(fit, which=4, cook.levels=cutoff)
- Influence Plot
influencePlot(fit, id.method="identify", main="Influence Plot", sub="Circle size is proportial to Cook's Distance" )
click to view Average variable plot click to view influenceplot click to view CooksD
NON-NORMALITY
- Normality of Residuals
- qq plot for studentized resid
qqPlot(fit, main="QQ Plot")
- distribution of studentized residuals
library(MASS) sresid <- studres(fit) hist(sresid, freq=FALSE,
main="Distribution of Studentized Residuals")
xfit<-seq(min(sresid),max(sresid),length=40) yfit<-dnorm(xfit) lines(xfit, yfit)
click to view qqplot2 click to view Studentized residuals
NON-CONSTANT ERROR VARIANCE
- Evaluate homoscedasticity
- non-constant error variance test
ncvTest(fit)
- plot studentized residuals vs. fitted values
spreadLevelPlot(fit)
click to view spreadlevel plot
MULTI-COLLINEARITY
- Evaluate Collinearity
vif(fit) *variance inflation factors sqrt(vif(fit)) > 2 *problem? NONLINEARITY
- Evaluate Nonlinearity
- component + residual plot
crPlots(fit)
- Ceres plots
ceresPlots(fit)
click to view component residual plots click to view ceres plot
NON-INDEPENDENCE OF ERRORS
- Test for Autocorrelated Errors
durbinWatsonTest(fit) ADDITIONAL DIAGNOSTIC HELP The gvlma( ) function in the gvlma package, performs a global validation of linear model assumptions as well separate evaluations of skewness, kurtosis, and heteroscedasticity.
- Global test of model assumptions
library(gvlma) gvmodel <- gvlma(fit) summary(gvmodel)
GOING FURTHER If you would like to delve deeper into regression diagnostics, two books written by John Fox can help: Applied regression analyses, linear models, and related methods and An R and S-Plus companion to applied regression.
Links
Testing the assumptions of linear regression by Robert Nau
Common Correlation and Reliability Analysis with SPSS for Windows by Robert A. Yaffee
Panda
P.A.N.D.A. Practical Analysis of Nutrition Data. Krishna Belbase UNICEF and Tulane University
