Multiple Regression Model
Donghyung Lee
2019-11-15
Last updated: 2019-11-15
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Knit directory: STA_463_563_Fall2019/
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Fit model with transformed predictor/response
Plastic hardness data, fit \(Y=\beta_0+\beta_1X+\epsilon\).
plastic=read.table("http://users.stat.ufl.edu/~rrandles/sta4210/Rclassnotes/data/textdatasets/KutnerData/Chapter%20%201%20Data%20Sets/CH01PR22.txt")
colnames(plastic)=c("Y","X")
lm(Y~X,data=plastic)
Call:
lm(formula = Y ~ X, data = plastic)
Coefficients:
(Intercept) X
168.600 2.034 lm(Y~X^2,data=plastic)
Call:
lm(formula = Y ~ X^2, data = plastic)
Coefficients:
(Intercept) X
168.600 2.034 What’s wrong with the above result?
Solutions:
Y2=(plastic$Y)^2
X2=(plastic$X)^2
lm(Y~X2,data=plastic)
Call:
lm(formula = Y ~ X2, data = plastic)
Coefficients:
(Intercept) X2
194.88253 0.03551 lm(Y~I(X^2),data=plastic)
Call:
lm(formula = Y ~ I(X^2), data = plastic)
Coefficients:
(Intercept) I(X^2)
194.88253 0.03551 Multiple Regression Fit (Lecture 13)
1. Data: The Prestige data frame is the Prestige of Canadian Occupations. It has 102 rows and 6 columns. The observations are occupations. First of all, we remove all observations that have missing values for varaible ``type’’.
#install.packages("car")
library(car)Loading required package: carData#?Prestige
mydata=Prestige[!is.na(Prestige$type), ]#remove the observations with missing values for "type"
unique(mydata$type)[1] prof bc wc
Levels: bc prof wcTo have an idea about the predictor variable and the response, we can view the enhanced scatterplot.
scatterplot(income~education|type, data=mydata,ylab="Income",
xlab="Education",main="Income vs Education")
Based on the plot, maybe there’s linear relationsihp between education and income. It seems the intercept of bc and prof are similar, different from that of the wc. We also want know whether the slope of the three groups are different (whether interaction term is necessary).
Alternative method using the ggplot:
library(ggplot2)
ggplot(data=mydata, aes(x=education, y=income)) +
geom_point(aes(colour = factor(type))) +
geom_smooth(aes(colour = factor(type)))`geom_smooth()` using method = 'loess' and formula 'y ~ x'
2. Generate the design matrix
X=model.matrix(~type,data=mydata)
#X
Y=as.matrix(mydata$income)
solve(t(X)%*%X)%*%t(X)%*%Y [,1]
(Intercept) 5374.136
typeprof 5185.315
typewc -321.832lm(income~type,data=mydata)#automatically, bc is the reference group.
Call:
lm(formula = income ~ type, data = mydata)
Coefficients:
(Intercept) typeprof typewc
5374.1 5185.3 -321.8 3. Mix continuous and categorical variable.
(1) Use income as response, education and type as predictor.
X=model.matrix(~education+type,data=mydata)
#X
Y=as.matrix(mydata$income)
solve(t(X)%*%X)%*%t(X)%*%Y [,1]
(Intercept) -2048.2080
education 887.9126
typeprof 102.1260
typewc -2685.8293lm(income~education+type,data=mydata)
Call:
lm(formula = income ~ education + type, data = mydata)
Coefficients:
(Intercept) education typeprof typewc
-2048.2 887.9 102.1 -2685.8 #If the data is in letters, R recognize it as a categorical factor.(2) If you don’t like bc be reference group, change it to wc.
new=relevel(mydata$type,ref="wc")
X=model.matrix(~education+new,data=mydata)
Y=as.matrix(mydata$income)
solve(t(X)%*%X)%*%t(X)%*%Y [,1]
(Intercept) -4734.0372
education 887.9126
newbc 2685.8293
newprof 2787.9553lm(income~education+new,data=mydata)
Call:
lm(formula = income ~ education + new, data = mydata)
Coefficients:
(Intercept) education newbc newprof
-4734.0 887.9 2685.8 2788.0 (3) Like constructing a linear regression model, you may also include interaction terms
X=model.matrix(~education+type+education*type,data=mydata)
Y=as.matrix(mydata$income)
solve(t(X)%*%X)%*%t(X)%*%Y [,1]
(Intercept) -1865.0362
education 866.0004
typeprof -3068.3645
typewc 3646.5441
education:typeprof 234.0166
education:typewc -569.2417lm(income~education+type+education*type,data=mydata)
Call:
lm(formula = income ~ education + type + education * type, data = mydata)
Coefficients:
(Intercept) education typeprof
-1865.0 866.0 -3068.4
typewc education:typeprof education:typewc
3646.5 234.0 -569.2 4. What if you want a categorical variable, but it’s coded in 1,2,3, etc.
(1) Directly use it, R will recognize it as numeric variable.
mydata2=cbind(mydata,c(rep(1,40),rep(2,38),rep(3,20)))
colnames(mydata2)[7]=c("group")
#mydata2
lm(income~group,data=mydata2)
Call:
lm(formula = income ~ group, data = mydata2)
Coefficients:
(Intercept) group
9889 -1643 (2) We can manually let R know it’s a categorical variable.
mydata2$group=factor(mydata2$group)
lm(income~group,data=mydata2)
Call:
lm(formula = income ~ group, data = mydata2)
Coefficients:
(Intercept) group2 group3
8891 -3643 -2641 #alternative way
groupf=factor(mydata2$group)
lm(income~groupf,data=mydata2)
Call:
lm(formula = income ~ groupf, data = mydata2)
Coefficients:
(Intercept) groupf2 groupf3
8891 -3643 -2641 #alternative way
lm(income~factor(group),data=mydata2)
Call:
lm(formula = income ~ factor(group), data = mydata2)
Coefficients:
(Intercept) factor(group)2 factor(group)3
8891 -3643 -2641
sessionInfo()R version 3.6.1 (2019-07-05)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] ggplot2_3.2.1 car_3.0-4 carData_3.0-2
loaded via a namespace (and not attached):
[1] zip_2.0.4 Rcpp_1.0.2 compiler_3.6.1
[4] pillar_1.4.2 cellranger_1.1.0 git2r_0.26.1
[7] workflowr_1.4.0 forcats_0.4.0 tools_3.6.1
[10] zeallot_0.1.0 digest_0.6.20 gtable_0.3.0
[13] evaluate_0.14 tibble_2.1.3 pkgconfig_2.0.2
[16] rlang_0.4.0 openxlsx_4.1.3 curl_4.0
[19] yaml_2.2.0 haven_2.1.1 xfun_0.9
[22] rio_0.5.16 withr_2.1.2 dplyr_0.8.3
[25] stringr_1.4.0 knitr_1.24 fs_1.3.1
[28] vctrs_0.2.0 hms_0.5.1 tidyselect_0.2.5
[31] rprojroot_1.3-2 grid_3.6.1 glue_1.3.1
[34] data.table_1.12.2 R6_2.4.0 readxl_1.3.1
[37] foreign_0.8-71 rmarkdown_1.15 purrr_0.3.2
[40] magrittr_1.5 whisker_0.3-2 scales_1.0.0
[43] backports_1.1.4 htmltools_0.3.6 assertthat_0.2.1
[46] abind_1.4-5 colorspace_1.4-1 labeling_0.3
[49] stringi_1.4.3 lazyeval_0.2.2 munsell_0.5.0
[52] crayon_1.3.4