Joint inference on a vector of estimable parameters
Data Cars2004nh
data(Cars2004nh, package = "mffSM")
head(Cars2004nh)
## vname type drive price.retail price.dealer price cons.city cons.highway
## 1 Chevrolet.Aveo.4dr 1 1 11690 10965 11327.5 8.4 6.9
## 2 Chevrolet.Aveo.LS.4dr.hatch 1 1 12585 11802 12193.5 8.4 6.9
## 3 Chevrolet.Cavalier.2dr 1 1 14610 13697 14153.5 9.0 6.4
## 4 Chevrolet.Cavalier.4dr 1 1 14810 13884 14347.0 9.0 6.4
## 5 Chevrolet.Cavalier.LS.2dr 1 1 16385 15357 15871.0 9.0 6.4
## 6 Dodge.Neon.SE.4dr 1 1 13670 12849 13259.5 8.1 6.5
## consumption engine.size ncylinder horsepower weight iweight lweight wheel.base length width
## 1 7.65 1.6 4 103 1075 0.0009302326 6.980076 249 424 168
## 2 7.65 1.6 4 103 1065 0.0009389671 6.970730 249 389 168
## 3 7.70 2.2 4 140 1187 0.0008424600 7.079184 264 465 175
## 4 7.70 2.2 4 140 1214 0.0008237232 7.101676 264 465 173
## 5 7.70 2.2 4 140 1187 0.0008424600 7.079184 264 465 175
## 6 7.30 2.0 4 132 1171 0.0008539710 7.065613 267 442 170
## ftype fdrive
## 1 personal front
## 2 personal front
## 3 personal front
## 4 personal front
## 5 personal front
## 6 personal front
dim(Cars2004nh)
## [1] 425 20
summary(Cars2004nh)
## vname type drive price.retail price.dealer
## Length:425 Min. :1.000 Min. :1.000 Min. : 10280 Min. : 9875
## Class :character 1st Qu.:1.000 1st Qu.:1.000 1st Qu.: 20370 1st Qu.: 18973
## Mode :character Median :1.000 Median :1.000 Median : 27905 Median : 25672
## Mean :2.219 Mean :1.692 Mean : 32866 Mean : 30096
## 3rd Qu.:3.000 3rd Qu.:2.000 3rd Qu.: 39235 3rd Qu.: 35777
## Max. :6.000 Max. :3.000 Max. :192465 Max. :173560
##
## price cons.city cons.highway consumption engine.size ncylinder
## Min. : 10078 Min. : 6.20 Min. : 5.100 Min. : 5.65 Min. :1.300 Min. :-1.000
## 1st Qu.: 19600 1st Qu.:11.20 1st Qu.: 8.100 1st Qu.: 9.65 1st Qu.:2.400 1st Qu.: 4.000
## Median : 26656 Median :12.40 Median : 9.000 Median :10.70 Median :3.000 Median : 6.000
## Mean : 31481 Mean :12.36 Mean : 9.142 Mean :10.75 Mean :3.208 Mean : 5.791
## 3rd Qu.: 37514 3rd Qu.:13.80 3rd Qu.: 9.800 3rd Qu.:11.65 3rd Qu.:3.900 3rd Qu.: 6.000
## Max. :183012 Max. :23.50 Max. :19.600 Max. :21.55 Max. :8.300 Max. :12.000
## NA's :14 NA's :14 NA's :14
## horsepower weight iweight lweight wheel.base length
## Min. :100.0 Min. : 923 Min. :0.0003067 Min. :6.828 Min. :226.0 Min. :363.0
## 1st Qu.:165.0 1st Qu.:1412 1st Qu.:0.0005542 1st Qu.:7.253 1st Qu.:262.0 1st Qu.:450.0
## Median :210.0 Median :1577 Median :0.0006341 Median :7.363 Median :272.0 Median :472.0
## Mean :216.8 Mean :1626 Mean :0.0006412 Mean :7.373 Mean :274.9 Mean :470.6
## 3rd Qu.:255.0 3rd Qu.:1804 3rd Qu.:0.0007082 3rd Qu.:7.498 3rd Qu.:284.0 3rd Qu.:490.0
## Max. :500.0 Max. :3261 Max. :0.0010834 Max. :8.090 Max. :366.0 Max. :577.0
## NA's :2 NA's :2 NA's :2 NA's :2 NA's :26
## width ftype fdrive
## Min. :163.0 personal:242 front:223
## 1st Qu.:175.0 wagon : 30 rear :110
## Median :180.0 SUV : 60 4x4 : 92
## Mean :181.1 pickup : 24
## 3rd Qu.:185.0 sport : 49
## Max. :206.0 minivan : 20
## NA's :28
To be able to compare a model fitted here with other models where also other covariates will be included, we restrict ourselves to a subset of the dataset where all variables consumption, lweight and engine.size are known.
isComplete <- complete.cases(Cars2004nh[, c("consumption", "lweight", "engine.size")])
sum(!isComplete)
## [1] 16
CarsNow <- subset(Cars2004nh, isComplete, select = c("consumption", "drive", "fdrive", "weight", "lweight", "engine.size"))
dim(CarsNow)
## [1] 409 6
summary(CarsNow)
## consumption drive fdrive weight lweight engine.size
## Min. : 5.65 Min. :1.000 front:212 Min. : 923 Min. :6.828 Min. :1.300
## 1st Qu.: 9.65 1st Qu.:1.000 rear :108 1st Qu.:1415 1st Qu.:7.255 1st Qu.:2.400
## Median :10.70 Median :1.000 4x4 : 89 Median :1577 Median :7.363 Median :3.000
## Mean :10.75 Mean :1.699 Mean :1622 Mean :7.371 Mean :3.178
## 3rd Qu.:11.65 3rd Qu.:2.000 3rd Qu.:1804 3rd Qu.:7.498 3rd Qu.:3.800
## Max. :21.55 Max. :3.000 Max. :2903 Max. :7.973 Max. :6.000
consumption on fdrive(ybar <- with(CarsNow, mean(consumption)))
## [1] 10.75134
(ybargr <- with(CarsNow, tapply(consumption, fdrive, mean)))
## front rear 4x4
## 9.741274 11.293981 12.498876
set.seed(20010911)
par(mfrow = c(1, 1), bty = BTY, mar = c(4, 4, 1, 1) + 0.1)
plot(CarsNow[, "drive"] + runif(nrow(CarsNow), -0.2, 0.2), CarsNow[, "consumption"], xaxt = "n", col = COL, bg = BGC, pch = PCH,
xlab = "Drive", ylab = "Consumption [l/100 km]", cex.lab = 1.2, cex.axis = 1.2)
points(1:3, ybargr, pch = 22, col = "darkgreen", bg = "seagreen", cex = 2)
axis(1, at = 1:3, labels = levels(CarsNow[, "fdrive"]), cex.axis = 1.2)
FCOL <- rainbow_hcl(3)
names(FCOL) <- levels(CarsNow[, "fdrive"])
par(mfrow = c(1, 1), bty = BTY, mar = c(4, 4, 1, 1) + 0.1)
plot(consumption ~ fdrive, data = CarsNow, col = FCOL, xlab = "Drive", ylab = "Consumption [l/100 km]", cex.lab = 1.2, cex.axis = 1.2)
A well-known one-way ANOVA is used below to evaluate the effect of fdrive on consumption.
a1 <- aov(consumption ~ fdrive, data = CarsNow)
summary(a1)
## Df Sum Sq Mean Sq F value Pr(>F)
## fdrive 2 519.9 259.94 78.92 <2e-16 ***
## Residuals 406 1337.4 3.29
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Covariate fdrive is categorical with three levels. So called reference group pseudocontrasts are used to include this covariate in a linear model.
levels(CarsNow[["fdrive"]]) ## Which group is the first one?
## [1] "front" "rear" "4x4"
mTrt <- lm(consumption ~ fdrive, data = CarsNow)
summary(mTrt)
##
## Call:
## lm(formula = consumption ~ fdrive, data = CarsNow)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.0913 -1.2489 -0.0440 0.9587 9.0511
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.7413 0.1247 78.149 < 2e-16 ***
## fdriverear 1.5527 0.2146 7.237 2.32e-12 ***
## fdrive4x4 2.7576 0.2292 12.030 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.815 on 406 degrees of freedom
## Multiple R-squared: 0.2799, Adjusted R-squared: 0.2764
## F-statistic: 78.91 on 2 and 406 DF, p-value: < 2.2e-16
vcov(mTrt)
## (Intercept) fdriverear fdrive4x4
## (Intercept) 0.01553763 -0.01553763 -0.01553763
## fdriverear -0.01553763 0.04603742 0.01553763
## fdrive4x4 -0.01553763 0.01553763 0.05254861
Matrix \(\mathbb{L}\) is chosen such that \(\theta\) provides the group means.
Lmu <- cbind(1, contr.treatment(3))
rownames(Lmu) <- levels(CarsNow[["fdrive"]])
colnames(Lmu) <- names(coef(mTrt))
print(Lmu)
## (Intercept) fdriverear fdrive4x4
## front 1 0 0
## rear 1 1 0
## 4x4 1 0 1
(muhat <- Lmu %*% coef(mTrt))
## [,1]
## front 9.741274
## rear 11.293981
## 4x4 12.498876
LSest is a function from package mffSM.library("mffSM")
(muCI <- LSest(mTrt, L = Lmu))
## Estimate Std. Error t value P value Lower Upper
## front 9.741274 0.1246500 78.14899 < 2.22e-16 9.496234 9.986314
## rear 11.293981 0.1746419 64.66937 < 2.22e-16 10.950666 11.637297
## 4x4 12.498876 0.1923824 64.96892 < 2.22e-16 12.120686 12.877066
(Lnow <- Lmu[1:2,])
## (Intercept) fdriverear fdrive4x4
## front 1 0 0
## rear 1 1 0
(mNow <- nrow(Lnow)) ## number of parameters to estimate
## [1] 2
(elCenter <- as.numeric(Lnow %*% coef(mTrt))) ## center of the ellipse
## [1] 9.741274 11.293981
(elShape <- Lnow %*% vcov(mTrt) %*% t(Lnow)) ## shape of the ellipse
## front rear
## front 0.01553763 0.00000000
## rear 0.00000000 0.03049979
(elRadius <- sqrt(mNow * qf(0.95, mNow, mTrt[["df.residual"]]))) ## radius of the ellipse
## [1] 2.456805
ellipse is a function from package car.library("car")
el <- ellipse(elCenter, elShape, elRadius, draw = FALSE)
#print(el)
plot(y ~ x, data = el, type = "l", xlab = rownames(Lnow)[1], ylab = rownames(Lnow)[2], col = "red3", lwd = 2)
points(elCenter[1], elCenter[2], pch = 21, col = "red3", bg = "orange", cex = 2)
confidenceEllipse is a function from package carconfidenceEllipse(mTrt, L = Lnow, col = "red3", fill = TRUE, center.cex = 2)
abline(v = c(muCI[["Lower"]][1], muCI[["Upper"]][1]), col = "blue", lwd = 2)
abline(h = c(muCI[["Lower"]][2], muCI[["Upper"]][2]), col = "blue", lwd = 2)
title(main = paste("(", paste(attr(muCI, "row.names")[c(1, 2)], collapse = ", "), ")", sep = ""))
par(bty = BTY, mar = c(4, 4, 4, 1) + 0.1)
layout(matrix(c(0,1,1,0, 2,2,3,3), nrow = 2, byrow = TRUE))
for (j in 1:2){
for (l in (j+1):3){
confidenceEllipse(mTrt, L = Lmu[c(j, l),], col = "red3", fill = TRUE, center.cex = 2)
abline(v = c(muCI[["Lower"]][j], muCI[["Upper"]][j]), col = "blue", lwd = 2)
abline(h = c(muCI[["Lower"]][l], muCI[["Upper"]][l]), col = "blue", lwd = 2)
title(main = paste("(", paste(attr(muCI, "row.names")[c(j, l)], collapse = ", "), ")", sep = ""))
}
}
par(mfrow = c(1, 1))
(mmu <- length(muhat))
## [1] 3
(el3DCenter <- as.numeric(Lmu %*% coef(mTrt))) ## center of the ellipsoid
## [1] 9.741274 11.293981 12.498876
(el3DShape <- Lmu %*% vcov(mTrt) %*% t(Lmu)) ## shape of the ellipsoid
## front rear 4x4
## front 1.553763e-02 0.00000000 3.469447e-18
## rear 0.000000e+00 0.03049979 0.000000e+00
## 4x4 3.469447e-18 0.00000000 3.701098e-02
(el3DRadius <- sqrt(mmu * qf(0.95, mmu, mTrt[["df.residual"]]))) ## radius of the ellipsoid
## [1] 2.807249
ellipse3d is a function from package rgl.library("rgl")
el3D <- ellipse3d(x = el3DShape, centre = el3DCenter, t = el3DRadius)
#print(el3D)
plot3d(el3D, xlab = rownames(Lmu)[1], ylab = rownames(Lmu)[2], zlab = rownames(Lmu)[3], type = "wire", col = "blue")
#rgl.snapshot("/home/komarek/teach/mff_2015/nmsa407_LinRegr/Lecture/RkoTutor/figPNG/LinRegr-NormalLM-02-06.png")
plot3d(el3D, xlab = rownames(Lmu)[1], ylab = rownames(Lmu)[2], zlab = rownames(Lmu)[3], col = "blue")
#rgl.snapshot("/home/komarek/teach/mff_2015/nmsa407_LinRegr/Lecture/RkoTutor/figPNG/LinRegr-NormalLM-02-07.png")