ANOVA Tables of Type I, II and III
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 lweight and fdriveconsumption on lweight by fdrivepar(mfrow = c(2, 2), bty = BTY, mar = c(5, 4, 3, 1) + 0.1)
for (dr in levels(CarsNow[, "fdrive"])){
plot(consumption ~ lweight, data = subset(CarsNow, fdrive == dr), pch = PCH, col = COL, bg = BGC,
xlab = "Log(weight) [log(kg)]", ylab = "Consumption [l/100 km]", main = dr,
xlim = range(CarsNow[, "lweight"]), ylim = range(CarsNow[, "consumption"]))
}
consumption on lweight by fdrive in one plotFCOL <- rainbow_hcl(3)
FCOL2 <- c("red3", "darkgreen", "darkblue")
FPCH <- c(21, 23, 24)
names(FCOL) <- names(FCOL2) <- names(FPCH) <- levels(CarsNow[, "fdrive"])
par(mfrow = c(1, 1), bty = BTY, mar = c(4, 4, 1, 1) + 0.1)
plot(consumption ~ lweight, data = CarsNow, pch = FPCH[fdrive], col = FCOL2[fdrive], bg = FCOL[fdrive],
xlab = "Log(weight) [log(kg)]", ylab = "Consumption [l/100 km]")
legend(6.9, 21, legend = levels(CarsNow[, "fdrive"]), title = "Drive", pch = FPCH, col = FCOL2, pt.bg = FCOL)
lweight and fdrive as covariatesmInter <- lm(consumption ~ fdrive + lweight + fdrive:lweight, data = CarsNow)
mAddit <- lm(consumption ~ fdrive + lweight, data = CarsNow)
mDrive <- lm(consumption ~ fdrive, data = CarsNow)
mWeight <- lm(consumption ~ lweight, data = CarsNow)
m0 <- lm(consumption ~ 1, data = CarsNow)
summary(mInter)
##
## Call:
## lm(formula = consumption ~ fdrive + lweight + fdrive:lweight,
## data = CarsNow)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4038 -0.6438 -0.1021 0.5672 4.3237
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -52.8047 2.5266 -20.900 < 2e-16 ***
## fdriverear 19.8445 5.1297 3.869 0.000128 ***
## fdrive4x4 -12.5366 4.6506 -2.696 0.007319 **
## lweight 8.5716 0.3461 24.763 < 2e-16 ***
## fdriverear:lweight -2.5890 0.6956 -3.722 0.000226 ***
## fdrive4x4:lweight 1.7837 0.6240 2.858 0.004480 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9404 on 403 degrees of freedom
## Multiple R-squared: 0.8081, Adjusted R-squared: 0.8057
## F-statistic: 339.4 on 5 and 403 DF, p-value: < 2.2e-16
summary(mAddit)
##
## Call:
## lm(formula = consumption ~ fdrive + lweight, data = CarsNow)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4064 -0.6649 -0.1323 0.5747 5.1533
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -52.5605 1.9627 -26.780 < 2e-16 ***
## fdriverear 0.6964 0.1181 5.897 7.83e-09 ***
## fdrive4x4 0.8787 0.1363 6.445 3.29e-10 ***
## lweight 8.5381 0.2688 31.762 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9726 on 405 degrees of freedom
## Multiple R-squared: 0.7937, Adjusted R-squared: 0.7922
## F-statistic: 519.5 on 3 and 405 DF, p-value: < 2.2e-16
fdrive onlysummary(mDrive)
##
## 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
lweight onlysummary(mWeight)
##
## Call:
## lm(formula = consumption ~ lweight, data = CarsNow)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.6544 -0.7442 -0.1526 0.5160 5.1616
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -58.2480 1.8941 -30.75 <2e-16 ***
## lweight 9.3606 0.2569 36.44 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.035 on 407 degrees of freedom
## Multiple R-squared: 0.7654, Adjusted R-squared: 0.7648
## F-statistic: 1328 on 1 and 407 DF, p-value: < 2.2e-16
summary(m0)
##
## Call:
## lm(formula = consumption ~ 1, data = CarsNow)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.1013 -1.1013 -0.0513 0.8987 10.7987
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.7513 0.1055 101.9 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.134 on 408 degrees of freedom
anova(mAddit, mInter)
## Analysis of Variance Table
##
## Model 1: consumption ~ fdrive + lweight
## Model 2: consumption ~ fdrive + lweight + fdrive:lweight
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 405 383.1
## 2 403 356.4 2 26.702 15.097 4.758e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(mInter)
## Analysis of Variance Table
##
## Response: consumption
## Df Sum Sq Mean Sq F value Pr(>F)
## fdrive 2 519.89 259.94 293.935 < 2.2e-16 ***
## lweight 1 954.26 954.26 1079.040 < 2.2e-16 ***
## fdrive:lweight 2 26.70 13.35 15.097 4.758e-07 ***
## Residuals 403 356.40 0.88
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mInter1 <- lm(consumption ~ fdrive + lweight + fdrive:lweight, data = CarsNow)
anova(mInter1)
## Analysis of Variance Table
##
## Response: consumption
## Df Sum Sq Mean Sq F value Pr(>F)
## fdrive 2 519.89 259.94 293.935 < 2.2e-16 ***
## lweight 1 954.26 954.26 1079.040 < 2.2e-16 ***
## fdrive:lweight 2 26.70 13.35 15.097 4.758e-07 ***
## Residuals 403 356.40 0.88
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mInter2 <- lm(consumption ~ lweight + fdrive + fdrive:lweight, data = CarsNow)
anova(mInter2)
## Analysis of Variance Table
##
## Response: consumption
## Df Sum Sq Mean Sq F value Pr(>F)
## lweight 1 1421.57 1421.57 1607.458 < 2.2e-16 ***
## fdrive 2 52.58 26.29 29.726 9.079e-13 ***
## lweight:fdrive 2 26.70 13.35 15.097 4.758e-07 ***
## Residuals 403 356.40 0.88
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova comes from package car.library("car")
Anova(mInter1, type = "II")
## Anova Table (Type II tests)
##
## Response: consumption
## Sum Sq Df F value Pr(>F)
## fdrive 52.58 2 29.726 9.079e-13 ***
## lweight 954.26 1 1079.040 < 2.2e-16 ***
## fdrive:lweight 26.70 2 15.097 4.758e-07 ***
## Residuals 356.40 403
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(mInter2, type = "II") ### the same results
## Anova Table (Type II tests)
##
## Response: consumption
## Sum Sq Df F value Pr(>F)
## lweight 954.26 1 1079.040 < 2.2e-16 ***
## fdrive 52.58 2 29.726 9.079e-13 ***
## lweight:fdrive 26.70 2 15.097 4.758e-07 ***
## Residuals 356.40 403
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(mInter1, type = "III")
## Anova Table (Type III tests)
##
## Response: consumption
## Sum Sq Df F value Pr(>F)
## (Intercept) 386.28 1 436.793 < 2.2e-16 ***
## fdrive 26.49 2 14.979 5.310e-07 ***
## lweight 542.30 1 613.216 < 2.2e-16 ***
## fdrive:lweight 26.70 2 15.097 4.758e-07 ***
## Residuals 356.40 403
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(mInter2, type = "III") ### the same results
## Anova Table (Type III tests)
##
## Response: consumption
## Sum Sq Df F value Pr(>F)
## (Intercept) 386.28 1 436.793 < 2.2e-16 ***
## lweight 542.30 1 613.216 < 2.2e-16 ***
## fdrive 26.49 2 14.979 5.310e-07 ***
## lweight:fdrive 26.70 2 15.097 4.758e-07 ***
## Residuals 356.40 403
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
fdrivemInter <- lm(consumption ~ fdrive + lweight + fdrive:lweight, data = CarsNow)
mInterSAS <- lm(consumption ~ fdrive + lweight + fdrive:lweight, data = CarsNow, contrasts = list(fdrive = contr.SAS))
mIntersum <- lm(consumption ~ fdrive + lweight + fdrive:lweight, data = CarsNow, contrasts = list(fdrive = contr.sum))
summary(mInter)
##
## Call:
## lm(formula = consumption ~ fdrive + lweight + fdrive:lweight,
## data = CarsNow)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4038 -0.6438 -0.1021 0.5672 4.3237
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -52.8047 2.5266 -20.900 < 2e-16 ***
## fdriverear 19.8445 5.1297 3.869 0.000128 ***
## fdrive4x4 -12.5366 4.6506 -2.696 0.007319 **
## lweight 8.5716 0.3461 24.763 < 2e-16 ***
## fdriverear:lweight -2.5890 0.6956 -3.722 0.000226 ***
## fdrive4x4:lweight 1.7837 0.6240 2.858 0.004480 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9404 on 403 degrees of freedom
## Multiple R-squared: 0.8081, Adjusted R-squared: 0.8057
## F-statistic: 339.4 on 5 and 403 DF, p-value: < 2.2e-16
summary(mInterSAS)
##
## Call:
## lm(formula = consumption ~ fdrive + lweight + fdrive:lweight,
## data = CarsNow, contrasts = list(fdrive = contr.SAS))
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4038 -0.6438 -0.1021 0.5672 4.3237
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -65.3414 3.9045 -16.735 < 2e-16 ***
## fdrive1 12.5366 4.6506 2.696 0.00732 **
## fdrive2 32.3811 5.9309 5.460 8.35e-08 ***
## lweight 10.3553 0.5192 19.943 < 2e-16 ***
## fdrive1:lweight -1.7837 0.6240 -2.858 0.00448 **
## fdrive2:lweight -4.3727 0.7961 -5.493 7.01e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9404 on 403 degrees of freedom
## Multiple R-squared: 0.8081, Adjusted R-squared: 0.8057
## F-statistic: 339.4 on 5 and 403 DF, p-value: < 2.2e-16
summary(mIntersum)
##
## Call:
## lm(formula = consumption ~ fdrive + lweight + fdrive:lweight,
## data = CarsNow, contrasts = list(fdrive = contr.sum))
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4038 -0.6438 -0.1021 0.5672 4.3237
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -50.3688 2.1489 -23.440 < 2e-16 ***
## fdrive1 -2.4360 2.5972 -0.938 0.349
## fdrive2 17.4085 3.3558 5.188 3.38e-07 ***
## lweight 8.3031 0.2894 28.696 < 2e-16 ***
## fdrive1:lweight 0.2684 0.3517 0.763 0.446
## fdrive2:lweight -2.3206 0.4529 -5.124 4.64e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9404 on 403 degrees of freedom
## Multiple R-squared: 0.8081, Adjusted R-squared: 0.8057
## F-statistic: 339.4 on 5 and 403 DF, p-value: < 2.2e-16
lweight.Anova(mInter, type = "III")
## Anova Table (Type III tests)
##
## Response: consumption
## Sum Sq Df F value Pr(>F)
## (Intercept) 386.28 1 436.793 < 2.2e-16 ***
## fdrive 26.49 2 14.979 5.310e-07 ***
## lweight 542.30 1 613.216 < 2.2e-16 ***
## fdrive:lweight 26.70 2 15.097 4.758e-07 ***
## Residuals 356.40 403
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(mInterSAS, type = "III")
## Anova Table (Type III tests)
##
## Response: consumption
## Sum Sq Df F value Pr(>F)
## (Intercept) 247.68 1 280.063 < 2.2e-16 ***
## fdrive 26.49 2 14.979 5.310e-07 ***
## lweight 351.72 1 397.714 < 2.2e-16 ***
## fdrive:lweight 26.70 2 15.097 4.758e-07 ***
## Residuals 356.40 403
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(mIntersum, type = "III")
## Anova Table (Type III tests)
##
## Response: consumption
## Sum Sq Df F value Pr(>F)
## (Intercept) 485.88 1 549.416 < 2.2e-16 ***
## fdrive 26.49 2 14.979 5.310e-07 ***
## lweight 728.22 1 823.440 < 2.2e-16 ***
## fdrive:lweight 26.70 2 15.097 4.758e-07 ***
## Residuals 356.40 403
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1