NMSA407 Linear Regression: Tutorial

Transformation of response: Regression with log-transformed response to stabilize the variance, Box-Cox transformation

Data Cars2004nh

Introduction

Load used data and calculate basic summaries

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

Weight in tons

Cars2004nh <- transform(Cars2004nh, weightTon = weight/1000)

Complete cases subset used here

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", "weightTon", "engine.size"))
dim(CarsNow)

## [1] 409   7

summary(CarsNow)

##   consumption        drive         fdrive        weight        lweight        weightTon    
##  Min.   : 5.65   Min.   :1.000   front:212   Min.   : 923   Min.   :6.828   Min.   :0.923  
##  1st Qu.: 9.65   1st Qu.:1.000   rear :108   1st Qu.:1415   1st Qu.:7.255   1st Qu.:1.415  
##  Median :10.70   Median :1.000   4x4  : 89   Median :1577   Median :7.363   Median :1.577  
##  Mean   :10.75   Mean   :1.699               Mean   :1622   Mean   :7.371   Mean   :1.622  
##  3rd Qu.:11.65   3rd Qu.:2.000               3rd Qu.:1804   3rd Qu.:7.498   3rd Qu.:1.804  
##  Max.   :21.55   Max.   :3.000               Max.   :2903   Max.   :7.973   Max.   :2.903  
##   engine.size   
##  Min.   :1.300  
##  1st Qu.:2.400  
##  Median :3.000  
##  Mean   :3.178  
##  3rd Qu.:3.800  
##  Max.   :6.000

Colors and plotting symbols for plots

COL3 <- c("red3", "darkgreen", "blue3")
LCOL3 <- c("red", "green3", "blue")
BG3 <- rainbow_hcl(3)
PCH3 <- c(21, 22, 23)
names(COL3) <- names(BG3) <- names(PCH3) <- levels(Cars2004nh[, "fdrive"])

Dependence of `consumption` on `weightTon` and `fdrive`

Basic scatterplot while distinguishing also the cars according to their drive

plot(consumption ~ weightTon, data = CarsNow, col = COL3[fdrive], bg = BG3[fdrive], pch = PCH3[fdrive],
     xlab = "Weight [t]", ylab = "Consumption [l/100 km]")
legend(1, 20, legend = levels(CarsNow[, "fdrive"]), pt.bg = BG3, col = COL3, pch = PCH3)

plot of chunk fig-CheckModelAssumpt-07-01

Linear model (three not necessarily parallel lines)

m1 <- lm(consumption ~ weightTon + fdrive + weightTon:fdrive, data = CarsNow)
summary(m1)

## 
## Call:
## lm(formula = consumption ~ weightTon + fdrive + weightTon:fdrive, 
##     data = CarsNow)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.2952 -0.5924 -0.0984  0.5009  3.5622 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            1.1824     0.3412   3.465 0.000588 ***
## weightTon              5.6996     0.2233  25.522  < 2e-16 ***
## fdriverear             3.8936     0.7062   5.514 6.29e-08 ***
## fdrive4x4              1.2177     0.6046   2.014 0.044677 *  
## weightTon:fdriverear  -1.9296     0.4330  -4.456 1.08e-05 ***
## weightTon:fdrive4x4   -0.3105     0.3437  -0.904 0.366797    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9191 on 403 degrees of freedom
## Multiple R-squared:  0.8167, Adjusted R-squared:  0.8144 
## F-statistic: 359.1 on 5 and 403 DF,  p-value: < 2.2e-16

Can we assume that the three lines are parallel?

anova(m1)

## Analysis of Variance Table
## 
## Response: consumption
##                   Df  Sum Sq Mean Sq  F value    Pr(>F)    
## weightTon          1 1445.71 1445.71 1711.481 < 2.2e-16 ***
## fdrive             2   54.01   27.00   31.968 1.299e-13 ***
## weightTon:fdrive   2   17.11    8.55   10.125 5.124e-05 ***
## Residuals        403  340.42    0.84                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Calculation of three fitted lines

lpred <- 500
wpred <- seq(0.9, 3.3, length = lpred)
dataPred <- data.frame(weightTon = rep(wpred, 3),
                       fdrive = gl(3, lpred, labels = levels(CarsNow[, "fdrive"])))
print(dataPred[1:10,])

##    weightTon fdrive
## 1  0.9000000  front
## 2  0.9048096  front
## 3  0.9096192  front
## 4  0.9144289  front
## 5  0.9192385  front
## 6  0.9240481  front
## 7  0.9288577  front
## 8  0.9336673  front
## 9  0.9384770  front
## 10 0.9432866  front

print(dataPred[(lpred + 1):(lpred + 10),])

##     weightTon fdrive
## 501 0.9000000   rear
## 502 0.9048096   rear
## 503 0.9096192   rear
## 504 0.9144289   rear
## 505 0.9192385   rear
## 506 0.9240481   rear
## 507 0.9288577   rear
## 508 0.9336673   rear
## 509 0.9384770   rear
## 510 0.9432866   rear

print(dataPred[(2*lpred + 1):(2*lpred + 10),])

##      weightTon fdrive
## 1001 0.9000000    4x4
## 1002 0.9048096    4x4
## 1003 0.9096192    4x4
## 1004 0.9144289    4x4
## 1005 0.9192385    4x4
## 1006 0.9240481    4x4
## 1007 0.9288577    4x4
## 1008 0.9336673    4x4
## 1009 0.9384770    4x4
## 1010 0.9432866    4x4

pm1 <- predict(m1, newdata = dataPred)

print(pm1[1:10])

##        1        2        3        4        5        6        7        8        9       10 
## 6.311984 6.339397 6.366810 6.394223 6.421635 6.449048 6.476461 6.503874 6.531286 6.558699

print(pm1[(lpred + 1):(lpred + 10)])

##      501      502      503      504      505      506      507      508      509      510 
## 8.468907 8.487039 8.505172 8.523304 8.541436 8.559568 8.577700 8.595832 8.613964 8.632097

print(pm1[(2*lpred + 1):(2*lpred + 10)])

##     1001     1002     1003     1004     1005     1006     1007     1008     1009     1010 
## 7.250234 7.276153 7.302072 7.327992 7.353911 7.379830 7.405750 7.431669 7.457588 7.483508

Plot of data together with fitted lines

plot(consumption ~ weightTon, data = CarsNow, col = COL3[fdrive], bg = BG3[fdrive], pch = PCH3[fdrive],
     xlab = "Weight [t]", ylab = "Consumption [l/100 km]")
for (l in 1:3){
  lines(wpred, pm1[((l-1)*lpred + 1):(l*lpred)], col = LCOL3[l], lwd = 2)
}  
legend(1, 20, legend = levels(CarsNow[, "fdrive"]), pt.bg = BG3, col = LCOL3, pch = PCH3, lty = 1, lwd = 2)

plot of chunk fig-CheckModelAssumpt-07-02

Basic residual plots for fitted model

library("mffSM")
plotLM(m1)

plot of chunk fig-CheckModelAssumpt-07-03

Test of homoscedasticity

Alternative of a residual variance changing with the response mean.

library("lmtest")
bptest(m1, varformula = ~fitted(m1))

## 
##  studentized Breusch-Pagan test
## 
## data:  m1
## BP = 3.5537, df = 1, p-value = 0.05941

Model with a log-transformed response

Linear model

m2 <- lm(log(consumption) ~ weightTon + fdrive + weightTon:fdrive, data = CarsNow)
summary(m2)

## 
## Call:
## lm(formula = log(consumption) ~ weightTon + fdrive + weightTon:fdrive, 
##     data = CarsNow)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.44734 -0.05304 -0.00676  0.05508  0.30350 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           1.39228    0.03155  44.129  < 2e-16 ***
## weightTon             0.57762    0.02065  27.975  < 2e-16 ***
## fdriverear            0.46467    0.06529   7.117 5.08e-12 ***
## fdrive4x4             0.34690    0.05590   6.205 1.36e-09 ***
## weightTon:fdriverear -0.23764    0.04003  -5.936 6.31e-09 ***
## weightTon:fdrive4x4  -0.16658    0.03178  -5.242 2.56e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.08498 on 403 degrees of freedom
## Multiple R-squared:  0.8162, Adjusted R-squared:  0.814 
## F-statistic:   358 on 5 and 403 DF,  p-value: < 2.2e-16

anova(m2)

## Analysis of Variance Table
## 
## Response: log(consumption)
##                   Df  Sum Sq Mean Sq  F value    Pr(>F)    
## weightTon          1 11.9938 11.9938 1661.012 < 2.2e-16 ***
## fdrive             2  0.5914  0.2957   40.953 < 2.2e-16 ***
## weightTon:fdrive   2  0.3405  0.1702   23.577 2.073e-10 ***
## Residuals        403  2.9100  0.0072                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Calculation of three fitted lines

pm2 <- predict(m2, newdata = dataPred)

Plot of data (on log-scale) together with fitted lines

plot(log(consumption) ~ weightTon, col = COL3[fdrive], bg = BG3[fdrive], pch = PCH3[fdrive], data = CarsNow,
     xlab = "Weight [t]", ylab = "Log(consumption) [log(l/100 km)]")
for (l in 1:3){
  lines(wpred, pm2[((l-1)*lpred + 1):(l*lpred)], col = LCOL3[l], lwd = 2)
}  
legend(1, 3, legend = levels(CarsNow[, "fdrive"]), pt.bg = BG3, col = LCOL3, pch = PCH3, lty = 1, lwd = 2)

plot of chunk fig-CheckModelAssumpt-07-04

Plot of data (on original scale) together with fitted lines

Fitted lines must be considered as predictions of consumption and not as estimates of the expected consumption.

plot(consumption ~ weightTon, col = COL3[fdrive], bg = BG3[fdrive], pch = PCH3[fdrive], data = CarsNow,
     xlab = "Weight [t]", ylab = "Consumption [l/100 km]")
for (l in 1:3){
  lines(wpred, exp(pm2[((l-1)*lpred + 1):(l*lpred)]), col = LCOL3[l], lwd = 2)
}  
legend(1, 20, legend = levels(CarsNow[, "fdrive"]), pt.bg = BG3, col = LCOL3, pch = PCH3, lty = 1, lwd = 2)

plot of chunk fig-CheckModelAssumpt-07-05

Basic residual plots for the fitted model

plotLM(m2)

Test of homoscedasticity

Alternative of a residual variance changing with the response mean.

bptest(m2, varformula = ~fitted(m2))

## 
##  studentized Breusch-Pagan test
## 
## data:  m2
## BP = 0.21273, df = 1, p-value = 0.6446

detach("package:lmtest")

Estimated differences (on log scale) between the three groups for `weight` = 1.5 t

LSE of regression coefficients

coef(m2)

##          (Intercept)            weightTon           fdriverear            fdrive4x4 
##            1.3922801            0.5776185            0.4646743            0.3468986 
## weightTon:fdriverear  weightTon:fdrive4x4 
##           -0.2376359           -0.1665767

Used pseudocontrast matrix to parameterize a categorical covariate `fdrive`

ldrive <- levels(CarsNow[, "fdrive"])

C <- contr.treatment(length(ldrive))
rownames(C) <- ldrive
colnames(C) <- names(coef(m2))[3:4]
print(C)

##       fdriverear fdrive4x4
## front          0         0
## rear           1         0
## 4x4            0         1

“Important” coefficients of linear combinations to calculate differences between the three groups if `weightTon` = 0

L.diff <- rbind(C[2,] - C[1,],
                C[3,] - C[1,],
                C[3,] - C[2,])
rownames(L.diff) <- paste(ldrive[c(2, 3, 3)], "-", ldrive[c(1, 1, 2)], sep = "")
print(L.diff)

##            fdriverear fdrive4x4
## rear-front          1         0
## 4x4-front           0         1
## 4x4-rear           -1         1

Coefficients of linear combinations to calculate differences between the three groups if `weightTon` = 1.5

x <- 1.5
L <- cbind(0, 0, L.diff, x*L.diff)
colnames(L) <- names(coef(m2))
print(L)

##            (Intercept) weightTon fdriverear fdrive4x4 weightTon:fdriverear weightTon:fdrive4x4
## rear-front           0         0          1         0                  1.5                 0.0
## 4x4-front            0         0          0         1                  0.0                 1.5
## 4x4-rear             0         0         -1         1                 -1.5                 1.5

No corrections for multiple comparison

LSest(m2, L)

##               Estimate Std. Error    t value    P value       Lower      Upper
## rear-front  0.10822048 0.01127677  9.5967640 < 2.22e-16  0.08605185 0.13038912
## 4x4-front   0.09703352 0.01402729  6.9174829 1.8142e-11  0.06945773 0.12460932
## 4x4-rear   -0.01118696 0.01599398 -0.6994485    0.48468 -0.04262901 0.02025508

Estimates exponentiated to get quantities which can be interpreted

ls2 <- LSest(m2, L)
ls2$Estimate <- exp(ls2$Estimate)
ls2$Lower <- exp(ls2$Lower)
ls2$Upper <- exp(ls2$Upper)
print(ls2)

##             Estimate Std. Error    t value    P value     Lower    Upper
## rear-front 1.1142934 0.01127677  9.5967640 < 2.22e-16 1.0898628 1.139272
## 4x4-front  1.1018973 0.01402729  6.9174829 1.8142e-11 1.0719267 1.132706
## 4x4-rear   0.9888754 0.01599398 -0.6994485    0.48468 0.9582668 1.020462

print(ls2, digits = 4)

##            Estimate Std. Error t value    P value  Lower Upper
## rear-front   1.1143    0.01128  9.5968 < 2.22e-16 1.0899 1.139
## 4x4-front    1.1019    0.01403  6.9175 1.8142e-11 1.0719 1.133
## 4x4-rear     0.9889    0.01599 -0.6994    0.48468 0.9583 1.020

NOTE: Standard errors do not correspond to the estimated ratios!!!

Hothorn-Bretz-Westfall multiple comparison

library("multcomp")
mcp2 <- glht(m2, linfct = L)
set.seed(221913282)
(smpc2 <- summary(mcp2))

## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Fit: lm(formula = log(consumption) ~ weightTon + fdrive + weightTon:fdrive, 
##     data = CarsNow)
## 
## Linear Hypotheses:
##                 Estimate Std. Error t value Pr(>|t|)    
## rear-front == 0  0.10822    0.01128   9.597   <1e-05 ***
## 4x4-front == 0   0.09703    0.01403   6.917   <1e-05 ***
## 4x4-rear == 0   -0.01119    0.01599  -0.699     0.76    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)

(ci2 <- confint(mcp2))

## 
##   Simultaneous Confidence Intervals
## 
## Fit: lm(formula = log(consumption) ~ weightTon + fdrive + weightTon:fdrive, 
##     data = CarsNow)
## 
## Quantile = 2.3396
## 95% family-wise confidence level
##  
## 
## Linear Hypotheses:
##                 Estimate lwr      upr     
## rear-front == 0  0.10822  0.08184  0.13460
## 4x4-front == 0   0.09703  0.06421  0.12985
## 4x4-rear == 0   -0.01119 -0.04861  0.02623

Estimates exponentiated to get quantities which can be interpreted

names(ci2)

## [1] "model"       "linfct"      "rhs"         "coef"        "vcov"        "df"         
## [7] "alternative" "type"        "confint"

exp(ci2$confint)

##             Estimate       lwr      upr
## rear-front 1.1142934 1.0852787 1.144084
## 4x4-front  1.1018973 1.0663213 1.138660
## 4x4-rear   0.9888754 0.9525552 1.026580
## attr(,"conf.level")
## [1] 0.95
## attr(,"calpha")
## [1] 2.339644

Box-Cox transformation

Original model with untransformed response

m1 <- lm(consumption ~ weightTon + fdrive + weightTon:fdrive, data = CarsNow)
summary(m1)

## 
## Call:
## lm(formula = consumption ~ weightTon + fdrive + weightTon:fdrive, 
##     data = CarsNow)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.2952 -0.5924 -0.0984  0.5009  3.5622 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            1.1824     0.3412   3.465 0.000588 ***
## weightTon              5.6996     0.2233  25.522  < 2e-16 ***
## fdriverear             3.8936     0.7062   5.514 6.29e-08 ***
## fdrive4x4              1.2177     0.6046   2.014 0.044677 *  
## weightTon:fdriverear  -1.9296     0.4330  -4.456 1.08e-05 ***
## weightTon:fdrive4x4   -0.3105     0.3437  -0.904 0.366797    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9191 on 403 degrees of freedom
## Multiple R-squared:  0.8167, Adjusted R-squared:  0.8144 
## F-statistic: 359.1 on 5 and 403 DF,  p-value: < 2.2e-16

Profile log-likelihood for \(\lambda\) in the Box-Cox model

library("MASS")
boxcox(m1)

plot of chunk fig-CheckModelAssumpt-07-07

Zoomed profile log-likelihood for \(\lambda\) in the Box-Cox model

boxcox(m1, lambda = seq(-0.1, 0.8, by = 0.05))

plot of chunk fig-CheckModelAssumpt-07-08

\(\lambda = 0\) is just outside the 95% confidence interval. Since it provides an interpretable model, it is still reasonable to use it.

Values of \(\lambda\) ('x') and a profile log-likelihood ('y')

boxcox(m1, lambda = seq(0.2, 0.4, by = 0.025), plotit = FALSE)

## $x
## [1] 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400
## 
## $y
## [1] -215.8314 -215.6701 -215.5455 -215.4575 -215.4061 -215.3915 -215.4135 -215.4722 -215.5676

NMSA407 Linear Regression: Tutorial

Introduction

Load used data and calculate basic summaries

Weight in tons

Complete cases subset used here

Colors and plotting symbols for plots

Dependence of consumption on weightTon and fdrive

Basic scatterplot while distinguishing also the cars according to their drive

Linear model (three not necessarily parallel lines)

Can we assume that the three lines are parallel?

Calculation of three fitted lines

Plot of data together with fitted lines

Basic residual plots for fitted model

Test of homoscedasticity

Model with a log-transformed response

Linear model

Calculation of three fitted lines

Plot of data (on log-scale) together with fitted lines

Plot of data (on original scale) together with fitted lines

Basic residual plots for the fitted model

Test of homoscedasticity

Estimated differences (on log scale) between the three groups for weight = 1.5 t

LSE of regression coefficients

Used pseudocontrast matrix to parameterize a categorical covariate fdrive

“Important” coefficients of linear combinations to calculate differences between the three groups if weightTon = 0

Coefficients of linear combinations to calculate differences between the three groups if weightTon = 1.5

No corrections for multiple comparison

Estimates exponentiated to get quantities which can be interpreted

Hothorn-Bretz-Westfall multiple comparison

Estimates exponentiated to get quantities which can be interpreted

Box-Cox transformation

Original model with untransformed response

Profile log-likelihood for \(\lambda\) in the Box-Cox model

Zoomed profile log-likelihood for \(\lambda\) in the Box-Cox model

Values of \(\lambda\) ('x') and a profile log-likelihood ('y')

Dependence of `consumption` on `weightTon` and `fdrive`

Estimated differences (on log scale) between the three groups for `weight` = 1.5 t

Used pseudocontrast matrix to parameterize a categorical covariate `fdrive`

“Important” coefficients of linear combinations to calculate differences between the three groups if `weightTon` = 0

Coefficients of linear combinations to calculate differences between the three groups if `weightTon` = 1.5