Fuel Economy
Jane Braun
October 23rd, 2014
Recipe 5: Completely Randomized Block “Pseudo Design”
Recipes for the Design of Experiments
Fuel Economy
Jane Braun
RPI
October 23rd Version 1.0
1. Setting
System under test
The data analyzed in this recipe is from one of Hadley Wickham’s datasets. In this recipe, the fuel economy dataset was used.
This dataset is the result of vehicle testing done at the Environmental Protection Agency’s National Vehicle and Fuel Emmisions laboratory in Ann Arbor, Michigan.
# Read in dataset as a .csv file.
vehicles <- read.csv("C:/Users/braunj6/Documents/Fall 2014/vehicles.csv")
# Remove unnecessary columns
vehicles <- vehicles[,c(1,5,15, 16, 23:32, 35, 44:50, 58:67)]
# Subset the variety of vehicle makes
vehicles4 = subset(vehicles, make %in% c("Toyota","Volkswagen", "Subaru", "BMW", "Dodge", "Ford", "Honda"))
write.table(vehicles4, "C:/Users/braunj6/Documents/Fall 2014/x.txt", sep="\t")
x <- read.delim("~/Fall 2014/x.txt")
# Recategorize data as factors
x$make <- as.factor(x$make)
x$VClass <- as.factor(x$VClass)
x$trany <- as.factor(x$trany)
x$drive <- as.factor(x$drive)
x$year <- as.factor(x$year)
x$cylinders <- as.factor(x$cylinders)
x$fuelType <- as.factor(x$fuelType)Factors and Levels
The data being examined originally consisted of 74 variables, with 35,557 observations. It was subsetted down to 32 variables with 11,405 observations.
head(x)## barrels08 city08 co2TailpipeGpm comb08 cylinders displ
## 3 12.19557 23 329.1481 27 4 2.2
## 4 29.95056 10 807.9091 11 8 5.2
## 5 17.33749 17 467.7368 19 4 2.2
## 6 14.96429 21 403.9545 22 4 1.8
## 7 13.18440 22 355.4800 25 4 1.8
## 8 13.73375 23 370.2917 24 4 1.6
## drive engId eng_dscr feScore fuelCost08
## 3 Front-Wheel Drive 2100 (FFS) -1 1800
## 4 Rear-Wheel Drive 2850 -1 4400
## 5 4-Wheel or All-Wheel Drive 66031 (FFS,TRBO) -1 2850
## 6 Front-Wheel Drive 66020 (FFS) -1 2200
## 7 Front-Wheel Drive 66020 (FFS) -1 1950
## 8 Front-Wheel Drive 57005 (FFS) -1 2000
## fuelCostA08 fuelType fuelType1 highway08 id lv2 lv4 make
## 3 0 Regular Regular Gasoline 33 100 0 0 Dodge
## 4 0 Regular Regular Gasoline 12 1000 0 0 Dodge
## 5 0 Premium Premium Gasoline 23 10000 0 14 Subaru
## 6 0 Regular Regular Gasoline 24 10001 0 15 Subaru
## 7 0 Regular Regular Gasoline 29 10002 0 15 Subaru
## 8 0 Regular Regular Gasoline 26 10003 0 13 Toyota
## model mpgData phevBlended trany UCity UCityA
## 3 Charger Y false Manual 5-spd 29.0000 0
## 4 B150/B250 Wagon 2WD N false Automatic 3-spd 12.2222 0
## 5 Legacy AWD Turbo N false Manual 5-spd 21.0000 0
## 6 Loyale N false Automatic 3-spd 27.0000 0
## 7 Loyale Y false Manual 5-spd 28.0000 0
## 8 Corolla Y false Automatic 3-spd 29.0000 0
## UHighway UHighwayA VClass year youSaveSpend guzzler trans_dscr
## 3 47.0000 0 Subcompact Cars 1985 1500 SIL
## 4 16.6667 0 Vans 1985 -11500
## 5 32.0000 0 Compact Cars 1993 -3750
## 6 33.0000 0 Compact Cars 1993 -500
## 7 41.0000 0 Compact Cars 1993 750
## 8 37.0000 0 Compact Cars 1993 500summary(x)## barrels08 city08 co2TailpipeGpm comb08
## Min. : 0.05989 Min. : 8.00 Min. : 0.0 Min. : 9.00
## 1st Qu.:14.32705 1st Qu.: 14.00 1st Qu.:386.4 1st Qu.: 16.00
## Median :16.48050 Median : 17.00 Median :444.4 Median : 20.00
## Mean :17.53171 Mean : 18.11 Mean :474.4 Mean : 20.29
## 3rd Qu.:20.58964 3rd Qu.: 21.00 3rd Qu.:555.4 3rd Qu.: 23.00
## Max. :36.60868 Max. :137.00 Max. :987.4 Max. :124.00
##
## cylinders displ drive
## 4 :4863 Min. :0.600 Front-Wheel Drive :4062
## 6 :3935 1st Qu.:2.200 Rear-Wheel Drive :3897
## 8 :2303 Median :3.000 4-Wheel or All-Wheel Drive:2215
## 5 : 115 Mean :3.183 All-Wheel Drive : 519
## 10 : 69 3rd Qu.:4.000 : 331
## (Other): 92 Max. :8.400 2-Wheel Drive : 218
## NA's : 28 NA's :28 (Other) : 163
## engId eng_dscr feScore fuelCost08
## Min. : 0 :4883 Min. :-1.000 Min. : 500
## 1st Qu.: 0 (FFS) :3744 1st Qu.:-1.000 1st Qu.:2100
## Median : 501 SIDI : 609 Median :-1.000 Median :2550
## Mean :11244 (FFS) CA model: 344 Mean :-0.356 Mean :2634
## 3rd Qu.: 3821 (FFS,TRBO) : 213 3rd Qu.:-1.000 3rd Qu.:3000
## Max. :66097 FFV : 162 Max. :10.000 Max. :5950
## (Other) :1450
## fuelCostA08 fuelType fuelType1
## Min. : 0.0 Regular :8510 Diesel : 255
## 1st Qu.: 0.0 Premium :2237 Electricity : 26
## Median : 0.0 Gasoline or E85: 277 Midgrade Gasoline: 24
## Mean : 100.2 Diesel : 255 Natural Gas : 47
## 3rd Qu.: 0.0 CNG : 47 Premium Gasoline :2239
## Max. :5400.0 Electricity : 26 Regular Gasoline :8814
## (Other) : 53
## highway08 id lv2 lv4
## Min. : 10.00 Min. : 12 Min. : 0.00 Min. : 0.000
## 1st Qu.: 19.00 1st Qu.: 8435 1st Qu.: 0.00 1st Qu.: 0.000
## Median : 24.00 Median :16768 Median : 0.00 Median : 0.000
## Mean : 24.01 Mean :17370 Mean : 1.84 Mean : 6.006
## 3rd Qu.: 28.00 3rd Qu.:26388 3rd Qu.: 0.00 3rd Qu.:13.000
## Max. :111.00 Max. :35849 Max. :36.00 Max. :53.000
##
## make model mpgData phevBlended
## BMW :1491 F150 Pickup 2WD : 200 N:6839 false:11393
## Dodge :2460 F150 Pickup 4WD : 178 Y:4566 true : 12
## Ford :2941 Jetta : 173
## Honda : 837 Mustang : 171
## Subaru : 793 Ranger Pickup 2WD: 169
## Toyota :1827 Civic : 152
## Volkswagen:1056 (Other) :10362
## trany UCity UCityA
## Automatic 4-spd:3160 Min. : 0.00 Min. : 0.0000
## Manual 5-spd :3017 1st Qu.: 17.78 1st Qu.: 0.0000
## Automatic 3-spd:1162 Median : 21.60 Median : 0.0000
## Automatic (S6) : 808 Mean : 22.81 Mean : 0.4976
## Automatic 5-spd: 757 3rd Qu.: 26.00 3rd Qu.: 0.0000
## Manual 6-spd : 695 Max. :196.40 Max. :181.5609
## (Other) :1806
## UHighway UHighwayA VClass
## Min. : 0.00 Min. : 0.0000 Compact Cars :1770
## 1st Qu.: 26.53 1st Qu.: 0.0000 Subcompact Cars :1756
## Median : 33.10 Median : 0.0000 Midsize Cars : 954
## Mean : 33.48 Mean : 0.6751 Standard Pickup Trucks : 939
## 3rd Qu.: 38.70 3rd Qu.: 0.0000 Small Station Wagons : 614
## Max. :159.10 Max. :152.1878 Sport Utility Vehicle - 4WD: 592
## (Other) :4780
## year youSaveSpend guzzler trans_dscr
## 1984 : 641 Min. :-19250 :11028 :7424
## 1985 : 550 1st Qu.: -4500 G: 216 CLKUP :2253
## 1987 : 428 Median : -2250 S: 9 SIL : 348
## 1986 : 403 Mean : -2668 T: 152 3MODE CLKUP: 279
## 1989 : 394 3rd Qu.: 0 Creeper : 263
## 1988 : 385 Max. : 8000 2MODE CLKUP: 201
## (Other):8604 (Other) : 637# As seen below, there are 11,405 observations of 32 variables, including: 3 factors, 2 blocking variables and 1 response variables.
str(x)## 'data.frame': 11405 obs. of 32 variables:
## $ barrels08 : num 12.2 30 17.3 15 13.2 ...
## $ city08 : int 23 10 17 21 22 23 23 23 23 18 ...
## $ co2TailpipeGpm: num 329 808 468 404 355 ...
## $ comb08 : int 27 11 19 22 25 24 26 25 26 21 ...
## $ cylinders : Factor w/ 8 levels "2","3","4","5",..: 3 6 3 3 3 3 3 3 3 3 ...
## $ displ : num 2.2 5.2 2.2 1.8 1.8 1.6 1.6 1.8 1.8 2 ...
## $ drive : Factor w/ 8 levels "","2-Wheel Drive",..: 6 8 4 6 6 6 6 6 6 6 ...
## $ engId : int 2100 2850 66031 66020 66020 57005 57005 57006 57006 59007 ...
## $ eng_dscr : Factor w/ 206 levels "","(16-VALVE) (FFS)",..: 27 1 39 27 27 27 27 27 27 27 ...
## $ feScore : int -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ...
## $ fuelCost08 : int 1800 4400 2850 2200 1950 2000 1850 1950 1850 2300 ...
## $ fuelCostA08 : int 0 0 0 0 0 0 0 0 0 0 ...
## $ fuelType : Factor w/ 12 levels "CNG","Diesel",..: 11 11 8 11 11 11 11 11 11 11 ...
## $ fuelType1 : Factor w/ 6 levels "Diesel","Electricity",..: 6 6 5 6 6 6 6 6 6 6 ...
## $ highway08 : int 33 12 23 24 29 26 31 30 30 26 ...
## $ id : int 100 1000 10000 10001 10002 10003 10004 10005 10006 10007 ...
## $ lv2 : int 0 0 0 0 0 0 0 0 0 0 ...
## $ lv4 : int 0 0 14 15 15 13 13 13 13 0 ...
## $ make : Factor w/ 7 levels "BMW","Dodge",..: 2 2 5 5 5 6 6 6 6 7 ...
## $ model : Factor w/ 795 levels "1-Ton Truck 2WD",..: 264 209 512 523 523 287 287 287 287 454 ...
## $ mpgData : Factor w/ 2 levels "N","Y": 2 1 1 1 2 2 2 2 2 1 ...
## $ phevBlended : Factor w/ 2 levels "false","true": 1 1 1 1 1 1 1 1 1 1 ...
## $ trany : Factor w/ 29 levels "","Auto(A1)",..: 26 17 26 17 26 17 26 18 26 18 ...
## $ UCity : num 29 12.2 21 27 28 ...
## $ UCityA : num 0 0 0 0 0 0 0 0 0 0 ...
## $ UHighway : num 47 16.7 32 33 41 ...
## $ UHighwayA : num 0 0 0 0 0 0 0 0 0 0 ...
## $ VClass : Factor w/ 29 levels "Compact Cars",..: 25 27 1 1 1 1 1 1 1 1 ...
## $ year : Factor w/ 32 levels "1984","1985",..: 2 2 10 10 10 10 10 10 10 10 ...
## $ youSaveSpend : int 1500 -11500 -3750 -500 750 500 1250 750 1250 -1000 ...
## $ guzzler : Factor w/ 4 levels "","G","S","T": 1 1 1 1 1 1 1 1 1 1 ...
## $ trans_dscr : Factor w/ 32 levels "","2LKUP","2MODE",..: 28 1 1 1 1 1 1 1 1 6 ...Continuous variables
The continuous variables in this dataset are the city gas mileage (in miles per gallon) - with a range from 17.78 to 196.40, and a mean of 22.81 mpg - and the highway gas mileage (in miles per gallon) - with a range from 26.53 to 159.10, and a mean of 33.10 mpg.
Response variables
The response variable being analyzed in this dataset is also one of the continuous variables - city gas mileage.
The Data
The data is organized into 32 columns, with 11,405 observations. It consists of both factors and response variables.
Randomization
This experiment was completed as a completely randomized block “pseudo design” with two blocking variables and 3 factors.
2. (Experimental) Design
How will the experiment be organized and conducted to test the hypothesis?
The experiment will use a two-factor analysis of variance. It will use various attributes as factors, such as drive type, transmission, V Class, fuel type and make, and will measure the variation in city gas mileage among these groups.
What is the rationale for this design?
The rationale for this type of design is that the goal was to analyze the difference in means between the groups. The ANOVA was set up to include interaction between the factors, with the blocking variables as make and fuel type. Therefore, we wanted to see what the variation was both among, and between groups.
Randomize: What is the Randomization Scheme?
The experiment was completed as a completely randomized design.
Replicate: Are there replicates and/or repeated measures?
The experiment had no repeated measures or replicates because they were observations from a study done by the EPA.
Block: Did you use blocking in the design?
Blocking was used for make and fuel type.
3. (Statistical) Analysis
(Exploratory Data Analysis) Graphics and descriptive summary
# Originally, make and drive type were selected as the blocking variables or this pseudo design. However, in order to be a "good" blocking variable, the within group variation must be smaller than the between group variation. Therefore, the IQR must be less than the difference between means of groups.
attach(x)
# The first blocking variable in question was the make of the car.
boxplot(x$UCity~x$make)
BMW <- subset(x, make =="BMW")
IQR(BMW$UCity)## [1] 4.13765mean(BMW$UCity)## [1] 21.26009Dodge <- subset(x, make =="Dodge")
IQR(Dodge$UCity)## [1] 8.1703mean(Dodge$UCity)## [1] 19.12284Ford <- subset(x, make =="Ford")
IQR(Ford$UCity)## [1] 7mean(Ford$UCity)## [1] 20.60479Honda <- subset(x, make =="Honda")
IQR(Honda$UCity)## [1] 10.4444mean(Honda$UCity)## [1] 31.09445mean(BMW$UCity) - mean(Dodge$UCity)## [1] 2.13725mean(BMW$UCity) - mean(Ford$UCity)## [1] 0.6553011mean(BMW$UCity) - mean(Honda$UCity)## [1] -9.834358# The IQRs are less than the difference between means for most of the groups. Therefore, make is a good blocking variable for this design.
# Testing Drive Type as a Blocking Variable
boxplot(UCity~drive)
FourW <- subset(x, drive =="4-Wheel Drive")
FW <- subset(x, drive =="Front-Wheel Drive")
RW <- subset(x, drive =="Rear-Wheel Drive")
AW <- subset(x, drive =="All-Wheel Drive")
IQR(FourW$UCity)## [1] 3.642825IQR(FW$UCity)## [1] 7.3685IQR(RW$UCity)## [1] 5.5IQR(AW$UCity)## [1] 5.18215mean(FourW$UCity)## [1] 18.99343mean(FW$UCity)## [1] 28.34302mean(RW$UCity)## [1] 18.78389mean(FourW$UCity) - mean(FW$UCity)## [1] -9.349586mean(FourW$UCity) - mean(RW$UCity)## [1] 0.2095462mean(FW$UCity) - mean(RW$UCity)## [1] 9.559132mean(FourW$UCity) - mean(AW$UCity)## [1] -4.867036# The IQRs for Drive Type are not generally less than the difference between means. Therefore, drive type is not a good blocking variable. Instead, we must try a different variable.
# Testing Fuel Type as a Block Variable
boxplot(UCity~fuelType)
Diesel <- subset(x, fuelType =="Diesel")
Electricity <- subset(x, fuelType =="Electricity")
Regular <- subset(x, fuelType =="Regular")
Midgrade <- subset(x, fuelType =="Midgrade")
Premium <- subset(x, fuelType =="Premium")
IQR(Diesel$UCity)## [1] 5.3057IQR(Electricity$UCity)## [1] 70.6604IQR(Regular$UCity)## [1] 9.121IQR(Midgrade$UCity)## [1] 2.3IQR(Premium$UCity)## [1] 3.9395mean(Diesel$UCity)## [1] 37.63458mean(Electricity$UCity)## [1] 116.7255mean(Regular$UCity)## [1] 22.5153mean(Midgrade$UCity)## [1] 18.15207mean(Premium$UCity)## [1] 21.59687mean(Diesel$UCity) - mean(Electricity$UCity)## [1] -79.09089mean(Diesel$UCity) - mean(Regular$UCity)## [1] 15.11928mean(Diesel$UCity) - mean(Midgrade$UCity)## [1] 19.48251mean(Diesel$UCity) - mean(Premium$UCity)## [1] 16.03771# The IQRs of the levels are generally less than the difference between means. Therefore, fuel type appears to be a sufficient blocking variable for the design.Testing
Analysis of Variance
The focus of this recipe was on a 3 factor ANOVA with 2 blocking variables.
#ANOVA Testing
#Comparing the average city gas mileage to three factors: drive type, transmission, and V class - blocked for make and fuel type.
# H0: There is no difference between the means of the samples. The variation in city gas mileage is not caused by anything other than randomization.
# Ha: The difference in variation of city gas mileage can caused by something other than randomization.
# The data set was too large to run an ANOVA without crashing my session of R Studio. Therefore, I took a random sample of 500 of the original 11,405. Although this is significantly smaller, this was the maximum that I could get to run.
x_sample <- x[sample(1:nrow(x), 500, replace=FALSE),]
model1 <- aov(UCity~ drive * trany * VClass + fuelType + make , data = x_sample)
summary(model1)## Df Sum Sq Mean Sq F value Pr(>F)
## drive 7 12022 1717.4 124.978 < 2e-16 ***
## trany 19 21680 1141.0 83.033 < 2e-16 ***
## VClass 27 2892 107.1 7.795 < 2e-16 ***
## fuelType 7 11678 1668.3 121.402 < 2e-16 ***
## make 6 1162 193.6 14.087 4.25e-14 ***
## drive:trany 32 671 21.0 1.526 0.0387 *
## drive:VClass 27 230 8.5 0.619 0.9328
## trany:VClass 65 815 12.5 0.912 0.6650
## drive:trany:VClass 15 106 7.1 0.515 0.9321
## Residuals 294 4040 13.7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The summary of the ANOVA gives the p-values for each individual factor, along with the p-value for the interactions between the factors. The null hypothesis states that the variation in the response variable, city gas mileage, cannot be explained by anything other than randomization. The p-values in this ANOVA support that this is not the case. The p-values for drive type, transmission, and V class are less than an alpha of 0.05. Therefore, it is likely that the variation in city gas mileage can be explained by these factors. The interactions between drive type and transmission, and transmission and v class are both significant. However, the p-value for the interaction between drive and v class, and the interaction between all three factors are not significant. Therefore, the interaction between these two/three factors may not be cause for variation. #### Tukey’s Range Tests
Tukey’s range tests are used alongside ANOVAs in order to find means that are significantly different from each other, by compairing pairs of means. This test compares the mean of each treatment level to the mean of the other treatment levels.
The null hypothesis for a Tukey test states that there is no difference between the means of a pair of data, while the alternative states that there is a significant difference between the means.
tukey1 <- TukeyHSD(aov(UCity ~ drive, data = x_sample))
tukey1## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = UCity ~ drive, data = x_sample)
##
## $drive
## diff
## 2-Wheel Drive- 11.96296000
## 4-Wheel Drive- -0.37720000
## 4-Wheel or All-Wheel Drive- -5.29873426
## All-Wheel Drive- 0.56039259
## Front-Wheel Drive- 4.35812571
## Part-time 4-Wheel Drive- -5.94042500
## Rear-Wheel Drive- -5.92915941
## 4-Wheel Drive-2-Wheel Drive -12.34016000
## 4-Wheel or All-Wheel Drive-2-Wheel Drive -17.26169426
## All-Wheel Drive-2-Wheel Drive -11.40256741
## Front-Wheel Drive-2-Wheel Drive -7.60483429
## Part-time 4-Wheel Drive-2-Wheel Drive -17.90338500
## Rear-Wheel Drive-2-Wheel Drive -17.89211941
## 4-Wheel or All-Wheel Drive-4-Wheel Drive -4.92153426
## All-Wheel Drive-4-Wheel Drive 0.93759259
## Front-Wheel Drive-4-Wheel Drive 4.73532571
## Part-time 4-Wheel Drive-4-Wheel Drive -5.56322500
## Rear-Wheel Drive-4-Wheel Drive -5.55195941
## All-Wheel Drive-4-Wheel or All-Wheel Drive 5.85912685
## Front-Wheel Drive-4-Wheel or All-Wheel Drive 9.65685997
## Part-time 4-Wheel Drive-4-Wheel or All-Wheel Drive -0.64169074
## Rear-Wheel Drive-4-Wheel or All-Wheel Drive -0.63042515
## Front-Wheel Drive-All-Wheel Drive 3.79773312
## Part-time 4-Wheel Drive-All-Wheel Drive -6.50081759
## Rear-Wheel Drive-All-Wheel Drive -6.48955200
## Part-time 4-Wheel Drive-Front-Wheel Drive -10.29855071
## Rear-Wheel Drive-Front-Wheel Drive -10.28728513
## Rear-Wheel Drive-Part-time 4-Wheel Drive 0.01126559
## lwr upr
## 2-Wheel Drive- -3.6733637 27.5992837
## 4-Wheel Drive- -30.3184804 29.5640804
## 4-Wheel or All-Wheel Drive- -14.7350649 4.1375964
## All-Wheel Drive- -10.0076123 11.1283975
## Front-Wheel Drive- -4.9238593 13.6401108
## Part-time 4-Wheel Drive- -22.8295849 10.9487349
## Rear-Wheel Drive- -15.2185199 3.3602011
## 4-Wheel Drive-2-Wheel Drive -43.6128074 18.9324874
## 4-Wheel or All-Wheel Drive-2-Wheel Drive -30.3208881 -4.2025005
## All-Wheel Drive-2-Wheel Drive -25.3015218 2.4963870
## Front-Wheel Drive-2-Wheel Drive -20.5529404 5.3432719
## Part-time 4-Wheel Drive-2-Wheel Drive -37.0538923 1.2471223
## Rear-Wheel Drive-2-Wheel Drive -30.8455137 -4.9387251
## 4-Wheel or All-Wheel Drive-4-Wheel Drive -33.6012866 23.7582181
## All-Wheel Drive-4-Wheel Drive -28.1341563 30.0093415
## Front-Wheel Drive-4-Wheel Drive -23.8940142 33.3646656
## Part-time 4-Wheel Drive-4-Wheel Drive -37.4807371 26.3542871
## Rear-Wheel Drive-4-Wheel Drive -34.1836914 23.0797725
## All-Wheel Drive-4-Wheel or All-Wheel Drive -0.2834012 12.0016549
## Front-Wheel Drive-4-Wheel or All-Wheel Drive 6.1635563 13.1501636
## Part-time 4-Wheel Drive-4-Wheel or All-Wheel Drive -15.1775652 13.8941837
## Rear-Wheel Drive-4-Wheel or All-Wheel Drive -4.1432790 2.8824287
## Front-Wheel Drive-All-Wheel Drive -2.1049410 9.7004072
## Part-time 4-Wheel Drive-All-Wheel Drive -21.7955891 8.7939540
## Rear-Wheel Drive-All-Wheel Drive -12.4038172 -0.5752868
## Part-time 4-Wheel Drive-Front-Wheel Drive -24.7347052 4.1376038
## Rear-Wheel Drive-Front-Wheel Drive -13.3615415 -7.2130287
## Rear-Wheel Drive-Part-time 4-Wheel Drive -14.4296322 14.4521633
## p adj
## 2-Wheel Drive- 0.2799631
## 4-Wheel Drive- 1.0000000
## 4-Wheel or All-Wheel Drive- 0.6812448
## All-Wheel Drive- 0.9999999
## Front-Wheel Drive- 0.8434141
## Part-time 4-Wheel Drive- 0.9625697
## Rear-Wheel Drive- 0.5219433
## 4-Wheel Drive-2-Wheel Drive 0.9314383
## 4-Wheel or All-Wheel Drive-2-Wheel Drive 0.0016948
## All-Wheel Drive-2-Wheel Drive 0.1989548
## Front-Wheel Drive-2-Wheel Drive 0.6286739
## Part-time 4-Wheel Drive-2-Wheel Drive 0.0863897
## Rear-Wheel Drive-2-Wheel Drive 0.0008104
## 4-Wheel or All-Wheel Drive-4-Wheel Drive 0.9995448
## All-Wheel Drive-4-Wheel Drive 1.0000000
## Front-Wheel Drive-4-Wheel Drive 0.9996432
## Part-time 4-Wheel Drive-4-Wheel Drive 0.9994955
## Rear-Wheel Drive-4-Wheel Drive 0.9989861
## All-Wheel Drive-4-Wheel or All-Wheel Drive 0.0740461
## Front-Wheel Drive-4-Wheel or All-Wheel Drive 0.0000000
## Part-time 4-Wheel Drive-4-Wheel or All-Wheel Drive 1.0000000
## Rear-Wheel Drive-4-Wheel or All-Wheel Drive 0.9993886
## Front-Wheel Drive-All-Wheel Drive 0.5112349
## Part-time 4-Wheel Drive-All-Wheel Drive 0.9010043
## Rear-Wheel Drive-All-Wheel Drive 0.0201676
## Part-time 4-Wheel Drive-Front-Wheel Drive 0.3708944
## Rear-Wheel Drive-Front-Wheel Drive 0.0000000
## Rear-Wheel Drive-Part-time 4-Wheel Drive 1.0000000plot(tukey1)
# The results of the Tukey Test display almost all p-values greater than an alpha, 0.05 between all pairs of drive groups - with the exception of Front-Wheel Drive vs. 4-Wheel or All-Wheel Drive, and Rear-wheel vs. front wheel drive. This shows that, for these 2 pairs, there is a noticeable difference in means between these pairs.
tukey2 <- TukeyHSD(aov(UCity ~ trany, data = x_sample))
tukey2## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = UCity ~ trany, data = x_sample)
##
## $trany
## diff
## Auto(AM-S6)- -95.2148000
## Auto(AV-S6)- -92.4990000
## Automatic (S4)- -100.9636000
## Automatic (S5)- -101.6643500
## Automatic (S6)- -101.9440176
## Automatic (S7)- -103.9684000
## Automatic (S8)- -100.7382200
## Automatic (variable gear ratios)- -69.4434273
## Automatic 3-spd- -105.6124925
## Automatic 4-spd- -104.8771916
## Automatic 5-spd- -105.0353125
## Automatic 6-spd- -104.6332800
## Automatic 6spd- -97.8146000
## Automatic 8-spd- -100.8148000
## Manual 3-spd- -107.8148000
## Manual 4-spd- -106.5370250
## Manual 4-spd Doubled- -94.8148000
## Manual 5-spd- -99.8976242
## Manual 6-spd- -98.2274692
## Auto(AV-S6)-Auto(AM-S6) 2.7158000
## Automatic (S4)-Auto(AM-S6) -5.7488000
## Automatic (S5)-Auto(AM-S6) -6.4495500
## Automatic (S6)-Auto(AM-S6) -6.7292176
## Automatic (S7)-Auto(AM-S6) -8.7536000
## Automatic (S8)-Auto(AM-S6) -5.5234200
## Automatic (variable gear ratios)-Auto(AM-S6) 25.7713727
## Automatic 3-spd-Auto(AM-S6) -10.3976925
## Automatic 4-spd-Auto(AM-S6) -9.6623916
## Automatic 5-spd-Auto(AM-S6) -9.8205125
## Automatic 6-spd-Auto(AM-S6) -9.4184800
## Automatic 6spd-Auto(AM-S6) -2.5998000
## Automatic 8-spd-Auto(AM-S6) -5.6000000
## Manual 3-spd-Auto(AM-S6) -12.6000000
## Manual 4-spd-Auto(AM-S6) -11.3222250
## Manual 4-spd Doubled-Auto(AM-S6) 0.4000000
## Manual 5-spd-Auto(AM-S6) -4.6828242
## Manual 6-spd-Auto(AM-S6) -3.0126692
## Automatic (S4)-Auto(AV-S6) -8.4646000
## Automatic (S5)-Auto(AV-S6) -9.1653500
## Automatic (S6)-Auto(AV-S6) -9.4450176
## Automatic (S7)-Auto(AV-S6) -11.4694000
## Automatic (S8)-Auto(AV-S6) -8.2392200
## Automatic (variable gear ratios)-Auto(AV-S6) 23.0555727
## Automatic 3-spd-Auto(AV-S6) -13.1134925
## Automatic 4-spd-Auto(AV-S6) -12.3781916
## Automatic 5-spd-Auto(AV-S6) -12.5363125
## Automatic 6-spd-Auto(AV-S6) -12.1342800
## Automatic 6spd-Auto(AV-S6) -5.3156000
## Automatic 8-spd-Auto(AV-S6) -8.3158000
## Manual 3-spd-Auto(AV-S6) -15.3158000
## Manual 4-spd-Auto(AV-S6) -14.0380250
## Manual 4-spd Doubled-Auto(AV-S6) -2.3158000
## Manual 5-spd-Auto(AV-S6) -7.3986242
## Manual 6-spd-Auto(AV-S6) -5.7284692
## Automatic (S5)-Automatic (S4) -0.7007500
## Automatic (S6)-Automatic (S4) -0.9804176
## Automatic (S7)-Automatic (S4) -3.0048000
## Automatic (S8)-Automatic (S4) 0.2253800
## Automatic (variable gear ratios)-Automatic (S4) 31.5201727
## Automatic 3-spd-Automatic (S4) -4.6488925
## Automatic 4-spd-Automatic (S4) -3.9135916
## Automatic 5-spd-Automatic (S4) -4.0717125
## Automatic 6-spd-Automatic (S4) -3.6696800
## Automatic 6spd-Automatic (S4) 3.1490000
## Automatic 8-spd-Automatic (S4) 0.1488000
## Manual 3-spd-Automatic (S4) -6.8512000
## Manual 4-spd-Automatic (S4) -5.5734250
## Manual 4-spd Doubled-Automatic (S4) 6.1488000
## Manual 5-spd-Automatic (S4) 1.0659758
## Manual 6-spd-Automatic (S4) 2.7361308
## Automatic (S6)-Automatic (S5) -0.2796676
## Automatic (S7)-Automatic (S5) -2.3040500
## Automatic (S8)-Automatic (S5) 0.9261300
## Automatic (variable gear ratios)-Automatic (S5) 32.2209227
## Automatic 3-spd-Automatic (S5) -3.9481425
## Automatic 4-spd-Automatic (S5) -3.2128416
## Automatic 5-spd-Automatic (S5) -3.3709625
## Automatic 6-spd-Automatic (S5) -2.9689300
## Automatic 6spd-Automatic (S5) 3.8497500
## Automatic 8-spd-Automatic (S5) 0.8495500
## Manual 3-spd-Automatic (S5) -6.1504500
## Manual 4-spd-Automatic (S5) -4.8726750
## Manual 4-spd Doubled-Automatic (S5) 6.8495500
## Manual 5-spd-Automatic (S5) 1.7667258
## Manual 6-spd-Automatic (S5) 3.4368808
## Automatic (S7)-Automatic (S6) -2.0243824
## Automatic (S8)-Automatic (S6) 1.2057976
## Automatic (variable gear ratios)-Automatic (S6) 32.5005904
## Automatic 3-spd-Automatic (S6) -3.6684748
## Automatic 4-spd-Automatic (S6) -2.9331740
## Automatic 5-spd-Automatic (S6) -3.0912949
## Automatic 6-spd-Automatic (S6) -2.6892624
## Automatic 6spd-Automatic (S6) 4.1294176
## Automatic 8-spd-Automatic (S6) 1.1292176
## Manual 3-spd-Automatic (S6) -5.8707824
## Manual 4-spd-Automatic (S6) -4.5930074
## Manual 4-spd Doubled-Automatic (S6) 7.1292176
## Manual 5-spd-Automatic (S6) 2.0463934
## Manual 6-spd-Automatic (S6) 3.7165484
## Automatic (S8)-Automatic (S7) 3.2301800
## Automatic (variable gear ratios)-Automatic (S7) 34.5249727
## Automatic 3-spd-Automatic (S7) -1.6440925
## Automatic 4-spd-Automatic (S7) -0.9087916
## Automatic 5-spd-Automatic (S7) -1.0669125
## Automatic 6-spd-Automatic (S7) -0.6648800
## Automatic 6spd-Automatic (S7) 6.1538000
## Automatic 8-spd-Automatic (S7) 3.1536000
## Manual 3-spd-Automatic (S7) -3.8464000
## Manual 4-spd-Automatic (S7) -2.5686250
## Manual 4-spd Doubled-Automatic (S7) 9.1536000
## Manual 5-spd-Automatic (S7) 4.0707758
## Manual 6-spd-Automatic (S7) 5.7409308
## Automatic (variable gear ratios)-Automatic (S8) 31.2947927
## Automatic 3-spd-Automatic (S8) -4.8742725
## Automatic 4-spd-Automatic (S8) -4.1389716
## Automatic 5-spd-Automatic (S8) -4.2970925
## Automatic 6-spd-Automatic (S8) -3.8950600
## Automatic 6spd-Automatic (S8) 2.9236200
## Automatic 8-spd-Automatic (S8) -0.0765800
## Manual 3-spd-Automatic (S8) -7.0765800
## Manual 4-spd-Automatic (S8) -5.7988050
## Manual 4-spd Doubled-Automatic (S8) 5.9234200
## Manual 5-spd-Automatic (S8) 0.8405958
## Manual 6-spd-Automatic (S8) 2.5107508
## Automatic 3-spd-Automatic (variable gear ratios) -36.1690652
## Automatic 4-spd-Automatic (variable gear ratios) -35.4337643
## Automatic 5-spd-Automatic (variable gear ratios) -35.5918852
## Automatic 6-spd-Automatic (variable gear ratios) -35.1898527
## Automatic 6spd-Automatic (variable gear ratios) -28.3711727
## Automatic 8-spd-Automatic (variable gear ratios) -31.3713727
## Manual 3-spd-Automatic (variable gear ratios) -38.3713727
## Manual 4-spd-Automatic (variable gear ratios) -37.0935977
## Manual 4-spd Doubled-Automatic (variable gear ratios) -25.3713727
## Manual 5-spd-Automatic (variable gear ratios) -30.4541969
## Manual 6-spd-Automatic (variable gear ratios) -28.7840420
## Automatic 4-spd-Automatic 3-spd 0.7353008
## Automatic 5-spd-Automatic 3-spd 0.5771800
## Automatic 6-spd-Automatic 3-spd 0.9792125
## Automatic 6spd-Automatic 3-spd 7.7978925
## Automatic 8-spd-Automatic 3-spd 4.7976925
## Manual 3-spd-Automatic 3-spd -2.2023075
## Manual 4-spd-Automatic 3-spd -0.9245325
## Manual 4-spd Doubled-Automatic 3-spd 10.7976925
## Manual 5-spd-Automatic 3-spd 5.7148682
## Manual 6-spd-Automatic 3-spd 7.3850232
## Automatic 5-spd-Automatic 4-spd -0.1581209
## Automatic 6-spd-Automatic 4-spd 0.2439116
## Automatic 6spd-Automatic 4-spd 7.0625916
## Automatic 8-spd-Automatic 4-spd 4.0623916
## Manual 3-spd-Automatic 4-spd -2.9376084
## Manual 4-spd-Automatic 4-spd -1.6598334
## Manual 4-spd Doubled-Automatic 4-spd 10.0623916
## Manual 5-spd-Automatic 4-spd 4.9795674
## Manual 6-spd-Automatic 4-spd 6.6497224
## Automatic 6-spd-Automatic 5-spd 0.4020325
## Automatic 6spd-Automatic 5-spd 7.2207125
## Automatic 8-spd-Automatic 5-spd 4.2205125
## Manual 3-spd-Automatic 5-spd -2.7794875
## Manual 4-spd-Automatic 5-spd -1.5017125
## Manual 4-spd Doubled-Automatic 5-spd 10.2205125
## Manual 5-spd-Automatic 5-spd 5.1376883
## Manual 6-spd-Automatic 5-spd 6.8078433
## Automatic 6spd-Automatic 6-spd 6.8186800
## Automatic 8-spd-Automatic 6-spd 3.8184800
## Manual 3-spd-Automatic 6-spd -3.1815200
## Manual 4-spd-Automatic 6-spd -1.9037450
## Manual 4-spd Doubled-Automatic 6-spd 9.8184800
## Manual 5-spd-Automatic 6-spd 4.7356558
## Manual 6-spd-Automatic 6-spd 6.4058108
## Automatic 8-spd-Automatic 6spd -3.0002000
## Manual 3-spd-Automatic 6spd -10.0002000
## Manual 4-spd-Automatic 6spd -8.7224250
## Manual 4-spd Doubled-Automatic 6spd 2.9998000
## Manual 5-spd-Automatic 6spd -2.0830242
## Manual 6-spd-Automatic 6spd -0.4128692
## Manual 3-spd-Automatic 8-spd -7.0000000
## Manual 4-spd-Automatic 8-spd -5.7222250
## Manual 4-spd Doubled-Automatic 8-spd 6.0000000
## Manual 5-spd-Automatic 8-spd 0.9171758
## Manual 6-spd-Automatic 8-spd 2.5873308
## Manual 4-spd-Manual 3-spd 1.2777750
## Manual 4-spd Doubled-Manual 3-spd 13.0000000
## Manual 5-spd-Manual 3-spd 7.9171758
## Manual 6-spd-Manual 3-spd 9.5873308
## Manual 4-spd Doubled-Manual 4-spd 11.7222250
## Manual 5-spd-Manual 4-spd 6.6394008
## Manual 6-spd-Manual 4-spd 8.3095558
## Manual 5-spd-Manual 4-spd Doubled -5.0828242
## Manual 6-spd-Manual 4-spd Doubled -3.4126692
## Manual 6-spd-Manual 5-spd 1.6701550
## lwr
## Auto(AM-S6)- -134.6215602
## Auto(AV-S6)- -131.9057602
## Automatic (S4)- -140.3703602
## Automatic (S5)- -130.6669402
## Automatic (S6)- -130.2156118
## Automatic (S7)- -138.0956555
## Automatic (S8)- -129.9630556
## Automatic (variable gear ratios)- -98.5472485
## Automatic 3-spd- -133.7389268
## Automatic 4-spd- -132.8317209
## Automatic 5-spd- -133.3321379
## Automatic 6-spd- -133.4119087
## Automatic 6spd- -137.2213602
## Automatic 8-spd- -140.2215602
## Manual 3-spd- -139.9902850
## Manual 4-spd- -135.5396152
## Manual 4-spd Doubled- -134.2215602
## Manual 5-spd- -127.8710467
## Manual 6-spd- -126.6230622
## Auto(AV-S6)-Auto(AM-S6) -36.6909602
## Automatic (S4)-Auto(AM-S6) -45.1555602
## Automatic (S5)-Auto(AM-S6) -35.4521402
## Automatic (S6)-Auto(AM-S6) -35.0008118
## Automatic (S7)-Auto(AM-S6) -42.8808555
## Automatic (S8)-Auto(AM-S6) -34.7482556
## Automatic (variable gear ratios)-Auto(AM-S6) -3.3324485
## Automatic 3-spd-Auto(AM-S6) -38.5241268
## Automatic 4-spd-Auto(AM-S6) -37.6169209
## Automatic 5-spd-Auto(AM-S6) -38.1173379
## Automatic 6-spd-Auto(AM-S6) -38.1971087
## Automatic 6spd-Auto(AM-S6) -42.0065602
## Automatic 8-spd-Auto(AM-S6) -45.0067602
## Manual 3-spd-Auto(AM-S6) -44.7754850
## Manual 4-spd-Auto(AM-S6) -40.3248152
## Manual 4-spd Doubled-Auto(AM-S6) -39.0067602
## Manual 5-spd-Auto(AM-S6) -32.6562467
## Manual 6-spd-Auto(AM-S6) -31.4082622
## Automatic (S4)-Auto(AV-S6) -47.8713602
## Automatic (S5)-Auto(AV-S6) -38.1679402
## Automatic (S6)-Auto(AV-S6) -37.7166118
## Automatic (S7)-Auto(AV-S6) -45.5966555
## Automatic (S8)-Auto(AV-S6) -37.4640556
## Automatic (variable gear ratios)-Auto(AV-S6) -6.0482485
## Automatic 3-spd-Auto(AV-S6) -41.2399268
## Automatic 4-spd-Auto(AV-S6) -40.3327209
## Automatic 5-spd-Auto(AV-S6) -40.8331379
## Automatic 6-spd-Auto(AV-S6) -40.9129087
## Automatic 6spd-Auto(AV-S6) -44.7223602
## Automatic 8-spd-Auto(AV-S6) -47.7225602
## Manual 3-spd-Auto(AV-S6) -47.4912850
## Manual 4-spd-Auto(AV-S6) -43.0406152
## Manual 4-spd Doubled-Auto(AV-S6) -41.7225602
## Manual 5-spd-Auto(AV-S6) -35.3720467
## Manual 6-spd-Auto(AV-S6) -34.1240622
## Automatic (S5)-Automatic (S4) -29.7033402
## Automatic (S6)-Automatic (S4) -29.2520118
## Automatic (S7)-Automatic (S4) -37.1320555
## Automatic (S8)-Automatic (S4) -28.9994556
## Automatic (variable gear ratios)-Automatic (S4) 2.4163515
## Automatic 3-spd-Automatic (S4) -32.7753268
## Automatic 4-spd-Automatic (S4) -31.8681209
## Automatic 5-spd-Automatic (S4) -32.3685379
## Automatic 6-spd-Automatic (S4) -32.4483087
## Automatic 6spd-Automatic (S4) -36.2577602
## Automatic 8-spd-Automatic (S4) -39.2579602
## Manual 3-spd-Automatic (S4) -39.0266850
## Manual 4-spd-Automatic (S4) -34.5760152
## Manual 4-spd Doubled-Automatic (S4) -33.2579602
## Manual 5-spd-Automatic (S4) -26.9074467
## Manual 6-spd-Automatic (S4) -25.6594622
## Automatic (S6)-Automatic (S5) -9.6359769
## Automatic (S7)-Automatic (S5) -23.5861329
## Automatic (S8)-Automatic (S5) -11.0048592
## Automatic (variable gear ratios)-Automatic (S5) 20.5895041
## Automatic 3-spd-Automatic (S5) -12.8562172
## Automatic 4-spd-Automatic (S5) -11.5622845
## Automatic 5-spd-Automatic (S5) -12.8032376
## Automatic 6-spd-Automatic (S5) -13.7609158
## Automatic 6spd-Automatic (S5) -25.1528402
## Automatic 8-spd-Automatic (S5) -28.1530402
## Manual 3-spd-Automatic (S5) -24.1370929
## Manual 4-spd-Automatic (S5) -16.2484268
## Manual 4-spd Doubled-Automatic (S5) -22.1530402
## Manual 5-spd-Automatic (S5) -6.6457561
## Manual 6-spd-Automatic (S5) -6.2876854
## Automatic (S7)-Automatic (S6) -22.2989932
## Automatic (S8)-Automatic (S6) -8.8182383
## Automatic (variable gear ratios)-Automatic (S6) 22.8350489
## Automatic 3-spd-Automatic (S6) -9.7911041
## Automatic 4-spd-Automatic (S6) -8.2101039
## Automatic 5-spd-Automatic (S6) -9.9542831
## Automatic 6-spd-Automatic (S6) -11.3263770
## Automatic 6spd-Automatic (S6) -24.1421765
## Automatic 8-spd-Automatic (S6) -27.1423765
## Manual 3-spd-Automatic (S6) -22.6532767
## Manual 4-spd-Automatic (S6) -13.9493166
## Manual 4-spd Doubled-Automatic (S6) -21.1423765
## Manual 5-spd-Automatic (S6) -3.3297246
## Manual 6-spd-Automatic (S6) -3.5429268
## Automatic (S8)-Automatic (S7) -18.3537915
## Automatic (variable gear ratios)-Automatic (S7) 13.1051403
## Automatic 3-spd-Automatic (S7) -21.7157919
## Automatic 4-spd-Automatic (S7) -20.7388829
## Automatic 5-spd-Automatic (S7) -21.3766919
## Automatic 6-spd-Automatic (S7) -21.6407299
## Automatic 6spd-Automatic (S7) -27.9734555
## Automatic 8-spd-Automatic (S7) -30.9736555
## Manual 3-spd-Automatic (S7) -29.2833544
## Manual 4-spd-Automatic (S7) -23.8507079
## Manual 4-spd Doubled-Automatic (S7) -24.9736555
## Manual 5-spd-Automatic (S7) -15.7859404
## Manual 6-spd-Automatic (S7) -14.7062331
## Automatic (variable gear ratios)-Automatic (S8) 19.1197908
## Automatic 3-spd-Automatic (S8) -14.4812787
## Automatic 4-spd-Automatic (S8) -13.2303944
## Automatic 5-spd-Automatic (S8) -14.3920708
## Automatic 6-spd-Automatic (S8) -15.2708118
## Automatic 6spd-Automatic (S8) -26.3012156
## Automatic 8-spd-Automatic (S8) -29.3014156
## Manual 3-spd-Automatic (S8) -25.4194286
## Manual 4-spd-Automatic (S8) -17.7297942
## Manual 4-spd Doubled-Automatic (S8) -23.3014156
## Manual 5-spd-Automatic (S8) -8.3087553
## Manual 6-spd-Automatic (S8) -7.8578538
## Automatic 3-spd-Automatic (variable gear ratios) -45.4013978
## Automatic 4-spd-Automatic (variable gear ratios) -44.1283239
## Automatic 5-spd-Automatic (variable gear ratios) -45.3309808
## Automatic 6-spd-Automatic (variable gear ratios) -46.2510066
## Automatic 6spd-Automatic (variable gear ratios) -57.4749940
## Automatic 8-spd-Automatic (variable gear ratios) -60.4751940
## Manual 3-spd-Automatic (variable gear ratios) -56.5207940
## Manual 4-spd-Automatic (variable gear ratios) -48.7250163
## Manual 4-spd Doubled-Automatic (variable gear ratios) -54.4751940
## Manual 5-spd-Automatic (variable gear ratios) -39.2093110
## Manual 6-spd-Automatic (variable gear ratios) -38.8064846
## Automatic 4-spd-Automatic 3-spd -3.6985759
## Automatic 5-spd-Automatic 3-spd -5.6609192
## Automatic 6-spd-Automatic 3-spd -7.1702061
## Automatic 6spd-Automatic 3-spd -20.3285419
## Automatic 8-spd-Automatic 3-spd -23.3287419
## Manual 3-spd-Automatic 3-spd -18.7390965
## Manual 4-spd-Automatic 3-spd -9.8326073
## Manual 4-spd Doubled-Automatic 3-spd -17.3287419
## Manual 5-spd-Automatic 3-spd 1.1633939
## Manual 6-spd-Automatic 3-spd 0.7131978
## Automatic 5-spd-Automatic 4-spd -5.5685997
## Automatic 6-spd-Automatic 4-spd -7.2908361
## Automatic 6spd-Automatic 4-spd -20.8919377
## Automatic 8-spd-Automatic 4-spd -23.8921377
## Manual 3-spd-Automatic 4-spd -19.1802926
## Manual 4-spd-Automatic 4-spd -10.0092763
## Manual 4-spd Doubled-Automatic 4-spd -17.8921377
## Manual 5-spd-Automatic 4-spd 1.6516072
## Manual 6-spd-Automatic 4-spd 0.7444151
## Automatic 6-spd-Automatic 5-spd -8.3173162
## Automatic 6spd-Automatic 5-spd -21.0761129
## Automatic 8-spd-Automatic 5-spd -24.0763129
## Manual 3-spd-Automatic 5-spd -19.6044513
## Manual 4-spd-Automatic 5-spd -10.9339876
## Manual 4-spd Doubled-Automatic 5-spd -18.0763129
## Manual 5-spd-Automatic 5-spd -0.3695739
## Manual 6-spd-Automatic 5-spd -0.5492806
## Automatic 6spd-Automatic 6-spd -21.9599487
## Automatic 8-spd-Automatic 6-spd -24.9601487
## Manual 3-spd-Automatic 6-spd -20.8047589
## Manual 4-spd-Automatic 6-spd -12.6957308
## Manual 4-spd Doubled-Automatic 6-spd -18.9601487
## Manual 5-spd-Automatic 6-spd -2.8688875
## Manual 6-spd-Automatic 6-spd -2.6289230
## Automatic 8-spd-Automatic 6spd -42.4069602
## Manual 3-spd-Automatic 6spd -42.1756850
## Manual 4-spd-Automatic 6spd -37.7250152
## Manual 4-spd Doubled-Automatic 6spd -36.4069602
## Manual 5-spd-Automatic 6spd -30.0564467
## Manual 6-spd-Automatic 6spd -28.8084622
## Manual 3-spd-Automatic 8-spd -39.1754850
## Manual 4-spd-Automatic 8-spd -34.7248152
## Manual 4-spd Doubled-Automatic 8-spd -33.4067602
## Manual 5-spd-Automatic 8-spd -27.0562467
## Manual 6-spd-Automatic 8-spd -25.8082622
## Manual 4-spd-Manual 3-spd -16.7088679
## Manual 4-spd Doubled-Manual 3-spd -19.1754850
## Manual 5-spd-Manual 3-spd -8.3580031
## Manual 6-spd-Manual 3-spd -7.4032190
## Manual 4-spd Doubled-Manual 4-spd -17.2803652
## Manual 5-spd-Manual 4-spd -1.7730811
## Manual 6-spd-Manual 4-spd -1.4150104
## Manual 5-spd-Manual 4-spd Doubled -33.0562467
## Manual 6-spd-Manual 4-spd Doubled -31.8082622
## Manual 6-spd-Manual 5-spd -4.3239512
## upr
## Auto(AM-S6)- -55.8080398
## Auto(AV-S6)- -53.0922398
## Automatic (S4)- -61.5568398
## Automatic (S5)- -72.6617598
## Automatic (S6)- -73.6724235
## Automatic (S7)- -69.8411445
## Automatic (S8)- -71.5133844
## Automatic (variable gear ratios)- -40.3396060
## Automatic 3-spd- -77.4860581
## Automatic 4-spd- -76.9226623
## Automatic 5-spd- -76.7384871
## Automatic 6-spd- -75.8546513
## Automatic 6spd- -58.4078398
## Automatic 8-spd- -61.4080398
## Manual 3-spd- -75.6393150
## Manual 4-spd- -77.5344348
## Manual 4-spd Doubled- -55.4080398
## Manual 5-spd- -71.9242018
## Manual 6-spd- -69.8318763
## Auto(AV-S6)-Auto(AM-S6) 42.1225602
## Automatic (S4)-Auto(AM-S6) 33.6579602
## Automatic (S5)-Auto(AM-S6) 22.5530402
## Automatic (S6)-Auto(AM-S6) 21.5423765
## Automatic (S7)-Auto(AM-S6) 25.3736555
## Automatic (S8)-Auto(AM-S6) 23.7014156
## Automatic (variable gear ratios)-Auto(AM-S6) 54.8751940
## Automatic 3-spd-Auto(AM-S6) 17.7287419
## Automatic 4-spd-Auto(AM-S6) 18.2921377
## Automatic 5-spd-Auto(AM-S6) 18.4763129
## Automatic 6-spd-Auto(AM-S6) 19.3601487
## Automatic 6spd-Auto(AM-S6) 36.8069602
## Automatic 8-spd-Auto(AM-S6) 33.8067602
## Manual 3-spd-Auto(AM-S6) 19.5754850
## Manual 4-spd-Auto(AM-S6) 17.6803652
## Manual 4-spd Doubled-Auto(AM-S6) 39.8067602
## Manual 5-spd-Auto(AM-S6) 23.2905982
## Manual 6-spd-Auto(AM-S6) 25.3829237
## Automatic (S4)-Auto(AV-S6) 30.9421602
## Automatic (S5)-Auto(AV-S6) 19.8372402
## Automatic (S6)-Auto(AV-S6) 18.8265765
## Automatic (S7)-Auto(AV-S6) 22.6578555
## Automatic (S8)-Auto(AV-S6) 20.9856156
## Automatic (variable gear ratios)-Auto(AV-S6) 52.1593940
## Automatic 3-spd-Auto(AV-S6) 15.0129419
## Automatic 4-spd-Auto(AV-S6) 15.5763377
## Automatic 5-spd-Auto(AV-S6) 15.7605129
## Automatic 6-spd-Auto(AV-S6) 16.6443487
## Automatic 6spd-Auto(AV-S6) 34.0911602
## Automatic 8-spd-Auto(AV-S6) 31.0909602
## Manual 3-spd-Auto(AV-S6) 16.8596850
## Manual 4-spd-Auto(AV-S6) 14.9645652
## Manual 4-spd Doubled-Auto(AV-S6) 37.0909602
## Manual 5-spd-Auto(AV-S6) 20.5747982
## Manual 6-spd-Auto(AV-S6) 22.6671237
## Automatic (S5)-Automatic (S4) 28.3018402
## Automatic (S6)-Automatic (S4) 27.2911765
## Automatic (S7)-Automatic (S4) 31.1224555
## Automatic (S8)-Automatic (S4) 29.4502156
## Automatic (variable gear ratios)-Automatic (S4) 60.6239940
## Automatic 3-spd-Automatic (S4) 23.4775419
## Automatic 4-spd-Automatic (S4) 24.0409377
## Automatic 5-spd-Automatic (S4) 24.2251129
## Automatic 6-spd-Automatic (S4) 25.1089487
## Automatic 6spd-Automatic (S4) 42.5557602
## Automatic 8-spd-Automatic (S4) 39.5555602
## Manual 3-spd-Automatic (S4) 25.3242850
## Manual 4-spd-Automatic (S4) 23.4291652
## Manual 4-spd Doubled-Automatic (S4) 45.5555602
## Manual 5-spd-Automatic (S4) 29.0393982
## Manual 6-spd-Automatic (S4) 31.1317237
## Automatic (S6)-Automatic (S5) 9.0766416
## Automatic (S7)-Automatic (S5) 18.9780329
## Automatic (S8)-Automatic (S5) 12.8571192
## Automatic (variable gear ratios)-Automatic (S5) 43.8523413
## Automatic 3-spd-Automatic (S5) 4.9599323
## Automatic 4-spd-Automatic (S5) 5.1366013
## Automatic 5-spd-Automatic (S5) 6.0613126
## Automatic 6-spd-Automatic (S5) 7.8230558
## Automatic 6spd-Automatic (S5) 32.8523402
## Automatic 8-spd-Automatic (S5) 29.8521402
## Manual 3-spd-Automatic (S5) 11.8361929
## Manual 4-spd-Automatic (S5) 6.5030768
## Manual 4-spd Doubled-Automatic (S5) 35.8521402
## Manual 5-spd-Automatic (S5) 10.1792077
## Manual 6-spd-Automatic (S5) 13.1614469
## Automatic (S7)-Automatic (S6) 18.2502285
## Automatic (S8)-Automatic (S6) 11.2298335
## Automatic (variable gear ratios)-Automatic (S6) 42.1661319
## Automatic 3-spd-Automatic (S6) 2.4541545
## Automatic 4-spd-Automatic (S6) 2.3437560
## Automatic 5-spd-Automatic (S6) 3.7716934
## Automatic 6-spd-Automatic (S6) 5.9478523
## Automatic 6spd-Automatic (S6) 32.4010118
## Automatic 8-spd-Automatic (S6) 29.4008118
## Manual 3-spd-Automatic (S6) 10.9117120
## Manual 4-spd-Automatic (S6) 4.7633019
## Manual 4-spd Doubled-Automatic (S6) 35.4008118
## Manual 5-spd-Automatic (S6) 7.4225115
## Manual 6-spd-Automatic (S6) 10.9760236
## Automatic (S8)-Automatic (S7) 24.8141515
## Automatic (variable gear ratios)-Automatic (S7) 55.9448052
## Automatic 3-spd-Automatic (S7) 18.4276070
## Automatic 4-spd-Automatic (S7) 18.9212997
## Automatic 5-spd-Automatic (S7) 19.2428669
## Automatic 6-spd-Automatic (S7) 20.3109699
## Automatic 6spd-Automatic (S7) 40.2810555
## Automatic 8-spd-Automatic (S7) 37.2808555
## Manual 3-spd-Automatic (S7) 21.5905544
## Manual 4-spd-Automatic (S7) 18.7134579
## Manual 4-spd Doubled-Automatic (S7) 43.2808555
## Manual 5-spd-Automatic (S7) 23.9274919
## Manual 6-spd-Automatic (S7) 26.1880946
## Automatic (variable gear ratios)-Automatic (S8) 43.4697947
## Automatic 3-spd-Automatic (S8) 4.7327338
## Automatic 4-spd-Automatic (S8) 4.9524512
## Automatic 5-spd-Automatic (S8) 5.7978858
## Automatic 6-spd-Automatic (S8) 7.4806918
## Automatic 6spd-Automatic (S8) 32.1484556
## Automatic 8-spd-Automatic (S8) 29.1482556
## Manual 3-spd-Automatic (S8) 11.2662686
## Manual 4-spd-Automatic (S8) 6.1321842
## Manual 4-spd Doubled-Automatic (S8) 35.1482556
## Manual 5-spd-Automatic (S8) 9.9899468
## Manual 6-spd-Automatic (S8) 12.8793553
## Automatic 3-spd-Automatic (variable gear ratios) -26.9367325
## Automatic 4-spd-Automatic (variable gear ratios) -26.7392048
## Automatic 5-spd-Automatic (variable gear ratios) -25.8527896
## Automatic 6-spd-Automatic (variable gear ratios) -24.1286988
## Automatic 6spd-Automatic (variable gear ratios) 0.7326485
## Automatic 8-spd-Automatic (variable gear ratios) -2.2675515
## Manual 3-spd-Automatic (variable gear ratios) -20.2219514
## Manual 4-spd-Automatic (variable gear ratios) -25.4621791
## Manual 4-spd Doubled-Automatic (variable gear ratios) 3.7324485
## Manual 5-spd-Automatic (variable gear ratios) -21.6990829
## Manual 6-spd-Automatic (variable gear ratios) -18.7615993
## Automatic 4-spd-Automatic 3-spd 5.1691776
## Automatic 5-spd-Automatic 3-spd 6.8152791
## Automatic 6-spd-Automatic 3-spd 9.1286310
## Automatic 6spd-Automatic 3-spd 35.9243268
## Automatic 8-spd-Automatic 3-spd 32.9241268
## Manual 3-spd-Automatic 3-spd 14.3344815
## Manual 4-spd-Automatic 3-spd 7.9835422
## Manual 4-spd Doubled-Automatic 3-spd 38.9241268
## Manual 5-spd-Automatic 3-spd 10.2663426
## Manual 6-spd-Automatic 3-spd 14.0568486
## Automatic 5-spd-Automatic 4-spd 5.2523579
## Automatic 6-spd-Automatic 4-spd 7.7786593
## Automatic 6spd-Automatic 4-spd 35.0171209
## Automatic 8-spd-Automatic 4-spd 32.0169209
## Manual 3-spd-Automatic 4-spd 13.3050758
## Manual 4-spd-Automatic 4-spd 6.6896095
## Manual 4-spd Doubled-Automatic 4-spd 38.0169209
## Manual 5-spd-Automatic 4-spd 8.3075275
## Manual 6-spd-Automatic 4-spd 12.5550296
## Automatic 6-spd-Automatic 5-spd 9.1213812
## Automatic 6spd-Automatic 5-spd 35.5175379
## Automatic 8-spd-Automatic 5-spd 32.5173379
## Manual 3-spd-Automatic 5-spd 14.0454763
## Manual 4-spd-Automatic 5-spd 7.9305626
## Manual 4-spd Doubled-Automatic 5-spd 38.5173379
## Manual 5-spd-Automatic 5-spd 10.6449504
## Manual 6-spd-Automatic 5-spd 14.1649671
## Automatic 6spd-Automatic 6-spd 35.5973087
## Automatic 8-spd-Automatic 6-spd 32.5971087
## Manual 3-spd-Automatic 6-spd 14.4417189
## Manual 4-spd-Automatic 6-spd 8.8882408
## Manual 4-spd Doubled-Automatic 6-spd 38.5971087
## Manual 5-spd-Automatic 6-spd 12.3401991
## Manual 6-spd-Automatic 6-spd 15.4405445
## Automatic 8-spd-Automatic 6spd 36.4065602
## Manual 3-spd-Automatic 6spd 22.1752850
## Manual 4-spd-Automatic 6spd 20.2801652
## Manual 4-spd Doubled-Automatic 6spd 42.4065602
## Manual 5-spd-Automatic 6spd 25.8903982
## Manual 6-spd-Automatic 6spd 27.9827237
## Manual 3-spd-Automatic 8-spd 25.1754850
## Manual 4-spd-Automatic 8-spd 23.2803652
## Manual 4-spd Doubled-Automatic 8-spd 45.4067602
## Manual 5-spd-Automatic 8-spd 28.8905982
## Manual 6-spd-Automatic 8-spd 30.9829237
## Manual 4-spd-Manual 3-spd 19.2644179
## Manual 4-spd Doubled-Manual 3-spd 45.1754850
## Manual 5-spd-Manual 3-spd 24.1923546
## Manual 6-spd-Manual 3-spd 26.5778805
## Manual 4-spd Doubled-Manual 4-spd 40.7248152
## Manual 5-spd-Manual 4-spd 15.0518827
## Manual 6-spd-Manual 4-spd 18.0341219
## Manual 5-spd-Manual 4-spd Doubled 22.8905982
## Manual 6-spd-Manual 4-spd Doubled 24.9829237
## Manual 6-spd-Manual 5-spd 7.6642612
## p adj
## Auto(AM-S6)- 0.0000000
## Auto(AV-S6)- 0.0000000
## Automatic (S4)- 0.0000000
## Automatic (S5)- 0.0000000
## Automatic (S6)- 0.0000000
## Automatic (S7)- 0.0000000
## Automatic (S8)- 0.0000000
## Automatic (variable gear ratios)- 0.0000000
## Automatic 3-spd- 0.0000000
## Automatic 4-spd- 0.0000000
## Automatic 5-spd- 0.0000000
## Automatic 6-spd- 0.0000000
## Automatic 6spd- 0.0000000
## Automatic 8-spd- 0.0000000
## Manual 3-spd- 0.0000000
## Manual 4-spd- 0.0000000
## Manual 4-spd Doubled- 0.0000000
## Manual 5-spd- 0.0000000
## Manual 6-spd- 0.0000000
## Auto(AV-S6)-Auto(AM-S6) 1.0000000
## Automatic (S4)-Auto(AM-S6) 1.0000000
## Automatic (S5)-Auto(AM-S6) 0.9999996
## Automatic (S6)-Auto(AM-S6) 0.9999988
## Automatic (S7)-Auto(AM-S6) 0.9999958
## Automatic (S8)-Auto(AM-S6) 1.0000000
## Automatic (variable gear ratios)-Auto(AM-S6) 0.1618949
## Automatic 3-spd-Auto(AM-S6) 0.9990143
## Automatic 4-spd-Auto(AM-S6) 0.9996096
## Automatic 5-spd-Auto(AM-S6) 0.9995867
## Automatic 6-spd-Auto(AM-S6) 0.9998214
## Automatic 6spd-Auto(AM-S6) 1.0000000
## Automatic 8-spd-Auto(AM-S6) 1.0000000
## Manual 3-spd-Auto(AM-S6) 0.9978933
## Manual 4-spd-Auto(AM-S6) 0.9979762
## Manual 4-spd Doubled-Auto(AM-S6) 1.0000000
## Manual 5-spd-Auto(AM-S6) 1.0000000
## Manual 6-spd-Auto(AM-S6) 1.0000000
## Automatic (S4)-Auto(AV-S6) 0.9999998
## Automatic (S5)-Auto(AV-S6) 0.9998932
## Automatic (S6)-Auto(AV-S6) 0.9997595
## Automatic (S7)-Auto(AV-S6) 0.9997380
## Automatic (S8)-Auto(AV-S6) 0.9999812
## Automatic (variable gear ratios)-Auto(AV-S6) 0.3433085
## Automatic 3-spd-Auto(AV-S6) 0.9832082
## Automatic 4-spd-Auto(AV-S6) 0.9905598
## Automatic 5-spd-Auto(AV-S6) 0.9905030
## Automatic 6-spd-Auto(AV-S6) 0.9946981
## Automatic 6spd-Auto(AV-S6) 1.0000000
## Automatic 8-spd-Auto(AV-S6) 0.9999998
## Manual 3-spd-Auto(AV-S6) 0.9790338
## Manual 4-spd-Auto(AV-S6) 0.9750344
## Manual 4-spd Doubled-Auto(AV-S6) 1.0000000
## Manual 5-spd-Auto(AV-S6) 0.9999931
## Manual 6-spd-Auto(AV-S6) 0.9999999
## Automatic (S5)-Automatic (S4) 1.0000000
## Automatic (S6)-Automatic (S4) 1.0000000
## Automatic (S7)-Automatic (S4) 1.0000000
## Automatic (S8)-Automatic (S4) 1.0000000
## Automatic (variable gear ratios)-Automatic (S4) 0.0183144
## Automatic 3-spd-Automatic (S4) 1.0000000
## Automatic 4-spd-Automatic (S4) 1.0000000
## Automatic 5-spd-Automatic (S4) 1.0000000
## Automatic 6-spd-Automatic (S4) 1.0000000
## Automatic 6spd-Automatic (S4) 1.0000000
## Automatic 8-spd-Automatic (S4) 1.0000000
## Manual 3-spd-Automatic (S4) 0.9999998
## Manual 4-spd-Automatic (S4) 1.0000000
## Manual 4-spd Doubled-Automatic (S4) 1.0000000
## Manual 5-spd-Automatic (S4) 1.0000000
## Manual 6-spd-Automatic (S4) 1.0000000
## Automatic (S6)-Automatic (S5) 1.0000000
## Automatic (S7)-Automatic (S5) 1.0000000
## Automatic (S8)-Automatic (S5) 1.0000000
## Automatic (variable gear ratios)-Automatic (S5) 0.0000000
## Automatic 3-spd-Automatic (S5) 0.9904584
## Automatic 4-spd-Automatic (S5) 0.9983220
## Automatic 5-spd-Automatic (S5) 0.9993782
## Automatic 6-spd-Automatic (S5) 0.9999872
## Automatic 6spd-Automatic (S5) 1.0000000
## Automatic 8-spd-Automatic (S5) 1.0000000
## Manual 3-spd-Automatic (S5) 0.9996647
## Manual 4-spd-Automatic (S5) 0.9935964
## Manual 4-spd Doubled-Automatic (S5) 0.9999989
## Manual 5-spd-Automatic (S5) 0.9999998
## Manual 6-spd-Automatic (S5) 0.9994671
## Automatic (S7)-Automatic (S6) 1.0000000
## Automatic (S8)-Automatic (S6) 1.0000000
## Automatic (variable gear ratios)-Automatic (S6) 0.0000000
## Automatic 3-spd-Automatic (S6) 0.8367576
## Automatic 4-spd-Automatic (S6) 0.9088461
## Automatic 5-spd-Automatic (S6) 0.9885323
## Automatic 6-spd-Automatic (S6) 0.9999143
## Automatic 6spd-Automatic (S6) 1.0000000
## Automatic 8-spd-Automatic (S6) 1.0000000
## Manual 3-spd-Automatic (S6) 0.9995381
## Manual 4-spd-Automatic (S6) 0.9711554
## Manual 4-spd Doubled-Automatic (S6) 0.9999968
## Manual 5-spd-Automatic (S6) 0.9985447
## Manual 6-spd-Automatic (S6) 0.9563661
## Automatic (S8)-Automatic (S7) 1.0000000
## Automatic (variable gear ratios)-Automatic (S7) 0.0000031
## Automatic 3-spd-Automatic (S7) 1.0000000
## Automatic 4-spd-Automatic (S7) 1.0000000
## Automatic 5-spd-Automatic (S7) 1.0000000
## Automatic 6-spd-Automatic (S7) 1.0000000
## Automatic 6spd-Automatic (S7) 1.0000000
## Automatic 8-spd-Automatic (S7) 1.0000000
## Manual 3-spd-Automatic (S7) 1.0000000
## Manual 4-spd-Automatic (S7) 1.0000000
## Manual 4-spd Doubled-Automatic (S7) 0.9999914
## Manual 5-spd-Automatic (S7) 0.9999999
## Manual 6-spd-Automatic (S7) 0.9999824
## Automatic (variable gear ratios)-Automatic (S8) 0.0000000
## Automatic 3-spd-Automatic (S8) 0.9599894
## Automatic 4-spd-Automatic (S8) 0.9870754
## Automatic 5-spd-Automatic (S8) 0.9940570
## Automatic 6-spd-Automatic (S8) 0.9996584
## Automatic 6spd-Automatic (S8) 1.0000000
## Automatic 8-spd-Automatic (S8) 1.0000000
## Manual 3-spd-Automatic (S8) 0.9982643
## Manual 4-spd-Automatic (S8) 0.9739487
## Manual 4-spd Doubled-Automatic (S8) 0.9999999
## Manual 5-spd-Automatic (S8) 1.0000000
## Manual 6-spd-Automatic (S8) 0.9999984
## Automatic 3-spd-Automatic (variable gear ratios) 0.0000000
## Automatic 4-spd-Automatic (variable gear ratios) 0.0000000
## Automatic 5-spd-Automatic (variable gear ratios) 0.0000000
## Automatic 6-spd-Automatic (variable gear ratios) 0.0000000
## Automatic 6spd-Automatic (variable gear ratios) 0.0661730
## Automatic 8-spd-Automatic (variable gear ratios) 0.0195490
## Manual 3-spd-Automatic (variable gear ratios) 0.0000000
## Manual 4-spd-Automatic (variable gear ratios) 0.0000000
## Manual 4-spd Doubled-Automatic (variable gear ratios) 0.1830817
## Manual 5-spd-Automatic (variable gear ratios) 0.0000000
## Manual 6-spd-Automatic (variable gear ratios) 0.0000000
## Automatic 4-spd-Automatic 3-spd 1.0000000
## Automatic 5-spd-Automatic 3-spd 1.0000000
## Automatic 6-spd-Automatic 3-spd 1.0000000
## Automatic 6spd-Automatic 3-spd 0.9999855
## Automatic 8-spd-Automatic 3-spd 1.0000000
## Manual 3-spd-Automatic 3-spd 1.0000000
## Manual 4-spd-Automatic 3-spd 1.0000000
## Manual 4-spd Doubled-Automatic 3-spd 0.9983726
## Manual 5-spd-Automatic 3-spd 0.0015997
## Manual 6-spd-Automatic 3-spd 0.0134295
## Automatic 5-spd-Automatic 4-spd 1.0000000
## Automatic 6-spd-Automatic 4-spd 1.0000000
## Automatic 6spd-Automatic 4-spd 0.9999967
## Automatic 8-spd-Automatic 4-spd 1.0000000
## Manual 3-spd-Automatic 4-spd 1.0000000
## Manual 4-spd-Automatic 4-spd 0.9999999
## Manual 4-spd Doubled-Automatic 4-spd 0.9993139
## Manual 5-spd-Automatic 4-spd 0.0000274
## Manual 6-spd-Automatic 4-spd 0.0104005
## Automatic 6-spd-Automatic 5-spd 1.0000000
## Automatic 6spd-Automatic 5-spd 0.9999961
## Automatic 8-spd-Automatic 5-spd 1.0000000
## Manual 3-spd-Automatic 5-spd 1.0000000
## Manual 4-spd-Automatic 5-spd 1.0000000
## Manual 4-spd Doubled-Automatic 5-spd 0.9992809
## Manual 5-spd-Automatic 5-spd 0.1027456
## Manual 6-spd-Automatic 5-spd 0.1108098
## Automatic 6spd-Automatic 6-spd 0.9999989
## Automatic 8-spd-Automatic 6-spd 1.0000000
## Manual 3-spd-Automatic 6-spd 1.0000000
## Manual 4-spd-Automatic 6-spd 1.0000000
## Manual 4-spd Doubled-Automatic 6-spd 0.9996753
## Manual 5-spd-Automatic 6-spd 0.7871709
## Manual 6-spd-Automatic 6-spd 0.5635291
## Automatic 8-spd-Automatic 6spd 1.0000000
## Manual 3-spd-Automatic 6spd 0.9999166
## Manual 4-spd-Automatic 6spd 0.9999492
## Manual 4-spd Doubled-Automatic 6spd 1.0000000
## Manual 5-spd-Automatic 6spd 1.0000000
## Manual 6-spd-Automatic 6spd 1.0000000
## Manual 3-spd-Automatic 8-spd 0.9999997
## Manual 4-spd-Automatic 8-spd 0.9999999
## Manual 4-spd Doubled-Automatic 8-spd 1.0000000
## Manual 5-spd-Automatic 8-spd 1.0000000
## Manual 6-spd-Automatic 8-spd 1.0000000
## Manual 4-spd-Manual 3-spd 1.0000000
## Manual 4-spd Doubled-Manual 3-spd 0.9968647
## Manual 5-spd-Manual 3-spd 0.9737119
## Manual 6-spd-Manual 3-spd 0.8968337
## Manual 4-spd Doubled-Manual 4-spd 0.9968507
## Manual 5-spd-Manual 4-spd 0.3504129
## Manual 6-spd-Manual 4-spd 0.2124177
## Manual 5-spd-Manual 4-spd Doubled 1.0000000
## Manual 6-spd-Manual 4-spd Doubled 1.0000000
## Manual 6-spd-Manual 5-spd 0.9999844plot(tukey2)
# The results of the Tukey Test display almost all p-values greater than an alpha, 0.05 between all pairs of transmission groups - with the exception of manual 5-speed vs. automatic 4-speed, manual 5-speed vs. automatic 3-speed, Front-Wheel Drive vs. 4-Wheel or All-Wheel Drive, Rear-wheel vs. front wheel drive, and mostly anything vs. automatic (variable gear ratios). This shows that, for these 5+ pairs, there is a noticeable difference in means between these pairs.
tukey3 <- TukeyHSD(aov(UCity ~ fuelType, data = x_sample))
tukey3## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = UCity ~ fuelType, data = x_sample)
##
## $fuelType
## diff lwr
## Diesel-CNG 29.2637875 12.671094
## Electricity-CNG 131.5074000 110.519118
## Gasoline or E85-CNG 8.1144000 -7.751250
## Midgrade-CNG 11.3000000 -9.688282
## Premium-CNG 12.5391107 -2.477484
## Regular-CNG 12.7936592 -2.085597
## Regular Gas and Electricity-CNG 45.5000000 19.794709
## Electricity-Diesel 102.2436125 85.650919
## Gasoline or E85-Diesel -21.1493875 -30.451449
## Midgrade-Diesel -17.9637875 -34.556481
## Premium-Diesel -16.7246768 -24.490476
## Regular-Diesel -16.4701283 -23.966912
## Regular Gas and Electricity-Diesel 16.2362125 -6.025222
## Gasoline or E85-Electricity -123.3930000 -139.258650
## Midgrade-Electricity -120.2074000 -141.195682
## Premium-Electricity -118.9682893 -133.984884
## Regular-Electricity -118.7137408 -133.592997
## Regular Gas and Electricity-Electricity -86.0074000 -111.712691
## Midgrade-Gasoline or E85 3.1856000 -12.680050
## Premium-Gasoline or E85 4.4247107 -1.634084
## Regular-Gasoline or E85 4.6792592 -1.030655
## Regular Gas and Electricity-Gasoline or E85 37.3856000 15.660664
## Premium-Midgrade 1.2391107 -13.777484
## Regular-Midgrade 1.4936592 -13.385597
## Regular Gas and Electricity-Midgrade 34.2000000 8.494709
## Regular-Premium 0.2545485 -2.271794
## Regular Gas and Electricity-Premium 32.9608893 11.848047
## Regular Gas and Electricity-Regular 32.7063408 11.690960
## upr p adj
## Diesel-CNG 45.856481 0.0000034
## Electricity-CNG 152.495682 0.0000000
## Gasoline or E85-CNG 23.980050 0.7757237
## Midgrade-CNG 32.288282 0.7263938
## Premium-CNG 27.555706 0.1805319
## Regular-CNG 27.672915 0.1521148
## Regular Gas and Electricity-CNG 71.205291 0.0000031
## Electricity-Diesel 118.836306 0.0000000
## Gasoline or E85-Diesel -11.847326 0.0000000
## Midgrade-Diesel -1.371094 0.0232765
## Premium-Diesel -8.958877 0.0000000
## Regular-Diesel -8.973345 0.0000000
## Regular Gas and Electricity-Diesel 38.497647 0.3414030
## Gasoline or E85-Electricity -107.527350 0.0000000
## Midgrade-Electricity -99.219118 0.0000000
## Premium-Electricity -103.951694 0.0000000
## Regular-Electricity -103.834485 0.0000000
## Regular Gas and Electricity-Electricity -60.302109 0.0000000
## Midgrade-Gasoline or E85 19.051250 0.9987292
## Premium-Gasoline or E85 10.483506 0.3396796
## Regular-Gasoline or E85 10.389173 0.2001202
## Regular Gas and Electricity-Gasoline or E85 59.110536 0.0000066
## Premium-Midgrade 16.255706 0.9999968
## Regular-Midgrade 16.372915 0.9999878
## Regular Gas and Electricity-Midgrade 59.905291 0.0015252
## Regular-Premium 2.780891 0.9999875
## Regular Gas and Electricity-Premium 54.073732 0.0000717
## Regular Gas and Electricity-Regular 53.721722 0.0000769plot(tukey3)
# The results of the Tukey Test display about half with p-values less than an alpha, 0.05 between pairs of fuel type groups - with the exceptions of Diesel vs. CNG, Gasoline or E85 vs. CNG, Premium vs. CNG, Regular vs. CNG, Premium vs. E85, Regular vs. E85, and Regular vs. premium. This shows that, for these 8 pairs with significant p-values, there is a noticeable difference in means between these pairs.Diagnostics/Model Adequacy Checking
par(mfrow = c(1,1))
qqnorm(residuals(model1))
qqline(residuals(model1))
The Q-Q Normality Plots of the residuals do not appear to be normal. When normal, the data points would lay linearly, and in this case, they do not seem to lie along the line.
interaction.plot(x_sample$drive, x_sample$trany, x_sample$UCity)
# The interaction plot between drive type and transmission seems to demonstrate that there is interaction between the groups. This is because there is a difference in slopes between the two lines.
interaction.plot(x_sample$drive, x_sample$VClass, x_sample$UCity)
# The interaction plot between drive type and v class also seems to demonstrate interaction between the two factors. The slopes of the two lines are different, and therefore infers interaction. They also cross at multiple points.
interaction.plot(x_sample$VClass, x_sample$trany, x_sample$UCity)
# The interaction plot between v class and transmission also seems to demonstrate interaction between the two factors. The slope of the two lines is different, and therefore infers interaction. plot(fitted(model1),residuals(model1))
The plots of the fitted values versus the residual values are supposed to appear symmetric over a residual of 0. The datapoints of the model appear to be relaively symmetric over the dynamic range, but are not ver uniform across the range. There is a large cluster on the lower part of the range.
4. Contingencies
Based on the model adequacy testing, it appears that the data is not normally distributed. Unfortunately, normality is an assumption for using ANOVA tests.In order to verify the results of the ANOVA test, a nonparametric analysis of multiple factors can be used.
A Kruskal-Wallis test can be used to evaluate whether groups’ distributions are identical, without the assumption that the data is normally distributed.
kruskal.test(UCity ~ drive, data = x_sample)##
## Kruskal-Wallis rank sum test
##
## data: UCity by drive
## Kruskal-Wallis chi-squared = 220.7897, df = 7, p-value < 2.2e-16# The p-value of this Kruskal-Wallis test is less than an alpha of 0.05. Therefore, you must reject the null - that variation in city gas mileage cannot be explained by anything other than randomization. Therefore, variation can be explained, in part, by drive.
kruskal.test(UCity ~ trany, data = x)##
## Kruskal-Wallis rank sum test
##
## data: UCity by trany
## Kruskal-Wallis chi-squared = 2037.493, df = 28, p-value < 2.2e-16# The p-value of this Kruskal-Wallis test is less than an alpha of 0.05. Therefore, you must reject the null - that variation in city gas mileage cannot be explained by anything other than randomization. Therefore, transmission type can help to explain variation in city gas mileage.
kruskal.test(UCity ~ fuelType, data = x)##
## Kruskal-Wallis rank sum test
##
## data: UCity by fuelType
## Kruskal-Wallis chi-squared = 828.2673, df = 11, p-value < 2.2e-16# The p-value of this Kruskal-Wallis test is less than an alpha of 0.05. Therefore, you must reject the null - that variation in city gas mileage cannot be explained by anything other than randomization. Instead, fuel type can help to explain variation in city gas mileage.4. References
Fuel economy data are the result of vehicle testing done at the Environmental Protection Agency’s National Vehicle and Fuel Emissions Laboratory in Ann Arbor, Michigan, and by vehicle manufacturers with oversight by EPA.