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Loading the data and libraries

If you use your personal computer an additional package mffSM is needed. The package is not available from CRAN but can be downloaded from this website. Download the package and use the following command for the installation:

install.packages("mffSM_1.1.zip", repos = NULL)
# or without necessity to download first 
install.packages("https://www2.karlin.mff.cuni.cz/~komarek/vyuka/2021_22/nmsa407/R/mffSM_1.2.tar.gz", repos = NULL)

Libraries can be accessed the following way. When in doubt, use help() or ? to show the documentation:

library("mffSM")
library("colorspace")
help(package="mffSM")

This lab session we will work with the data set on car vehicles that were on the market in the U.S. in 2004:

data(Cars2004nh, package = "mffSM")

Quick overview and summary of the data:

dim(Cars2004nh)
## [1] 425  20
str(Cars2004nh)
## 'data.frame':    425 obs. of  20 variables:
##  $ vname       : chr  "Chevrolet.Aveo.4dr" "Chevrolet.Aveo.LS.4dr.hatch" "Chevrolet.Cavalier.2dr" "Chevrolet.Cavalier.4dr" ...
##  $ type        : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ drive       : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ price.retail: int  11690 12585 14610 14810 16385 13670 15040 13270 13730 15460 ...
##  $ price.dealer: int  10965 11802 13697 13884 15357 12849 14086 12482 12906 14496 ...
##  $ price       : num  11328 12194 14154 14347 15871 ...
##  $ cons.city   : num  8.4 8.4 9 9 9 8.1 8.1 9 8.7 9 ...
##  $ cons.highway: num  6.9 6.9 6.4 6.4 6.4 6.5 6.5 7.1 6.5 7.1 ...
##  $ consumption : num  7.65 7.65 7.7 7.7 7.7 7.3 7.3 8.05 7.6 8.05 ...
##  $ engine.size : num  1.6 1.6 2.2 2.2 2.2 2 2 2 2 2 ...
##  $ ncylinder   : int  4 4 4 4 4 4 4 4 4 4 ...
##  $ horsepower  : int  103 103 140 140 140 132 132 130 110 130 ...
##  $ weight      : int  1075 1065 1187 1214 1187 1171 1191 1185 1182 1182 ...
##  $ iweight     : num  0.00093 0.000939 0.000842 0.000824 0.000842 ...
##  $ lweight     : num  6.98 6.97 7.08 7.1 7.08 ...
##  $ wheel.base  : int  249 249 264 264 264 267 267 262 262 262 ...
##  $ length      : int  424 389 465 465 465 442 442 427 427 427 ...
##  $ width       : int  168 168 175 173 175 170 170 170 170 170 ...
##  $ ftype       : Factor w/ 6 levels "personal","wagon",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ fdrive      : Factor w/ 3 levels "front","rear",..: 1 1 1 1 1 1 1 1 1 1 ...
head(Cars2004nh)
##                         vname type drive price.retail price.dealer price cons.city cons.highway consumption
## 1          Chevrolet.Aveo.4dr    1     1        11690        10965 11328       8.4          6.9        7.65
## 2 Chevrolet.Aveo.LS.4dr.hatch    1     1        12585        11802 12194       8.4          6.9        7.65
## 3      Chevrolet.Cavalier.2dr    1     1        14610        13697 14154       9.0          6.4        7.70
## 4      Chevrolet.Cavalier.4dr    1     1        14810        13884 14347       9.0          6.4        7.70
## 5   Chevrolet.Cavalier.LS.2dr    1     1        16385        15357 15871       9.0          6.4        7.70
## 6           Dodge.Neon.SE.4dr    1     1        13670        12849 13260       8.1          6.5        7.30
##   engine.size ncylinder horsepower weight   iweight lweight wheel.base length width    ftype fdrive
## 1         1.6         4        103   1075 0.0009302   6.980        249    424   168 personal  front
## 2         1.6         4        103   1065 0.0009390   6.971        249    389   168 personal  front
## 3         2.2         4        140   1187 0.0008425   7.079        264    465   175 personal  front
## 4         2.2         4        140   1214 0.0008237   7.102        264    465   173 personal  front
## 5         2.2         4        140   1187 0.0008425   7.079        264    465   175 personal  front
## 6         2.0         4        132   1171 0.0008540   7.066        267    442   170 personal  front
summary(Cars2004nh)
##     vname                type          drive       price.retail     price.dealer        price       
##  Length:425         Min.   :1.00   Min.   :1.00   Min.   : 10280   Min.   :  9875   Min.   : 10078  
##  Class :character   1st Qu.:1.00   1st Qu.:1.00   1st Qu.: 20370   1st Qu.: 18973   1st Qu.: 19601  
##  Mode  :character   Median :1.00   Median :1.00   Median : 27905   Median : 25672   Median : 26656  
##                     Mean   :2.22   Mean   :1.69   Mean   : 32866   Mean   : 30096   Mean   : 31481  
##                     3rd Qu.:3.00   3rd Qu.:2.00   3rd Qu.: 39235   3rd Qu.: 35777   3rd Qu.: 37514  
##                     Max.   :6.00   Max.   :3.00   Max.   :192465   Max.   :173560   Max.   :183013  
##                                                                                                     
##    cons.city     cons.highway    consumption     engine.size     ncylinder       horsepower      weight    
##  Min.   : 6.2   Min.   : 5.10   Min.   : 5.65   Min.   :1.30   Min.   :-1.00   Min.   :100   Min.   : 923  
##  1st Qu.:11.2   1st Qu.: 8.10   1st Qu.: 9.65   1st Qu.:2.40   1st Qu.: 4.00   1st Qu.:165   1st Qu.:1412  
##  Median :12.4   Median : 9.00   Median :10.70   Median :3.00   Median : 6.00   Median :210   Median :1577  
##  Mean   :12.4   Mean   : 9.14   Mean   :10.75   Mean   :3.21   Mean   : 5.79   Mean   :217   Mean   :1626  
##  3rd Qu.:13.8   3rd Qu.: 9.80   3rd Qu.:11.65   3rd Qu.:3.90   3rd Qu.: 6.00   3rd Qu.:255   3rd Qu.:1804  
##  Max.   :23.5   Max.   :19.60   Max.   :21.55   Max.   :8.30   Max.   :12.00   Max.   :500   Max.   :3261  
##  NA's   :14     NA's   :14      NA's   :14                                                   NA's   :2     
##     iweight            lweight       wheel.base      length        width          ftype       fdrive   
##  Min.   :0.000307   Min.   :6.83   Min.   :226   Min.   :363   Min.   :163   personal:242   front:223  
##  1st Qu.:0.000554   1st Qu.:7.25   1st Qu.:262   1st Qu.:450   1st Qu.:175   wagon   : 30   rear :110  
##  Median :0.000634   Median :7.36   Median :272   Median :472   Median :180   SUV     : 60   4x4  : 92  
##  Mean   :0.000641   Mean   :7.37   Mean   :275   Mean   :471   Mean   :181   pickup  : 24              
##  3rd Qu.:0.000708   3rd Qu.:7.50   3rd Qu.:284   3rd Qu.:490   3rd Qu.:185   sport   : 49              
##  Max.   :0.001083   Max.   :8.09   Max.   :366   Max.   :577   Max.   :206   minivan : 20              
##  NA's   :2          NA's   :2      NA's   :2     NA's   :26    NA's   :28

Data subsampling and preprocessing

Exclude cars where the consumption is missing:

Cars2004nh <- Cars2004nh[!is.na(Cars2004nh$consumption),]

Create a working subsample:

set.seed(123456789)
idx <- c(sample((1:nrow(Cars2004nh))[Cars2004nh$ftype=="personal"], size=100), 
         sample((1:nrow(Cars2004nh))[Cars2004nh$ftype!="personal"], size=120))
Cars2004nh <- Cars2004nh[idx,]

Next, we derive some additional variables. First, binary factor fdrive2, which distinguishes whether the drive is front or other (rear or 4x4):

Cars2004nh <- transform(Cars2004nh, 
                        fdrive2 = factor(1*(fdrive == "front") + 2*(fdrive != "front"), 
                                         labels = c("front", "other")))
with(Cars2004nh, table(fdrive, fdrive2))
##        fdrive2
## fdrive  front other
##   front   102     0
##   rear      0    61
##   4x4       0    57

Alternatively, it could be done by changing labels:

Cars2004nh <- transform(Cars2004nh, 
                        fdrive2 = factor(fdrive, 
                                         levels = c("front", "rear", "4x4"),
                                         labels = c("front", "other", "other")))

Next, we derive binary indicator cons10 of having a consumption <= 10 l/100 km and then create a factor with No (0) and Yes (1) labels:

Cars2004nh <- transform(Cars2004nh, 
                        cons10  = 1 * (consumption <= 10), # 0 or 1
                        fcons10 = factor(1 * (consumption <= 10), 
                                         labels = c("No", "Yes")))

summary(Cars2004nh)
##     vname                type          drive      price.retail     price.dealer        price       
##  Length:220         Min.   :1.00   Min.   :1.0   Min.   : 10760   Min.   : 10144   Min.   : 10452  
##  Class :character   1st Qu.:1.00   1st Qu.:1.0   1st Qu.: 20969   1st Qu.: 19468   1st Qu.: 20313  
##  Mode  :character   Median :2.00   Median :2.0   Median : 26991   Median : 24876   Median : 25900  
##                     Mean   :2.53   Mean   :1.8   Mean   : 32516   Mean   : 29759   Mean   : 31138  
##                     3rd Qu.:4.00   3rd Qu.:3.0   3rd Qu.: 38931   3rd Qu.: 35494   3rd Qu.: 37500  
##                     Max.   :6.00   Max.   :3.0   Max.   :128420   Max.   :119600   Max.   :124010  
##                                                                                                    
##    cons.city     cons.highway    consumption     engine.size     ncylinder       horsepower      weight    
##  Min.   : 6.5   Min.   : 5.30   Min.   : 5.90   Min.   :1.30   Min.   :-1.00   Min.   :103   Min.   : 923  
##  1st Qu.:11.2   1st Qu.: 8.10   1st Qu.: 9.69   1st Qu.:2.40   1st Qu.: 4.00   1st Qu.:170   1st Qu.:1440  
##  Median :12.4   Median : 9.00   Median :10.70   Median :3.00   Median : 6.00   Median :210   Median :1584  
##  Mean   :12.6   Mean   : 9.37   Mean   :10.98   Mean   :3.23   Mean   : 5.81   Mean   :214   Mean   :1646  
##  3rd Qu.:13.8   3rd Qu.:10.20   3rd Qu.:12.04   3rd Qu.:3.92   3rd Qu.: 6.00   3rd Qu.:243   3rd Qu.:1838  
##  Max.   :23.5   Max.   :19.60   Max.   :21.55   Max.   :6.00   Max.   :12.00   Max.   :493   Max.   :2903  
##                                                                                              NA's   :2     
##     iweight            lweight       wheel.base      length        width          ftype       fdrive   
##  Min.   :0.000345   Min.   :6.83   Min.   :226   Min.   :381   Min.   :163   personal:100   front:102  
##  1st Qu.:0.000544   1st Qu.:7.27   1st Qu.:262   1st Qu.:450   1st Qu.:175   wagon   : 19   rear : 61  
##  Median :0.000632   Median :7.37   Median :272   Median :470   Median :180   SUV     : 41   4x4  : 57  
##  Mean   :0.000635   Mean   :7.38   Mean   :276   Mean   :468   Mean   :181   pickup  : 18              
##  3rd Qu.:0.000694   3rd Qu.:7.52   3rd Qu.:285   3rd Qu.:488   3rd Qu.:185   sport   : 29              
##  Max.   :0.001083   Max.   :7.97   Max.   :366   Max.   :556   Max.   :206   minivan : 13              
##  NA's   :2          NA's   :2                    NA's   :18    NA's   :19                              
##   fdrive2        cons10      fcons10  
##  front:102   Min.   :0.000   No :144  
##  other:118   1st Qu.:0.000   Yes: 76  
##              Median :0.000            
##              Mean   :0.345            
##              3rd Qu.:1.000            
##              Max.   :1.000            
##

Two sample problem

In this section, we review the methods for two-sample problems, i.e. exploring the relationship between continuous variable and binary variable. In particular, we are going to explore the following question:

Does the consumption depend on the drive while distinguishing only two categories: front and other?

Exploration

We start with an exploration phase. The most straightforward visualization tool for this problem is boxplot, the default plot for numeric ~ factor:

plot(consumption ~ fdrive2, data = Cars2004nh)

With some effort you can make this plot more compelling:

COL <- c("yellowgreen", "goldenrod")
names(COL) <- levels(Cars2004nh$fdrive2)

par(mfrow = c(1,1), mar = c(4,4,1.5,0.5))
plot(consumption ~ fdrive2, data = Cars2004nh, col = COL,
     xlab = "Drive of the vehicle", ylab = "Consumption [l/100 km]",
     main = "Consumption vs. Front drive",
     cex.main = 1.0,
     xaxt = "n")
axis(1, at = 1:nlevels(Cars2004nh$fdrive2), labels = c("Front drive", "Other"))
Boxplots of consumption for two categories of drive.

Boxplots of consumption for two categories of drive.

Think about other ways of displaying the two different distributions. Here are the sample characteristics:

with(Cars2004nh, tapply(consumption, fdrive2, summary))
## $front
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    5.90    8.62    9.80    9.87   10.70   15.25 
## 
## $other
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    8.95   10.40   11.45   11.94   13.40   21.55
with(Cars2004nh, tapply(consumption, fdrive2, sd, na.rm = TRUE))
## front other 
## 1.869 2.071
X = subset(Cars2004nh, fdrive2 == "front")$consumption
Y = subset(Cars2004nh, fdrive2 == "other")$consumption

Formal statistical test(s)

For each test refresh the following:

  • What is the underlying model in each test?
  • What are the assumptions?

Standard two-sample t-test (assuming equal variances and normality)

t.test(consumption ~ fdrive2, data = Cars2004nh, var.equal = TRUE)
## 
##  Two Sample t-test
## 
## data:  consumption by fdrive2
## t = -7.7, df = 218, p-value = 4e-13
## alternative hypothesis: true difference in means between group front and group other is not equal to 0
## 95 percent confidence interval:
##  -2.600 -1.545
## sample estimates:
## mean in group front mean in group other 
##               9.866              11.939
mean(X) - mean(Y)
## [1] -2.073

Welch two-sample t-test (not assuming equal variances)

t.test(consumption ~ fdrive2, data = Cars2004nh)  
## 
##  Welch Two Sample t-test
## 
## data:  consumption by fdrive2
## t = -7.8, df = 218, p-value = 3e-13
## alternative hypothesis: true difference in means between group front and group other is not equal to 0
## 95 percent confidence interval:
##  -2.596 -1.549
## sample estimates:
## mean in group front mean in group other 
##               9.866              11.939

Wilcoxon two-sample rank test

wilcox.test(consumption ~ fdrive2, data = Cars2004nh, conf.int = TRUE)
## 
##  Wilcoxon rank sum test with continuity correction
## 
## data:  consumption by fdrive2
## W = 2638, p-value = 7e-13
## alternative hypothesis: true location shift is not equal to 0
## 95 percent confidence interval:
##  -2.40 -1.35
## sample estimates:
## difference in location 
##                   -1.9
median(X) - median(Y)
## [1] -1.65
XYdiff = outer(X, Y, FUN="-")
median(XYdiff)
## [1] -1.9

Kolmogorov-Smirnov test

ks.test(consumption ~ fdrive2, data = Cars2004nh)
## Warning in ks.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): p-value will be approximate in the presence
## of ties
## 
##  Asymptotic two-sample Kolmogorov-Smirnov test
## 
## data:  consumption by fdrive2
## D = 0.44, p-value = 2e-09
## alternative hypothesis: two-sided
par(mfrow = c(1,1), mar = c(4,4,1.5,0.5))
plot(ecdf(X), main="ECDF")
plot(ecdf(Y), col=2, add=TRUE)

More sample problem

What if there are more than two groups?

Does the consumption depend on the drive (distinguishing front/rear/4x4)?

Exploration

Let’s start with boxplots:

plot(consumption ~ fdrive, data = Cars2004nh)

with(Cars2004nh, tapply(consumption, fdrive, summary))
## $front
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    5.90    8.62    9.80    9.87   10.70   15.25 
## 
## $rear
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    8.95   10.40   11.05   11.30   11.80   15.25 
## 
## $`4x4`
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    9.65   10.55   12.25   12.62   14.05   21.55
with(Cars2004nh, tapply(consumption, fdrive, sd, na.rm = TRUE))
## front  rear   4x4 
## 1.869 1.440 2.413

Formal statistical test(s)

For each test refresh the following:

  • What is the underlying model in each test?
  • What are the assumptions?

Standard ANOVA

summary(aov(consumption ~ fdrive, data = Cars2004nh))  
##              Df Sum Sq Mean Sq F value  Pr(>F)    
## fdrive        2    287   143.3    38.7 4.2e-15 ***
## Residuals   217    803     3.7                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

ANOVA without assuming equal variances

oneway.test(consumption ~ fdrive, data = Cars2004nh)   
## 
##  One-way analysis of means (not assuming equal variances)
## 
## data:  consumption and fdrive
## F = 32, num df = 2, denom df = 122, p-value = 5e-12

Kruskal-Wallis Rank Sum Test

kruskal.test(consumption ~ fdrive, data = Cars2004nh)  
## 
##  Kruskal-Wallis rank sum test
## 
## data:  consumption by fdrive
## Kruskal-Wallis chi-squared = 57, df = 2, p-value = 5e-13

Alternatively: the same test statistics all in the standard ANOVA above

summary(m0 <- lm(consumption ~ fdrive, data=Cars2004nh))
## 
## Call:
## lm(formula = consumption ~ fdrive, data = Cars2004nh)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -3.966 -1.354 -0.094  1.127  8.927 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    9.866      0.190   51.80   <2e-16 ***
## fdriverear     1.434      0.311    4.60    7e-06 ***
## fdrive4x4      2.757      0.318    8.67    1e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.92 on 217 degrees of freedom
## Multiple R-squared:  0.263,  Adjusted R-squared:  0.256 
## F-statistic: 38.7 on 2 and 217 DF,  p-value: 4.15e-15
anova(m0)
## Analysis of Variance Table
## 
## Response: consumption
##            Df Sum Sq Mean Sq F value  Pr(>F)    
## fdrive      2    287   143.3    38.7 4.2e-15 ***
## Residuals 217    803     3.7                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
oneway.test(consumption ~ fdrive, data = Cars2004nh, var.equal=TRUE)   
## 
##  One-way analysis of means
## 
## data:  consumption and fdrive
## F = 39, num df = 2, denom df = 217, p-value = 4e-15

Follow these steps to compute the sum of squares and degrees of freedom needed for ANOVA with equal variances:

X = subset(Cars2004nh, fdrive == "front")$consumption
Y = subset(Cars2004nh, fdrive == "rear")$consumption
Z = subset(Cars2004nh, fdrive == "4x4")$consumption
nX = length(X)
nY = length(Y)
nZ = length(Z)
n = nX+nY+nZ
mX = mean(X)
mY = mean(Y)
mZ = mean(Z)
m = mean(c(X,Y,Z))
(SS_between = nX*(m - mX)^2+nY*(m-mY)^2+nZ*(m-mZ)^2)
## [1] 286.6
(SS_error = sum((X - mX)^2)+sum((Y-mY)^2)+sum((Z-mZ)^2))
## [1] 803.1
(SS_total = sum((c(X,Y,Z)-m)^2))
## [1] 1090
(Fstat = SS_between/(3-1) / SS_error*(n - 3))
## [1] 38.72

Conditional mean / Regression

Now we review the basic methods for describing the relationship between two continuous variables.

Exploration

Why are boxplots not reasonable choice? Use scatterplot instead, default for plot between numeric and numeric variable.

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
plot(consumption ~ length, data = Cars2004nh, col = "red3", bg = "orange", pch = 23)

Histograms to inspect the shape of marginal distributions:

### Histograms 
par(mfrow = c(1, 2), mar = c(4,4,0.5,0.5))
hist(Cars2004nh$consumption, prob = TRUE, main = "", col = "seagreen", 
     xlab = "Consumption [l/100 km]")
hist(Cars2004nh$length, prob = TRUE, main = "", col = "seagreen", 
     xlab = "Length [cm]")

with(Cars2004nh, cor(consumption, length))
## [1] NA
with(Cars2004nh, cor(consumption, length, use = "complete.obs"))
## [1] 0.4312
with(Cars2004nh, cor(consumption, length, method = "spearman", use = "complete.obs"))
## [1] 0.4562

Formal statistical test(s)

For each test refresh the following:

  • What is the underlying model in each test?
  • What are the assumptions?

Test based on Pearson’s correlation coefficient

with(Cars2004nh, cor.test(consumption, length))
## 
##  Pearson's product-moment correlation
## 
## data:  consumption and length
## t = 6.8, df = 200, p-value = 1e-10
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.3117 0.5373
## sample estimates:
##    cor 
## 0.4312

Test based on Spearman’s correlation coefficient

with(Cars2004nh, cor.test(consumption, length, method = "spearman"))
## Warning in cor.test.default(consumption, length, method = "spearman"): Cannot compute exact p-value with ties
## 
##  Spearman's rank correlation rho
## 
## data:  consumption and length
## S = 746963, p-value = 9e-12
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##    rho 
## 0.4562

How would you assess the relationship between the dependent variable and independent covariates?

Here are some graphical tools to examine the (non)linearity of the relationship:

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
plot(consumption ~ length, data = Cars2004nh, col = "red3", bg = "orange", pch = 23)
grid <- quantile(Cars2004nh$length, seq(0, 1, length = 20), na.rm = T)

xGrid <- NULL 
yMean <- NULL 
for(i in 2:length(grid)){
  tempData <- Cars2004nh$consumption[Cars2004nh$length >= grid[i - 1] & Cars2004nh$length < grid[i]]
  lines(rep(mean(tempData, na.rm = T), 2) ~ c(grid[i - 1], grid[i]), col = "red")
  points((grid[i - 1] + grid[i])/2, mean(tempData, na.rm = T), 
         pch = 23, bg = "red", cex = 1.2)
  
  xGrid <- c(xGrid, (grid[i - 1] + grid[i])/2)
  yMean <- c(yMean, mean(tempData, na.rm = T))
  
  lines(rep(grid[i], 2), c(min(Cars2004nh$consumption), max(Cars2004nh$consumption)), lty = 3)
}
### which interpolates as: 
lines(yMean ~ xGrid, col = "darkred", lwd = 3)

Less groups without for cycle:

grid <- quantile(Cars2004nh$length, seq(0, 1, length = 11), na.rm = T)
Cars2004nh$flength <- cut(Cars2004nh$length, breaks = grid, right = FALSE)
xGrid <- (grid[-1] + grid[-length(grid)]) / 2
yMean <- tapply(Cars2004nh$consumption, Cars2004nh$flength, mean)

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
plot(consumption ~ length, data = Cars2004nh, type = "n")
segments(x0 = grid, y0 = min(Cars2004nh$consumption), y1 = max(Cars2004nh$consumption), lty = 3)
points(consumption ~ length, data = Cars2004nh, col = "red3", bg = "orange", pch = 23)
segments(x0 = grid[-length(grid)], x1 = grid[-1], y0 = yMean, col = "red")
lines(x = xGrid, y = yMean, col = "darkred", lwd = 3)
points(x = xGrid, y = yMean, pch = 23, bg = "red", cex = 1.2)

Category for every value of length that appears in the data:

par(mfrow = c(1,1), mar = c(4,4,0.5,0.5))
plot(consumption ~ length, data = Cars2004nh, col = "red3", bg = "orange", pch = 23)

grid <- as.numeric(names(table(Cars2004nh$length)))
xGrid <- NULL 
yMean <- NULL 
for (i in 2:(length(grid) - 1)){
  tempData <- Cars2004nh$consumption[Cars2004nh$length >= grid[i - 1] & Cars2004nh$length <= grid[i + 1]]
  points(grid[i], mean(tempData, na.rm = T), pch = 23, bg = "red", cex = 1.2)
  
  xGrid <- c(xGrid, grid[i])
  yMean <- c(yMean, mean(tempData, na.rm = T))
}

### which now interpolates as: 
lines(yMean ~ xGrid, col = "darkred", lwd = 3)

Local trend by lowess curve:

par(mfrow = c(1,3), mar = c(4,4,1.5,0.5))
for(f in c(1/10, 1/5, 1/2)){
  plot(consumption ~ length, data = Cars2004nh, col = "red3", bg = "orange", 
       pch = 23, main = paste0("f = ", f))
  tempData <- Cars2004nh[!is.na(Cars2004nh$length),]
  lines(lowess(tempData$consumption ~ tempData$length, f = f), col = "red3", lwd = 3)
}

How a straight line should be fitted into the data?

Contingency table

We use them to capture the relationship between two categorical variables.

Does the fact that the car is economical (in the U.S. sense, consumption <= 10) depend on the drive (distinguishing only front and other)?

Exploration

The default plot for factor ~ factor is the spineplot that compares the proportions of the first factor within each category of the second factor separately:

par(mfrow = c(1,1), mar = c(4,4,1,2.5))
plot(fcons10 ~ fdrive2, data = Cars2004nh, col = diverge_hcl(2))

Contingency table:

(TAB2 <- with(Cars2004nh, table(fdrive2, fcons10)))
##        fcons10
## fdrive2 No Yes
##   front 46  56
##   other 98  20

Proportion table:

round(prop.table(TAB2, margin = 1), 2)
##        fcons10
## fdrive2   No  Yes
##   front 0.45 0.55
##   other 0.83 0.17

Formal statistical test(s)

For each test refresh the following:

  • What is the underlying model in each test?
  • What are the assumptions?

Chi-square test using continuity correction to be more conservative

chisq.test(TAB2)
## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  TAB2
## X-squared = 33, df = 1, p-value = 8e-09

Chi-square test without continuity correction

(chi = chisq.test(TAB2, correct = FALSE))
## 
##  Pearson's Chi-squared test
## 
## data:  TAB2
## X-squared = 35, df = 1, p-value = 4e-09

How to obtain the test results manually:

# Expected observed counts under the null hypothesis 
print(chi$expected)  
##        fcons10
## fdrive2    No   Yes
##   front 66.76 35.24
##   other 77.24 40.76
n = sum(TAB2)         # number of observations
R = rowSums(TAB2)/n   # observed proportion in rows
C = colSums(TAB2)/n   # observed proportion in columns
n * outer(R, C)       # expected counts under independence
##          No   Yes
## front 66.76 35.24
## other 77.24 40.76
(chi$observed - chi$expected)/sqrt(chi$expected)  # residuals
##        fcons10
## fdrive2     No    Yes
##   front -2.541  3.498
##   other  2.363 -3.252
chi$residuals
##        fcons10
## fdrive2     No    Yes
##   front -2.541  3.498
##   other  2.363 -3.252
(ts<-sum(chi$res^2))    # test statistic
## [1] 34.85
pchisq(ts, df=1, lower.tail=FALSE) # p-value
## [1] 3.559e-09

Test based on difference between proportions. Notice the equivalency to chisq.test. What can you learn from the output?

(x <- TAB2[, "Yes"])
## front other 
##    56    20
(n <- apply(TAB2, 1, sum))
## front other 
##   102   118
prop.test(x, n)
## 
##  2-sample test for equality of proportions with continuity correction
## 
## data:  x out of n
## X-squared = 33, df = 1, p-value = 8e-09
## alternative hypothesis: two.sided
## 95 percent confidence interval:
##  0.2525 0.5066
## sample estimates:
## prop 1 prop 2 
## 0.5490 0.1695
prop.test(x, n, correct = FALSE)
## 
##  2-sample test for equality of proportions without continuity correction
## 
## data:  x out of n
## X-squared = 35, df = 1, p-value = 4e-09
## alternative hypothesis: two.sided
## 95 percent confidence interval:
##  0.2616 0.4975
## sample estimates:
## prop 1 prop 2 
## 0.5490 0.1695

Reconstruction of the test statistic:

p2 = x/n              # proportion estimates in groups
p = sum(x) / sum(n)   # common proportion estimate
(ts <- (diff(p2) / sqrt(p*(1-p)*sum(1/n)))^2) # test statistic
## other 
## 34.85
pchisq(ts, df=1, lower.tail=FALSE)
##     other 
## 3.559e-09

What is tested here?

prop.test(x, n, alternative = "greater")
## 
##  2-sample test for equality of proportions with continuity correction
## 
## data:  x out of n
## X-squared = 33, df = 1, p-value = 4e-09
## alternative hypothesis: greater
## 95 percent confidence interval:
##  0.2714 1.0000
## sample estimates:
## prop 1 prop 2 
## 0.5490 0.1695

Individual work

Two sample problem

  • Verify the assumptions for each two-sample test.
  • For instance, a formal test for equal variances in two groups or visual assessment of normality of the two samples.
  • How would you visualize the sample means and medians which are tested in the two-sample problems?

More sample problem

  • Instead of using three group specific means, consider a reformulation by using the conditional mean given the groups. What is the analogy with the regression formulation?

Contingency tables

  • Try also for categorical variables with more than 2 groups.
  • Explore the following pairs: (fcons10, fdrive) and (fdrive,ftype).

For practical purposes one sided alternative can be tested instead. Which alternative makes more sense? What exactly do we test here? Reconstruct the test statistics and the corresponding p-values.

t.test(consumption ~ fdrive2, data = Cars2004nh, alternative = "less", mu = -2)
## 
##  Welch Two Sample t-test
## 
## data:  consumption by fdrive2
## t = -0.27, df = 218, p-value = 0.4
## alternative hypothesis: true difference in means between group front and group other is less than -2
## 95 percent confidence interval:
##    -Inf -1.634
## sample estimates:
## mean in group front mean in group other 
##               9.866              11.939
t.test(consumption ~ fdrive2, data = Cars2004nh, alternative = "less", mu = -2, 
       var.equal = TRUE)
## 
##  Two Sample t-test
## 
## data:  consumption by fdrive2
## t = -0.27, df = 218, p-value = 0.4
## alternative hypothesis: true difference in means between group front and group other is less than -2
## 95 percent confidence interval:
##    -Inf -1.631
## sample estimates:
## mean in group front mean in group other 
##               9.866              11.939
wilcox.test(consumption ~ fdrive2, data = Cars2004nh, alternative = "less", 
            mu = -2, conf.int = TRUE)
## 
##  Wilcoxon rank sum test with continuity correction
## 
## data:  consumption by fdrive2
## W = 6253, p-value = 0.7
## alternative hypothesis: true location shift is less than -2
## 95 percent confidence interval:
##   -Inf -1.45
## sample estimates:
## difference in location 
##                   -1.9

Does the fact that the car is economical (in the U.S. sense, consumption<=10) depend on the drive (while distinguishing front, rear or 4x4)?

  • What are we testing by the following test?
  • What is the model behind this test?
  • Does the model seem to be good enough to be useful?
  • Which artefacts of the model might be relatively safely ignored and why?
  • What is the conclusion?
chisq.test(TAB3)
## 
##  Pearson's Chi-squared test
## 
## data:  TAB3
## X-squared = 36, df = 2, p-value = 2e-08
print(chisq.test(TAB3)$expected)
##        fcons10
## fdrive     No   Yes
##   front 66.76 35.24
##   rear  39.93 21.07
##   4x4   37.31 19.69
print(chisq.test(TAB3)$residuals)
##        fcons10
## fdrive      No    Yes
##   front -2.541  3.498
##   rear   1.278 -1.759
##   4x4    2.078 -2.860
print(chisq.test(TAB3)$stdres)
##        fcons10
## fdrive      No    Yes
##   front -5.903  5.903
##   rear   2.557 -2.557
##   4x4    4.107 -4.107

Does the drive (front/rear/4x4) depend on the type of the car (personal/wagon/SUV/pickup/sport/minivan)?

  • What are we testing by the following test?
  • What is the model behind this test?
  • Does the model seem to be good enough to be useful?
  • Which artefacts of the model might be relatively safely ignored and why?
  • What is the conclusion?
  • Why did we get the warning message? What can be done about it?
chisq.test(TAB4)
## Warning in chisq.test(TAB4): Chi-squared approximation may be incorrect
## 
##  Pearson's Chi-squared test
## 
## data:  TAB4
## X-squared = 111, df = 10, p-value <2e-16
print(chisq.test(TAB4)$expected)
## Warning in chisq.test(TAB4): Chi-squared approximation may be incorrect
##           fdrive
## ftype       front   rear    4x4
##   personal 46.364 27.727 25.909
##   wagon     8.809  5.268  4.923
##   SUV      19.009 11.368 10.623
##   pickup    8.345  4.991  4.664
##   sport    13.445  8.041  7.514
##   minivan   6.027  3.605  3.368