# Chapter 8 When assumptions are not met: non-parametric alternatives

## 8.1 Introduction

Linear models do not apply to every data set. As discussed in Chapter 7, sometimes the assumptions of linear models are not met. One of the assumptions is linearity or additivity. Additivity requires that one unit change in variable $$X$$ leads to the same amount of change in $$Y$$, no matter what value $$X$$ has. For bivariate relationships this leads to a linear shape. But sometimes you can only expect that $$Y$$ will change in the same direction, but you don’t believe that this amount is the same for all values of $$X$$. This is the case for example with an ordinal dependent variable. Suppose we wish to model the relationship between the age of a mother and an aggression score in her 7-year-old child. Suppose aggression is measured on a three-point ordinal scale: ‘not aggressive,’ ‘sometimes aggressive,’ ‘often aggressive.’ Since we do not know the quantitative differences between these three levels, there are many graphs we could draw for a given data set.

Suppose we have the data set given in Table 8.1. If we want to make a scatter plot, we could arbitrarily choose the values 1, 2, and 3 for the three categories, respectively. We would then get the plot in Figure 8.1. But since the aggression data are ordinal, we could also choose the arbitrary numeric values 0, 2, and 3, which would yield the plot in Figure 8.2.

Table 8.1: Aggression in children and age of the mother.
AgeMother Aggression
32 Sometimes aggressive
31 Often aggressive
32 Often aggressive
30 Not aggressive
31 Sometimes aggressive
30 Sometimes aggressive
31 Not aggressive
31 Often aggressive
31 Not aggressive
30 Sometimes aggressive
32 Often aggressive
32 Often aggressive
31 Sometimes aggressive
30 Sometimes aggressive
31 Not aggressive

As you can see from the least squares regression lines in Figures 8.1 and 8.2, when we change the way in which we code the ordinal variable into a numeric one, we also see the best fitting regression line changing. This does not mean though, that ordinal data cannot be modelled linearly. Look at the example data in Table 8.2 where aggression is measured with a 7-point scale. Plotting these data in Figure 8.3 using the values 1 through 7, we see a nice linear relationship. So even when the values 1 thru 7 are arbitrarily chosen, a linear model can be a good model for a given data set with one or more ordinal variables. Whether the interpretation makes sense is however up to the researcher.

Table 8.2: Aggression in children on a 7-point Likert scale and age of the mother.
AgeMother Aggression
35 6
32 4
35 6
36 5
33 3
30 1
32 4
32 2
34 4
30 2
32 3
31 2
32 3
31 3
38 7

So with ordinal data, always check that your data indeed conform to a linear model, but realise at the same time that you’re assuming a quantitative and additive relationship between the variables that may or may not make sense. If you believe that a quantitative analysis is meaningless then you may consider a non-parametric analysis that we discuss in this chapter.

Another instance where we favour a non-parametric analysis over a linear model one, is when the assumption of normally distributed residuals is not tenable. For instance, look again at Figure 8.1 where we regressed aggression in the child on the age of its mother. Figure 8.4 shows a histogram of the residuals. Because of the limited number of possible values in the dependent variable (1, 2 and 3), the number of possible values for the residuals is also very restricted, which leads to a very discrete distribution. The histogram looks therefore far removed from a continuous symmetric, bell-shaped distribution, which is a violation of the normality assumption.

Every time we see a distribution of residuals that is either very skew, or has very few different values, we should consider a non-parametric analysis. Note that the shape of the distribution of the residuals is directly related to what scale values we choose for the ordinal categories. By changing the values we change the regression line, and that directly affects the relative sizes of the residuals.

First, we will discuss a non-parametric alternative for two numeric variables. We will start with Spearman’s $$\rho$$ (rho, pronouned ‘row’), also called Spearman’s rank-order correlation coefficient $$r_s$$. Next we will discuss an alternative to $$r_s$$, Kendall’s $$\tau$$ (tau, pronounced ‘taw’). After that, we will discuss the combination of numeric and categorical variables, when comparing groups.

## 8.2 Spearman’s $$\rho$$ (rho)

Suppose we have 10 students and we ask their teachers to rate their performance. One teacher rates them on geography and the other teacher rates them on history. We first let them score the performance on an ordinal scale with the following labels: 5 = not sufficient, 6 = sufficient, 7 = reasonably good, 8 = good, 9 = very good, 10 = exceptionally good. We call this an ordinal scale because we can’t really quantify the difference between ‘reasonably good’ and ‘good,’ for instance, but we do know that ‘sufficient’ is better than ‘not sufficient,’ ‘reasonably good’ is better than ‘sufficient,’ ‘good’ is better than ‘reasonably good,’ etcetera. The data are in Table 8.3.

Table 8.3: Student performance ratings on geography and history.
student geography history
1 5 6
2 6 5
3 10 7
4 9 8
5 8 9

Because the data are ordinal, it feels a bit strange to put these data into a quantitative linear model. What is sometimes done is to rank the data: assign the lowest observed value as rank 1, the next lowest observed value as rank 2, etcetera. If we do that for the student data for each teacher separately we get the ranks in Table 8.4.

Table 8.4: Student performance ratings and their rankings on geography and history.
student geography history rank.geography rank.history
1 5 6 1 2
2 6 5 2 1
3 10 7 5 3
4 9 8 4 4
5 8 9 3 5

We see that student 3 is ranked highest on geography (rank of 5) and that student 5 is the brightest student in history (rank 5). And although the original ratings for student 4 are different for geography and history, the ranks are the same (both rank 4). We acknowledge here that for a given teacher, a rating of 9 is better than a rating of 8, but we can’t say anything about the difference in performance between a rating of 9 on geography and a rating of 8 on history (maybe history is much easier than geography?). These rankings only tell us something about the relative differences within a teacher.

Still, we want to say something about the relationship between these two ratings. Generally we see that a low rank on geography is associated with a low rank on history (students 1 and 2). Figure 8.5 plots the ranks and plots the least squares regression line.

What we could do is simply taking these rankings and compute the correlation, that is, Pearson’s correlation (see Chapter 4). If we do that, we get a correlation of 0.5. Because this Pearson correlation is based on ranks, we call it Spearman’s correlation.

But you may rightly ask yourself, why do we call it Spearman’s correlation when it is simply Pearson’s correlation? That’s due to some history. Back in the day of clever mathematicians like Pearson and Spearman, every computation had to be done by hand. The more complex the formula for a statistic, the more time consuming it is to compute. The regular formula for a Pearson correlation between two numeric variables is, as you may recall from Chapter 4,

$\rho_{XY}= \frac{\sigma_{XY}}{\sigma_X\sigma_Y}= \frac{\sum(X_i-\bar{X})(Y_i-\bar{Y})}{\sqrt{\sum(X_i-\bar{X})^2\sum(X_i-\bar{X})^2}}$

Spearman was the one that spotted that if these $$X$$ and $$Y$$-values are ranks (that is, the lowest value is 1 and the highest value is equal to $$n$$, the sample size), you can use a simpler formula to compute the Pearson correlation. If two variables $$X$$ and $$Y$$ are ranks, the correlation formula can be simplified to

$$$r_s = 1 - \frac{6 \sum (X_i - Y_i)^2 }{n^3-n} \tag{8.1}$$$

This is called the Spearman rank-order correlation coefficient $$r_s$$, or Spearman’s $$\rho$$ (the Greek letter ‘rho’).

## 8.3 Spearman’s rho in R

When we let R compute $$r_s$$ for us, it automatically ranks the data for us. Let’s look at the mpg data on 234 cars from the ggplot2 package again. Suppose we want to treat the variables cyl (the number of cylinders) and year (year of the model) as ordinal variables, and we want to look whether the ranking on the cyl variable is related to the ranking on the year variable. We use the function rcorr() from the Hmisc package to compute Spearman’s rho:

library(Hmisc)
rcorr(mpg$cyl, mpg$year, type = "spearman")
##      x    y
## x 1.00 0.12
## y 0.12 1.00
##
## n= 234
##
##
## P
##   x      y
## x        0.0685
## y 0.0685

Since the Spearman correlation is the same as the Pearson correlation on ranked variables, we could alternatively do

mpg <- mpg %>%
mutate(cyl_ranked = rank(cyl),
year_ranked = rank(year))
rcorr(mpg$cyl_ranked, mpg$year_ranked, type = "pearson")
##      x    y
## x 1.00 0.12
## y 0.12 1.00
##
## n= 234
##
##
## P
##   x      y
## x        0.0685
## y 0.0685

Note that the results are identical.

In the output you will see a correlation matrix very similar to the one for a Pearson correlation. Spearman’s rho is equal to 0.12. You will also see whether the correlation is significantly different from 0, indicated by a $$p$$-value. If the $$p$$-value is very small, you may conclude that on the basis of these data, the correlation in the population is not equal to 0, ergo, in the population there is a relationship between the year (ranked) a car was produced and the number of cylinders (ranked).

"We tested the null-hypothesis that the number of cylinders is not related to the year of the model. For that we used a Spearman correlation that quantifies the extent to which the ranking of the cars in terms of cylinders is related to the ranking of the cars in terms of year. Results showed that the null-hypothesis could not be rejected at an $$\alpha$$ of 0.05, $$\rho(n = 234) = .12, p = .07$$."

Extra material

Ranking and linear models

A Spearman correlation is the same as a Pearson correlation, except that the data are ranked before the calculation of the correlation. Because the Pearson correlation is closely tied to the linear model, we can also use a linear model to get a Spearman correlation.

To see that, let’s have R compute ranks for two variables in the R built-in dataset USArrests which contains statistics about violent crime rates in 1973 by US state. We focus on the variables Assault (number of assault arrests per 100,000) and Rape (number of rape arrests per 100,000). In the figure below we see the relationship between these two variables (before ranking).

USArrests %>%
ggplot(aes(x = Rape, y = Assault)) +
geom_point()

The plot suggests there is a positive correlation between the number of assault arrests and the number of rape arrests. You may spot it from the above plot: there seems to be a violation of the homogeneity assumption. It becomes clearer when we fit a linear model and look at a residuals plot.

model <- USArrests %>%
lm(Assault ~ Rape, data = .)

USArrests %>%
ggplot(aes(x = pred, y = resid)) +
geom_point()

We see there is more variance in the residuals for lower predicted values than for higher predicted values. There also seems to be non-normal skewness in the distribution of the residuals (relatively large positive residuals, relatively small negative residuals).

USArrests %>%
ggplot(aes(x = resid)) +
geom_histogram()

This violation of the assumptions of a linear model has consequences for the appropriateness of using the Pearson correlation and interpreting the $$p$$-value.

Let’s see what we can gain if we transform the values into ranks. Let’s zoom in on some of the states with the highest number of assault arrests.

USArrests <- USArrests %>%
mutate(Assault_ranked = rank(Assault),
Rape_ranked = rank(Rape))
USArrests %>%
select(Rape, Rape_ranked, Assault, Assault_ranked) %>%
arrange(Assault) %>%
tail(5)
##                Rape Rape_ranked Assault Assault_ranked
## New Mexico     32.1          45     285             46
## Arizona        31.0          43     294             47
## Maryland       27.8          40     300             48
## Florida        31.9          44     335             49
## North Carolina 16.1          16     337             50

We see that North Carolina is ranked 50th on Assault, which means it has the highest number of assault arrests per capita of the 50 American states. It is however ranked 16th on Rape, which means there are 34 states with more rape arrests per capita.

With ranks of the variables, we get the following plot:

USArrests %>%
ggplot(aes(x = Rape_ranked, y = Assault_ranked)) +
geom_point()

This looks better, which we can also see with a residuals plot

model2 <- USArrests %>%
lm(Assault_ranked ~ Rape_ranked, data = .)

USArrests %>%
ggplot(aes(x = pred, y = resid)) +
geom_point()

and a histogram of the residuals.

USArrests %>%
ggplot(aes(x = resid)) +
geom_histogram()

This is a more symmetrical histogram than when using the original values. So we see that sometimes when we analyse ranks instead of the original values, we can have a more appropriate linear model.

In terms of interpretation, there is of course a difference. In a linear model with the original variables, the interpretation of the slope coefficient is the increase in number of assault arrests per capita with a change of 1 in the number of rape arrests (i.e., moving from a state with $$x$$ rape arrests to a state with $$x+1$$ rape arrests, the predicted number of assault arrests increases by the slope). When analysing ranks, the slope coefficient represents the increase in ranks in the numbers of assault arrests, with a change of 1 rank in the number of rape arrests (i.e., when we move from a state that is ranked $$x$$ for rape arrests to a state that is ranked $$x+1$$, the expected increase in the assault rank is equal to the slope).

Because the two correlation coefficients are closely related, and because Pearson’s correlation is closely tied to linear models (see Chapter 4), we can also relate Spearman’s correlation to the linear model.

Remember from Chapter 4 that Pearson’s correlation is equivalent to the slope coefficient of a linear model if both the dependent and the independent variable are standardised, that is, if they are scaled in such a way that the standard deviations are 1. Thus, if we want to have the Spearman correlation, we can do that also by simply running a linear model. We not only have to standardise the data, but before we do, also turn the values into ranks:

USArrests %>%
mutate(Rape_transformed = rank(Rape) %>% scale(),  # first rank, then scale
Assault_transformed = rank(Assault) %>% scale()) %>%
lm(Assault_transformed ~ Rape_transformed, data = .) %>%
tidy()
## # A tibble: 2 × 5
##   term             estimate std.error statistic       p.value
##   <chr>               <dbl>     <dbl>     <dbl>         <dbl>
## 1 (Intercept)      3.76e-17     0.100  3.76e-16 1
## 2 Rape_transformed 7.14e- 1     0.101  7.07e+ 0 0.00000000569

We see a slope coefficient (Spearman correlation) of 0.714, with an associated $$p$$-value of 0.00000000569.

We get the same results when using rcorr() on the original variables:

rcorr(USArrests$Assault, USArrests$Rape, type = "spearman") 
##      x    y
## x 1.00 0.71
## y 0.71 1.00
##
## n= 50
##
##
## P
##   x  y
## x     0
## y  0
#zooming in on the correlations, more decimals
rcorr(USArrests$Assault, USArrests$Rape, type = "spearman")$r ## x y ## x 1.0000000 0.7143681 ## y 0.7143681 1.0000000 # zooming in on the p-values, more decimals rcorr(USArrests$Assault, USArrests$Rape, type = "spearman")$P
##                   x                 y
## x                NA 0.000000005688701
## y 0.000000005688701                NA

In summary, Spearman’s correlation is the same as Pearson’s correlation except that it is computed using the ranks of the values. It can be determined by ranking the data and subsequently determine the Pearson correlation, but also by applying a linear model on the standardised ranks. Ranking can be helpful when the assumptions of the linear model are not tenable with the original values.

## 8.4 Kendall’s rank-order correlation coefficient $$\tau$$

If you want to study the relationship between two variables, of which at least one is ordinal, you can either use Spearman’s $$r_s$$ or Kendall’s $$\tau$$ (tau, pronounced ‘taw’ as in ‘law’). However, if you have three variables, and you want to know whether there is a relationship between variables $$A$$ and $$B$$, over and above the effect of variable $$C$$, you can use an extension of Kendall’s $$\tau$$. Note that this is very similar to the idea of multiple regression: a coefficient for variable $$X_1$$ in multiple regression with two predictors is the effect of $$X_1$$ on $$Y$$ over and above the effect of $$X_2$$ on $$Y$$. The logic of Kendall’s $$\tau$$ is also based on rank orderings, but it involves a different computation. Let’s look at the student data again with the teachers’ rankings of ten students on two subjects in Table 8.5.

Table 8.5: Student rankings on geography and history, now ordered according to the ranking for geography.
student rank.geography rank.history
1 1 2
2 2 1
5 3 5
4 4 4
3 5 3

From this table we see that the history teacher disagrees with the geography teacher that student 8 is brighter than student 10. She also disagrees with her colleague that student 1 is brighter than student 2. If we do this for all possible pairs of students, we can count the number of times that they agree and we can count the number of times they disagree. The total number of possible pairs is equal to $$10 \choose 2$$ $$= n(n-1)/2 = 90/2= 45$$ (see Chapter 3). This is a rather tedious job to do, but it can be made simpler if we reshuffle the data a bit. We put the students in a new order, such that the brightest student in geography comes first, and the dullest last. This also changes the order in the variable history. We then get the data in Table 8.5. We see that the geography teacher believes that student 9 outperforms all 9 other students. On this, the history teacher agrees, as she also ranks student 9 first. This gives us 9 agreements. Moving down the list, we see that the geography teacher believes student 8 outperforms 8 other students. However, we see that the history teacher believes student 8 only outperforms 7 other students. This results in 7 agreements and 1 disagreement. So now in total we have $$9+7=16$$ agreements and 1 disagreements. If we go down the whole list in the same way, we will find that there are in total 41 agreements and 4 disagreements.

Table 8.6: Student rankings on geography and history, now ordered according to the ranking for geography, with number of agreements.
student rank.geography rank.history number
1 1 2 9
2 2 1 7
5 3 5 7
4 4 4 5
3 5 3 5
1 1 2 3
2 2 1 2
5 3 5 2
4 4 4 1
3 5 3 0

The computation is rather tedious. There is a trick to do it faster. Now focus on Table 8.5 but start in the column of the history teacher. Start at the top row and count the number of rows beneath it with a rank higher than the rank in the first row. The rank in the first row is 1, and all other ranks beneath it are higher, so the number of ranks is 9. We plug that value in the last column in Table 8.6. Next we move to row 2. The rank is 3. We count the number of rows below row 2 with a rank higher than 3. Rank 2 is lower, so we are left with 7 rows and we again plug 7 in the last column of Table 8.6. Then we move on to row 3, with rank 2. There are 7 rows left, and all of them have a higher rank. So the number is 7. Then we move on to row 4. It has rank 5. Of the 6 rows below it, only 5 have a higher rank. Next, row 5 shows rank 4. Of the 5 rows below it, all 5 show a higher rank. Row 6 shows rank 7. Of the 4 rows below it, only 3 show a higher rank. Row 7 shows rank 8. Of the three rows below it, only 2 show a higher rank. Row 8 shows rank 6. Both rows below it show a higher rank. And row 9 shows rank 9, and the row below it shows a higher rank so that is 1. Finally, when we add up the values in the last column in Table 8.6, we find 41. This is the number of agreements. The number of disagreements can be found by reasoning that the total number of pairs equals the number of pairs that can be formed using a total number of 10 objects: $$10 \choose 2$$ = $$10(10-1)/2 = 45$$. In this case we have 45 possible pairs. Of these there are 41 agreements, so there must be $$45-41=4$$ disagreements. We can then fill in the formula to compute Kendall’s $$\tau$$:

\begin{aligned} \tau = \frac { agreements - disagreements }{total number of pairs} = \frac{37} {45} = 0.82 \end{aligned}

This $$\tau$$-statistic varies between -1 and 1 and can therefore be seen as a non-parametric analogue of a Pearson correlation. Here, the teachers more often agree than disagree, and therefore the correlation is positive. A negative correlation means that the teachers more often disagree than agree on the relative brightness of their students.

As said, the advantage of Kendall’s $$\tau$$ over Spearman’s $$r_s$$ is that Kendall’s $$\tau$$ can be extended to cover the case that you wish to establish the strength of the relationships of two variables $$A$$ and $$B$$, over and above the relationship with $$C$$. The next section shows how to do that in R.

## 8.5 Kendall’s $$\tau$$ in R

Let’s again use the mpg data on 234 cars. We can compute Kendall’s $$\tau$$ for the variables cyl and year using the Kendall package:

library(Kendall)
Kendall(mpg$cyl, mpg$year)
## tau = 0.112, 2-sided pvalue =0.068798

As said, Kendall’s $$\tau$$ can also be used if you want to control for a third variable (or even more variables). This can be done with the ppcor package. Because this package has its own function select(), you need to be explicit about which function from which package you want to use. Here you want to use the select() function from the dplyr package (part of the tidyverse suite of packages).

library(ppcor)
mpg %>%
dplyr::select(cyl, year, cty) %>%
pcor(method = "kendall") 
## $estimate ## cyl year cty ## cyl 1.0000000 0.1642373 -0.7599993 ## year 0.1642373 1.0000000 0.1210952 ## cty -0.7599993 0.1210952 1.0000000 ## ##$p.value
##                                                                             cyl
## cyl  0.000000000000000000000000000000000000000000000000000000000000000000000000
## year 0.000189654827628346264994235736978112072392832487821578979492187500000000
## cty  0.000000000000000000000000000000000000000000000000000000000000000000770657
##              year
## cyl  0.0001896548
## year 0.0000000000
## cty  0.0059236967
##                                                                             cty
## cyl  0.000000000000000000000000000000000000000000000000000000000000000000770657
## year 0.005923696652933938683327497187747212592512369155883789062500000000000000
## cty  0.000000000000000000000000000000000000000000000000000000000000000000000000
##
## $statistic ## cyl year cty ## cyl 0.000000 3.732412 -17.271535 ## year 3.732412 0.000000 2.751975 ## cty -17.271535 2.751975 0.000000 ## ##$n
## [1] 234
##
## $gp ## [1] 1 ## ##$method
## [1] "kendall"

In the output, we see that the Kendall correlation between cyl and year, controlled for cty, equals 0.16, with an associated $$p$$-value of 0.00019.

We can report this in the following way:

"The null-hypothesis was tested that there is no correlation between cyl and year, when one controls for cty. This was tested with a Kendall correlation coefficient. We found that the hypothesis of a Kendall correlation of 0 in the population of cars (controlling for cty) could be rejected, $$\tau(n = 234) = .16, p < .001$$."

## 8.6 Kruskal-Wallis test for group comparisons

Now that we have discussed relationships between ordinal and numeric variables, let’s have a look at the case where we also have categorical variables.

Suppose we have three groups of students that go on a field trip together: mathematicians, psychologists and engineers. Each can pick a rain coat, with five possible sizes: ‘extra small,’ ‘small,’ ‘medium,’ ‘large’ or ‘extra large.’ We want to know if preferred size is different in the three populations, so that teachers can be better prepared in the future. Now we have information about size, but this knowledge is not numeric: we do not know the difference in size between ‘medium’ and ‘large,’ only that ‘large’ is larger than ‘medium.’ We have ordinal data, so computing a mean is impossible here. Even if we would assign values like 1= ‘extra small,’ 2=‘small,’ 3= ‘medium,’ etcetera, the mean would be rather meaningless as these values are arbitrary. So instead of focussing on means, we can focus on medians: the middle values. For instance, the median value for our sample of mathematicians could be ‘medium,’ for our sample of psychologists ‘small,’ and for our sample of engineers ‘large.’ Our question might then be whether the median values in the three populations are really different.

This can be assessed using the Kruskal-Wallis test. Similar to Spearman’s $$r_s$$ and Kendall’s $$\tau$$, the data are transformed into ranks. This is done for all data at once, so for all students together.

For example, if we had the data in Table 8.7, we could transform the variable size into ranks, from smallest to largest. Student 1 has size ‘extra small’ so he or she gets the rank 1. Next, both student 4 and student 6 have size ‘small,’ so they should get ranks 2 and 3. However, because there is no reason to prefer one student over the other, we give them both the mean of ranks 2 and 3, so they both get the rank 2.5. Next in line is student 3 with size ‘medium’ and (s)he gets rank 4. Next in line is student 5 with size ‘large’ (rank 5) and last in line is student 2 with size ‘extra large’ (rank 6).

Table 8.7: Field trip data.
student group size rank
1 math extra small 1.0
2 math extra large 6.0
3 psych medium 4.0
4 psych small 2.5
5 engineer large 5.0
6 math small 2.5

Next, we could compute the average rank per group. The group with the smallest sizes would have the lowest average rank, etcetera. Under the null-hypothesis, if the distribution of size were the same in all three groups, the average ranks would be about the same.

$$H_0$$: All groups have the same average rank.

If the average rank is very different across groups, this is an indication that size is not distributed equally among the three groups. In order to have a proper statistical test for this null-hypothesis, a rather complex formula is used to compute the so-called $$KW$$-statistic, see Castellan & Siegel (1988), that you don’t need to know. The distribution of this $$KW$$-statistic under the null-hypothesis is known, so we know what extreme values are, and consequently can compute $$p$$-values. This computation can be done in R.

## 8.7 Kruskal-Wallis test in R

Let’s look at the mpg data again. It contains data on cars from 7 different types. Suppose we want to know whether the different types show differences in the distribution of city miles per gallon. Figure 8.6 shows a box plot: it seems that indeed the distributions of cty are very different for different car types. But these differences could be due to sampling: maybe by accident, the pick-up trucks in this sample happened to have a relatively low mileage, and that the differences in the population of all cars are non-existing. To test this hypothesis, we can make it more specific and we can use a Kruskal-Wallis test to test the null-hypothesis that the average rank of cty is the same in all groups.

We run the following R code:

mpg %>%
kruskal.test(cty ~ class, data = .)
##
##  Kruskal-Wallis rank sum test
##
## data:  cty by class
## Kruskal-Wallis chi-squared = 149.53, df = 6, p-value <
## 0.00000000000000022
mpg$cty %>% length() ## [1] 234 mpg$class %>% length()
## [1] 234

The output allows us to report the following:

"The null-hypothesis that city miles per gallon is distributed equally for all types of cars was tested using a Kruskal-Wallis test with an $$\alpha$$ of 0.05. Results showed that the null-hypothesis could be rejected, $$\chi^2(6, N = 234) = 149.53$$, $$p < .001$$."

## 8.8 Take-away points

• When a distribution of residuals looks very far removed from a normal distribution, consider using a nonparametric method of analysis.

### Key concepts

• Pearson’s $$\rho$$
• Spearman’s rank-order correlation coefficient $$r_s$$
• Kendall’s $$\tau$$
• Kruskall-Wallis test