Mann-Whitney-Wilcoxon.knit

Mann-Whitney U & Wilcoxon signed-rank tests

When we have an ordinal dependent variable or a continuous variable that does not meet the assumptions of the t-test, we can look to the Mann-Whitney U test (also known as the Wilcoxon rank sum test). If the groups are paired then the Wilcoxon signed-rank test replaces the paired t-test as a non-parametric alternative. This means that while these tests do make the assumption that the two samples come from similarly shaped (symmetrical) distributions, those distributions do not need to be normally distributed (parametric).

Before committing to the Mann-Whitney U or the Wilcoxon signed-rank test we should determine if the following assumptions are valid:

The dependent variable is ordinal or continuous and the independent variable has two groups
The observations are independent and randomly sampled
Both groups have symmetrical distributions

We can then test the null and alternative hypotheses:

H₀: The probability that a randomly drawn observation from one population will be greater than a randomly drawn observation from the second observation is 50%. Or, the two samples belong to the same population with the same median. M₁ = M₂
H_A: The two samples have different medians and thus do not come from the same population. M₁ ≠ M₂

Fitting the Mann-Whitney U test

In this example we will use the mtcars data set that is provided in base R. The data set includes 10 aspects of automobile design and performance for 32 automobile models built between 1973 and 1974. We will specifically be interested in whether there are differences in miles per gallon (mpg) between the two types of transmission (am), automatic and manual. Specifying these two columns specifically in the summary() function we can get an idea of how each look like.

summary(mtcars[, c("am", "mpg")])

##        am              mpg       
##  Min.   :0.0000   Min.   :10.40  
##  1st Qu.:0.0000   1st Qu.:15.43  
##  Median :0.0000   Median :19.20  
##  Mean   :0.4062   Mean   :20.09  
##  3rd Qu.:1.0000   3rd Qu.:22.80  
##  Max.   :1.0000   Max.   :33.90

Because am is coded as a number rather than a factor the summary() function calculates the mean and quantiles, which are not very useful to us. For the mpg variable we do see that it appears to be continuous and there are no missing data or miscoded values. Plotting the data as below, we might hypothesize that there is a difference between automatic and manual transmission and the miles per gallon for the automobiles.

To statistically determine if there are differences between these two types of transmissions we will use the Mann-Whitney U test. In R, we can perform a Mann-Whitney test using the wilcox.test() function with the continuous variable (mpg) on the left and the categorical variable (mpg) on the right of the ~. We will also set the conf.int option to TRUE so that the 95% confidence intervale and differences in location are calculated. Although am is coded as a numerical variable, because it only consists of 0’s and 1’s the wilcox.test() function will recognize the two groups and we do not need to change the variable type to a factor.

wilcox.test(mpg ~ am, data = mtcars, conf.int = TRUE)

## Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot compute exact p-value
## with ties

## Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot compute exact confidence
## intervals with ties

## 
##  Wilcoxon rank sum test with continuity correction
## 
## data:  mpg by am
## W = 42, p-value = 0.001871
## alternative hypothesis: true location shift is not equal to 0
## 95 percent confidence interval:
##  -11.699942  -2.900042
## sample estimates:
## difference in location 
##              -6.799963

Note that we get warning messages with the output from the test This is because we have some values with tied rankings and the sample sizes lower than 50, which causes the exact p-value and confidence intervals to be unable to be calculated. There are different ways we can avoid this issue, such as increasing the sample size to 50 or greater, but in many cases that solution would not be feasible. Instead, the estimated p-value based on a normal approximation calculated by the wilcox.test() is good enough, and we can conclude that automobiles with automatic transmissions have a lower miles per gallon than those with manual transmissions. The median difference between automatic and manual miles per gallon is somewhere between -2.9 and -11.7 mpg with 95% confidence, and the median of the difference between automatic and manual transmissions is -6.8 mpg (note that this is not the difference in medians).

Fitting the Wilcoxon signed-rank test

In the case that our data is paired the Mann-Whitney U test is no longer viable. Instead, we should apply the Wilcoxon signed-rank test which in R uses the same wilcox.test() function but requires us to set the paired option to TRUE.

summary(sleep)

##      extra        group        ID   
##  Min.   :-1.600   1:10   1      :2  
##  1st Qu.:-0.025   2:10   2      :2  
##  Median : 0.950          3      :2  
##  Mean   : 1.540          4      :2  
##  3rd Qu.: 3.400          5      :2  
##  Max.   : 5.500          6      :2  
##                          (Other):8

wilcox.test(extra ~ group, sleep, paired = TRUE, conf.int = TRUE)

## Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot compute exact p-value
## with ties

## Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot compute exact confidence
## interval with ties

## Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot compute exact p-value
## with zeroes

## Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot compute exact confidence
## interval with zeroes

## 
##  Wilcoxon signed rank test with continuity correction
## 
## data:  extra by group
## V = 0, p-value = 0.009091
## alternative hypothesis: true location shift is not equal to 0
## 95 percent confidence interval:
##  -2.949921 -1.050018
## sample estimates:
## (pseudo)median 
##      -1.400031

Again we see that there are warning messages, this time both that the samples have ties while the sample sizes are <50 and that there are 0’s in our data set. Many times this is unavoidable without increases our sample size, but again the estimated p-value is suitable for this analysis.

With a p-value less than 0.05 (p-value = 0.009091) we can conclude that the medians are not equal and that Drug #2 has a greater effect on the change in hours of sleep compared to Drug #1. The median difference in the change of hours of sleep for Drug #1 compared to Drug #2 is somewhere between -2.95 and -1.05 with 95% confidence.

Full code block

# Print summary statistics for the am and mpg variables from the mtcars data set
summary(mtcars[, c("am", "mpg")])

# Perform a Mann-Whitney test with miles per gallon as the dependent and transmission as the
# independent variable
wilcox.test(mpg ~ am, data = mtcars, conf.int = TRUE)

# Print summary statistics for the sleep data set
summary(sleep)

# Perform a Wilcoxon signed-rank test with the change in hours of sleep as the dependent and the
# type of trug as the independent variable
wilcox.test(extra ~ group, sleep, paired = TRUE, conf.int = TRUE)