Wilcoxon Rank Sum

Introduction

• We last discussed assumptions on t-tests

  • Dependent / paired t-test: normality

  • Independent two-sample t-test: normality and variance

• If we break the normality assumption, we must look to nonparametric methods.

Introduction

• The t-tests we have already learned are considered parametric methods.

  • There is a distributional assumption on the test.

• Nonparametric methods do not have distributional assumptions.

  • We typically transform the data to their ranks and then perform calculations.

• Why don’t we always use nonparametric methods?

  • They are often less efficient: a larger sample size is required to achieve the same probability of a Type I error.

  • They discard useful information :(

Ranking Data

• In the nonparametric tests we will be learning, the data will be ranked.

• Let us first consider a simple example, x: \ 1, 7, 10, 2, 6, 8

• Our first step is to reorder the data:x: \ 1, 2, 6, 7, 8, 10

• Then, we replace with the ranks:R: \ 1, 2, 3, 4, 5, 6

Ranking Data

• What if all data values are not unique?

  • We will assign the average ranks.

• For example, x: \ 9, 8, 8, 0, 3, 4, 4, 8

• Let’s reorder:x: \ 0, 3, 4, 4, 8, 8, 8, 9

• Rank ignoring ties:R: \ 1, 2, 3, 4, 5, 6, 7, 8

• Now, the final rank:R: \ 1, 2, 3.5, 3.5, 6, 6, 6, 8

Wilcoxon Rank Sum

• We will use the wilcox.test() function to perform the test,

wilcox.test(continuous_variable ~ grouping_variable,
            data = dataset_name,
            alternative = "alternative",
            mu = hypothesized_value,
            exact = FALSE)

• Like before, R will use the group that is “first” in the grouping variable.

  • “First” is in terms of numeric or alphabetical.

Wilcoxon Rank Sum

• When exposed to an infection, a person typically develops antibodies. The extent to which the antibodies respond can be measured by looking at a person’s titer, which is a measure of the number of antibodies present. The higher the titer is, the more antibodies that are present.

• The following data represent the titers of 11 ill people and 11 healthy people exposed to the tularemia virus in Vermont.

• Is the level of titer in the ill group greater than the level of titer in the healthy group? Use the \alpha = 0.10 level of significance.

titer_levels <- tibble(level = c(640, 160, 1280, 320, 80, 640, 640, 160, 1280, 640, 160, 
                                  10, 320, 160, 160, 320, 320, 10, 320, 320, 80, 640),
                       group = c(rep("ill",11), rep("healthy",11)))

• Recall the R syntax,

wilcox.test(continuous_variable ~ grouping_variable,
            data = dataset_name,
            alternative = "alternative",
            mu = hypothesized_value,
            exact = FALSE)

Wilcoxon Rank Sum

• Is the level of titer in the ill group greater than the level of titer in the healthy group?

wilcox.test(level ~ group, 
            data = titer_levels,
            alternative = "less",
            exact = FALSE)

    Wilcoxon rank sum test with continuity correction

data:  level by group
W = 35, p-value = 0.04657
alternative hypothesis: true location shift is less than 0

Wrap Up

• Today we reviewed the Wilcoxon rank sum test.
  • Nonparametric alternative to the two-sample t-test.
• Next lecture:
  • Wilcoxon signed rank