Mann-Whitney U test

A/B Testing in R

Lauryn Burleigh

Data Scientist

Mann-Whitney U

Time to eat Cheese versus Pepperoni pizza
Not normal distribution
Non-parametric
- Distribution shape not assumed
- Mann-Whitney U test

Two left skewed histograms. Pepperoni in pink with a mean of 8 and Cheese in blue with a mean of 6.2.

Assumptions

Same distribution shapes
Assesses difference in medians
Normal distribution: mean = median
Non-normal distribution: median more appropriate
Extra assumptions: more powerful test

Null hypothesis: no difference in the median time to eat cheese and pepperoni pizza

library(ggplot2)
ggplot(pizza, aes(x = Time, 
                  fill = Topping)) +
       geom_histogram() + 
       facet_grid(Topping~.)

Two left skewed histograms. Pepperoni in pink with a mean of 8 and Cheese in blue with a mean of 6.2.

Sample Size

library(pwr)
pwr.2p2n.test(h = 0.40, 
              sig.level = 0.05, 
              power = 0.8, n1 = 100)

    difference of proportion power 
 calculation for binomial distribution
              h = 0.4
             n1 = 100
             n2 = 96.29156
      sig.level = 0.05
          power = 0.8
    alternative = two.sided
NOTE: different sample sizes

pwr.2p2n.test(h = 0.40, 
              sig.level = 0.05, 
              power = 0.8, n1 = 110)

    difference of proportion power 
 calculation for binomial distribution
              h = 0.4
             n1 = 110
             n2 = 88.54092
      sig.level = 0.05
          power = 0.8
    alternative = two.sided
NOTE: different sample sizes

Expected effect size h: rank-biserial correlation r
- See how subjects in groups rank

Test

wilcox.test(Time ~ Topping, 
            data = Pizza)

y ~ x
- y: data
- x: group

    Wilcoxon rank sum test with 
    continuity correction
data:  Enjoyment by Topping
W = 6051, p-value = 0.01026
alternative hypothesis: true location 
shift is not equal to 0

Effect size and power

Effect size

library(effectsize)
rank_biserial(Time ~ Topping, 
              data = pizza)

r (rank biserial) |         95% CI
<----------------------------------
0.21              | [0.05, 0.36]

Small: 0.1
Medium: 0.3
Large: 0.5

1 - 0.14 = 0.86 probability of Type II error

Power Analysis

library(pwr)
pwr.2p2n.test(h = 0.21, sig.level = 0.01, 
              n1 = 100, n2 = 100)

     difference of proportion power calculation for binomial distribution 

              h = 0.21
             n1 = 100
             n2 = 100
      sig.level = 0.01
          power = 0.1376818
    alternative = two.sided

NOTE: different sample sizes

Let's practice!

A/B Testing in R