Two-Sample Medians

Introduction: Topics

• We previously have reviewed estimation and basic statistical inference.
  • Point estimation
  • Confidence intervals
  • Hypothesis testing
• Today we will focus on the following scenarios today:
  • Two-sample medians, independent data (M_1-M_2)
  • Two-sample medians, dependent data (M_d)

Introduction: Data

• The Wonderbolts have opened a new Reserve Camp, where 30 cadets spend two weeks training for a shot at the main squad.

• On arrival, each cadet is assigned to one of two training squadrons:

  • Thunder Squadron
  • Lightning Squadron

• The two squadrons train separately and compete against each other in a series of trials.

  • Independent data

• Every cadet is also measured once at the start of camp and once again at the end, after two weeks of identical drills.

  • Dependent data

Introduction: Data

• Eyeballing the dataset,

The Normality Assumption

• All t-tests assume approximate normality of the data.

  • In the case of one-sample t-tests, the measure of interest must somewhat follow a normal distribution.

  • In the case of two-sample t-tests, the measure of interest in each group must somewhat follow a normal distribution.

• Note that a paired t-test is technically a one-sample t-test, so we will examine normality of the difference.

Normality Assumption: Quantile-Quantile Plots

• There are formal tests for normality (see article here), however, we will not use them.
  

• Instead, we will assess normality using a quantile-quantile (Q-Q) plot.

  • A Q-Q plot helps us visually check if our data follows a specific distribution (here, the normal).

• How do we read Q-Q plots?

  • Each dot represents one observation in our dataset.

  • If the data follow a normal distribution, we will observe that the dots fall roughly along a straight diagonal line.

  • We focus on the “middle” of the graph.

Normality Assumption: Quantile-Quantile Plots

Normality Assumption: Quantile-Quantile Plots

Normality Assumption: Quantile-Quantile Plots

Normality Assumption: Quantile-Quantile Plots

Normality Assumption (R)

• We will check this graphically.

• For independent data,

dataset_name |> independent_qq(outcome = outcome_variable,
                               grouping = grouping_variable)

• For dependent data,

dataset_name |> dependent_qq(col1 = first_measurement,
                             col2 = second_measurement)

Example 1

• Every cadet runs a timed sprint course as part of camp trials.

• The coaches want us to determine if the Thunder Squadron and Lightning Squadron are finishing the sprint course at different times.

• Looking at the data,

Example 1

• Checking the normality assumption,

wonderbolt_camp |> independent_qq(outcome = sprint_time_sec,
                                  grouping = squadron)

Example 2

• At camp, judges score each cadet’s aerial loop maneuver on a 0–100 accuracy scale during trials.

• Based on her gut feeling, the head judge believes that, on average, the Thunder Squadron is performing more accurate loop maneuvers.

• Looking at the data,

Example 2

• Checking the normality assumption,

wonderbolt_camp |> independent_qq(outcome = loop_accuracy_pct,
                                  grouping = squadron)

Example 3

• Each cadet’s aerial accuracy is scored when they arrive at camp, and scored again at the end after two weeks of drills.

• Coaches want to know whether cadets are actually flying more accurately by the end of camp.

• Looking at the data,

Example 3

• Checking the normality assumption,

wonderbolt_camp |> dependent_qq(col1 = accuracy_score_pre,
                                col2 = accuracy_score_post)

Example 4

• Each cadet’s recovery time after a hard trial is timed at the start of camp and again at the end.

• Coaches want to know whether cadets are recovering faster by the end of camp.

• Looking at the data,

Example 4

• Checking the normality assumption,

wonderbolt_camp |> dependent_qq(col1 = recovery_time_min_pre,
                                col2 = recovery_time_min_post)

Introduction: Two Independent Medians

• The Wilcoxon Rank Sum test is a nonparametric alternative to the two-sample t-test.

• Instead of comparing group means, we will now turn to comparing the ranks of the data.

• 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

Two Independent Medians: Ranking Data

• What if all data values are not unique? We will assign the average rank for that group.

• 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

Hypothesis Testing: Two Independent Medians

• Hypotheses: Two Tailed
  • H_0: \ M_1-M_2 = M_0
  • H_1: \ M_1-M_2 \ne M_0
• Hypotheses: Left Tailed
  • H_0: \ M_1-M_2 \ge M_0
  • H_1: \ M_1-M_2 < M_0
• Hypotheses: Right Tailed
  • H_0: \ M_1-M_2 \le M_0
  • H_1: \ M_1-M_2 > M_0

Hypothesis Testing: Two Independent Medians

T_0 = \sum R_{\text{1}} - \frac{n_1(n_1+1)}{2}

• T = \sum R_{\text{sample 1}} - \frac{n_1(n_1+1)}{2}

• where

  • \sum R_1 is the sum of the ranks for the first group
  • n_1 is the sample size of the first group

• Note that p = (calculated by R :))

Two Independent Medians (R)

• We will use the independent_medians function from library(ssstats) to perform the necessary calculations for the hypothesis test.

• Generic syntax:

dataset_name %>% independent_medians(continuous = continuous_variable,
                                     grouping = grouping_variable,
                                     alternative = "two" | "greater" | "less",
                                     m = hypothesized_difference,
                                     alpha = specified_alpha)

Example 1

• The coaches want us to determine if the Thunder Squadron and Lightning Squadron are finishing the sprint course at different times.

• We will use the independent_medians() function,

wonderbolt_camp |> independent_medians(outcome = sprint_time_sec,
                                       grouping = squadron, 
                                       alternative = "two",
                                       m = 0,
                                       alpha = 0.05) 

Example 1

• Running the code,

Independent medians

M[Lightning Squadron] = 45.05
M[Thunder Squadron] = 41.35
Hodges-Lehmann estimate of median difference: 2.8

95% CI for M[Lightning Squadron] − M[Thunder Squadron]: (0.4, 5.5)

Hypotheses:
H₀: M[Lightning Squadron] − M[Thunder Squadron] = 0
H₁: M[Lightning Squadron] − M[Thunder Squadron] ≠ 0
Test statistic: W = 168
p-value: 0.019
Conclusion: Reject the null hypothesis (p = 0.019 < α = 0.050)

Example 1

• Hypotheses:
  • H_0: \ M_{\text{L}} - M_{\text{T}} = 0
  • H_1: \ M_{\text{L}} - M_{\text{T}} \ne 0
• Test Statistic and p-Value
  • W_0 = 168, p = 0.019
• Rejection Region
  • Reject H_0 if p < \alpha; \alpha = 0.05
• Conclusion and interpretation
  • Reject H_0. There is sufficient evidence to suggest that there is a difference between the Lightning and Thunder squadrons.

Introduction: Two Dependent Medians

• The Wilcoxon Signed Rank test is a nonparametric alternative to the dependent t-test.

• Instead of examining the mean of the difference, we will now turn to examining the ranks of the differences.

• Before ranking, we will find the difference between the paired observations and eliminate any 0 differences.

  • Note 1: elimniating 0 differences is the big difference between the other tests!
  • Note 2: because we are eliminating 0 differences, this means that our sample size will update to the number of pairs with a non-0 difference.

Two Dependent Medians: Ranking Data

• When ranking, we the differences are ranked based on the absolute value of the difference.

• Then, ranks can be identified as “positive” or “negative” based on the direction of the difference.

• In this (very basic) example, we started with n=3, but reduced to n=2 due to the 0 difference of the third observation.

Hypothesis Testing: Two Dependent Medians

• Hypotheses: Two Tailed
  • H_0: \ M_d=M_0
  • H_1: \ M_d \ne M_0
• Hypotheses: Left Tailed
  • H_0: \ M_d \ge M_0
  • H_1: \ M_d < M_0
• Hypotheses: Right Tailed
  • H_0: \ M_d \le M_0
  • H_1: \ M_d > M_0
• Note! M_d = M_1 - M_2

Hypothesis Testing: Two Dependent Medians

• Test Statistic

T_0 = \begin{cases} R_+ = \text{sum of positive ranks} & \text{if left-tailed} \\ R_- = \text{sum of negative ranks} & \text{if right-tailed} \\ \min(R_+, R_-) & \text{if two-tailed} \end{cases}

Two Dependent Medians (R)

• We will use the dependent_medians function from library(ssstats) to perform statistical inference.

• Generic syntax:

dataset_name |> dependent_medians(col1 = first_variable,
2                                  col2 = second_variable,
3                                  alternative = "two" | "less" | "greater",
4                                  m = hypothesized_difference,
5                                  alpha = alpha_level)

Example 4

• Each cadet’s recovery time after a hard trial is timed at the start of camp and again at the end.

• Coaches want to know whether cadets are recovering faster by the end of camp.

• We will use the dependent_means() function,

wonderbolt_camp |> dependent_medians(col1 = recovery_time_min_post,
                                     col2 = recovery_time_min_pre,
                                     alternative = "less", 
                                     m = 0,
                                     alpha = 0.05) 

Example 4

• Running the code,

Paired (dependent) medians

Complete pairs: 30
Effective pairs (non-zero differences): 30

M[recovery_time_min_post] = 7.6
M[recovery_time_min_pre] = 12.6
Hodges-Lehmann estimate of median difference: -2.4

95% CI for M[recovery_time_min_post] − M[recovery_time_min_pre]: (-3.3, -1.95)

Hypotheses:
H₀: M[recovery_time_min_post] − M[recovery_time_min_pre] ≥ 0
H₁: M[recovery_time_min_post] − M[recovery_time_min_pre] < 0
Test statistic: T = 0
p-value: < 0.001
Conclusion: Reject the null hypothesis (p = < 0.001 < α = 0.050)

Example 4

• Hypotheses:
  • H_0: \ M_{\text{post}} - M_{\text{pre}} \ge 0
  • H_1: \ M_{\text{post}} - M_{\text{pre}} < 0
• Test Statistic and p-Value
  • T_0 = 0, p < 0.001
• Rejection Region
  • Reject H_0 if p < \alpha; \alpha = 0.05
• Conclusion and interpretation
  • Reject H_0. There is sufficient evidence to suggest that cadets are recovering faster at the end of camp.

Let’s Practice! 1

• Spectators toss bits to cadets who pull off a standout run during trials.

• Camp organizers want to know whether one squadron is out-earning the other in prize bits.

• Looking at the data,

Let’s Practice! 2

• Spectators tip cadets with bits after their runs. The amount tipped is recorded once early in camp and once again near the end.

• Camp organizers want to know whether cadets are earning more in tips as they improve over camp.

• Looking at the data,

Let’s Practice! 3

• A sound crew records the peak crowd noise (in decibels) during each cadet’s run.

• Camp organizers are curious whether one squadron draws a louder crowd reaction than the other.

• Looking at the data,

Let’s Practice! 4

• The camp medic tracks how many days each cadet spends recovering from an injury over the two-week camp.

• Coaches want to know whether one squadron’s training style is leading to more injury days than the other’s.

• Looking at the data,

Let’s Practice! 5

• Each cadet’s wing-flap rate while hovering in place is measured on arrival, then measured again after two weeks of drills.

• Coaches want to know whether training actually changes a cadet’s wing-flap rate.

• Looking at the data,

Let’s Practice! 6

• Each cadet rates how confident they feel about their flying on a self-reported scale, once at the start of camp and once at the end.

• Camp staff want to know whether two weeks of training changes how confident cadets feel.

• Looking at the data,

Wrap Up

• Today’s lecture:
  • Independent Wilcoxon (rank sum)
  • Dependent Wilcoxon (signed rank)
• Monday: R lab – bring a laptop!