Two-Sample Medians

Introduction: Topics

• Last week, we talked about checking the assumptions on t-tests.
  • If we break the variance assumption of the two-sample t-test \to use the two-sample t-test for unequal variances.
  • What happens when we break the normality assumption?
• Nonparametric alternatives
  • Independent medians (M_1-M_2)
  • Dependent medians (M_d)

Introduction: Nonparametrics

• 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 :(

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 :))

Hypothesis Testing: Two Independent Medians (R)

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

• Generic syntax:

dataset_name %>% independent_median_HT(continuous = continuous_variable,
                                       grouping = grouping_variable,
                                       alternative = "alternative_direction",
                                       m = hypothesized difference,
                                       alpha = specified_alpha)

Hypothesis Testing: Two Independent Medians

• Recall our example:
  • Perform the appropriate hypothesis test to determine if the above target pegasi are eating 5 or more apples than the below target pegasi. Test at the \alpha=0.05 level.
• How should we change the following code?

dataset_name %>% independent_median_HT(continuous = continuous_variable,
                                       grouping = grouping_variable,
                                       alternative = "alternative_direction",
                                       m = hypothesized_difference,
                                       alpha = specified_alpha)

Hypothesis Testing: Two Independent Medians

• Recall our example:
  • Perform the appropriate hypothesis test to determine if the above target pegasi are eating 5 or more apples than the below target pegasi. Test at the \alpha=0.05 level.
• Our updated code,

wing_flap %>% independent_median_HT(continuous = apples,
                                    grouping = target,
                                    alternative = "greater",
                                    m = 5,
                                    alpha = 0.05)

Hypothesis Testing: Two Independent Medians

• Running the code,

wing_flap %>% independent_median_HT(continuous = apples,
                                    grouping = target,
                                    alternative = "greater",
                                    m = 5,
                                    alpha = 0.05)

Wilcoxon Rank Sum Test for two independent medians
Null: H₀: M₁ - M₂ ≤ 5
Alternative: H₁: M₁ - M₂ > 5
Test statistic: T = 135.5
p-value: p < 0.001
Conclusion: Reject the null hypothesis (p = < 0.001 < α = 0.05)

Hypothesis Testing: Two Independent Medians

• Hypotheses:
  • H_0: \ M_{\text{above}} - M_{\text{below}} \le 5
  • H_1: \ M_{\text{above}} - M_{\text{below}} > 5
• Test Statistic and p-Value
  • T_0 = 135.5, p < 0.001
• Rejection Region
  • Reject H_0 if p < \alpha; \alpha = 0.05
• Conclusion and interpretation
  • Reject H_0 (p \text{ vs } \alpha \to p < 0.001 < 0.05). There is sufficient evidence to suggest that ponies above target eat 5 more apples than those below target.

Two Independent Groups: Means vs. Medians

• t-test for independent means:

wing_flap %>% independent_mean_HT(continuous = apples, 
                                  grouping = target,
                                  mu = 5, 
                                  alternative = "greater", 
                                  alpha = 0.05)

• Wilcoxon rank sum for independent medians:

wing_flap %>% independent_median_HT(continuous = apples,
                                    grouping = target,
                                    m = 5,
                                    alternative = "greater",
                                    alpha = 0.05)

Two Independent Groups: Means vs. Medians

• t-test for independent means:

Two-sample t-test for two independent means and equal variance:
Null: H₀: μ₁ − μ₂ ≤ 5
Alternative: H₁: μ₁ − μ₂ > 5
Test statistic: t(23) = 5.445
p-value: p < 0.001
Conclusion: Reject the null hypothesis (p = < 0.001 < α = 0.05)

• Wilcoxon rank sum for independent medians:

Wilcoxon Rank Sum Test for two independent medians
Null: H₀: M₁ - M₂ ≤ 5
Alternative: H₁: M₁ - M₂ > 5
Test statistic: T = 135.5
p-value: p < 0.001
Conclusion: Reject the null hypothesis (p = < 0.001 < α = 0.05)

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}

• p-Value is calculated by R.

Hypothesis Testing: Two Dependent Medians (R)

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

• Generic syntax:

dataset_name %>% dependent_median_HT(col1 = first_group,
                                     col2 = second_group,
                                     alternative = "alternative_direction",
                                     m = hypothesized_value,
                                     alpha = specified_alpha)

Hypothesis Testing: Two Dependent Medians

• Perform the appropriate hypothesis test to determine if there is a difference in wing-flap rate pre- and post-training. Test at the \alpha=0.01 level.

• How should we change the following code?

dataset_name %>% dependent_median_HT(col1 = first_group,
                                     col2 = second_group,
                                     alternative = "alternative_direction",
                                     m = hypothesized_value,
                                     alpha = specified_alpha)

Hypothesis Testing: Two Dependent Medians

• Perform the appropriate hypothesis test to determine if there is a difference in wing-flap rate pre- and post-training. Test at the \alpha=0.01 level.

• Our updated code,

wing_flap %>% dependent_median_HT(col1 = pre_training_wfr,
                                  col2 = post_training_wfr,
                                  alternative = "two",
                                  m = 0,
                                  alpha = 0.01)

Hypothesis Testing: Two Dependent Medians

• Running the code,

wing_flap %>% dependent_median_HT(col1 = pre_training_wfr,
                                  col2 = post_training_wfr,
                                  alternative = "two",
                                  m = 0,
                                  alpha = 0.01)

Wilcoxon Signed-Rank Test for the median of differences:
Null: H₀: M_d = 0
Alternative: H₁: M_d ≠ 0
Test statistic: T = 188
p-value: p = 0.501
Conclusion: Fail to reject the null hypothesis (p = 0.5011 ≥ α = 0.01)

Hypothesis Testing: Two Dependent Medians

• Hypotheses:
  • H_0: \ M_{\text{d}} = 0, where M_{\text{d}} = M_{\text{pre}}-M_{\text{post}}
  • H_1: \ M_{\text{d}} \ne 0
• Test Statistic and p-Value
  • T_0 = 188, p = 0.501
• Rejection Region
  • Reject H_0 if p < \alpha; \alpha = 0.01
• Conclusion and interpretation
  • Fail to reject H_0 (p \text{ vs } \alpha \to p = 0.501 > 0.01). There is not sufficient evidence to suggest that there is a difference in wing-flap rate.

Two Dependent Groups: Means vs. Medians

• t-test for dependent means:

wing_flap %>% dependent_mean_HT(col1 = pre_training_wfr,
                                col2 = post_training_wfr,
                                alternative = "two",
                                mu = 0,
                                alpha = 0.01)

• Wilcoxon signed for dependent medians:

wing_flap %>% dependent_median_HT(col1 = pre_training_wfr,
                                  col2 = post_training_wfr,
                                  alternative = "two",
                                  m = 0,
                                  alpha = 0.01)

Two Dependent Groups: Means vs. Medians

• t-test for dependent means:

Paired t-test for the mean of differences:
Null: H₀: μ_d = 0
Alternative: H₁: μ_d ≠ 0
Test statistic: t(24) = 0.859
p-value: p = 0.399
Conclusion: Fail to reject the null hypothesis (p = 0.3991 ≥ α = 0.01)

• Wilcoxon signed for dependent medians:

Wilcoxon Signed-Rank Test for the median of differences:
Null: H₀: M_d = 0
Alternative: H₁: M_d ≠ 0
Test statistic: T = 188
p-value: p = 0.501
Conclusion: Fail to reject the null hypothesis (p = 0.5011 ≥ α = 0.01)

Wrap Up

• Today’s lecture:
  • Wilcoxon rank sum (nonparametric equivalent to independent t)
  • Wilcoxon signed rank (nonparametric equivalent to dependent t)
• Next class:
  • R lab: Wilcoxons
  • Quiz: Wilcoxons
• Next week:
  • Review of Module 1
    • Quiz for Module 1!
  • Project 1 work time