Healthcare
Helpm31990
9781284108217_CH10_SLID.pptChapter 10
Nonparametric Tests
Learning Objectives (1 of 2)
- Compare and contrast parametric and nonparametric tests
- Identify multiple applications where nonparametric approaches are appropriate
- Perform and interpret the Mann–Whitney U test
- Perform and interpret the Sign test and Wilcoxon Signed Rank test
Learning Objectives (2 of 2)
- Compare and contrast the Sign test and Wilcoxon Signed Rank test
- Perform and interpret the Kruskal–Wallis test
- Identify the appropriate nonparametric hypothesis testing procedure based on type of outcome variable and number of samples
Nonparametric Tests
- Appropriate when outcome is continuous but not normally distributed
- Rank scores (e.g., disease stage)
- Continuous but subject to extremes
- Continuous but there are limits of detection (on high or low end of scale)
General Approach
- Rank data
- Perform analysis on ranks
- Follow same 5-step procedure for hypothesis testing
Ranking Data
- Raw data
7 5 9 3 0 2
- Ordered data
0 2 3 5 7 9
- Ranked data
1 2 3 4 5 6
Ranking Data with Ties
- Raw data
7 7 9 3 0 2
- Ordered data
0 2 3 7 7 9
- Ranked data
1 2 3 4.5 4.5 6
Assign mean rank to ties,
Sum of ranks = n(n + 1)/2
Tests with Two Independent Samples: Mann–Whitney U Test (1 of 2)
- Continuous outcome that is not assumed to follow a normal distribution
- Two independent samples
H0: Two populations are equal
H1: Two populations are not equal
Tests with Two Independent Samples: Mann–Whitney U Test (2 of 2)
- Test statistic is U = min(U1, U2),
where R1 and R2 are the sums of the ranks in groups 1 and 2.
- Reject H0 if U ≤ critical value in Table 5.
Example 10.1.
Mann–Whitney U Test (1 of 4)
- A Phase II clinical trial is run to investigate efficacy of a new drug for asthma in children.
- Outcome is number of episodes of shortness of breath over a 1-week period.
Placebo 7 5 6 4 12
Drug 3 6 4 2 1
Example 10.1.
Mann–Whitney U Test (2 of 4)
H0: The two populations are equal
H1: The two populations are not equal
a = 0.05
- Test statistic is U.
- Rank data in pooled sample (n = 10), and compute R1 and R2.
Example 10.1.
Mann–Whitney U Test (3 of 4)
Example 10.1.
Mann–Whitney U Test (4 of 4)
- Test statistic is U = 3.
- Reject H0 if U ≤ 2 (Table 5).
- Do not reject H0 because 3 > 2. We do not have significant evidence to show that the two populations are not equal.
Tests with Matched Samples:
Sign Test (1 of 2)
- Continuous outcome measured in matched or paired samples; differences are not assumed to follow a normal distribution.
- Matched or paired samples
H0: Median difference is zero
H1: Median difference >, <, or ≠ 0
Tests with Matched Samples:
Sign Test (2 of 2)
- Test statistic is the smaller of the number of positive or negative signs (of differences).
- Reject H0 if the smaller of the number of positive of negative signs ≤ critical value in Table 6.
- A new chemotherapy treatment is proposed for patients with breast cancer. Investigators want to assess tolerability of treatment.
- Outcome is quality of life (QOL) measured on an ordinal scale (1 = poor, 2 = fair, 3 = good, 4 = very good, 5 = excellent) both before and after treatment.
Example 10.5.
Sign Test (1 of 5)
- Observed data
Example 10.5.
Sign Test (2 of 5)
- Difference scores
Example 10.5.
Sign Test (3 of 5)
- Signs of the difference scores
NOTE: Randomly assign “+” or “–” when there are zeros.
Example 10.5.
Sign Test (4 of 5)
Example 10.5.
Sign Test (5 of 5)
- Test statistic is 3.
- Reject H0 if the smaller of the number of positive or negative signs ≤ 2 (Table 6).
- Do not reject H0 because 3 > 2. We do not have significant evidence to show that there is a difference in QOL measured before versus after chemotherapy treatment.
Tests with Matched Samples: Wilcoxon Signed Rank Test (1 of 2)
- Continuous outcome measured in matched or paired samples, differences are not assumed to follow a normal distribution.
- Matched or paired samples
H0: Median difference is zero
H1: Median difference >, <, or ≠ 0
Tests with Matched Samples: Wilcoxon Signed Rank Test (2 of 2)
- Test statistic is W, the smaller of W+ and W–, the sums of the positive and negative ranks of the differences scores.
- Reject H0 if W ≤ critical value in Table 7.
Example 10.7.
Wilcoxon Signed Rank Test (1 of 6)
- A study is run to evaluate the effectiveness of a new exercise program to reduce systolic blood pressure (SBP) in patients with pre-hypertension; n = 15 patients participate and have SBP measured before and after 6 weeks on the program.
- Is there a significant difference in SBP after participating in the program?
- Observed data
Example 10.7.
Wilcoxon Signed Rank Test (2 of 6)
- Difference scores
Example 10.7.
Wilcoxon Signed Rank Test (3 of 6)
- Ranks of the difference scores
Example 10.7.
Wilcoxon Signed Rank Test (4 of 6)
- Signed ranks of the difference scores
W+ = 89 W– = 31
Example 10.7.
Wilcoxon Signed Rank Test (5 of 6)
- Test statistic is W = 31.
- Reject H0 if W ≤ 25 (Table 7).
- Do not reject H0 because 31 > 25. We do not have significant evidence to show that the median difference in SBP is not zero.
Example 10.7.
Wilcoxon Signed Rank Test (6 of 6)
Tests with More Than Two Independent Samples: Kruskal–Wallis Test (1 of 2)
- Continuous outcome that is not assumed to follow a normal distribution
- k (k > 2) independent samples
H0: k population medians are equal
H1: k population medians are not all equal
- Test statistic is H,
where k = number of groups, N = total sample size, nj = sample size in jth group, Rj = sum of the ranks in jth group.
- Reject H0 if H ≥ critical value in Table 8.
Tests with More Than Two Independent Samples: Kruskal–Wallis Test (2 of 2)
Example 10.8.
Kruskal–Wallis Test (1 of 4)
- A clinical study is run to assess differences in albumin levels in patients following 5%, 10%, and 15% protein diets.
H0: The three population medians are equal
H1: The three population medians are not equal
a = 0.05
- Test statistic is H.
- Rank data in pooled sample (n = 12), and compute R1, R2, and R3.
Example 10.8.
Kruskal–Wallis Test (2 of 4)
Example 10.8.
Kruskal–Wallis Test (3 of 4)
- R1 = 7.5, R2 = 30.5, R3 = 40.
- Test statistic is H.
- Reject H0 if H ≥ 5.656 (Table 8).
Example 10.8.
Kruskal–Wallis Test (4 of 4)
Tests with More Than Two Independent Samples: Kruskal Wallis Test
- Reject H0 because 7.52 > 5.656.
- We have statistically significant evidence to show that there is a difference in median albumin levels among the three diets.
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