Chapter 19: Nonparametric Methods
Interactive Self-Study Guide — KIN 610
Ovande Furtado Jr., PhD. 
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1. The Logic of Ranks (Parametric vs Nonparametric)
Most nonparametric tests bypass distributional constraints by relying on rank transformation. Instead of crunching the raw numbers, the procedure translates the lowest raw score into Rank 1, the next lowest into Rank 2, and so on.
Consider an extreme outlier in strength data: If you have raw data [45, 48, 51, 60, 480], the 480 breaks parametric computations by drastically shifting the mean and inflating the variance.
However, when nonparametrically converting this data to ranks, it becomes [1, 2, 3, 4, 5]. The massive gap between 60 and 480 is neutralized. The test statistic derives mathematical significance purely from the ordering, effectively filtering the noise.
(Note that tied ranks are given the average of the rank numbers they overlap!).
2. Categorical Association: Chi-Square
Goodness-of-Fit vs. Independence
Goodness-of-Fit Explores whether our sample’s frequency counts deviate from a theoretical expectation. (e.g., Expected 50/50 sex split, observed 40/60). \[df = k - 1 \quad \text{(where k is number of categories)}\]
Test of Independence Explores whether two categorical variables associate dynamically in a contingency table. (e.g., Does Gender associate with Intervention Type?). The key assumption here is that all expected cell counts must be at least 5. If a condition dips below 5, you risk violating the chi-square distributional approximation.
3. Rank-Order Correlation: Spearman ρ
When determining if two variables correlate, Pearson r demands they follow a strict linear trend (a straight line). But sometimes relationships curve slightly—they are monotonic (one variable generally goes up as the other goes up, though not at a perfectly constant rate).
To test monotonic relationships (or when avoiding outlier interference), we cast the data into ranks and run Spearman ρ. Just like Pearson, it yields a standardized coefficient stretching between -1 to 1.
4. Testing Group Differences (Two Groups)
When we replace the Student t-Test family, we shift into specialized nonparametrics utilizing the sums of ranks.
Mann-Whitney U Test (Independent Samples)
Replaces the Independent Samples t-test. It pools all participants across both groups, orders them from lowest to highest, and asks: Is Group A consistently taking up the higher ranks while Group B sits at the bottom ranks?
Its primary effect size is the rank-biserial r.
Wilcoxon Signed-Rank Test (Paired Samples)
Replaces the Paired t-test. Instead of evaluating independent groups, it calculates the raw differences between each subject’s Pre vs. Post measurements. It then strips the signs (+/-) away, ranks the magnitudes of change, and then reattaches the signs to compute positive rank sums vs. negative rank sums.
(Note: There is also the simpler Sign Test. However, the Sign Test only calculates the simple direction of change ignoring magnitude entirely. The Wilcoxon signed-rank uses more information and has much higher statistical power, assuming magnitude ranks are interpretable!)
5. Multiple Groups & The Decision Tree
What if you have three or more groups (ANOVA)?
- Kruskal-Wallis Test: Replaces One-Way ANOVA (Between Subjects). Yields an H statistic.
- Friedman’s Test: Replaces Repeated Measures ANOVA (Within Subjects). Yields a \(\chi^2\) statistic. Effect size is Kendall’s W.
6. Complex Factorial Nonparametrics & Pitfalls
When managing Factorial ANOVA models featuring 2 or more Crossed factors, you may need the Scheirer-Ray-Hare test to process independent categorical associations nonparametrically (though note it often severely lacks power). Similarly, if you need to use a Covariate (ANCOVA) but violate homogeneity assumptions, Quade’s Rank ANCOVA acts as the alternative structural mechanism.
In all of these options, resist the fatal interpretational pitfall: Do not conflate a non-significant p-value directly with “zero difference.” With small datasets, rank transformations cast out extreme variance and often lead to lower statistical power outright. A rejected H1 only signifies insufficient evidence under power limits, so reporting specific effect sizes visually helps discern reality.
7. Flashcards: Key Terms
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