W5 L2 Notes on Nonparametric Alternatives: Spearman & Wilcoxon

Spearman Correlation (ρ or rₛ)

Purpose

  • Used when parametric assumptions for Pearson’s correlation are violated (e.g., non-normal data, outliers).

  • Assesses strength of monotonic relationships (increasing/decreasing but not necessarily linear).

Key Features

  • Based on ranks, not raw scores.

  • Captures monotonicity (relationship always goes in one direction, though may plateau).

  • Requires:

    • At least one continuous variable.

    • The other can be continuous or dichotomous.

    • Relationship must be monotonic (check with scatterplot).

Comparison with Pearson

  • Pearson: raw data, assumes linearity, normal distribution.

  • Spearman: ranks, monotonicity, fewer assumptions.

How to Report

  • Format: rₛ(df) = value, p = value.

Examples:

  • rₛ(38) = 0.34, p = 0.009 → weak positive correlation (TV hours ↑, fatigue ↑).

  • rₛ(196) = -0.75, p < 0.001 → strong negative correlation (TV hours ↑, sleep ↓).

Effect Size

  • rₛ itself is the effect size (like Pearson’s r).

When to Use

  • Non-normal data, small samples, or presence of outliers.

  • When variables show a monotonic (not necessarily linear) relationship.

Wilcoxon Tests (Mann–Whitney U)

Purpose

  • Nonparametric alternative to the independent-samples t-test.

  • Used when distributions are non-normal or when outliers make parametric tests inappropriate.

  • Focus here: between-groups comparison (independent samples).

Terminology

  • “Wilcoxon rank-sum test” = “Mann–Whitney U test” (two independent groups).

  • Reports the W statistic.

Interpretation

  • Tests whether group distributions differ (not strictly their means).

  • Robust to non-normality and outliers.

How to Report

  • Format: W = value, p = value.

Examples:

  • Fatigue: W = 110, p < 0.001 → Heavy TV group significantly more fatigued.

  • Sleep: W = 54, p = 0.037 → Heavy TV group slept significantly less.

Effect Size

  • This course: not required for Wilcoxon (debated in literature).

When to Use

  • Two independent groups, outcome variable not normally distributed.

  • Small samples or outlier-prone data.

Key Takeaways

  • Spearman (ρ, rₛ): For monotonic associations with non-normal data or outliers.

  • Wilcoxon (Mann–Whitney U): For group comparisons when t-test assumptions are violated.

  • Always visualise data (scatterplots, boxplots) before choosing tests.