When rock-data no behave, Ugg no panic

Sometimes the numbers no act nice. They lopsided like broken spear. Herd too small. Or they no even numbers — they categories, like "red berry / blue berry / green berry" or "mammoth run away: yes / no." The old tests all whisper same lie: "data make nice bell-shape hill, spread is fair, big enough herd." When that lie break, Ugg reach for different rocks.

Non-parametric test — test that no assume the data make a nice bell-hill shape. It work on the order of things (who bigger, who smaller — the ranks) or on counts in piles. Fewer promises needed. Ugg use these only when data refuse to behave.

Big secret: every scary bell-shape test — the parametric kind — have a non-parametric cave-cousin. Learn the pairs, half the hunt is done.

The picking-rock map (carve on cave wall first!)

Ugg carve this 2×3 grid. Level of measurement = what kind of numbers the outcome is. Design = same tribe measured twice (within-subjects, paired) or two different tribes (between-subjects, independent).

• Nice numbers (interval/ratio, bell-shape): between → Independent t-test; within → Paired t-test.
• Order-only numbers (ordinal, ranks): between → Mann-Whitney U; within → Wilcoxon signed-rank.
• Pile-counts (categorical): between → Chi-square test; within → Binomial sign test (or McNemar for 2×2).

More than two herds? Independent → one-way ANOVA (nice) or Kruskal-Wallis (rank). Same tribe, many conditions → RM-ANOVA (nice) or Friedman's test (rank). Two-plus pile-kinds crossed → Chi-square for independence.

Three questions pick the rock

1. What kind of numbers is the outcome (DV)? True-measure (interval/ratio), order-only (ordinal), or pile (categorical/nominal). 2. How many herds or conditions? Two, or more than two. 3. Same tribe or different tribes? Within or between.

One more sniff: are the nice-test promises kept? Bell-shape, equal spread (homogeneity of variance), no crazy outliers, herd big enough. Promises kept AND true numbers → parametric (more power — catch real effect with fewer hunters). Promises broken → non-parametric. Ugg slogan: "lose power for robustness." Rank test throw away how far apart, keep only order — that why weaker when bell-hill real (about 5% weaker for Wilcoxon vs t). Ugg warn: no drop to rank test for no reason — you miss real mammoth.

Chi-square — the count-pile rock

When data is counts in piles, Ugg use chi-square (χ²). Same formula both jobs:

O is what Ugg see in a pile, E is what Ugg expect. For each pile: take see-minus-expect, square it, divide by expect. Add all piles. Big χ² means see and expect far apart.

Chi-square goodness-of-fit — do seen counts match a claimed shape? Like berry-maker say "13% brown, 24% blue…" and Ugg count real berries to check. H₀: seen matches expected. df = k − 1 (k piles, minus one because total count already fixed).

Chi-square test for independence — are two pile-kinds tangled together? Make a contingency table (grid of counts). Expected count in each box:

That come from H₀ "no tangle": chance-of-row times chance-of-col, times herd size. df = (r − 1)(c − 1). Why? Fix the edge-totals (marginals); then only $(r-1)(c-1)$ boxes free to fill, rest forced. 2×2 → df=1. 2×3 → df=2.

χ² only say tangle there or not there — not how strong. So report effect size. Phi $\phi =\sqrt{{\chi }^2/n}$ for 2×2. Cramér's V $V=\sqrt{{\chi }^2/(n\cdot \min (r-1,c-1))}$ for bigger grids. Both run 0 to 1: about 0.1 small, 0.3 middle, 0.5 big.

Ugg warn about χ²:(1) Each hunter counted in exactly one box — between-subjects only; paired yes/no → use McNemar.(2) Counts only, never means or percents.(3) Every box expect ≥ 5; below that the χ² magic break — use Fisher's exact (2×2, exact hypergeometric counting, no approximation) or squish piles together.(4) Never say χ² prove one thing cause other — only association.

The rank cave-cousins

Mann-Whitney U — non-parametric independent t. Throw both herds' values in one line, rank them all, add ranks per herd. Then

It test stochastic dominance — does one herd tend to sit higher? (Ugg warn: NOT means! No say "compares means.") Good for ordinal, non-bell numbers, small herd with outliers.

Wilcoxon signed-rank — non-parametric paired t. For each pair take the difference, rank the absolute differences, add ranks of the plus-differences → W. Tests if differences sit balanced around zero. (Ugg warn: no mix up Mann-Whitney and Wilcoxon — one different tribes, one same tribe paired!)

Kruskal-Wallis — non-parametric one-way ANOVA. Rank across many herds, get H. Says "at least one herd higher/lower." Then Dunn's post-hoc (with correction) to find which.

Friedman's test — non-parametric RM-ANOVA. Same tribe, many conditions: rank within each hunter across conditions, compare the rank-sums.

Paired piles and last rocks

McNemar's test — same tribe, two yes/no measures (before vs after). Only the discordant boxes matter — b is yes→no count, c is no→yes count:

Same-answer boxes ignored, they no carry change. Need ≥ 25 discordant pairs, else use exact binomial version. Binomial sign test — simplest paired rock: count plus/minus signs of differences, test against $P(+)=0.5$. Spearman ρ — Pearson r done on ranks; outlier-tough correlation for monotone climbs.

Report rank tests with the statistic (U, W, H, χ², ρ), the p-value, AND the effect size $r=Z/\sqrt{N}$. Example: "Mann-Whitney U = 40, p = 0.03, r ≈ 0.45 (moderate)."

Ugg remember

• Every parametric test have non-parametric cousin. Mann-Whitney = independent t; Wilcoxon signed-rank = paired t; Kruskal-Wallis = one-way ANOVA; Friedman = RM-ANOVA; McNemar = paired χ² 2×2.
• Pick rock by 3 questions: number-kind (interval / ordinal / categorical), how many groups (2 or more), same-or-different tribe — plus the filter "are bell-shape promises kept?"
• χ² = Σ(O−E)²/E. df: goodness-of-fit = k−1; independence = (r−1)(c−1). Need expect ≥ 5 per box, or use Fisher's exact.
• Effect size always: φ (2×2) or Cramér's V (bigger) for χ²; $r=Z/\sqrt{N}$ for rank tests. χ² show association only, never cause.
• Non-parametric weaker when bell-shape true — drop down only for ordinal data, ugly unfixable outliers, or small skewed herd. Rank test compares stochastic dominance, not means.