Three berry-piles, one big question

Ugg have two berry-piles, ask "same size?" — easy, use t-test (two-pile mean-compare). But Ugg have THREE pile. Four. Five! Now what?

Lazy caveman run t-test on every pair. Ugg warn: no do this! Every test roll dice, dice sometimes lie — more test, more lie. Count the pairs: $\left(\genfrac{}{}{0ex}{}{k}{2}\right)=\frac{k(k-1)}{2}$ — 3 piles make 3 pairs, 4 piles make 6, 5 piles make 10. Chance one lie sneak in grows:

Three test → lie-chance jump to 14 in 100. Ten test → 40 in 100! A false yell — say "different!" when truly same — is called Type I error.

One number to judge all piles — the F

Smart caveman use ANOVA (Analysis of Variance — one test for 3+ pile-means at once, one α). Design: one thing Ugg change (factor, the IV) with $k\ge 3$ settings, one number Ugg measure (DV), and each tribe-member in ONE pile only (between-subjects).

Big guess (null, H₀): all pile-means same, ${\mu }_1={\mu }_2=\cdots ={\mu }_k$. Rival guess (H₁): at least one differ — but H₁ no say WHICH. This called omnibus (one big yes/no, no finger-point yet).

Trick that make Ugg grin: to compare MEAN, look at VARIANCE. Two wiggles live in the data. Between-group variance — how far pile-means sit from the whole-tribe mean; this the SIGNAL (fire working?). Within-group variance — how much members wiggle inside own pile; this the NOISE (always there). Ugg make ratio, the F-ratio:

Fire do nothing → signal like noise → F sit near 1. Fire work → signal big, noise same → F grow big. F just signal-over-noise.

Break the big rock in two

Heart of ANOVA: the total wiggle, the sum of squares, split CLEAN with no crumb left:

Signal rock ${\text{SS}}_B={\sum }_j{n}_j({\bar{x}}_j-{\bar{x}}_{\text{grand}}{)}^2$; noise rock ${\text{SS}}_W={\sum }_j{\sum }_i(x_{ij}-{\bar{x}}_j{)}^2$. Why clean? Write each stone as (stone − pile-mean) + (pile-mean − grand-mean), square, sum — the middle cross-piece vanish because stones balance around own pile-mean (${\sum }_i(x_{ij}-{\bar{x}}_j)=0$). Two rocks, no leftover.

Count freedom-bones (degrees of freedom): $d{f}_B=k-1$, $d{f}_W=N-k$. Grind SS by df to get mean square: $\text{MSB}={\text{SS}}_B/d{f}_B$ and $\text{MSW}={\text{SS}}_W/d{f}_W$. Then $F=\text{MSB}/\text{MSW}$.

Read the F-bone right

The F-distribution lean to one side, ONE tail (a ratio never go below zero). F under 1 → signal weaker than noise, no fire. F big enough that $p<\alpha$ → throw out H₀, yell "different!". Ugg warn: F is one-tail — no report two-tail p. And F no tell DIRECTION, only that SOME pile differ.

Write the kill this exact way: $F(d{f}_B,d{f}_W)=\text{value},p=\text{value},{\eta }^2=\text{value}$. Like $F(2,87)=4.61,p=.013,{\eta }^2=.10$.

That last stone, eta-squared, is the effect size — how BIG the fire, not just "is there fire":

Share of all wiggle the factor explain. Bands: .01 small, .06 medium, .14 big. Ugg warn: NEVER report F without effect size — F alone only half the hunt! For many-factor hunt use partial η² $={\text{SS}}_{\text{effect}}/({\text{SS}}_{\text{effect}}+{\text{SS}}_{\text{error}})$ to weigh one factor alone.

F yelled "different" — but WHICH pile?

F omnibus, no point finger. To find guilty pile, run a post-hoc test (after-look at all pairs, still guarding the lie-rate). Standard for equal-size piles: Tukey HSD. A pair count as different only if the mean-gap beat:

where q the studentized range bone. Gap bigger than HSD → real difference. Other after-looks: Bonferroni (compare each p to $\alpha /m$; simple, careful, few pairs), Games-Howell (unequal pile-size or unequal wiggle), Scheffé (most careful, any contrast), Dunnett (each pile vs one control). Know the pairs BEFORE the hunt? Use a planned contrast — cheaper on lie-cost. Ugg warn: pick the route before collecting berries; switch after = cheating. And no run post-hoc when the omnibus F say "no different" — that fishing, not hunting.

Rules before Ugg trust the F

ANOVA lean on three:(1) normality — each pile bell-shaped (Shapiro-Wilk, Q-Q).(2) homogeneity of variance — piles wiggle the same (Levene; rule of thumb: biggest wiggle under 4× smallest = fine).(3) independence — each member in one pile only.Broke? Non-bell + tiny tribe → Kruskal-Wallis (rank test). Unequal wiggle → Welch's ANOVA (no assume equal wiggle). Big tribe (n > 25 per pile) → bell-shape matter less (CLT save Ugg).

Same tribe try everything — repeated-measures

Same people try all k things? That repeated-measures (RM) ANOVA — each person is own control. Split finer:

Pull the person-wiggle OUT of the error rock → smaller bottom in $F={\text{MS}}_{\text{Between}}/{\text{MS}}_{\text{Error}}$ → bigger F, MORE power, same tribe size.

But new rule: sphericity — the wiggle of pair-differences equal across all pairs (within-person cousin of homogeneity). Check with Mauchly's test (H₀: sphericity holds; $p<.05$ → broke). Ugg warn: no skip Mauchly — broke sphericity puff up the lie-rate. Broke? No switch test — apply a fix that shrink df by multiply with ε: Greenhouse-Geisser (careful, $\epsilon <0.75$) or Huynh-Feldt (softer, $\epsilon >0.75$). F stay same; only the target-distribution move. Still scared? Use the Friedman test (rank-based RM cousin).

The cousin tribe (know their faces)

• ANCOVA — ANOVA plus a sneaky continuous covariate that muddy the IV. Wipe the covariate's straight-line effect first, THEN run ANOVA on the cleaned DV: cleaner question, smaller error, more power. New rule: homogeneity of regression slopes (covariate-DV line same in every pile).
• Factorial ANOVA — two-plus factors. Test the main effect of each (one IV averaged over the other) and the interaction (does A's fire depend on B?). Parallel lines in the plot = no interaction; crossing lines = interaction. Ugg warn: read interaction FIRST — it can flip the main-effect story.
• MANOVA — two-plus DIFFERENT things measured at once (Pillai's trace, Wilks' lambda), guarding the lie-rate across the whole DV set.
• Mixed ANOVA — one between-factor and one within-factor together.

Big exam trap: RM = same thing many times. MANOVA = different things once. Carve that on the cave wall.

Ugg remember

• F = MSB/MSW = between-wiggle over within-wiggle. Near 1 = no fire; big = fire. One-tail, no direction. The rock split clean: ${\text{SS}}_{\text{Total}}={\text{SS}}_B+{\text{SS}}_W$; $d{f}_B=k-1$, $d{f}_W=N-k$.
• Never report F without effect size. ${\eta }^2={\text{SS}}_B/{\text{SS}}_{\text{Total}}$; bands .01 / .06 / .14. Full kill-report: $F(d{f}_B,d{f}_W),p,{\eta }^2$.
• F is omnibus — it say "someone differ", not who. Post-hoc (Tukey HSD) point the finger, but ONLY after a significant F, never before, and never double-correct Tukey with Bonferroni.
• RM-ANOVA pull subject-wiggle out of error → more power, but must pass sphericity (Mauchly); if broke, correct (Greenhouse-Geisser / Huynh-Feldt), don't switch tests.
• Many t-tests = many lies ($1-(1-\alpha {)}^m$); one ANOVA, one α. Always check the three rules — normality, equal variance, independence — before trusting the bone.