Sums of Squares — Three-Way Partition

SS_total = Σ(Yijk − Ȳ...)²

Between-Subjects partition
  SS_A      = b·n · Σ_i (Ȳi.. − Ȳ...)²
  SS_S(A)   = b   · Σ_i Σ_k (Ȳi.k − Ȳi..)²        ← subjects-within-groups (error 1)

Within-Subjects partition
  SS_B      = a·n · Σ_j (Ȳ.j. − Ȳ...)²
  SS_AB     = n   · Σ_i Σ_j (Ȳij. − Ȳi.. − Ȳ.j. + Ȳ...)²
  SS_B×S(A) = SS_total − SS_A − SS_S(A) − SS_B − SS_AB   ← within-subjects error (error 2)

Degrees of Freedom

df_A       = a − 1                       (between groups)
df_S(A)    = a(n − 1)                    (subjects within groups)
df_B       = b − 1                       (within-subjects levels)
df_AB      = (a − 1)(b − 1)              (interaction)
df_B×S(A)  = a(n − 1)(b − 1)             (within-subjects error)

F-Statistics

F_A   = MS_A   / MS_S(A)        →  tests Group main effect
F_B   = MS_B   / MS_B×S(A)      →  tests within-subjects (Time) main effect
F_AB  = MS_AB  / MS_B×S(A)      →  tests Group × Time interaction

Effect Size — Partial Eta Squared

η²_p(A)  = SS_A  / (SS_A  + SS_S(A))
η²_p(B)  = SS_B  / (SS_B  + SS_B×S(A))
η²_p(AB) = SS_AB / (SS_AB + SS_B×S(A))

Benchmarks (Cohen, 1988): 0.01 small · 0.06 medium · 0.14 large

Sphericity Corrections

Greenhouse–Geisser ε̂ = (Σλi)² / [(b−1) · Σλi²]
Huynh–Feldt ε̃        = min(1, n·(b−1)·ε̂ − 2) / [(b−1)·(n−1 − (b−1)·ε̂)]

Corrected df:  df1' = ε·df1   df2' = ε·df2
Where λi are the eigenvalues of the within-subjects covariance matrix.

Where:

a         = number of between-subjects groups
b         = number of within-subjects levels
n         = subjects per group  (assumed balanced)
Yijk      = score for subject k in group i at level j
Ȳ...      = grand mean
Ȳi..      = mean of group i
Ȳ.j.      = mean of within-subjects level j
Ȳij.      = mean of cell (i, j)
Ȳi.k      = mean for subject k in group i across within levels
SS        = sum of squares
MS        = mean square = SS / df
ε         = sphericity epsilon (1 = perfect sphericity)

• Independence of observations between groups. Subjects in different between-subjects groups must be independent.
• Normality of residuals within each cell. Each combination of group × time should have approximately normally distributed scores. Robust to mild violations when group sizes are equal and n > 15.
• Homogeneity of between-subjects variances. Check with Levene's test or compare cell SDs. Largest/smallest SD ratio < 2 is acceptable.
• Sphericity (within-subjects). Tested by Mauchly's W when b ≥ 3. When violated, apply Greenhouse–Geisser or Huynh–Feldt corrections (both reported above).
• Balanced design preferred. Equal n per group simplifies the SS partition. This tool assumes balanced n.
• If assumptions fail: Consider linear mixed-effects models (R lme4) or robust ANOVA (Wilcox WRS2 package).

This free mixed-design ANOVA tool is designed for studies that combine a between-subjects factor (independent groups) with a within-subjects factor (repeated measures on the same subjects). It is one of the most commonly used designs in clinical trials, education research, sports science, and experimental psychology.

Use this test when ALL of these are true:

• You have at least 2 independent groups (between-subjects factor)
• Each subject is measured 2 or more times on the same outcome (within-subjects factor)
• Your dependent variable is continuous (interval or ratio)
• You want to test both main effects and their interaction
• Sample is approximately balanced and residuals are roughly normal
• Don't use if all measurements are on the same single group → use Repeated-Measures ANOVA
• Don't use if groups are measured only once → use One-Way ANOVA
• Don't use if data are heavily skewed and n is small → use a nonparametric mixed model or linear mixed-effects model

Real-world examples

1. Medical research. Three drug groups (Placebo, Drug A, Drug B) measured at Pre, Mid, and Post — does blood pressure change differently across drugs over time?
2. Education. Traditional vs Flipped classroom measured Q1 → Q4 — do learning trajectories diverge?
3. Sports science. Control vs HIIT training measured at Week 0, 4, and 8 — does HIIT improve sprint time faster than control?
4. Psychology. CBT vs Mindfulness vs Waitlist measured Pre, Post, and Follow-up — which therapy produces the largest reduction in anxiety, and does that benefit persist?

Sample size guidance

For 80% power to detect a medium interaction effect (Cohen's f = 0.25) at α = 0.05, you typically need n ≈ 24 per group with 2 groups and 3 time points (≈ 48 total). For small interaction effects (f = 0.10), expect n ≥ 80 per group.

Decision tree

Two or more groups → measured once → One-Way / Two-Way ANOVA
One group → measured repeatedly → Repeated-Measures ANOVA
Multiple groups → measured repeatedly → Mixed-Design ANOVA (this tool)
Non-normal data, complex random effects → Linear Mixed-Effects Model (LMM)

1. Step 1 — Enter your data. Use the Type / Paste tab (default). Each group is one row; within each group, paste comma-separated values per subject, one subject per line. For three time points, each subject row should be exactly 3 numbers (e.g., 52, 48, 55).
2. Step 2 — Edit group names. Click any group name field to rename it (e.g., from "Group 1" to "Placebo").
3. Step 3 — Set within-subjects level names. Enter comma-separated labels matching the number of values per row (e.g., Pre, Mid, Post).
4. Step 4 — Pick a sample dataset (optional). Five built-in datasets pre-fill the form so you can see the tool in action.
5. Step 5 — Configure alpha. 0.05 is standard. Use 0.01 for more conservative inference.
6. Step 6 — Run analysis. Click the green button. Nothing runs automatically on page load.
7. Step 7 — Read summary cards. Three F-tests are shown: Group, Time, and Group × Time. Color = significance.
8. Step 8 — Inspect the full ANOVA table. Each row gives df, SS, MS, F, p, and partial η².
9. Step 9 — Check four visualizations. Interaction plot, error-bar chart, individual trajectories, and box plots together give a complete picture.
10. Step 10 — Export. Click Download Doc for a plain-text report or Download PDF for a print-ready PDF.