Three-Way ANOVA Calculator | Free 3 Factor Calculator

A three-way ANOVA calculator tests whether three categorical factors, and their interactions, change the mean of one numeric outcome.

It splits the total variation into seven effects: three main effects (A, B, C), three two-way interactions (A×B, A×C, B×C) and one three-way interaction (A×B×C). Each effect gets its own F test: F = MS_effect ÷ MS_error. In the worked example below, Drug gives F(1, 24) = 234.26, p < .001, partial η² = .91. This page works as a 3 factor calculator for any design from 2×2×2 upward.

Rule of thumb: read the three-way interaction first. If A×B×C is significant, interpret the two-way interactions separately at each level of the third factor. If it is not, move down to the two-way interactions, then to the main effects.

A three-way ANOVA is an analysis of variance with three independent categorical factors and one continuous outcome. It asks two kinds of question at once. First, does each factor change the mean on its own? Second, does the effect of one factor depend on the level of another factor? The second kind of question is called an interaction, and it is usually the reason people run a three-way design instead of three separate one-way tests.

The seven effects it tests:

• Main effect of A, for example: does blood pressure differ between Drug and Placebo, averaged over sex and age?
• Main effect of B and main effect of C, the same question for the other two factors.
• A×B, A×C, B×C, the two-way interactions. A×C asks: is the drug effect bigger in older patients than in younger ones?
• A×B×C, the three-way interaction. It asks: does the A×C pattern itself change between males and females?

A worked one-liner: in a 2×2×2 blood pressure trial with 4 patients per cell (N = 32), the error mean square is 7.8125 and the drug mean square is 1830.125, so F = 1830.125 ÷ 7.8125 = 234.26 on 1 and 24 degrees of freedom.

1.1 Is a three-way ANOVA a "factor calculator"?

Yes, in the statistics sense. In ANOVA, a factor is a categorical variable such as treatment, sex or season, and each factor has levels. This page is a 3 factor calculator for experiments. It is not a tool for finding the factors of a number such as 36 (1, 2, 3, 4, 6, 9, 12, 18, 36). That is a different, arithmetic meaning of the same word.

Who uses it: clinical researchers (drug by sex by age group), agronomists (fertilizer by irrigation by crop variety), psychologists (condition by task by time), ecologists (light by soil by water in greenhouse trials), and marketers (channel by region by plan).

4.1 Your result in one paragraph

Across N = ___ observations in a ___ design, ___ of the seven effects were significant at α = ___. The largest effect was ___ (partial η² = ___).

4.2 Read the table from the bottom up

The ANOVA table lists main effects first, but you should read it in the opposite order. Start with the three-way interaction, then the two-way interactions, then the main effects. The reason is simple. A higher-order interaction changes the meaning of every lower-order effect it contains. If Drug × Age is significant, the phrase "the drug lowers blood pressure by 15 mmHg" is only an average. The true effect might be 12 mmHg in young patients and 18 mmHg in old ones.

4.3 The three-way interaction

The three-way interaction was F(___, ___) = ___, p = ___.

A significant A×B×C means the two-way interaction between two factors is different at different levels of the third factor. To follow it up, split the data by the third factor and run a two-way ANOVA inside each level, using the pooled error term from this analysis. A non-significant three-way interaction lets you interpret the two-way interactions as consistent patterns.

4.4 Two-way interactions and simple effects

A significant two-way interaction, say A×C, means the lines on the interaction plot are not parallel. The follow-up is a simple effects analysis: test the effect of A at each level of C. In the worked example, the drug lowers blood pressure by 12.25 mmHg in young patients and by 18.00 mmHg in old patients. Both simple effects are significant, but the difference between them is what makes the interaction significant.

Look at Chart 1. Lines that cross indicate a disordinal interaction, where the ranking of groups flips. Lines that diverge without crossing indicate an ordinal interaction, where one group is always higher but by different amounts.

4.5 Main effects, and when to trust them

Main effects: ___.

A main effect is the difference between marginal means, averaged over the other two factors. When a factor is not involved in any significant interaction, its main effect is a clean summary. When it is involved in one, report the main effect but describe it as an average and point readers to the simple effects.

4.6 Effect sizes: partial eta squared, eta squared and omega squared

The p-value tells you whether an effect is likely to be real. The effect size tells you how big it is. Partial η² is SS_effect divided by (SS_effect + SS_error), so each effect is judged against error only. This is what SPSS reports. η² divides by the total SS, so all seven values add to at most 1. Partial ω² corrects the upward bias of partial η² and is the better estimate for small samples. With large samples all three give similar answers.

Cohen (1988) bands for partial η² are about .01 small, .06 medium and .14 large. Treat these as a rough guide. In a tightly controlled lab study, .14 may be ordinary. In a field study with noisy measures, .03 may matter.

4.7 Assumption checks for your data

Shapiro-Wilk on the residuals: W = ___, p = ___. Levene's test (median-centred): F(___, ___) = ___, p = ___.

ANOVA is fairly robust to mild non-normality when cells are equal and n is 15 or more per cell. It is much less robust to unequal variances combined with unequal cell sizes. If the largest cell SD is more than about twice the smallest and the cell sizes also differ, the F tests can be badly wrong in either direction.

4.8 Post hoc tests for factors with three or more levels

A significant main effect for a two-level factor needs no post hoc test, because there is only one comparison. For a factor with three or more levels, the F test tells you that some means differ but not which ones. This calculator runs Bonferroni-adjusted pairwise comparisons on the estimated marginal means, using the pooled error term. The adjusted p multiplies each raw p by the number of comparisons.

4.9 Decision guide

4.10 Common mistakes this calculator prevents

• Running three one-way ANOVAs instead of one three-way ANOVA, which misses every interaction.
• Using Type I sums of squares on unbalanced data without saying so. The setting is explicit here, and the output states which type was used.
• Reporting eta squared and calling it partial eta squared. Both are labelled clearly in the table.
• Forgetting a factor combination with no data. The calculator stops and names the problem instead of returning a wrong answer.
• Testing normality on the raw outcome instead of the residuals. The Shapiro-Wilk test here runs on the residuals, as it should.

▶ Run the analysis above to auto-fill all five examples with your results.

5.1 APA 7th edition

5.2 Thesis or dissertation

5.3 Plain-language summary

5.4 Structured abstract or conference poster

5.5 Replication or pre-registration

5.6 Reporting checklist

• Design stated as a × b × c with factor names and levels.
• Cell sizes, cell means and SDs in a table.
• All seven F tests with both df, exact p and partial η².
• Follow-up tests for every significant interaction.
• Assumption checks and the SS type.
• An interaction plot with error bars that are labelled as SE or 95% CI.

The formulas below are written for a balanced design with a levels of A, b levels of B, c levels of C and n observations per cell. For unbalanced designs, the calculator gets the same quantities by fitting an effect-coded regression model and comparing the full model with the model that drops each term (Type III) or adds the terms in order (Type I).

This flowchart shows how a three-way ANOVA is calculated, step by step, from raw rows to seven F tests. Every box uses the same worked example as the Excel, R and Python sections: systolic blood pressure for 32 patients in a Drug × Sex × Age trial, 4 patients per cell.

Check any box yourself: the Excel sheet in Section 9, the R script in Section 10 and the Python script in Section 11 all return SS_error = 187.5 and F(1, 24) = 8.464 for Drug × Age on this dataset.

1. Arrange your data in long format. One row per observation, four columns: Factor A, Factor B, Factor C and the numeric outcome. The header row names the factors, for example Drug,Sex,Age,SystolicBP.
2. Paste it into the box. Commas, tabs and semicolons all work. A block copied from Excel or Google Sheets pastes as tab-separated text and is read correctly.
3. Or upload a file. On the Upload tab choose a CSV or Excel file, pick the sheet, then click one column for each role: Factor A, Factor B, Factor C and Outcome. Press Load selected columns.
4. Or type it in. The Manual Entry tab gives you a four-column table with editable headers. Add rows as needed, then press Use this table.
5. Try a sample dataset. Ten built-in datasets cover a clean 2×2×2 trial, 3-level factors with post hoc tests, a real three-way interaction, a null result, a small sample, an unbalanced survey and a skewed dataset that fails the assumptions.
6. Choose the significance level. 0.05 is standard. Use 0.01 when you run many analyses or need stronger evidence.
7. Choose the sums of squares type. Keep Type III to match SPSS and most journals. Choose Type I to match R's aov() and anova(). For equal cell sizes both give the same answer.
8. Press Run Three-Way ANOVA. Nothing is computed until you do. Changing the data or a setting clears old results so you never read stale numbers.
9. Read from the bottom of the table up. Check the three-way interaction first, then the two-way interactions, then the main effects, and look at Chart 1 to see the pattern behind any interaction.
10. Copy your write-up and export. Section 5 fills five reporting styles with your numbers. Download Doc gives a text report and Download PDF gives a print-ready copy.

Excel's Analysis ToolPak has "Anova: Single Factor" and "Anova: Two-Factor With Replication", but no three-way ANOVA. You can still get the exact answer with ordinary formulas, because every sum of squares is built from averages. This section builds the full table for the blood pressure example, with the spreadsheet shown at each stage.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases: unequal cell sizes and 3+ levels

The averaging method above is exact only for a balanced design, where every cell has the same n. With unequal n, the subtraction method mixes up the effects and gives neither Type I nor Type III answers. You have two honest options in Excel.

1. Regression with effect coding. Code each two-level factor as +1 and −1, build product columns for every interaction, and run Data → Data Analysis → Regression (ToolPak must be enabled first via File → Options → Add-ins). Refit without each term's columns; the rise in residual SS is that term's Type III SS.
2. Use this calculator or R or Python for the final numbers, and use Excel only for descriptive tables and charts.

For a factor with 3 levels, the main effect SS formula is the same, with three squared terms instead of two, and df = 2. Effect coding needs 2 columns for that factor instead of 1.

9.4 Why your Excel answer might differ from R, Python or SPSS

9.5 Charting the interaction in Excel

1. Build a small table of cell means: Drug levels down the rows, one column per Age level. Make one table per Sex level.
2. Select one table and choose Insert → Charts → Line with Markers.
3. Repeat for the second Sex level and place the charts side by side with the same y-axis range (Format Axis → Bounds).
4. Add error bars with + → Error Bars → More Options → Custom and point at your SE cells. Say in the caption that the bars are SE.

Chart 1 in Section 3 draws this plot for you, with every B and C combination as its own line.

9.6 Excel errors you will probably hit

R fits a three-way ANOVA in one line, aov(y ~ drug * sex * age), but that line gives Type I sums of squares. The script below uses base R only and builds the Type III table directly, checks both assumptions, prints a plain-English verdict and saves one publication-ready interaction plot at 300 dpi.

10.1 The complete script

R, base only, saves three_way_anova_plot.png

# ============================================================
# Three-way ANOVA with Type III sums of squares and one figure
# StatsUnlock - https://statsunlock.com/three-way-anova-calculator/
# Base R only. No packages to install.
# ============================================================

set.seed(2026)   # nothing here is random, but keep runs reproducible

# ---- 1. Your data (long format: one row per patient) -------
# To use a file instead:
#   bp <- read.csv("my_data.csv", stringsAsFactors = TRUE)
#   bp <- bp[!is.na(bp$y), ]            # drop missing outcomes
bp <- data.frame(
  drug = rep(c("Drug", "Placebo"), each = 16),
  sex  = rep(rep(c("Male", "Female"), each = 8), times = 2),
  age  = rep(rep(c("Young", "Old"), each = 4), times = 4),
  y    = c(120, 121, 119, 117,  126, 125, 128, 133,
           115, 115, 119, 118,  125, 122, 125, 127,
           131, 134, 129, 131,  143, 148, 145, 150,
           133, 131, 123, 130,  144, 144, 139, 142)
)
bp$drug <- factor(bp$drug, levels = c("Drug", "Placebo"))
bp$sex  <- factor(bp$sex,  levels = c("Male", "Female"))
bp$age  <- factor(bp$age,  levels = c("Young", "Old"))
bp <- bp[!is.na(bp$y), ]

# ---- 2. Fit with sum-to-zero contrasts ---------------------
# Sum contrasts are REQUIRED for correct Type III tests.
options(contrasts = c("contr.sum", "contr.poly"))
fit <- lm(y ~ drug * sex * age, data = bp)

# ---- 3. Type III table: drop each term's columns in turn ----
X      <- model.matrix(fit)
assign <- attr(X, "assign")
labs   <- attr(terms(fit), "term.labels")
sse    <- sum(resid(fit)^2)
dfe    <- df.residual(fit)
mse    <- sse / dfe
tab <- do.call(rbind, lapply(seq_along(labs), function(k) {
  reduced <- lm.fit(X[, assign != k, drop = FALSE], bp$y)
  ss  <- sum(reduced$residuals^2) - sse
  df1 <- sum(assign == k)
  F   <- (ss / df1) / mse
  data.frame(term = labs[k], SS = ss, df = df1, F = F,
             p = pf(F, df1, dfe, lower.tail = FALSE),
             partial_eta_sq = ss / (ss + sse))
}))
tab <- rbind(tab, data.frame(term = "Residuals", SS = sse, df = dfe,
                             F = NA, p = NA, partial_eta_sq = NA))
print(tab, digits = 4, row.names = FALSE)

# ---- 4. Assumption checks ----------------------------------
sw <- shapiro.test(resid(fit))
cell <- interaction(bp$drug, bp$sex, bp$age)
dev  <- abs(bp$y - ave(bp$y, cell, FUN = median))
lev  <- anova(lm(dev ~ cell))          # Levene, median-centred
cat(sprintf("\nShapiro-Wilk on residuals: W = %.4f, p = %.4f\n", sw$statistic, sw$p.value))
cat(sprintf("Levene (median): F(%d, %d) = %.4f, p = %.4f\n",
            lev$Df[1], lev$Df[2], lev$`F value`[1], lev$`Pr(>F)`[1]))

# ---- 5. Plain-English verdict ------------------------------
alpha <- 0.05
cat("\nVerdict:\n")
for (i in seq_along(labs)) {
  cat(sprintf("  %-16s F(%d, %d) = %6.2f, p = %.4f, partial eta^2 = %.3f -> %s\n",
      gsub(":", " x ", labs[i]), tab$df[i], dfe, tab$F[i], tab$p[i],
      tab$partial_eta_sq[i],
      ifelse(tab$p[i] < alpha, "SIGNIFICANT", "not significant")))
}

# ---- 6. One figure: interaction plot, one panel per sex -----
png("three_way_anova_plot.png", width = 2700, height = 1260, res = 300)
par(mfrow = c(1, 2), mar = c(4.5, 4.5, 3, 1))
for (s in levels(bp$sex)) {
  d <- bp[bp$sex == s, ]
  interaction.plot(d$drug, d$age, d$y, type = "b", pch = 19, lwd = 2,
                   col = c("#2563eb", "#c2410c"), lty = 1,
                   ylim = range(tapply(bp$y, cell, mean)) + c(-3, 3),
                   xlab = "Treatment", ylab = "Systolic BP (mmHg)",
                   trace.label = "Age", main = s, legend = (s == "Female"))
  grid(col = "grey85")
}
invisible(dev.off())
cat("\nSaved three_way_anova_plot.png at 300 dpi\n")

10.2 What each part does

10.3 What the figure shows

Two panels, male on the left and female on the right. Each panel plots mean blood pressure for Drug and Placebo, with a blue line for young patients and an orange line for old patients. In both panels the orange line is steeper than the blue one: the drug helps older patients more. That is the Drug × Age interaction. The two panels look alike, which is why the three-way interaction is not significant.

10.4 Common R problems and fixes

10.5 One-liners

# Type I table (equal to Type III when cells are balanced)
summary(aov(y ~ drug * sex * age, data = bp))

# Type III with the car package: install.packages("car")
car::Anova(lm(y ~ drug * sex * age, data = bp), type = 3)

# Cell means table
aggregate(y ~ drug + sex + age, data = bp, FUN = mean)

# Simple effect of drug at each age, pooled error
emmeans::emmeans(lm(y ~ drug * sex * age, data = bp), pairwise ~ drug | age)

Python fits a three-way ANOVA with statsmodels. The script builds the long-format DataFrame, fits the full factorial model with sum contrasts, prints the Type III table with partial eta squared, runs Shapiro-Wilk and Levene, prints a plain-English verdict, and saves one 300 dpi interaction plot.

Install once with pip install pandas numpy scipy statsmodels matplotlib.

11.1 The complete script

Python 3, saves three_way_anova_plot.png

# ============================================================
# Three-way ANOVA with Type III sums of squares and one figure
# StatsUnlock - https://statsunlock.com/three-way-anova-calculator/
# pip install pandas numpy scipy statsmodels matplotlib
# ============================================================
import numpy as np
import pandas as pd
import statsmodels.formula.api as smf
from statsmodels.stats.anova import anova_lm
from scipy import stats
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt

np.random.seed(2026)  # nothing here is random, but keep runs reproducible
pd.set_option("display.width", 160)

# ---- 1. Your data (long format: one row per patient) --------
# To use a file instead:
#   df = pd.read_csv("my_data.csv")
#   df = df.dropna(subset=["bp"])        # drop missing outcomes
cells = {
    ("Drug", "Male", "Young"):      [120, 121, 119, 117],
    ("Drug", "Male", "Old"):        [126, 125, 128, 133],
    ("Drug", "Female", "Young"):    [115, 115, 119, 118],
    ("Drug", "Female", "Old"):      [125, 122, 125, 127],
    ("Placebo", "Male", "Young"):   [131, 134, 129, 131],
    ("Placebo", "Male", "Old"):     [143, 148, 145, 150],
    ("Placebo", "Female", "Young"): [133, 131, 123, 130],
    ("Placebo", "Female", "Old"):   [144, 144, 139, 142],
}
df = pd.DataFrame([(d, s, a, y) for (d, s, a), v in cells.items() for y in v],
                  columns=["drug", "sex", "age", "bp"])
df = df.dropna(subset=["bp"])

# ---- 2. Fit the model with sum-to-zero contrasts -------------
# Sum contrasts are REQUIRED for correct Type III tests.
model = smf.ols("bp ~ C(drug, Sum) * C(sex, Sum) * C(age, Sum)", data=df).fit()
table = anova_lm(model, typ=3).drop(index="Intercept")
sse = table.loc["Residual", "sum_sq"]
table["partial_eta_sq"] = table["sum_sq"] / (table["sum_sq"] + sse)
table.loc["Residual", "partial_eta_sq"] = np.nan
print(table.round(4))

# ---- 3. Assumption checks -----------------------------------
sw = stats.shapiro(model.resid)
groups = [g["bp"].values for _, g in df.groupby(["drug", "sex", "age"])]
lev = stats.levene(*groups, center="median")
print(f"\nShapiro-Wilk on residuals: W = {sw.statistic:.4f}, p = {sw.pvalue:.4f}")
print(f"Levene (median): F = {lev.statistic:.4f}, p = {lev.pvalue:.4f}")

# ---- 4. Plain-English verdict --------------------------------
alpha = 0.05
names = {"C(drug, Sum)": "Drug", "C(sex, Sum)": "Sex", "C(age, Sum)": "Age"}
dfe = int(table.loc["Residual", "df"])
print("\nVerdict:")
for term, row in table.drop(index="Residual").iterrows():
    label = term
    for k, v in names.items():
        label = label.replace(k, v)
    label = label.replace(":", " x ")
    word = "SIGNIFICANT" if row["PR(>F)"] < alpha else "not significant"
    print(f"  {label:<18} F({int(row['df'])}, {dfe}) = {row['F']:.2f}, "
          f"p = {row['PR(>F)']:.4f}, partial eta^2 = {row['partial_eta_sq']:.3f} -> {word}")

# ---- 5. One figure: interaction plot, one panel per sex -------
means = df.groupby(["sex", "age", "drug"])["bp"].agg(["mean", "sem"]).reset_index()
fig, axes = plt.subplots(1, 2, figsize=(9, 4.2), sharey=True)
colors = {"Young": "#2563eb", "Old": "#c2410c"}
for ax, sex in zip(axes, ["Male", "Female"]):
    for age in ["Young", "Old"]:
        m = means[(means.sex == sex) & (means.age == age)].set_index("drug").loc[["Drug", "Placebo"]]
        ax.errorbar(m.index, m["mean"], yerr=m["sem"], marker="o", capsize=4,
                    lw=2, color=colors[age], label=age)
    ax.set_title(sex)
    ax.set_xlabel("Treatment")
    ax.grid(alpha=0.3)
axes[0].set_ylabel("Systolic BP (mmHg), mean ± SE")
axes[1].legend(title="Age")
fig.suptitle("Drug x Sex x Age: three-way ANOVA")
fig.tight_layout()
fig.savefig("three_way_anova_plot.png", dpi=300)
print("\nSaved three_way_anova_plot.png at 300 dpi")

11.2 What each part does

11.3 What the figure shows

The same two-panel interaction plot as the R version, with standard error bars on every mean. Steeper orange lines than blue lines in both panels show the Drug × Age interaction. Similar panels for male and female show the absent three-way interaction.

11.4 Common Python problems and fixes

11.5 One-liners

# Type I table (the default)
anova_lm(smf.ols("bp ~ drug * sex * age", data=df).fit())

# Cell means and SDs
df.groupby(["drug", "sex", "age"])["bp"].agg(["mean", "std", "count"])

# pingouin, if installed: pip install pingouin
import pingouin as pg
pg.anova(data=df, dv="bp", between=["drug", "sex", "age"], ss_type=3)

12.1 Degrees of freedom for common designs

Conclusion: the three-way interaction uses the product of the three main-effect df, so larger designs spend a lot of df on it.

12.2 Critical F values at α = .05

Conclusion: if your F is above the value for your df, p is below .05. The worked example's Drug × Age F of 8.46 is well above 4.26.

12.3 Effect size benchmarks

Conclusion: compare with effects published in your own field before you call an effect small or large.

12.4 Type I, Type II and Type III sums of squares

Conclusion: with equal cell sizes all three types give identical answers.

12.5 Approximate n per cell for 80% power, α = .05

Conclusion: medium interactions need about 11 to 16 per cell. Small interactions need very large samples, so plan with G*Power or the pwr package.

12.6 Which test should you use?

Conclusion: the three-way ANOVA assumes independent observations, so repeated measurements need a different model.

Eight worked three-way ANOVAs, each with its ANOVA summary, an interaction plot and a ready-to-use reporting sentence. Seven of them are built-in sample datasets, so you can load them in Section 2 and reproduce every number.

Study design: a three-way between-subjects ANOVA needs every unit measured once, in exactly one combination of three categorical factors, on one continuous outcome.

1. Fix the three factors and their levels before you start. Write them down, for example Drug (Drug, Placebo), Sex (Male, Female), Age (Young, Old). Adding a level halfway through breaks the design.
2. Check you can fill every cell. Multiply the levels: 2 × 3 × 2 = 12 cells. Each cell needs data, so rare combinations (for example, old patients in a paediatric ward) can make the design impossible.
3. Plan n per cell with a power analysis. Use Table 12.5. Medium interactions need about 11 to 16 per cell for 80% power. Aim for equal n in every cell.
4. Randomise manipulated factors. Assign units to Drug or Placebo at random within each Sex and Age group. Measured factors like sex cannot be randomised, so say so in the paper.
5. Keep units independent. One value per person, plot or animal. Repeated readings of the same unit need a repeated measures or mixed model instead.
6. Blind the measurement where you can. The person recording the outcome should not know the treatment.
7. Use one instrument and one protocol across all cells, and record units and resolution.
8. Record missing values as blanks, never as zero. A fake zero lowers a cell mean and inflates its variance.
9. Write the outlier and transformation rules in advance, for example "values beyond 3 SD of the cell mean are checked against source records".
10. Store the data in long format: one row per unit, one column per factor, one column for the outcome, exactly as the layout below shows.

Minimum sample: at least 2 values in every cell so the error term exists, and in practice 10 or more per cell. Unequal cell sizes are allowed; choose Type III sums of squares for them.

Pre-registration: record the factors, levels, n per cell, alpha, SS type, the effect you expect (main effect or interaction) and how you will follow up an interaction, before you collect data.

Common collection mistakes:

• Leaving one factor combination empty, which makes the three-way interaction untestable.
• Collecting far more data in one cell than the others, which makes Type I and Type III answers diverge.
• Measuring the same person in several conditions and treating the rows as independent.
• Spelling a level two ways ("male" and "Male"), which creates a fourth level by accident.
• Recording the outcome in different units in different cells.

Use a three-way ANOVA when your research question is about how three factors work together, not just one at a time.

Use it when:

• ✓ You have exactly three categorical factors, each with 2 or more levels.
• ✓ The outcome is continuous: a measurement, a score or a time.
• ✓ Each unit appears in one cell only (between-subjects).
• ✓ You expect, or want to rule out, interactions between the factors.
• ✓ Every factor combination has data.

Do not use it when:

• ✗ The same units are measured more than once. Use repeated measures or mixed ANOVA.
• ✗ The outcome is a count or a yes or no. Use Poisson or logistic regression with interaction terms.
• ✗ The outcome is ordinal or heavily skewed and cannot be transformed. Use an aligned rank transform ANOVA.
• ✗ You have only two factors. Use the two-way ANOVA.
• ✗ You have several correlated outcomes. Use MANOVA.
• ✗ A factor is continuous, like age in years. Keep it continuous and use regression or ANCOVA instead of cutting it into groups.

Real-world examples:

1. Clinical trial: does a drug's effect on blood pressure depend on sex and age group?
2. Agronomy: fertilizer by irrigation by crop variety on yield per hectare.
3. Psychology: caffeine by sleep deprivation by task difficulty on reaction time.
4. Ecology: light by soil by watering regime on seedling height in a greenhouse.

Design guidance: keep the number of levels small. A 2×2×2 design needs 8 cells; a 3×3×3 needs 27. Every extra level multiplies the sample you need.

Decision tree: one factor, use a one-way ANOVA. Two factors, use a two-way ANOVA. Three factors, all between-subjects, continuous outcome, use this three-way ANOVA. Any factor within-subjects, use a mixed model. Non-normal outcome that transformation cannot fix, use the aligned rank transform.

Assumptions

1. Independence of observations. Check the design: one value per unit, no clusters. If violated: the error term is too small, so p-values are too small and false positives rise. This is the most serious violation.
2. Normal residuals within each cell. Check the Shapiro-Wilk test and the Q-Q plot. If violated: with equal n and 15 or more per cell, the effect is small. With small, skewed cells, p-values can be off in either direction.
3. Equal variances across cells. Check Levene's test and the largest to smallest SD ratio. If violated with unequal n: when the larger cells have the larger variances, the test becomes too conservative; when the smaller cells have the larger variances, it becomes too liberal and false positives rise.
4. A continuous outcome on an interval or ratio scale. Check what the numbers mean. If violated: ordinal scores give means that may not be meaningful.
5. Fixed, categorical factors. Check that the levels are the ones you care about, not a random sample of levels. If violated: random factors need a mixed model; the F denominators change.
6. No empty cells. Check the cell table. If violated: the full model cannot be estimated and the calculator stops.

Limitations

• It handles between-subjects factors only. For repeated measurements use a repeated measures ANOVA or a linear mixed model.
• It fits one outcome. For several correlated outcomes use MANOVA.
• It does not include covariates. Use ANCOVA or regression to adjust for a continuous variable.
• Post hoc tests use Bonferroni, which is simple and safe but conservative. Tukey HSD or Holm in R or Python give slightly more power.
• Simple effects for interactions are described but not computed automatically. Use emmeans in R for full simple effects tables.
• It does not offer Welch-type or robust ANOVA. For badly unequal variances use the WRS2 package in R or a log transform.
• Power drops fast for higher-order interactions. A non-significant three-way interaction in a small study is weak evidence that none exists.

▶ Run the analysis above and this paragraph will summarise your own result: ___.

A three-way ANOVA calculator answers a question that simpler tests cannot: does the effect of one factor depend on the other two? It splits the variation in one outcome into seven parts, three main effects, three two-way interactions and one three-way interaction, and tests each against the same pooled error term.

The most useful habit is to read the table from the bottom up. A significant three-way interaction changes the meaning of every two-way interaction it contains, and a significant two-way interaction turns the related main effects into averages. In the worked example the drug lowers blood pressure in everyone, but by 12 mmHg in young patients and 18 mmHg in old ones. That difference is the finding, and a one-way analysis would have missed it.

The second habit is to check the design before the statistics. Every factor combination needs data, observations must be independent, and unequal cell sizes mean you must choose and report the sums of squares type. Type III is the usual choice and matches SPSS.

Load your data or one of the ten samples into this three-way ANOVA calculator, run it, look at the interaction plot, and copy the APA sentence from Section 5. Whenever you have three factors to test together, a 3 factor calculator like this one is the right place to start.

These peer-reviewed sources support the methods used in this three-way ANOVA calculator: factorial designs, interactions, sums of squares for unbalanced data, effect sizes, assumption tests and multiple comparisons.

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8. Levine, T. R., & Hullett, C. R. (2002). Eta squared, partial eta squared, and misreporting of effect size in communication research. Human Communication Research, 28(4), 612–625. https://doi.org/10.1111/j.1468-2958.2002.tb00828.x
9. Richardson, J. T. E. (2011). Eta squared and partial eta squared as measures of effect size in educational research. Educational Research Review, 6(2), 135–147. https://doi.org/10.1016/j.edurev.2010.12.001
10. Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159. https://doi.org/10.1037/0033-2909.112.1.155
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13. Gelman, A. (2005). Analysis of variance: Why it is more important than ever. The Annals of Statistics, 33(1), 1–53. https://doi.org/10.1214/009053604000001048
14. Kim, H.-Y. (2014). Statistical notes for clinical researchers: Two-way analysis of variance (ANOVA), exploring possible interaction between factors. Restorative Dentistry & Endodontics, 39(2), 143–147. https://doi.org/10.5395/rde.2014.39.2.143
15. Glass, G. V., Peckham, P. D., & Sanders, J. R. (1972). Consequences of failure to meet assumptions underlying the fixed effects analyses of variance and covariance. Review of Educational Research, 42(3), 237–288. https://doi.org/10.3102/00346543042003237
16. Blanca, M. J., Alarcón, R., Arnau, J., Bono, R., & Bendayan, R. (2017). Non-normal data: Is ANOVA still a valid option? Psicothema, 29(4), 552–557. https://doi.org/10.7334/psicothema2016.383
17. Wobbrock, J. O., Findlater, L., Gergle, D., & Higgins, J. J. (2011). The aligned rank transform for nonparametric factorial analyses using only ANOVA procedures. Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, 143–146. https://doi.org/10.1145/1978942.1978963
18. Bender, R., & Lange, S. (2001). Adjusting for multiple testing: When and how? Journal of Clinical Epidemiology, 54(4), 343–349. https://doi.org/10.1016/S0895-4356(00)00314-0
19. Armstrong, R. A. (2014). When to use the Bonferroni correction. Ophthalmic and Physiological Optics, 34(5), 502–508. https://doi.org/10.1111/opo.12131
20. Weissgerber, T. L., Milic, N. M., Winham, S. J., & Garovic, V. D. (2015). Beyond bar and line graphs: Time for a new data presentation paradigm. PLOS Biology, 13(4), e1002128. https://doi.org/10.1371/journal.pbio.1002128