One-way and Factorial ANOVA

Regression models for comparing means across designed groups

Statistics

Regression

ANOVA

Experimental Design

One-way and factorial analysis of variance as linear models, with sums of squares, interactions, assumptions, contrasts, diagnostics, and bioinformatics design guidance.

Author

Jiahong Sun

Published

October 15, 2026

Modified

October 15, 2026

1 The Question ANOVA Answers

Analysis of variance (ANOVA) compares systematic differences in group means with variation among independent units receiving the same conditions. Despite its name, its primary targets are usually mean contrasts. Variance is the accounting device used to quantify signal and residual noise.

For outcome $Y_{ij}$ from independent unit $j$ in group $i$, the one-way model is

\[ Y_{ij}=\mu+\alpha_i+\varepsilon_{ij}, \qquad \operatorname{E}(\varepsilon_{ij}\mid i)=0. \]

The population group means $\mu_i=\operatorname{E}(Y\mid i)$, or prespecified contrasts such as $\mu_2-\mu_1$, are the estimands. The sample means and fitted contrasts are estimates. ANOVA does not turn an observational group label into a cause; causal interpretation comes from random assignment or defensible identification assumptions.

The experimental unit is the smallest unit independently assigned to a condition or sampled from the target population. Repeated wells, reads, images, cells, or technical runs from one specimen may improve measurement, but they do not create new independent biological replicates.

2 One-way ANOVA as a Linear Model

With $g$ groups, the cell-means parameterization is

\[ Y_{ij}=\mu_i+\varepsilon_{ij}. \]

An equivalent regression uses an intercept and $g-1$ indicator columns. The coefficient values depend on the chosen coding, but fitted group means and the same estimable contrasts do not. This is why scientific conclusions should be reported as group means and contrasts rather than as unexplained “main-effect” coefficients.

For a balanced design with $n_i=n_0$ and total $N=gn_0$, the total corrected sum of squares decomposes as

\[ \underbrace{\sum_i\sum_j(Y_{ij}-\bar Y_{\cdot\cdot})^2}_{SS_{\mathrm{total}}} = \underbrace{n_0\sum_i(\bar Y_{i\cdot}-\bar Y_{\cdot\cdot})^2}_{SS_{\mathrm{group}}} + \underbrace{\sum_i\sum_j(Y_{ij}-\bar Y_{i\cdot})^2}_{SS_{\mathrm{error}}}. \]

The equality follows by writing $Y_{ij}-\bar Y_{\cdot\cdot}=(Y_{ij}-\bar Y_{i\cdot})+ (\bar Y_{i\cdot}-\bar Y_{\cdot\cdot})$; the cross-product vanishes because residuals sum to zero within each group. In matrix terms, it is a projection decomposition from least-squares geometry.

Under independent Gaussian errors with common variance $\sigma^2$, the omnibus hypothesis

\[ H_0:\mu_1=\cdots=\mu_g \]

is tested by

\[ F=\frac{SS_{\mathrm{group}}/(g-1)} {SS_{\mathrm{error}}/(N-g)} \sim F_{g-1,N-g}\quad\text{under }H_0. \]

The numerator measures improvement over an intercept-only model; the denominator estimates within-group noise. Rejecting the omnibus null says that the vector of means is not constant. It does not identify which differences matter, measure their size, or establish biological importance.

3 Factorial Designs and Interactions

A factorial experiment studies two or more factors together. For factors $A$ and $B$, a two-way fixed-effects model is

\[ Y_{ijk}=\mu+\alpha_i+\beta_j+(\alpha\beta)_{ij}+\varepsilon_{ijk}. \]

The interaction term allows the contrast between levels of $A$ to vary across levels of $B$. For two-level factors, an interaction contrast is a difference in differences:

\[ (\mu_{11}-\mu_{01})-(\mu_{10}-\mu_{00}). \]

If interaction is scientifically plausible, marginal main effects can hide the pattern. Report cell means, conditional contrasts, and an interaction plot with uncertainty. A nonsignificant interaction is not evidence that effects are identical; the design may simply be imprecise.

Factorial designs are efficient because every unit informs more than one factor. They also distinguish a biological interaction from a nuisance block or batch effect. Blocking reduces residual variation when blocks are known before randomization. Treating a block as a biological factor changes the question, so the design and estimands must be declared first.

4 Contrasts, Omnibus Tests, and Multiplicity

A contrast is a linear function

\[ L=\sum_{i=1}^{g}c_i\mu_i, \qquad \sum_i c_i=0. \]

Examples include one treatment versus control, a linear dose trend, or the average of two treatments versus another pair. Prespecified contrasts often answer the scientific question more directly and with more power than an omnibus test followed by every pairwise comparison.

For fitted coefficient vector $\hat{\boldsymbol\beta}$ and contrast vector $\mathbf c$,

\[ \widehat L=\mathbf c^\mathsf{T}\hat{\boldsymbol\beta}, \qquad \widehat{\operatorname{Var}}(\widehat L) =\mathbf c^\mathsf{T}\widehat{\operatorname{Var}}(\hat{\boldsymbol\beta})\mathbf c. \]

Report $\widehat L$, an interval, the outcome scale, and the multiplicity strategy. Tukey procedures suit all pairwise comparisons under their conditions; Dunnett procedures suit several treatments versus one control. Planned contrasts need not be delayed until an omnibus test rejects, but the full family of claims still determines the appropriate error control.

5 Balanced and Unbalanced Data

In a balanced, complete factorial design, orthogonal model terms yield the same sums of squares regardless of fitting order. With unequal cell sizes, missing cells, or correlated predictors, “the sum of squares for factor $A$” is not unique.

Sequential sums of squares compare each term with those entered before it.
Partial sums of squares compare a term after adjusting for specified other terms.
Different software labels such comparisons Type I, II, or III, but the label is less informative than the exact nested models or contrasts tested.

Type III tests depend on parameterization and may target averages over cell combinations that are poorly represented or absent. Empty cells can make some main effects or interactions non-estimable. Inspect the design matrix, cell counts, rank, and the weighting used for marginal means. State the hypothesis as a contrast or explicit reduced-versus-full model comparison.

6 Assumptions, Diagnostics, and Alternatives

The classical fixed-effects ANOVA $F$ reference law uses independent Gaussian errors with common variance and a correctly specified mean structure.

Issue	Why it matters	Check and response
Wrong experimental unit	Degrees of freedom and standard errors become artificially small	Reconstruct assignment and nesting; aggregate or model clusters
Unequal variances	Classical pooled standard errors may be wrong	Plot group spreads; use Welch’s one-way test, variance modeling, or robust covariance when justified
Dependence	Replicates contribute less information than assumed	Use blocked, repeated-measures, cluster-aware, or mixed models
Skewness or heavy tails	Small-sample $F$ calibration and means may be unstable	Check residuals and influence; consider transformation, robust methods, or design-respecting randomization inference
Sparse or empty cells	Contrasts may rely on extrapolation or be non-estimable	Show cell counts, rank, and contrast weights; simplify claims
Many post hoc comparisons	False-positive risk increases	Prespecify priorities and control the relevant error family

ANOVA is often reasonably robust to moderate nonnormality in balanced designs, but balance does not rescue pseudoreplication, severe outliers, or dependence. Rank-based tests change the estimand and generally do not provide a drop-in factorial analysis. Permutation tests require transformations that are valid under the design and null being tested.

7 Worked Bioinformatics Design

Suppose 36 mice are randomized equally to three drugs, with sex used as a second factor and RNA-seq pathway score as the outcome. The factorial model estimates six drug-by-sex cell means. Primary estimands might be each drug-versus-control contrast averaged equally across sexes and the drug-by-sex interactions.

The mouse is the experimental unit. Two library preparations and thousands of reads from one mouse are technical observations, not extra randomized units. Sequencing lane can be included as a design or processing factor if it is not confounded with drug. If every control library was sequenced in one lane and every treated library in another, drug and lane effects cannot be separated by ANOVA.

Statistically, a drug contrast estimates a difference in mean pathway score on the declared scale. Biologically, it supports a treatment-associated change in that pathway under this organism, tissue, dose, and time point; it does not by itself demonstrate a molecular mechanism or generalize to other settings.

8 Reporting and Connections

Report the design, experimental unit, allocation, cell counts, outcome scale, model formula, prespecified contrasts, estimated means and contrasts with intervals, multiplicity adjustment, residual and influence checks, missing-data handling, and sensitivity analyses. An ANOVA table alone is not a scientific result.

Categorical predictors and contrasts develops coding-invariant group comparisons.
Interaction terms and effect modification explains conditional effects and effect-scale dependence.
Linear-model assumptions and diagnostics links each regression assumption to the claim it supports.
Regression uncertainty and coefficient interpretation derives uncertainty for coefficients, contrasts, and predictions.
Family-wise error rate distinguishes weak and strong control over families of tests.
ANCOVA and regression-based ANOVA adds continuous covariates and explicit nested-model comparisons.

ANOVA is a linear model for means and contrasts, not merely a table of sums of squares.
The independent experimental unit determines the information and degrees of freedom.
In factorial designs, interactions make effects conditional on other factors.
With imbalance, define hypotheses through contrasts or nested models instead of relying on an unexplained sums-of-squares type.
Effect estimates, intervals, diagnostics, and biological context matter more than an omnibus p-value alone.

9 References

Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boyd. https://psychclassics.yorku.ca/Fisher/Methods/
Box, G. E. P., Hunter, J. S., and Hunter, W. G. (2005). Statistics for Experimenters (2nd ed.). Wiley.
Searle, S. R. (1971). Linear Models. Wiley.
Langsrud, O. (2003). ANOVA for unbalanced data: Use Type II instead of Type III sums of squares. Statistics and Computing, 13, 163–167. https://doi.org/10.1023/A:1023260610025
R Core Team. aov: fit an analysis of variance model; anova.lm: ANOVA tables for fitted linear models. R statistical software documentation. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/aov.html

--- title: "One-way and Factorial ANOVA" subtitle: "Regression models for comparing means across designed groups" description: "One-way and factorial analysis of variance as linear models, with sums of squares, interactions, assumptions, contrasts, diagnostics, and bioinformatics design guidance." author: "Jiahong Sun" date: "2026-10-15" date-modified: "2026-10-15" categories: [Statistics, Regression, ANOVA, Experimental Design] toc: true number-sections: true code-fold: show draft: false --- # The Question ANOVA Answers Analysis of variance (ANOVA) compares systematic differences in group means with variation among independent units receiving the same conditions. Despite its name, its primary targets are usually mean contrasts. Variance is the accounting device used to quantify signal and residual noise. For outcome $Y_{ij}$ from independent unit $j$ in group $i$, the one-way model is $$ Y_{ij}=\mu+\alpha_i+\varepsilon_{ij}, \qquad \operatorname{E}(\varepsilon_{ij}\mid i)=0. $$ The population group means $\mu_i=\operatorname{E}(Y\mid i)$, or prespecified contrasts such as $\mu_2-\mu_1$, are the estimands. The sample means and fitted contrasts are estimates. ANOVA does not turn an observational group label into a cause; causal interpretation comes from random assignment or defensible identification assumptions. The experimental unit is the smallest unit independently assigned to a condition or sampled from the target population. Repeated wells, reads, images, cells, or technical runs from one specimen may improve measurement, but they do not create new independent biological replicates. # One-way ANOVA as a Linear Model With $g$ groups, the cell-means parameterization is $$ Y_{ij}=\mu_i+\varepsilon_{ij}. $$ An equivalent regression uses an intercept and $g-1$ indicator columns. The coefficient values depend on the chosen coding, but fitted group means and the same estimable contrasts do not. This is why scientific conclusions should be reported as group means and contrasts rather than as unexplained "main-effect" coefficients. For a balanced design with $n_i=n_0$ and total $N=gn_0$, the total corrected sum of squares decomposes as $$ \underbrace{\sum_i\sum_j(Y_{ij}-\bar Y_{\cdot\cdot})^2}_{SS_{\mathrm{total}}} = \underbrace{n_0\sum_i(\bar Y_{i\cdot}-\bar Y_{\cdot\cdot})^2}_{SS_{\mathrm{group}}} + \underbrace{\sum_i\sum_j(Y_{ij}-\bar Y_{i\cdot})^2}_{SS_{\mathrm{error}}}. $$ The equality follows by writing $Y_{ij}-\bar Y_{\cdot\cdot}=(Y_{ij}-\bar Y_{i\cdot})+ (\bar Y_{i\cdot}-\bar Y_{\cdot\cdot})$; the cross-product vanishes because residuals sum to zero within each group. In matrix terms, it is a projection decomposition from [least-squares geometry](least-squares-and-regression-geometry.qmd). Under independent Gaussian errors with common variance $\sigma^2$, the omnibus hypothesis $$ H_0:\mu_1=\cdots=\mu_g $$ is tested by $$ F=\frac{SS_{\mathrm{group}}/(g-1)} {SS_{\mathrm{error}}/(N-g)} \sim F_{g-1,N-g}\quad\text{under }H_0. $$ The numerator measures improvement over an intercept-only model; the denominator estimates within-group noise. Rejecting the omnibus null says that the vector of means is not constant. It does not identify which differences matter, measure their size, or establish biological importance. # Factorial Designs and Interactions A factorial experiment studies two or more factors together. For factors $A$ and $B$, a two-way fixed-effects model is $$ Y_{ijk}=\mu+\alpha_i+\beta_j+(\alpha\beta)_{ij}+\varepsilon_{ijk}. $$ The interaction term allows the contrast between levels of $A$ to vary across levels of $B$. For two-level factors, an interaction contrast is a difference in differences: $$ (\mu_{11}-\mu_{01})-(\mu_{10}-\mu_{00}). $$ If interaction is scientifically plausible, marginal main effects can hide the pattern. Report cell means, conditional contrasts, and an interaction plot with uncertainty. A nonsignificant interaction is not evidence that effects are identical; the design may simply be imprecise. Factorial designs are efficient because every unit informs more than one factor. They also distinguish a biological interaction from a nuisance block or batch effect. Blocking reduces residual variation when blocks are known before randomization. Treating a block as a biological factor changes the question, so the design and estimands must be declared first. # Contrasts, Omnibus Tests, and Multiplicity A contrast is a linear function $$ L=\sum_{i=1}^{g}c_i\mu_i, \qquad \sum_i c_i=0. $$ Examples include one treatment versus control, a linear dose trend, or the average of two treatments versus another pair. Prespecified contrasts often answer the scientific question more directly and with more power than an omnibus test followed by every pairwise comparison. For fitted coefficient vector $\hat{\boldsymbol\beta}$ and contrast vector $\mathbf c$, $$ \widehat L=\mathbf c^\mathsf{T}\hat{\boldsymbol\beta}, \qquad \widehat{\operatorname{Var}}(\widehat L) =\mathbf c^\mathsf{T}\widehat{\operatorname{Var}}(\hat{\boldsymbol\beta})\mathbf c. $$ Report $\widehat L$, an interval, the outcome scale, and the multiplicity strategy. Tukey procedures suit all pairwise comparisons under their conditions; Dunnett procedures suit several treatments versus one control. Planned contrasts need not be delayed until an omnibus test rejects, but the full family of claims still determines the appropriate error control. # Balanced and Unbalanced Data In a balanced, complete factorial design, orthogonal model terms yield the same sums of squares regardless of fitting order. With unequal cell sizes, missing cells, or correlated predictors, "the sum of squares for factor $A$" is not unique. - **Sequential sums of squares** compare each term with those entered before it. - **Partial sums of squares** compare a term after adjusting for specified other terms. - Different software labels such comparisons Type I, II, or III, but the label is less informative than the exact nested models or contrasts tested. Type III tests depend on parameterization and may target averages over cell combinations that are poorly represented or absent. Empty cells can make some main effects or interactions non-estimable. Inspect the design matrix, cell counts, rank, and the weighting used for marginal means. State the hypothesis as a contrast or explicit reduced-versus-full model comparison. # Assumptions, Diagnostics, and Alternatives The classical fixed-effects ANOVA $F$ reference law uses independent Gaussian errors with common variance and a correctly specified mean structure. | Issue | Why it matters | Check and response | |---|---|---| | Wrong experimental unit | Degrees of freedom and standard errors become artificially small | Reconstruct assignment and nesting; aggregate or model clusters | | Unequal variances | Classical pooled standard errors may be wrong | Plot group spreads; use Welch's one-way test, variance modeling, or robust covariance when justified | | Dependence | Replicates contribute less information than assumed | Use blocked, repeated-measures, cluster-aware, or mixed models | | Skewness or heavy tails | Small-sample $F$ calibration and means may be unstable | Check residuals and influence; consider transformation, robust methods, or design-respecting randomization inference | | Sparse or empty cells | Contrasts may rely on extrapolation or be non-estimable | Show cell counts, rank, and contrast weights; simplify claims | | Many post hoc comparisons | False-positive risk increases | Prespecify priorities and control the relevant error family | ANOVA is often reasonably robust to moderate nonnormality in balanced designs, but balance does not rescue pseudoreplication, severe outliers, or dependence. Rank-based tests change the estimand and generally do not provide a drop-in factorial analysis. Permutation tests require transformations that are valid under the design and null being tested. # Worked Bioinformatics Design Suppose 36 mice are randomized equally to three drugs, with sex used as a second factor and RNA-seq pathway score as the outcome. The factorial model estimates six drug-by-sex cell means. Primary estimands might be each drug-versus-control contrast averaged equally across sexes and the drug-by-sex interactions. The mouse is the experimental unit. Two library preparations and thousands of reads from one mouse are technical observations, not extra randomized units. Sequencing lane can be included as a design or processing factor if it is not confounded with drug. If every control library was sequenced in one lane and every treated library in another, drug and lane effects cannot be separated by ANOVA. Statistically, a drug contrast estimates a difference in mean pathway score on the declared scale. Biologically, it supports a treatment-associated change in that pathway under this organism, tissue, dose, and time point; it does not by itself demonstrate a molecular mechanism or generalize to other settings. # Reporting and Connections Report the design, experimental unit, allocation, cell counts, outcome scale, model formula, prespecified contrasts, estimated means and contrasts with intervals, multiplicity adjustment, residual and influence checks, missing-data handling, and sensitivity analyses. An ANOVA table alone is not a scientific result. - [Categorical predictors and contrasts](categorical-predictors-and-contrasts.qmd) develops coding-invariant group comparisons. - [Interaction terms and effect modification](interaction-terms-and-effect-modification.qmd) explains conditional effects and effect-scale dependence. - [Linear-model assumptions and diagnostics](linear-model-assumptions-and-diagnostics.qmd) links each regression assumption to the claim it supports. - [Regression uncertainty and coefficient interpretation](regression-uncertainty-and-coefficient-interpretation.qmd) derives uncertainty for coefficients, contrasts, and predictions. - [Family-wise error rate](../statistical-inference/family-wise-error-rate.qmd) distinguishes weak and strong control over families of tests. - [ANCOVA and regression-based ANOVA](ancova-and-regression-based-anova.qmd) adds continuous covariates and explicit nested-model comparisons. 1. ANOVA is a linear model for means and contrasts, not merely a table of sums of squares. 2. The independent experimental unit determines the information and degrees of freedom. 3. In factorial designs, interactions make effects conditional on other factors. 4. With imbalance, define hypotheses through contrasts or nested models instead of relying on an unexplained sums-of-squares type. 5. Effect estimates, intervals, diagnostics, and biological context matter more than an omnibus p-value alone. # References - Fisher, R. A. (1925). *Statistical Methods for Research Workers*. Oliver and Boyd. <https://psychclassics.yorku.ca/Fisher/Methods/> - Box, G. E. P., Hunter, J. S., and Hunter, W. G. (2005). *Statistics for Experimenters* (2nd ed.). Wiley. - Searle, S. R. (1971). *Linear Models*. Wiley. - Langsrud, O. (2003). ANOVA for unbalanced data: Use Type II instead of Type III sums of squares. *Statistics and Computing*, 13, 163--167. <https://doi.org/10.1023/A:1023260610025> - R Core Team. `aov`: fit an analysis of variance model; `anova.lm`: ANOVA tables for fitted linear models. *R statistical software documentation*. <https://stat.ethz.ch/R-manual/R-devel/library/stats/html/aov.html>