Computes the full two-factor within-subjects ANOVA table, main
effects A, B, and the \(A \times B\) interaction
with each effect tested against its own residual stratum and with
Greenhouse-Geisser, Huynh-Feldt, and lower-bound sphericity
adjustments applied per effect. Returns a tidy long-form
data.frame that composes with the rest of DMAR.
anova_within_two_way(data, outcome, factor_A, factor_B, subject)A data.frame with rows for each of the three
effects (A, B, A:B) crossed with each
sphericity adjustment (none, Greenhouse-Geisser,
Huynh-Feldt, lower_bound). Columns: effect,
adjustment, F_value, df_1, df_2,
p_value, epsilon, partial_eta_squared. When
an effect has too few subjects for its epsilon to be estimable
(\(n - 1\) smaller than the effect's numerator degrees of
freedom; see Details), the Greenhouse-Geisser and Huynh-Feldt rows
for that effect carry NA in epsilon, df_1,
df_2, and p_value, and a single warning names the
condition; the unadjusted and lower-bound rows are unaffected.
A data.frame in long format, with one row per
subject-by-cell observation.
Character name of the response column in data.
Character name of the first within-subjects factor.
Character name of the second within-subjects factor.
Character name of the subject-id column.
Ken Kelley [email protected]
Design. The two within-subjects factors A (with \(a\) levels) and B (with \(b\) levels) are fully crossed; every subject contributes \(a \cdot b\) observations. Each of the three fixed effects is tested against its own subject-by-effect residual stratum:
\(F_A = MS_A / MS_{A:S}\)
\(F_B = MS_B / MS_{B:S}\)
\(F_{A:B} = MS_{A:B} / MS_{A:B:S}\)
Sphericity. Each effect's univariate F-ratio assumes sphericity of its corresponding subject-by-effect residual covariance matrix. Three adjustments are reported per effect: Greenhouse-Geisser (Greenhouse & Geisser, 1959), Huynh-Feldt (Huynh & Feldt, 1976), and the lower bound \(\epsilon = 1 / df\), where \(df\) is the effect's numerator degrees of freedom; this is the smallest value \(\epsilon\) can attain, reached under maximal departure from sphericity.
Subjects needed to estimate epsilon. The Greenhouse-Geisser
epsilon for an effect with \(q\) numerator degrees of freedom is
estimated from the sample covariance matrix of \(q\) orthonormal
contrasts among the effect's cell means, a different matrix for each
effect (Maxwell, Delaney, & Kelley, 2027, Chapters 11 and 12). That
matrix has rank at most \(n - 1\), so when \(n - 1 < q\) it is
necessarily singular; the same rank deficiency makes the
multivariate approach to a within-subjects design mathematically
impossible when \(n < a\) (Maxwell, Delaney, & Kelley, 2027,
Chapter 13). The Greenhouse-Geisser formula still returns a number
in that case, but the number is an artifact of the rank deficiency
rather than an estimate: it cannot exceed \((n - 1)/q\) no matter
what the population epsilon is, even under exact sphericity, where
the population value is 1. Rather than report a value the design
cannot support, the function reports NA for the
Greenhouse-Geisser and Huynh-Feldt rows of any effect with
\(n - 1 < q\) and issues a single warning naming the condition
(car::Anova likewise declines to report the corrections for
an effect whose error matrix is singular). The unadjusted row and
the lower-bound row remain: the lower bound \(1/q\) is Geisser and
Greenhouse's a priori bound on epsilon, valid no matter how badly
sphericity is violated, and it requires no estimate of the
covariance matrix (Maxwell, Delaney, & Kelley, 2027, Chapter 11).
Per-effect partial \(\eta^2\). Computed as \(SS_\mathrm{effect} / (SS_\mathrm{effect} + SS_\mathrm{effect,\, error})\) using the appropriate subject-by-effect residual sum of squares.
Balanced data assumed. The implementation assumes a fully
balanced design (every subject observed once in every cell). When
the design is unbalanced, the function errors and recommends a
mixed-effects fit via lmer.
Sums of squares are unambiguous here. Because the design is balanced, the within-subjects factors are orthogonal and the Type I, Type II, and Type III sums of squares for each effect coincide. A Type toggle is therefore not meaningful, and the reported decomposition is unambiguous: the sum of squares attributed to each effect does not depend on the order in which terms enter the model (Maxwell, Delaney, & Kelley, 2027, Chapter 12).
Greenhouse, S. W., & Geisser, S. (1959). On methods in the analysis of profile data. Psychometrika, 24(2), 95--112.
Huynh, H., & Feldt, L. S. (1976). Estimation of the Box correction for degrees of freedom from sample data in randomized block and split-plot designs. Journal of Educational Statistics, 1(1), 69--82.
Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapters 11--13.)
anova_within, mauchly_test,
epsilon_corrections
Other within-subjects analysis:
anova_within(),
epsilon_corrections(),
mauchly_test(),
pairwise_within(),
plot_trajectories_fitted()
# 1. Balanced 3x2 within-subjects design simulated for illustration.
set.seed(113)
n_sub <- 12
grid <- expand.grid(subject = factor(1:n_sub),
A = factor(c("a1", "a2", "a3")),
B = factor(c("b1", "b2")))
grid$y <- with(grid,
3 * (A == "a2") + 1 * (A == "a3") +
2 * (B == "b2") + 1 * ((A == "a3") & (B == "b2")) +
rnorm(nrow(grid), 0, 1) +
rep(rnorm(n_sub, 0, 1.5), times = 6))
anova_within_two_way(grid, outcome = "y", factor_A = "A",
factor_B = "B", subject = "subject")
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