DMAR (version 1.0.0)

manova_split_plot: Mixed-Design Multivariate ANOVA With All Four Test Statistics

Description

Computes the multivariate analysis of variance for a mixed-design (one between-subjects factor and one within-subjects factor), returning Wilks's \(\Lambda\), Pillai's trace, Hotelling-Lawley trace, and Roy's largest root, each with the associated F-approximation, degrees of freedom, and p-value. The three effects, between-subjects (A), within-subjects (B), and the interaction (\(A \times B\)), are tested separately. Wraps Anova and returns the result in the tidy DMAR style.

Usage

manova_split_plot(data, within, between, ss_type = 3L)

Value

A data.frame with rows for each of the three effects crossed with each of the four multivariate statistics, plus one trailing row recording the sum-of-squares type. Columns:

effect, statistic_name, statistic_value,

F_approx, df_1, df_2, p_value. The final row has effect == "sum_of_squares_type" and carries the chosen type (1, 2, or 3) in its numeric

statistic_value; its remaining numeric columns are

NA, so the statistic_value column stays numeric.

Arguments

data

A data.frame in wide format, one row per subject, with the within-subjects measurements in separate columns and the between-subjects factor as one additional column.

within

Character vector of column names holding the repeated measures values (one column per level of the within- subjects factor). Must be in the canonical level order.

between

Character name of the between-subjects factor column in data.

ss_type

The sum-of-squares type for the between-subjects effects, passed through to Anova. Accepts the integers 1, 2, or 3 or the equivalent Roman- numeral strings "I", "II", or "III", and defaults to 3L (Type III). The chosen type is recorded in the returned table (see Value). Type I (1 or "I") is not available for this mixed design, because Anova computes only Type II and Type III sums of squares for the multivariate repeated measures path; requesting it raises an error.

Author

Ken Kelley [email protected]

Details

The four statistics. For an effect with \(H\) and \(E\) hypothesis- and error-cross-products matrices:

• Wilks's \(\Lambda = \det(E) / \det(E + H)\)

• Pillai's trace \(V = \mathrm{tr}(H (E + H)^{-1})\)

• Hotelling-Lawley trace \(T_0^2 = \mathrm{tr}(H E^{-1})\)

• Roy's largest root \(\theta = \lambda_1(H E^{-1})\)

When to use which. Pillai's trace is the most robust to departures from the multivariate normal / homogeneous-covariance assumptions. Wilks's \(\Lambda\) is the most widely reported. Roy's largest root is the most powerful when the alternative concentrates on a single dimension. The four statistics agree exactly when the effect has 1 numerator degree of freedom.

Sum-of-squares type. The between-subjects effects are computed by Anova using the sum-of-squares type selected through ss_type (Type III by default). Type II conditions each effect on the others that do not contain it, and Type III conditions each effect on every other effect in the model; for a single between-subjects factor the two coincide, and they can differ once additional between-subjects terms are present. Type I, the sequential decomposition, is not available here: Anova computes only Type II and Type III for the multivariate repeated measures path. The type in force is reported in the returned table so the analysis is self-documenting.

Dependency. Requires the car package on CRAN.

References

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 7 on higher-order designs and Chapter 14.)

Rencher, A. C., & Christensen, W. F. (2012). Methods of multivariate analysis (4th ed.). Wiley.

See Also

Anova, manova, anova_within_two_way

Other hypothesis tests: adjusted_means(), ancova(), anova_within(), ci_dunnett(), ci_scheffe(), ci_tukey_kramer(), compare_cov_structures(), contrast_test(), correlations_test(), equivalence_r(), equivalence_smd(), factorial_anova(), mauchly_test(), mixed_anova(), obrien_test(), pairwise_within(), randomization_test(), randomization_test_paired(), regions_of_significance(), simple_effects_AB(), summary_t_test(), welch_t()

Other mixed models: R2_mixed_effects(), R2_mixed_effects_decomposition(), icc_lmer(), mixed_anova(), ss_aipe_mixed_effects(), ss_aipe_mixed_effects_sensitivity(), ss_power_mixed_effects(), ss_power_split_plot_anova()

Examples

# Two groups of ten measured at three times. Both groups start at the
# same place and rise over time, and group B rises twice as fast, so the
# data generating means differ in slope as well as in level.
set.seed(113)
n_per <- 10
d <- data.frame(
  subject = factor(1:(2 * n_per)),
  group   = factor(rep(c("A", "B"), each = n_per)),
  t1      = c(rnorm(n_per, 0,   1), rnorm(n_per, 0,   1)),
  t2      = c(rnorm(n_per, 0.4, 1), rnorm(n_per, 0.8, 1)),
  t3      = c(rnorm(n_per, 0.8, 1), rnorm(n_per, 1.6, 1))
)

# Rows are the between-subjects effect (labeled A), the within-subjects
# effect (labeled B), and their interaction, each with all four
# multivariate criteria, followed by a row recording the sum-of-squares
# type. The multivariate tests make no sphericity assumption, which is
# what recommends them over the univariate repeated measures F and its
# epsilon corrections.
manova_split_plot(d, within = c("t1", "t2", "t3"), between = "group")