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Mauchly's Test of Sphericity for a One-Way Within-Subjects Design

Source: R/mauchly_test.R
mauchly_test.Rd

Tests the null hypothesis that the covariance matrix of the orthonormal contrasts among the \(k\) repeated measurements is proportional to the identity (the sphericity assumption underlying univariate repeated measures F-tests).

Usage

mauchly_test(x, id = NULL, time = NULL, outcome = NULL)

Arguments

x

Either an \(n \times k\) numeric matrix or data.frame (rows = subjects, columns = repeated measurements); or a long-format data.frame together with id, time, and outcome column names.

id

Column name in x identifying the subject when x is in long format (NULL otherwise).

time

Column name in x identifying the within-subjects factor level when x is in long format (NULL otherwise).

outcome

Column name in x identifying the dependent variable when x is in long format (NULL otherwise).

Value

A one-row data.frame with columns W (Mauchly's statistic), statistic (the chi square approximation), df, p_value, n_subjects, n_levels, and method.

Details

Sphericity is the assumption that the variances of all pairwise differences among the \(k\) levels are equal, equivalently, that the covariance matrix \(\Sigma_C\) of any orthonormal set of \(k - 1\) contrasts among the levels is proportional to the identity. Mauchly's (1940) test statistic is $$W = \frac{\det(\hat\Sigma_C)}{\bigl(\mathrm{tr}(\hat\Sigma_C) / (k - 1)\bigr)^{k - 1}},$$ and the chi square approximation $$X^2 = -\,m \,\log W \quad \mathrm{with}\ m = (n - 1) - \frac{2(k - 1)^2 + (k - 1) + 2}{6\,(k - 1)}$$ has approximately \((k - 1)k/2 - 1\) degrees of freedom under \(H_0\). The reported p-value uses Box's (1949) second-order correction, a weighted combination of the chi square tails on \((k - 1)k/2 - 1\) and \((k - 1)k/2 + 3\) degrees of freedom, which improves the first-order approximation in small samples; this matches mauchly.test.

When sphericity is rejected, the univariate F test is liberal; correct using the Greenhouse-Geisser, Huynh-Feldt, or lower-bound epsilon adjustments via epsilon_corrections or directly via anova_within.

The test is only defined for \(k \ge 3\); with \(k = 2\), sphericity is trivially true and the function returns W = 1, p = 1.

References

Mauchly, J. W. (1940). Significance test for sphericity of a normal \(n\)-variate distribution. Annals of Mathematical Statistics, 11(2), 204–209.

Box, G. E. P. (1949). A general distribution theory for a class of likelihood criteria. Biometrika, 36(3/4), 317–346.

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 11 for sphericity in within-subjects designs.)

See also

epsilon_corrections, anova_within

Other within-subjects analysis: anova_within(), anova_within_two_way(), epsilon_corrections(), pairwise_within(), plot_trajectories_fitted()

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(), manova_split_plot(), mixed_anova(), obrien_test(), pairwise_within(), randomization_test(), randomization_test_paired(), regions_of_significance(), simple_effects_AB(), summary_t_test(), welch_t()

Author

Ken Kelley kkelley@nd.edu

Examples

# Wide-format example: simulated within-subjects data with 4 levels.
set.seed(113)
Y <- matrix(rnorm(20 * 4), nrow = 20)
mauchly_test(Y)
#>  W     statistic df p_value n_subjects n_levels method                      
#>  0.597 9.13      5  0.1045  20         4        Mauchly's test of sphericity

# Long-format example using built-in nlme::Orthodont (4 ages per subject).
mauchly_test(nlme::Orthodont, id = "Subject", time = "age",
             outcome = "distance")
#>  W     statistic df p_value n_subjects n_levels method                      
#>  0.758 6.85      5  0.2326  27         4        Mauchly's test of sphericity