One Way Within-Subjects ANOVA With Sphericity Diagnostics and Corrections

Performs the univariate one-way within-subjects F test together with Mauchly's test of sphericity and the three standard \(\varepsilon\)-corrected p-values (Greenhouse-Geisser, Huynh-Feldt, and lower-bound). Returns everything in a single tidy data.frame so the user can decide which adjustment to report.

Usage

anova_within(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 data.frame with one row per reported F test: adjustment ("none", "Greenhouse-Geisser", "Huynh-Feldt", "lower_bound"), F_value, df_1, df_2, p_value, and epsilon (the correction factor used; NA for the unadjusted row). attr(<output>, "mauchly") contains the row from mauchly_test, and the partial \(\eta^2\) is attached as attr(<output>, "partial_eta_squared").

Details

The unadjusted within-subjects F statistic is the same regardless of sphericity; corrections shrink the numerator and denominator degrees of freedom by a factor of \(\hat\varepsilon \in [1/(k - 1),\, 1]\), and the p-value is recomputed against the adjusted reference F distribution. When Mauchly's test rejects, prefer the Huynh-Feldt-corrected p-value (less conservative than Greenhouse-Geisser).

For multi-factor within-subjects designs or mixed designs, fit the model with stats::aov(... + Error(id/within)) or with lme4::lmer() directly.

References

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

See also

mauchly_test, epsilon_corrections, aov

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

Other hypothesis tests: adjusted_means(), ancova(), ci_dunnett(), ci_scheffe(), ci_tukey_kramer(), compare_cov_structures(), contrast_test(), correlations_test(), equivalence_r(), equivalence_smd(), factorial_anova(), manova_split_plot(), mauchly_test(), 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

# Simulated within-subjects data with no real effect.
set.seed(113)
Y <- matrix(rnorm(20 * 4), nrow = 20)
anova_within(Y)
#>  adjustment         F_value df_1 df_2 p_value epsilon
#>  none               0.618   3    57   0.6062  <NA>   
#>  Greenhouse-Geisser 0.618   2.36 44.9 0.5696  0.787  
#>  Huynh-Feldt        0.618   2.72 51.7 0.5911  0.906  
#>  lower_bound        0.618   1    19   0.4415  0.333  

# Built-in within-subjects example: nlme::Orthodont (distance ~ age).
res <- anova_within(nlme::Orthodont,
                    id = "Subject", time = "age", outcome = "distance")
res
#>  adjustment         F_value df_1 df_2 p_value  epsilon
#>  none               38      3    78   < 0.0001 <NA>   
#>  Greenhouse-Geisser 38      2.63 68.4 < 0.0001 0.877  
#>  Huynh-Feldt        38      2.95 76.8 < 0.0001 0.984  
#>  lower_bound        38      1    26   < 0.0001 0.333  
attr(res, "mauchly")
#>  W     statistic df p_value n_subjects n_levels method                      
#>  0.758 6.85      5  0.2326  27         4        Mauchly's test of sphericity
attr(res, "partial_eta_squared")
#> [1] 0.5940013