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Analysis of Multivariate Data and Repeated Measures Designs with the R Package MANOVA.RM
Sarah Friedrich, Frank Konietschke, Markus Pauly
TL;DR
Conventional methods for multivariate and repeated-measures data may rely on distributional or covariance assumptions that are difficult to justify, while accessible software for robust inference is needed. The paper presents MANOVA.RM, an R package implementing inference methods and a graphical interface; its procedures are designed to address covariance heterogeneity, with MATS also addressing covariance singularity.
Problem
Existing procedures may assume multivariate normality or specific covariance structures, motivating accessible software for inference when those assumptions are unsuitable.
Source: §1 · p. 3Method
MANOVA.RM implements Wald-type, ANOVA-type, and modified ANOVA-type tests with asymptotic, approximation, or resampling critical values for MANOVA and repeated-measures designs.
Source: §2 · p. 5, §4 · pp. 21–22Results
The procedures are designed to address covariance heterogeneity, and MATS is constructed to accommodate covariance singularity.
Source: §4 · p. 22Takeaways & Limitations
MANOVA.RM provides a flexible toolkit for hypotheses about factorial MANOVA effects and whole- and sub-plot effects in repeated-measures designs.
Source: §4 · p. 22The model assumes independent, identically distributed errors within groups, and WTS procedures additionally require positive-definite covariance matrices and finite fourth moments.
Source: §2 · p. 4
Abstract
from arXiv · showhide
The numerical availability of statistical inference methods for a modern and robust analysis of longitudinal- and multivariate data in factorial experiments is an essential element in research and education. While existing approaches that rely on specific distributional assumptions of the data (multivariate normality and/or characteristic covariance matrices) are implemented in statistical software packages, there is a need for user-friendly software that can be used for the analysis of data that do not fulfill the aforementioned assumptions and provide accurate p-value and confidence interval estimates. Therefore, newly developed statistical methods for the analysis of repeated measures designs and multivariate data that neither assume multivariate normality nor specific covariance matrices have been implemented in the freely available R-package MANOVA.RM. The package is equipped with a graphical user interface for plausible applications in academia and other educational purpose. Several motivating examples illustrate the application of the methods.
1 Introduction
The introduction motivates unified inference for multivariate and repeated-measures data when conventional distributional or covariance assumptions may be unjustifiable. It presents MANOVA.RM as a freely available R package implementing robust tests and accessible analyses for these designs.
1 Introduction
- Repeated-measures designs compare the same quantity across conditions or times, whereas MANOVA analyzes different, non-combinable quantities measured on the same subjects.The paper distinguishes their questions of interest: repeated-measures analyses can test sub-plot effects, while MANOVA typically tests effects of observed factors on multivariate outcomes.Source: §1 · p. 2
- Conventional procedures may rely on multivariate normality or specific covariance structures that are difficult to verify in small samples with many dimensions.The paper notes that such assumptions may lead to inflated type-I error rates.Source: §1 · p. 3
- MANOVA.RM implements Wald-type, ANOVA-type, and modified ANOVA-type statistics for factorial MANOVA and repeated-measures analyses.The introduction identifies these methods as suitable for the multivariate Behrens-Fisher setting, without assuming equal group covariance matrices.Source: §1 · p. 3
- The package offers flexible factorial testing, with sampling distributions approximated by resampling techniques that can be applied to small samples.Its methods are described as appropriate for the Behrens-Fisher situation.Source: §1 · p. 3
- MANOVA.RM is freely available and presents p-values, test statistics, descriptive summaries, and component-wise two-sided confidence intervals.Its MANOVA function also supports multivariate contrasts and confidence-ellipsoid plots.Source: §1 · p. 3
- The paper illustrates the package with examples and describes an optional graphical user interface for its use.The paper is organized to present the statistical model and inference procedures before examples and discussion.Source: §1 · p. 4
2 Statistical model and inference methods
The paper formulates MANOVA and repeated-measures analyses in a common multivariate linear-model framework, then describes test statistics and resampling procedures for varied factorial designs. It also specifies which procedures and design structures have particular conditions or scope boundaries.
2 Statistical model and inference methods
- The model represents each multivariate observation as a group mean plus an error vector; errors are independent and identically distributed within groups, while WTS procedures additionally require positive-definite covariance matrices and finite fourth moments.The observations may be arranged in higher-way factorial structures by subdividing group or measurement indices.Source: §2 · p. 4
- For repeated-measures tests, wild permutation pools components without preserving group membership or dependencies and is reported to control type-I error well in simulations.This permutation procedure is restricted to repeated-measures designs; applying it to MANOVA would mix differently scaled components.Source: §2 · p. 5
- The repeated-measures ATS uses an enhanced F-approximation and can be applied with singular covariance matrices, although it is known to be rather conservative.ATS is not invariant to scale transformations, limiting its use to repeated-measures settings.Source: §2 · p. 5
- MATS provides a MANOVA method applicable with singular covariance matrices and is invariant under component-wise scale transformations.The paper also describes asymptotic model-based and wild bootstrap approaches for improving finite-sample performance.Source: §2 · p. 5
2.1 Special Designs and Hypotheses
- The methods cover one-way and crossed two-way factorial effects, including main effects and interactions, as well as hierarchically nested designs.Only balanced hierarchically nested two-way designs are implemented; higher-way nested and arbitrary crossed layouts are also supported.Source: §2.1 · p. 6
- Repeated-measures and split-plot designs allow hypotheses about sub-plots, including profile tests of parallel or flat response patterns.The paper also notes that more complex factorial structures can be represented among repeated measurements.Source: §2.1 · pp. 6–7
3 Examples
The paper demonstrates the RM and MANOVA functions through examples and notes that the package also offers an optional graphical user interface. The examples cover both repeated-measures designs and multivariate data.
3 Examples
- The paper uses several examples to demonstrate the RM and MANOVA functions.The examples cover repeated-measures designs and multivariate data.Source: §3 · p. 7
- An optional graphical user interface accompanies the package.The interface is based on RGtk2.Source: §3 · p. 7
- The paper explains the GUI in detail in Section 3.3.Source: §3 · p. 7
3.1 Repeated Measures Designs
The RM function supports repeated-measures analysis with summaries, confidence intervals, and plotting. Its examples show significant effects in the oxygen-consumption data but only whole-plot effects in the EEG data.
3.1 Repeated Measures Designs
- RM accepts long-format data and returns test results, descriptive statistics, and confidence intervals for factor-level combinations.The user specifies the subject variable and number of sub-plot factors; confidence intervals can use t-quantiles or resampled-WTS quantiles.Source: §3.1 · p. 7
3.1.1 Example 1: One whole-plot and two sub-plot factors
- In the oxygen-consumption example, all factors and the Group-by-Time interaction are significant.For small samples, the text recommends resampling because the asymptotic WTS can be liberal and the ATS conservative; permutation is not implemented for ATS.Source: §3.1.1 · p. 9
3.1.2 Example 2: Two sub-plot and two whole-plot factors
- In the EEG example, sex and diagnosis are significant at α = 0.01, while sub-plot factors and interactions are not.The model includes two whole-plot and two sub-plot factors.Source: §3.1.2 · pp. 9, 12
3.1.3 Plotting
- RM plots display means with t-quantile confidence intervals and allow factor selection and customization in higher-way layouts.The examples plot sex, sex-by-diagnosis, and sex-by-diagnosis-by-feature effects.Source: §3.1.3 · p. 12
3.2 MANOVA Design
MANOVA and MANOVA.wide support multivariate analyses of crossed or nested designs, with resampling and confidence-region tools. Examples illustrate differing test decisions, joint outcomes, and nested-design input options.
3.2 MANOVA Design
- MANOVA and MANOVA.wide support crossed or nested multivariate designs; MATS also applies with singular covariance matrices and is scale-invariant.Only balanced nested designs with up to three factors are implemented, and designs combining crossed and nested factors are not supported.Source: §3.2 · p. 12
- MANOVA uses long-format responses and requires a subject variable, whereas MANOVA.wide takes wide-format responses without one.The wide-format formula binds the response variables with cbind.Source: §3.2 · p. 13, §3.2.1 · pp. 15–16
3.2.1 Data Example MANOVA: Two crossed factors
- In the EEG example, MATS detects a significant sex effect that the parametric-bootstrap WTS does not.Because some groups were very small, the example restricts analysis to two factors at a time.Source: §3.2.1 · pp. 13, 17
3.2.2 Confidence Regions
- For the water data, mortality and hardness differ significantly between northern and southern towns.The example analyzes measurements from 61 cities in England and Wales.Source: §3.2.2 · pp. 17–18
- The confidence-region output gives an ellipsoid center and axes; plotting is available only for two-dimensional outcomes.The water example uses mortality and hardness as its two outcomes.Source: §3.2.2 · pp. 17–18
3.2.3 Nested design
- The nested-design example sets nested.levels.unique to TRUE and obtains the same test statistics from wide- and long-format analyses.The reported statistics are 12.296 for season and 15.064 for season:site.Source: §3.2.3 · pp. 18, 20
3.3 Graphical user interface
The package provides GUI entry points for repeated-measures and multivariate analyses in long or wide format. Users can configure analysis and plotting options through the interface.
3.3 Graphical user interface
- GUI.RM(), GUI.MANOVA(), and GUI.MANOVAwide() launch interfaces for repeated-measures, long-format multivariate, and wide-format multivariate analyses.The GUI requires RGtk2 to be installed.Source: §3.3 · p. 21
- Users can specify data, formula, iterations, significance level, and resampling method in the GUI.The available resampling choices are permutation for RM, parametric bootstrap, and wild bootstrap.Source: §3.3 · p. 21
- The plotting GUI offers a one-way layout without factor selection and a two-way layout with interaction plotting.Source: Figure 5 · p. 21
4 Discussion and Outlook
MANOVA.RM implements robust inference procedures for factorial MANOVA and repeated-measures designs, with a graphical interface and planned extensions. Its methods avoid standard distributional and covariance assumptions and address covariance heterogeneity or singularity.
4 Discussion and Outlook
- Resampling-based critical values are recommended for small to moderate samples among the package’s Wald-, ANOVA-, and modified ANOVA-type tests.The tests use critical values derived from asymptotic considerations, approximations, or resampling.Source: §4 · p. 21
- The methods require neither multivariate normality nor specific covariance structures and are constructed to handle covariance heterogeneity across groups or covariance singularity with MATS.Source: §4 · pp. 21–22
- The toolbox supports main and interaction effects in factorial MANOVA and whole- and sub-plot effects in repeated-measures designs with complex factorial structures.Source: §4 · p. 22
- A GUI lets users specify data location, formula, and resampling approach, while covariates and subsequent multiple comparisons are described as possible future additions.Covariate implementation was under theoretical investigation, and regular package updates were planned.Source: §4 · p. 22, Figure 4 · p. 22