Multivariate Outcome — Distribution Diagnostics & Hypothesis Testing

Table of Contents

• Scope Boundary
• Workflow
• Decision Tree
• Required Inputs
• Code Structure
• Reporting Standards
• Hypothesis Tests
• Example Dataset
• Method Status
• Minimal Smoke Test
• Cross-Skill Interface

Open-source skill.

Scope Boundary

Use this skill when:

• There are multiple continuous outcomes that should be tested jointly before decomposing into separate univariate models.
• A baseline Hotelling / MANOVA analysis is appropriate before dimension reduction or multivariate modeling.

Do not use this skill when:

• There is only one dependent variable.
• The data are repeated / longitudinal, SEM-based, or primarily exploratory dimension-reduction from the outset.

Workflow

Read each step file in workflow/ before executing that step.

Step	Responsibility	Executor	Document	Input	Output
Collect	Collect Inputs	Main Agent	workflow/step01-collect-inputs.md	User input	Structured input summary
Diagnose	Check Distribution	Main Agent	workflow/step02-check-distribution.md	Prior step output	PART 1 code block
Test	Run Primary Test	Main Agent	workflow/step03-run-primary-test.md	Prior step output	PART 2-3 code blocks

Decision Tree

1. CHECK MULTIVARIATE DISTRIBUTION
   ├── Mardia skewness p >= .05 AND Mardia kurtosis p >= .05 → multivariate normal
   └── Either p < .05 → note violation, Pillai's Trace is robust

2. CHECK COVARIANCE EQUALITY
   ├── Box's M p >= .001 (strict alpha) → covariance matrices equal
   └── Box's M p < .001 → note violation, Pillai's Trace is robust

3. GROUP COMPARISON
   ├── 2 groups → Hotelling's T² + F approximation + individual ANOVAs
   └── 3+ groups → One-way MANOVA (all 4 statistics) + discriminant follow-up

Required Inputs

Role	What to collect
Outcomes (Y₁..Yₖ)	2+ continuous variable names, what they measure
Group variable	What defines groups, how many levels
Predictors	For recommendation block (not executed)
Covariates	For recommendation block (not executed)

Code Structure

PART 0: Setup & Data Loading
PART 1: Multivariate Distribution Diagnostics → plot_01_scattermatrix.png
PART 2: Primary Multivariate Test             → plot_02_groupmeans.png
PART 3: Recommendation Block                  → text listing additional analyses available

Reporting Standards

1. p-values: "< .001" not "0.000"; exact to 3 decimals otherwise
2. Effect sizes: partial eta-squared per DV in follow-up ANOVAs
3. 95% CIs: always for mean differences in univariate follow-ups
4. Degrees of freedom: always with F statistics
5. Sample size: final analytic N
6. Decimal places: 2 for M/SD, 3 for p and effect sizes
7. Non-significance: "not statistically significant at alpha = .05" — never "no effect"
8. MANOVA statistics: report all four (Pillai's V, Wilks' Lambda, Hotelling-Lawley T, Roy's theta) with F approximation and p
9. Box's M: report with F approximation and p, use strict alpha (.001). Caveat: Box's M is notoriously sensitive to deviations from multivariate normality — a significant result may reflect non-normality rather than true covariance heterogeneity. When Box's M rejects AND Mardia/Henze–Zirkler also indicate non-normality, do NOT conclude covariance heterogeneity on Box's M alone. Instead: (a) report Pillai's Trace as the primary MANOVA statistic (most robust to assumption violations), and (b) consider log/Box–Cox transformation or a robust MANOVA variant before interpreting Box's M as a substantive result.

Hypothesis Tests

Scenario	Test	Follow-Up
2 independent groups	Hotelling's T²	Individual Welch's t per DV
3+ independent groups	One-way MANOVA	Individual ANOVAs per DV

Paired/repeated multivariate → vera-data-repeated-reviewing. Single DV → vera-data-continuous-reviewing.

Example Dataset

R built-in iris: outcomes = Sepal.Length, Sepal.Width, Petal.Length, Petal.Width; group = Species. Python: from sklearn.datasets import load_iris.

Method Status

Status	Methods
Implemented in this skill	Multivariate normality / covariance checks, Hotelling's T-squared, one-way MANOVA, and univariate follow-up triggers
Implemented downstream in vera-data-multivariate-generating	MANCOVA, DFA, profile analysis, canonical correlation, PCA, multivariate regression, and exploratory trees
Out of scope in this open-source baseline	Repeated multivariate models and latent-variable multivariate frameworks

Minimal Smoke Test

• Smoke-test prompt: "Run vera-data-multivariate-reviewing on iris, using the four measurement columns as outcomes and Species as the grouping variable. Produce the standard baseline artifacts."

Cross-Skill Interface

Output:
├── code_r      → .R script
├── code_python → .py script
├── figures/    → 2 PNGs (scattermatrix + group means)
└── recommendations → text block (additional analyses available)

Next step: Invoke vera-data-multivariate-generating from this skillset to run the full pipeline (additional tests, subgroup analysis, modeling, manuscript generation). See ../../CROSS-SKILL-INTERFACE.md for the shared handoff contract.