Count 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:

• The outcome is a non-negative count or rate and the first task is distributional triage plus one primary group comparison.
• Exposure / offset handling can be represented as a simple baseline choice between count and rate formulations.

Do not use this skill when:

• The outcome is bounded binary, continuous, ordered categorical, survival time, or repeated within subject.
• The question already requires zero-inflated, hurdle, or multivariable count modeling as the primary analysis.

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 COUNT TYPE
   ├── Exposure variable exists → rate model path (offset = log(exposure))
   └── No exposure variable → count model path

2. CHECK DISTRIBUTION
   ├── Variance ≈ Mean (ratio < 1.5) → Poisson
   ├── Variance >> Mean (ratio ≥ 1.5) → Negative Binomial
   └── Excess zeros (>20%) → flag for Zero-Inflated models
   # 1.5 dispersion threshold: common rule of thumb; equivalent to a Pearson-chi²/df ≈ 1.5
   # cutoff. Some sources use 1.2; others use a test-based decision (overdispersion LR test).
   # 1.5 is chosen here as a conservative trigger that flags the full analysis skill.
   # 20% zero threshold: empirical convention — below this, Poisson/NB handle zeros adequately;
   # above this, zero-inflation is often driven by a distinct generating process and warrants ZIP/ZINB.

3. GROUP COMPARISON
   ├── 2 groups → Poisson/NB rate test + IRR + Mann-Whitney U
   └── 3+ groups → Poisson/NB regression with group factor + LR test + Kruskal-Wallis

Required Inputs

Role	What to collect
Outcome (Y)	Variable name, what it counts, units
Group variable	What defines groups, how many levels
Exposure/offset	Time at risk, population, area — or none
Predictors	For recommendation block (not executed)
Covariates	For recommendation block (not executed)

Code Structure

PART 0: Setup & Data Loading
PART 1: Distribution Diagnostics → plot_01_count_distribution.png
PART 2: Primary Hypothesis Test  → plot_02_mean_counts_[var].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: incidence rate ratio (IRR) with 95% CI — always alongside p
3. Overdispersion: report variance/mean ratio with interpretation
4. Rate: "X.XX events per [unit] of exposure" when applicable
5. Degrees of freedom: always with chi-square and LR test statistics
6. Sample size: final analytic N
7. Decimal places: 2 for means/SDs, 3 for p and IRR, 2 for rate ratios
8. Non-significance: "not statistically significant at α = .05" — never "no effect"

Hypothesis Tests

Scenario	Equidispersed	Overdispersed
2 independent groups	Poisson rate test	Negative binomial test
3+ independent groups	Poisson regression + LR test	NB regression + LR test

Nonparametric confirmation: Mann-Whitney U (2 groups) or Kruskal-Wallis (3+).

Paired/repeated designs → vera-data-repeated-reviewing.

Example Dataset

R built-in warpbreaks: outcome = breaks (count), predictors: wool (A/B), tension (L/M/H). Python: sm.datasets.get_rdataset("warpbreaks").data (with offline fallback to bundled examples/warpbreaks.csv).

Method Status

Status	Methods
Implemented in this skill	Overdispersion and zero-inflation triage, Poisson / negative-binomial primary testing, incidence-rate-ratio reporting
Implemented downstream in vera-data-count-generating	ZIP, ZINB, hurdle models, subgroup rate-ratio analysis, and exploratory tree-based count comparisons
Out of scope in this open-source baseline	Longitudinal count models, causal count estimands, and count families not explicitly named here

Minimal Smoke Test

• Smoke-test prompt: "Run vera-data-count-reviewing on warpbreaks, using breaks as the outcome and wool as the primary grouping variable. Produce the standard baseline artifacts."

Cross-Skill Interface

Output:
├── code_r      → .R script
├── code_python → .py script
├── figures/    → 2 PNGs (count distribution + mean counts bar chart)
└── recommendations → text block (additional analyses available)

Next step: Invoke vera-data-count-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.