PCM3 Model Design And Phylogenetic Regression¶

PCM3 follows one primate analysis through categorical coding, a log response, intercept removal, interactions, predictor standardization, and Martins phylogenetic covariance. Each operation changes either the estimand or its uncertainty. The Evidence Book has decomposed that path into eight governed claims, but no bundle yet retains comparable lecture and Bijux observations; every current bundle verdict is not_comparable.

Why PCM3 Matters¶

PCM3 tests whether a multivariable comparative analysis retains meaning while its design is deliberately re-expressed. It separates two questions that are often confused:

Scientific question	Quantity that must remain identifiable
how does log longevity vary with group size within each dimorphism category?	category-specific intercepts and slopes derived from the exact design columns
how does phylogenetic covariance change uncertainty and fit for that same mean structure?	Martins covariance parameterization, tree identity, likelihood, and diagnostics

The sequence is valuable evidence architecture even before numerical closure: each formula transformation has its own claim address, required outputs, and comparison keys. The present record proves that the proposed reproduction is finite and auditable. It does not prove that any coefficient, test, or covariance estimate has been reproduced.

The First Comparable Artifact Is The Design, Not The Coefficient Table¶

PCM3 should close from the population outward. The first numerical lane must retain the ordered admitted species, response vector, factor levels, contrast matrix, transformed predictor columns, and exclusions for both executions. Those records establish whether a coefficient key refers to the same vector.

Only after design identity aligns should the lane compare coefficients, uncertainty, likelihood, and residuals. Beginning with printed coefficient tables would hide exactly the differences PCM3 is designed to expose: another reference level, a different complete-case population, a differently centered predictor, or a test of another linear combination.

A useful first closure is therefore modest but decisive: reproduce one baseline GLS design and fit with a complete row and column ledger. That result would not close the interaction, standardization, or Martins-covariance claims, but it would establish the shared foundation those later bundles consume.

Separate Reparameterization From A New Estimand¶

An intercept model with one reference-group coefficient and a no-intercept model with one coefficient per group can span the same fitted means. Their raw coefficient tables are not row-wise comparable; the correct comparison first maps both tables to shared group intercepts or shared predictions. Likewise, centering a predictor relocates the intercept while preserving fitted values when the rest of the design is unchanged.

An interaction, log response, altered complete-case population, or different marginal/sequential hypothesis is not merely a new display basis. It changes the target quantity or tested statement. PCM3 needs the design and hypothesis ledgers to classify which kind of change occurred before any tolerance is applied.

flowchart LR
    raw["longevity ~ group size<br/>+ dimorphism"]
    log["log longevity<br/>same main effects"]
    groups["no intercept<br/>group-specific levels"]
    interaction["group-specific<br/>slopes"]
    scaled["standardized group size<br/>recentered intercept"]
    pgls["Martins covariance<br/>estimated on the tree"]

    raw --> log --> groups --> interaction --> scaled --> pgls

The Source Model Sequence¶

The checked-in source program names the intended statistical operations. Its formulas make the coefficient semantics concrete:

Source fit	Formula or operation	Scientific role
`gls2`	`longevity ~ social_group_size + sex_dimorphism`	baseline continuous and categorical GLS
`gls3`	`log(longevity) ~ social_group_size + sex_dimorphism`	response-scale repair and marginal/sequential tests
`gls4`	`log(longevity) ~ sex_dimorphism - 1 + social_group_size`	group-specific intercept estimates
`gls5`	main effects plus `social_group_size:sex_dimorphism`	conditional group slopes
`gls6`	`sex_dimorphism + social_group_size:sex_dimorphism - 1`	explicit group intercepts and slopes
`gls7`	interaction model using standardized group size	intercept at mean group size and slopes per standard deviation
`pgls5`	scaled interaction model with estimated `corMartins`	same design under phylogenetic residual covariance

This table describes the source analysis, not verified Bijux results. No coefficient, p-value, likelihood, covariance parameter, or diagnostic from this sequence is currently a governed parity observation.

Governed Claim Ledger¶

Bundle	Governed claim	Current verdict	Observation gap
`evidence-001`	source intake and workspace alignment	`not_comparable`	aligned source objects, taxa, and row identity
`evidence-002`	baseline GLS with a categorical predictor	`not_comparable`	design columns, coefficients, fit, and residuals
`evidence-003`	transformed response and model tests	`not_comparable`	transformed rows plus marginal and sequential tests
`evidence-004`	intercept-free group parameterization	`not_comparable`	group estimates under the same factor ordering
`evidence-005`	interaction coefficient interpretation	`not_comparable`	aligned interaction columns and conditional slopes
`evidence-006`	scaling and intercept semantics	`not_comparable`	center, scale, and projected coefficients
`evidence-007`	Martins covariance phylogenetic regression	`not_comparable`	covariance identity, fitted parameter, and model outputs
`evidence-008`	post-correction diagnostics and tests	`not_comparable`	comparable residual and hypothesis-test observations

Each bundle contains an explicit claim, source locators, required artifact inventory, limitations, and reviewer summary. Each checks.json currently has an empty primary_output_paths list. Those files make the absence auditable; they do not turn contract structure into scientific evidence.

Read Coefficients From The Design Matrix¶

Let x be group size and B indicate one dimorphism category. With an intercept and interaction, the two fitted lines are:

reference category:  beta_0 + beta_x * x
category B:          (beta_0 + beta_B) + (beta_x + beta_x:B) * x

beta_B is a difference at x = 0, not an average group effect. beta_x:B is a difference between slopes, not the slope for category B. After standardization, zero denotes the recorded mean of group size and one unit denotes the recorded standard deviation. After removing the intercept, the category coefficients become category-specific intercepts rather than contrasts against a reference.

Therefore, coefficient parity requires the design matrix—not merely the printed formula—to agree. Factor level order, contrast encoding, missing-value exclusions, centering, scaling, and term-column mapping are part of the result.

See How Each Design Changes The Estimand¶

For a two-level factor with indicator B, the relevant row vectors make the changes visible:

Model form	Design row	Meaning changed by the operation
baseline main effects	`[1, x, B]`	one reference intercept, one shared slope, one category contrast
no intercept	`[1-B, B, x]`	separate category intercepts replace the reference contrast
interaction	`[1, x, B, x*B]`	the category difference may vary with group size
no-intercept interaction	`[1-B, B, x(1-B), xB]`	each category owns an explicit intercept and slope
standardized interaction	replace `x` with `z = (x-center)/scale`	intercepts move to the recorded center; slopes become effects per recorded scale unit

Column order alone is not the contract: the evidence needs stable term keys and the actual column values after exclusions. Two tools can print the same formula while using different reference levels, missing-row populations, or standardization denominators. In that state, numerical coefficient comparison has no shared estimand and must remain not_comparable.

Marginal And Sequential Tests Are Different Questions¶

The source requests both sequential and marginal ANOVA tables. A sequential test evaluates a term after terms entered before it; its result depends on term order. A marginal test evaluates a term conditional on the other retained terms under the tool’s declared hypothesis convention. Their labels, degrees of freedom, and test statistics cannot be interchanged.

An executable comparison must retain the tested hypothesis matrix or an equivalent term-level definition. Matching p-values without matching the hypothesis is not parity.

Phylogenetic Covariance Changes The Fit, Not The Design¶

The final source model keeps the scaled interaction design and adds a Martins covariance initialized at 0.5 and estimated on primatetree. The coefficient names can remain identical while estimates, uncertainty, likelihood, and residuals change because observations no longer receive independent weight.

A comparable execution must align:

the trait rows and exact species-to-tip ordering;
tree topology, branch lengths, rooting, and scale;
the Martins covariance definition, bounds, initialization, and fixed versus estimated treatment;
maximum-likelihood convention and optimizer outcome;
normalized versus response-scale residual definitions.

Runtime support for PGLS and interaction coefficients shows that these components exist. It does not establish that this lecture model was executed under the same identity.

Comparable Execution Requirements¶

flowchart TB
    source["Retain source outputs<br/>and analysis population"]
    bijux["Execute Bijux with<br/>the same model identity"]
    normalize["Map terms, scales,<br/>tests and residual types"]
    compare["Record observations<br/>and tolerances"]
    adjudicate["Issue one verdict<br/>per governed claim"]

    source --> normalize
    bijux --> normalize --> compare --> adjudicate

For each bundle, closure requires primary reference and Bijux outputs, their input identities, a normalization rule where names or encodings differ, and machine-adjudicated observations. The evidence must include failures and missing values; a table of successful coefficients alone cannot establish the denominator.

Close A Fit As A Dependency Packet¶

Each fitted model should produce one joined packet whose components can be reviewed independently and invalidated together when an upstream identity changes.

flowchart LR
    population["Ordered analytical<br/>population"] --> design["Response and<br/>design matrix"]
    tree["Tree and branch<br/>identity"] --> covariance["Covariance<br/>matrix and parameters"]
    design --> fit["Fit state<br/>and coefficients"]
    covariance --> fit
    fit --> hypotheses["Hypothesis tests<br/>and uncertainty"]
    fit --> diagnostics["Residual and<br/>influence rows"]
    hypotheses --> claim["Comparable observations<br/>and claim verdict"]
    diagnostics --> claim

Packet component	Required review fields
population	included and excluded species, source rows, order, missingness reason
design	response vector, transformations, factor/contrast metadata, stable columns, rank and aliases
covariance	independence or tree-derived identity, parameterization, bounds, fitted state, matrix order
fit	coefficient keys, estimates, uncertainty, likelihood convention, convergence and warnings
hypotheses	null/alternative or hypothesis matrix, statistic, degrees of freedom, tail and term order
diagnostics	residual type and scale, fitted values, leverage/influence rules, complete row denominator

If a design or covariance component changes, child coefficient, test, and diagnostic observations become stale together. This dependency packet is the minimum durable object needed to convert PCM3 from a design dossier into numerical evidence.

Key Every Comparable Observation¶

A coefficient or test becomes comparable only after its observation key resolves the analytical object on both sides. Printed term labels are display text and are insufficient as join keys.

Key component	Coefficient observation	Hypothesis-test observation
population	ordered included species and excluded-row ledger	the same fitted population for every tested model
response	source column, transform, units, and missing policy	response and transform shared by null and alternative fits
design	formula, contrast matrix, factor levels, column vector identity	tested column set or explicit hypothesis matrix
covariance	independence or tree/covariance digest, scale, and fitted parameter state	covariance convention held fixed across compared fits
fit	likelihood convention, optimizer state, coefficient key, and uncertainty definition	model pair or term-order identity, statistic, degrees of freedom, and tail rule
projection	source-to-Bijux term mapping and any scale transformation	source-to-Bijux test mapping without relabeling marginal as sequential

If one key component cannot be aligned, retain both outputs as separate observations and keep the claim not_comparable. Do not tune a tolerance for values that have not first been shown to estimate or test the same quantity.

What Can Be Cited Now¶

PCM3 can be cited as a governed decomposition of a multivariable comparative analysis and as an exact statement of the comparison work still required. It cannot be cited for numerical correspondence, recovered biological effects, or successful Martins-covariance reproduction.

not_comparable is the evidence conclusion. It is neither a failed numerical comparison nor an informal promise that parity exists elsewhere. Keeping that boundary explicit protects future results from inheriting claims that have never been observed.