PCM3: Primate Continuous And Categorical Regression

PCM3 moves from single-predictor comparative models to multivariable design. Continuous predictors, categorical contrasts, interactions, transformations, and phylogenetic covariance are evaluated as separate sources of meaning rather than compressed into one PGLS label.

flowchart TB
    response["Longevity response"]
    continuous["Continuous predictor"]
    category["Categorical predictor<br/>and reference level"]
    interaction["Continuous × category"]
    covariance["Martins phylogenetic<br/>covariance"]
    interpretation["Conditional slopes,<br/>contrasts, diagnostics"]

    response --> interpretation
    continuous --> interaction --> interpretation
    category --> interaction
    covariance --> interpretation

The Design-Matrix Question In One View

Dimension	PCM3 contract
scientific object	primate longevity regressions whose rows, response scale, factor levels, contrasts, interaction columns, standardization, and Martins covariance must resolve together
central idea	a familiar formula label is not an estimand; coding and transformation determine what every intercept, slope, contrast, and test means
evidence role	decomposes a multivariable comparison into eight independently closable claims so a coefficient cannot hide a population, design, covariance, or hypothesis mismatch
present conclusion	all eight manifests are not_comparable; they retain source, inputs, claim boundaries, checks, and explicit closure requirements but no governed primary numerical outputs
relation to PCM1–PCM2	inherits the need for a governed primate population and covariance identity, then adds design-matrix identity as a separate comparison gate
forbidden inference	runtime support for regression or PGLS does not establish lecture coefficients, p-values, likelihoods, diagnostics, or parity

PCM3 is a governed comparison design, not a weak numerical result. Its evidence value is that it makes the missing denominator and estimand observable before any future values are allowed into a parity ledger.

Scientific Questions

• How does categorical coding change the intercept and reported contrasts?
• How should group-specific slopes be recovered from an interaction model?
• What changes when the response is transformed or the predictor standardized?
• Which differences remain after residual covariance follows the phylogeny?

An interaction coefficient is not an isolated effect. It modifies a conditional slope under a specific reference coding. Standardization changes the location represented by the intercept. Those semantics must match before coefficients from different implementations can be compared.

Coefficient Labels Are Coordinates, Not The Model Itself

A fitted mean can be expressed through different but equivalent design bases. Changing the factor reference level, removing the intercept, or centering a predictor changes the coefficient coordinates and their labels even when the model predicts the same value for every admitted row. By contrast, filtering rows, changing a response transform, or testing a different linear combination changes the scientific question.

PCM3 therefore requires both a prediction-level identity and a coefficient-to-design-column identity. Similar predictions cannot prove that named coefficients have the same meaning, and similar coefficient labels cannot prove that the fitted population or hypothesis is the same.

What PCM3 Is Designed To Distinguish

PCM3 turns one multivariable analysis into eight bounded questions because changes that look cosmetic in formula notation change the estimand.

Design choice	Quantity that becomes interpretable	Identity needed for evidence
continuous plus categorical main effects	reference-group intercept, common slope, and group contrast at x = 0	admitted rows, response scale, factor levels, reference level, design columns
log-transformed response	multiplicative rather than additive response interpretation	transform rule, zero handling, units, back-transform policy
intercept-free group parameterization	one directly estimated intercept per group	contrast/design matrix and coefficient mapping, not label similarity
continuous-by-category interaction	reference slope and group-specific slope differences	centered/scaled predictor, reference group, interaction column
standardized predictor	intercept at the predictor mean and slopes per standard deviation	training population, mean, scale, and application rule
Martins phylogenetic covariance	coefficients and tests under tree-structured residual dependence	tree/taxon order, covariance definition, fitted parameter, likelihood convention

The evidence question is therefore not “did both tools fit PGLS?” It is whether they fitted the same population, design matrix, covariance, estimand, and hypothesis before any coefficient or p-value was compared.

Design Identity Must Be Inspectable, Not Inferred

A future PCM3 execution needs a machine-readable design identity before it needs a coefficient table. Formula text is only one input to that identity.

Design layer	Minimum retained evidence	Failure hidden by formula text alone
row population	ordered species keys, source-row keys, exclusions, and response values	different complete-case populations
transformation	response transform, zero/domain policy, predictor center and scale, and fitted/application population	different intercept location or units
categorical coding	raw levels, normalized levels, reference level, and contrast matrix	reversed or renamed contrasts
numerical design	stable column keys, ordered column values, rank, and aliased-column state	same label attached to different vectors
covariance	tree digest, ordered taxa, branch scale, Martins definition, bounds, and fitted parameter state	same mean design fitted under different dependence
hypothesis	explicit tested column combination or hypothesis matrix, term order, and degrees of freedom	marginal and sequential tests mislabeled as equivalent

The strongest join key is a fingerprint over these governed components plus their independently reviewable rows—not a hash with no readable decomposition. Only after both executions resolve to the same design identity do numerical tolerances become meaningful.

Read Coefficients From The Design Matrix

For a model with continuous predictor x, group indicator B, and interaction x:B, the fitted mean is interpreted as:

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

beta_B is the group difference at the point represented by x = 0. beta_x:B is the change in slope for group B. Centering or standardizing x therefore changes the intercept and group contrast without changing the raw observations. A comparison must align the reference level and transformation before interpreting coefficient differences.

Evidence Decomposition

Bundle	Evaluated surface	Manifest verdict
evidence-001	source intake, workspace reload, and shared data alignment	not_comparable
evidence-002	baseline GLS with continuous and categorical predictors	not_comparable
evidence-003	log response, residual diagnostics, marginal and sequential tests	not_comparable
evidence-004	intercept-free group estimates	not_comparable
evidence-005	continuous-by-categorical interaction	not_comparable
evidence-006	predictor scaling and intercept semantics	not_comparable
evidence-007	categorical phylogenetic regression with Martins covariance	not_comparable
evidence-008	post-correction diagnostics and hypothesis tests	not_comparable

All eight bundles have governed structure and review artifacts. Their bundle verdicts are not_comparable; the work to create lecture-specific comparisons remains open. Evidence infrastructure is complete enough to localize the missing comparisons, but it is not a passing parity result.

Current Verdict

All eight bundle manifests record not_comparable. The repository has the source programs, identified inputs, claim decomposition, runtime mapping, checks, and review locations, but it does not yet have governed observation-level lecture outputs under aligned factor coding, transformations, covariance, and hypothesis-test semantics.

What The Present Record Contributes

Retained evidence	Decision it enables now	Decision it does not enable
checked lecture source and provenance	identify the intended formulas, objects, and source boundary	claim that the lecture execution completed in the governed lane
governed table and tree derivatives	inspect exact candidate inputs and their identities	claim that both implementations admitted identical rows and design columns
eight separated claim contracts	prevent baseline, interaction, transform, and phylogenetic tests from being collapsed	assign a favorable numerical verdict to any claim
executable contract wrappers and zero-primary-output manifests	verify that absence is explicit and machine-readable	treat wrapper completion as regression execution
closure requirements	run a future comparison without redefining its population or estimands	publish coefficients, likelihoods, p-values, or diagnostic correspondence today

PCM3 is evidence of a governed comparison design and its current absence of primary observations. That is useful evidence about readiness and scope, not numerical evidence about regression correspondence.

No Numerical Headline Is Supported

PCM3 has no governed coefficient, p-value, likelihood, or model-comparison observation that can be presented as lecture parity. The checked-in reviewer summaries establish record completeness and expose the absent execution boundary; they are not primary scientific outputs.

The runtime can express multivariable regression, but capability proximity is not evidence that the eight PCM3 lecture claims were executed or adjudicated.

Retained Record And Missing Evidence

Evidence component	Current state	Consequence
lecture source and copied study inputs	retained with provenance	the intended study and source boundary can be inspected
eight claim identities and bundle manifests	retained	the comparison denominator and required surfaces are explicit
nearby comparative runtime implementations	identified by governed code paths	implementation relevance is traceable but does not prove execution
governed primary R outputs	absent from every bundle	no reference coefficient, test, or diagnostic can enter comparison
governed primary Bijux outputs	absent from every bundle	no repository observation can enter comparison
aligned observation rules and tolerances	not instantiated with paired outputs	no numerical correspondence verdict can be calculated

All eight manifests currently have zero governed primary output paths. That is the earliest owning reason for not_comparable; reviewer summaries and rich runtime APIs cannot close it.

Claim Closure Contract

Claim family	Reference record required	Bijux record required	Comparison identity
baseline continuous and categorical GLS	fitted population, design columns, coefficients, uncertainty, likelihood	same fields under the same row exclusions	factor levels, reference group, formula, test convention
transformed response and model tests	transform, fitted models, marginal and sequential tables	matching transformed response and hypotheses	response scale, degrees of freedom, test ordering
intercept-free group estimates	group-specific parameterization and uncertainty	same design columns and covariance	no-intercept coding and group order
interaction model	conditional slopes and contrasts	main and interaction terms plus reconstructed slopes	centering, reference group, term mapping
predictor scaling	center, scale, and transformed coefficients	same scaling constants and coefficient projection	numerical transform and intercept meaning
phylogenetic regression	tree, Martins covariance, fitted parameters, coefficient table	identical taxa, tree scale, covariance, and design	covariance convention and likelihood scale
post-correction diagnostics	residual definition and hypothesis tests	corresponding diagnostics and tests	residual scale, leverage, test family

Each bundle remains not_comparable until its row has both primary outputs and a declared observation rule. Structural completeness alone cannot satisfy a scientific comparison.

Close Dependencies Before Coefficients

PCM3 coefficients are downstream observations. They cannot close while the analysis population, design, covariance, or hypothesis identity remains open.

flowchart LR
    population["Species and row<br/>population"]
    transform["Response and predictor<br/>transformations"]
    design["Factor levels, contrasts<br/>and design columns"]
    covariance["Independent or Martins<br/>covariance identity"]
    fit["Coefficients, uncertainty<br/>and likelihood"]
    tests["Marginal, sequential<br/>and diagnostic tests"]
    claim["Claim observation<br/>and verdict"]

    population --> transform --> design --> fit
    population --> covariance --> fit
    fit --> tests --> claim

An observation row must point to the immutable parent records it consumed. Reusing the same formula text does not establish the same design; reusing the same species names does not establish the same ordered population; and a matching coefficient cannot close a claim whose tested hypothesis differs.

The bundle denominator remains all claims and all eligible observations selected before execution. Failed fits, missing terms, aliased columns, and non-comparable tests remain in that denominator rather than disappearing from the successful coefficient table.

Closing a bundle requires more than executing a neighboring regression API. The comparison must reconstruct the same design matrix, analysis population, coefficient names, test type, and phylogenetic covariance, then retain the reference and Bijux observations with an explicit tolerance or structural rule.

Verdict And Resolution State

not_comparable is the current evidence conclusion, not a placeholder for failed and not a runtime skip. It records that aligned lecture-level observations do not yet exist for the proposed comparison. Open follow-through describes the work needed to create and adjudicate those observations. Keep both facts visible without calling the follow-through itself a verdict.

Inputs And Provenance

PCM3 reuses the primate source workspace and governed tree/trait derivatives while carrying its own dataset registry and provenance. See the dataset catalog, provenance record, and runtime mapping.

Review Priorities

• Match factor levels, reference category, design matrix, and missing-value exclusions before comparing coefficients.
• Compare transformed response and standardized predictor conventions exactly.
• Reconstruct conditional group slopes from main and interaction terms.
• Distinguish marginal from sequential hypothesis tests.
• Confirm that the phylogenetic fit uses the same taxa and response as the non-phylogenetic comparator.

For every coefficient table, retain the term-to-design-column mapping. For every hypothesis-test table, identify whether the test is marginal, sequential, or based on nested-model comparison. Equal labels across tools do not guarantee equal hypotheses.

Current Scope

PCM3 supplies a complete claim decomposition and executable review locations for the multivariable lecture. It does not yet claim lecture-level numerical parity for any of the eight slices. The closure work is comparison and verdict adjudication, not inventing broader runtime capability by implication.

PCM3 defines the model semantics and evidence needed for a defensible comparison while making no numerical correspondence claim at this revision.

See model design and phylogenetic regression for the detailed interpretation contract.