PCM2 Model Comparisons And PGLS

PCM2 asks a deceptively simple question: what changes when the residuals of a primate regression are allowed to covary along a phylogeny? The answer is tracked from an independence model through Pagel-λ covariance, transformed trees, continuous-mode fits, and likelihood-ratio comparisons. The shared response, predictors, taxa, and source tree make each change scientifically attributable—but only where the recorded observations actually agree.

PCM2 Is A Controlled Model Contrast

The study holds the biological observations fixed while changing how evolutionary dependence enters the model. That structure gives each evidence family a distinct job:

Evidence family	Deliberate change	Question it can answer
baseline GLS versus Pagel-lambda PGLS	independent residuals become tree-correlated residuals	how do the coefficient and uncertainty change under phylogenetic covariance?
lambda-zero versus fitted-lambda likelihood	one covariance parameter changes within a shared model family	does the registered fitted covariance improve likelihood over the zero-signal boundary?
Brownian, OU, early-burst, and late-burst transforms	branch-time weighting changes	do implementations construct and fit the same evolutionary-time parameterization?
residual diagnostics	fitted residual definition and scale are inspected	does remaining structure have the same numerical meaning across implementations?
ancestral modes	internal-node process projection changes	do clade-keyed conditional reconstructions correspond under the named mode?

This design makes disagreements informative. A stable baseline fit beside a reversed early-/late-burst transform points to the transform boundary rather than the shared trait table. It does not prove the cause, but it narrows where resolution must begin.

Use Disagreement To Locate The Broken Contract

PCM2’s surfaces form a dependency chain, so the earliest disagreement limits which later comparisons are interpretable:

Observation pattern	Boundary localized first
baseline coefficients disagree	population, design, likelihood, or ordinary regression implementation
baseline agrees but Pagel-λ fit disagrees	tree ordering, covariance transform, λ bounds, or phylogenetic likelihood
both regressions agree but residual summaries disagree	residual definition, scale, ordering, or diagnostic computation
Brownian mode agrees but early-burst transform reverses	rate-through-time parameter direction or transform convention
fitted modes align but likelihood-ratio result disagrees	nesting, parameter count, likelihood convention, or test calculation

This localization is a scientific benefit of the dossier. It preserves the accepted parent observations while refusing unsupported children. A later failure must not be used to erase an earlier match, and an earlier match must not be used to waive a broken downstream dependency.

Why The Slope Can Change Without Changing The Data

Ordinary GLS gives the residual vector an independence structure. Under PGLS, closely related species contribute correlated information, so the fit no longer treats every tip as an equally independent replicate. The coefficient is re-estimated under that covariance; it is not obtained by subtracting a phylogenetic component from the baseline slope.

This distinction is why PCM2 retains both accepted fits. The baseline result describes the association under independent residuals, while the Pagel-λ result describes it under fitted tree-structured residuals. Their difference is scientifically meaningful only because population, response, predictor, and mean design remain fixed.

Correspondence And Model Adequacy Are Separate Decisions

Parity establishes whether implementations answer the same registered calculation. Model assessment asks whether that calculation is scientifically adequate for the data and question.

Correspondence state	Adequacy state	Defensible action
established	acceptable under declared diagnostics	interpret the model within its population, assumptions, and uncertainty
established	diagnostics or assumptions fail	retain the parity result but refuse or qualify the scientific model
unresolved mismatch	apparently adequate in one implementation	investigate correspondence before transferring the result across implementations
not comparable	unknown or unassessed	preserve the design boundary; make neither a parity nor adequacy claim

Matching coefficients cannot rescue residual misspecification, and favorable diagnostics in one runtime cannot resolve a transform mismatch. PCM2 therefore retains implementation verdicts, model comparisons, and diagnostic outcomes as linked records rather than reducing them to one score.

flowchart LR
    inputs["75 species<br/>shared traits and tree"]
    gls["Baseline GLS<br/>independent residuals"]
    pgls["Pagel-λ PGLS<br/>estimated covariance"]
    transforms["OU and time-dependent<br/>branch transforms"]
    modes["Brownian · OU<br/>early burst"]
    ledger["56 scalar observations<br/>44 favorable · 12 unresolved"]

    inputs --> gls --> ledger
    inputs --> pgls --> ledger
    inputs --> transforms --> modes --> ledger

Hold The Estimand Fixed Before Comparing Values

A coefficient can be compared only after the population, response, design column, and residual covariance are identified. The baseline and PGLS slope both describe social_group_size, but they are estimates under different error models.

Component	Baseline GLS	Pagel-λ PGLS	Comparison consequence
admitted observations	governed 75-species population	same governed population	population change is not the explanation for the slope difference
mean structure	shared response and predictor formula	shared response and predictor formula	coefficient labels address the same design column
residual covariance	independence model	tree-derived covariance with fitted λ	uncertainty, weighting, coefficient, and likelihood can all change
model identity	baseline likelihood and diagnostics	λ bounds, transformed covariance, likelihood and diagnostics	both records must travel with any coefficient comparison

The observed slope change is scientifically interpretable because the shared parts and changed covariance are explicit. Similar column names from an unreconciled dataset or different likelihood convention would not support the same comparison.

The Regression Result

The non-phylogenetic and phylogenetic fits use the same substantive formula, but not the same residual model. That difference materially changes the social-group-size coefficient:

Quantity	R reference	Bijux observation	Evidence state
baseline intercept	263.916162624481	263.916162626577	matched
baseline social-group-size slope	3.591155530518	3.591155530471	matched
baseline R²	0.257169890829	0.257169890829	matched
baseline log likelihood	−456.310146562924	−456.310146563131	matched
estimated Pagel λ	0.768656451899	0.768657155570	matched with tolerance
Pagel-λ PGLS intercept	249.95843496	249.95843742	matched with tolerance
Pagel-λ PGLS slope	1.662148237	1.662147437	matched with tolerance
Pagel-λ PGLS slope p-value	0.0168886063	0.0168886611	matched with tolerance
Pagel-λ PGLS log likelihood	−444.069148162235	−444.069148162290	matched with tolerance

The slope falls from approximately 3.59 to 1.66 after phylogenetic covariance is fitted. This is the central PCM2 result: coefficient meaning depends on the declared covariance, not just the formula printed above the table. The fitted relationship remains associational; neither model establishes causality.

The numerical difference between slopes is not itself a parity observation: the two slopes come from intentionally different scientific models. Parity is assessed within each model by comparing the R and Bijux estimates. Scientific interpretation then compares the accepted baseline and phylogenetic models as a controlled contrast.

Model Evidence Ledger

Bundle	Governed comparison	Manifest verdict
evidence-002	baseline GLS coefficients and fit	matched
evidence-003	Pagel-λ PGLS	matched_with_tolerance
evidence-006	OU, early-burst, and late-burst tree transformations	matched
evidence-007	Brownian, OU, and early-burst intercept fits	matched_with_tolerance
evidence-008	pairwise likelihood-ratio comparisons	matched_with_tolerance

These are claim-scoped manifest verdicts. They do not supersede the aggregate observation record. The 56-row scalar ledger contains 8 matched, 36 matched_with_tolerance, and 12 mismatch_unexplained observations.

Focused Bundle Verdicts Do Not Erase Scalar Mismatches

The favorable manifests and unresolved scalar rows coexist because they are different levels of evidence. A bundle verdict summarizes its declared claim and checks; the aggregate ledger preserves every registered numerical observation, including observations outside or stricter than those checks.

Surface	Reference observation	Bijux observation	Integrity consequence
early-burst transform, parameter 2	total branch length 87.5076762141263	4.1181	direction or sign convention is not aligned
late-burst transform, parameter −2	total branch length 4.11805690444875	87.5077	values are reversed relative to the reference labels
early-burst fit	rate change 10.6718757865625	0	mode parameter is unresolved
early-burst fit	log likelihood −461.253095658179	−476.4234	difference is 15.170304341821
Brownian versus early-burst	LRT statistic 30.340684573422	0	the governed comparison does not agree

Two OU branch-length totals also remain mismatch_unexplained under an exact zero-tolerance rule: 11.8428761446323 versus 11.8429, and 3.20350911964606 versus 3.2035. Those discrepancies look like output precision, but the record does not classify them as rounding matches. Readers must not silently change the adjudication rule after seeing the values.

The early- and late-burst reversal is evidence of a probable parameterization boundary, not proof of its cause. Resolution requires an explicit mapping of the reference rate-through-time convention to the Bijux transform, followed by regenerated observations and adjudication.

Likelihood-Ratio Discipline

For models fitted to the same observations under the same likelihood convention, the statistic is

LR = 2 × (logLik_complex − logLik_simple)

The Brownian-versus-OU statistic agrees: 30.340684116704 in the R reference and 30.3406 in the Bijux observation. Brownian-versus-early-burst does not. Consequently, evidence-008 may be cited for its claim-scoped matched_with_tolerance verdict, but it cannot support a statement that all pairwise mode comparisons agree.

A defensible likelihood-ratio claim also records nesting, degrees of freedom, parameter boundaries, optimization status, and whether models use maximum or restricted likelihood. AIC ranks fitted candidates under a penalty; it is not a substitute name for the likelihood-ratio test.

Reconstruct A Model Comparison

Retain the following identity with every coefficient or fit statistic:

• the response transformation, formula, factor coding, included taxa, and row order;
• the tree, branch scale, covariance transform, parameter bounds, and root treatment;
• coefficient estimates and uncertainty, log likelihood, parameter count, and optimizer outcome;
• the residual definition and diagnostic scale;
• the comparison pair, test statistic, degrees of freedom, and acceptance tolerance.

Without that identity, a matching decimal may refer to a different estimand. With it, the baseline and Pagel-λ regression result is reproducible, while the early-burst family remains visibly unresolved.

Do not select only the model family with favorable correspondence. The candidate denominator is the declared baseline, covariance, transformed-tree, and continuous-mode roster. A model may be excluded from a biological conclusion for a predeclared adequacy reason, but an unresolved parity row must remain in the evidence record and cannot disappear because another candidate fits better.

Inspect the complete scalar-parity-table.json before making a PCM2-wide parity statement. Continue with signal diagnostics and open boundaries for the residual, ancestral, and uncovered comparison surfaces.