PCM5: Cichlid Phylogenetic Generalized Linear Mixed Models

PCM5 extends the cichlid analysis beyond Gaussian responses. It covers posterior-derived phylogenetic signal, ancestral node effects, Poisson parasite counts, binary-style infection outcomes, probability-scale summaries, and repeated observations with both species and phylogenetic random effects.

flowchart TB
    data["Repeated cichlid<br/>observations"]
    gaussian["Gaussian phylogenetic<br/>mixed model"]
    poisson["Poisson count model"]
    binary["Binary-style infection<br/>model"]
    repeated["Species + phylogeny<br/>random effects"]
    derived["Signal, ancestral states,<br/>probabilities, variance partitions"]

    data --> gaussian --> derived
    data --> poisson --> derived
    data --> binary --> derived
    binary --> repeated --> derived

The Derived-Scale Question In One View

Dimension	PCM5 contract
scientific object	cichlid Gaussian, Poisson, infection, ancestral, and repeated-observation posterior models linked to draw-wise signal, probability, and variance-partition quantities
central idea	family, link, residual convention, random-effect denominator, and inverse-link transformation determine the scale on which every posterior quantity is meaningful
evidence role	requires each derived value to remain a child of identified model, chain, draw, operand, tree/node, and transformation records
present conclusion	all nine manifests are not_comparable, use bundle_contract_only, and report governed_primary_output_count: 0; the generalized posterior comparison has not run
relation to PCM4	consumes the Gaussian mixed-model discipline, then adds model-family separation, repeated-measure structure, and draw-wise derivation lineage
forbidden inference	a logistic helper, plotted probability, similar posterior mean, or large draw count cannot substitute for an aligned governed posterior denominator

PCM5 is the evidence sequence's scale-discipline study. Its current record says exactly how future posterior claims must be derived and joined; it does not provide those posterior observations today.

What PCM5 Adds To PCM4

PCM4 establishes the contract for a Gaussian phylogenetic mixed model. PCM5 changes the response distribution, observation unit, link scale, and derived posterior questions. Each change creates a new evidence obligation.

Extension	Scientific question	Evidence that must be added
Gaussian variance → phylogenetic signal ratio	what share of modeled variance is phylogenetic under the declared convention?	same-draw variance components and complete denominator
species summary → repeated observations	how are within-species records separated from species and phylogenetic heterogeneity?	observation keys, species levels, tree levels, and all random-effect components
Gaussian response → Poisson counts	how does expected parasite abundance change on the log-linked count scale?	count coding, offset/exposure, link, dispersion treatment, priors, posterior draws
infection categories → binary-style latent model	how does infection probability change under the source category and residual convention?	category order, link/latent scale, residual convention, coefficient draws
coefficient posterior → probability curve	what response-scale probability is implied across predictor values?	paired coefficient draws, predictor grid, inverse link, draw-wise summaries
tip/random effects → ancestral node effects	what node-linked posterior quantities follow from the exact tree?	tree and node identity, parameter scale, parent draws, intervals, diagnostics

PCM5 is therefore a dependency graph, not a larger coefficient table. A derived probability, signal ratio, or variance share is trustworthy only when its parent model, chain, draw, parameter, and transformation identities remain linked.

Observation Grain Changes The Scientific Population

PCM5 moves between species summaries and repeated records. Those populations cannot share a denominator merely because they contain the same species names.

Grain	Unit represented by one row	Dependence that must be modeled or retained
species-level summary	one aggregated species response	aggregation members and rule, tree-linked species covariance
parasite count record	one declared count observation	exposure or offset, sampling unit, overdispersion policy, species/tip link
infection composition	infected and not-infected outcomes for one sampling unit	trial/composition denominator, category coding, link and latent residual convention
repeated observation	one measurement nested within species	record identity, within-species repetition, species effect, phylogenetic effect
ancestral node quantity	one model-derived node effect or response	exact tree, descendant identity, parent posterior, parameter and scale

Collapsing repeated records changes the response and removes a variance layer; expanding a species summary without source records invents replication. Every model and derived quantity must state which grain it consumes and how that population maps to species and tree identities.

Why Model Families Stay Separate

Gaussian, Poisson, and binary-style models operate on different likelihood and link scales. A coefficient on the log or latent scale is not already a change in expected count or infection probability. Residual variance conventions also differ, which changes posterior phylogenetic-signal calculations.

Repeated observations add another variance component. Species identity and phylogenetic relatedness must not be merged into one unnamed random effect.

The Same Predictor Has Different Meanings Across Families

A slope in a Gaussian model describes an additive response-scale change. A Poisson log-link slope describes a multiplicative expected-count ratio after exponentiation. A binary-style latent or logit slope changes log odds and must be transformed, with its category convention, to describe probability. Equal coefficient names or signs do not make these effects numerically comparable.

PCM5 therefore compares model identity before posterior values. The response family, link, exposure or trial denominator, observation grain, random effects, and residual convention decide what a coefficient or variance component means. Only then can draw-wise response-scale quantities be constructed and reviewed.

Preserve The Scale Of Every Quantity

Model component	Linear-predictor scale	Response-scale interpretation
Gaussian response	identity	expected response changes directly with the coefficient
Poisson count	log	exponentiated coefficient is a multiplicative expected-count ratio
binary-style outcome	logit or source-declared latent scale	inverse-link transformation yields probability; averaging and transforming do not generally commute
phylogenetic random effect	model-specific latent scale	variance contribution depends on covariance scaling and family convention
species random effect	model-specific latent scale	repeated-observation heterogeneity distinct from phylogenetic covariance

Posterior intervals should be transformed draw by draw when a response-scale summary is required. Transforming only posterior means can conceal asymmetry and does not preserve interval coverage.

The Preserved Source Model Sequence

The preserved source moves from a Gaussian phylogenetic model into ancestral, count, infection, and repeated-observation questions. Each object changes a scientific assumption or derives a quantity from named posterior columns:

Source object	Model or derivation	Scientific role
m2	Gaussian fixed effect plus phylogenetic treetip random effect	establishes the continuous-response phylogenetic model
m2PhyloSig	treetip / (treetip + units) for every retained draw	derives Gaussian phylogenetic signal
m3	intercept-only Gaussian model with pr = TRUE	retains tip and node effects for ancestral reconstruction
blupsm3	intercept-plus-effect posterior modes and HPD intervals	maps posterior ancestral summaries to explicit effect identities
m5	intercept-only Poisson phylogenetic model	models parasite counts on their declared family scale
m6	intercept-only multinomial2 model of infected and not_infected	models infection composition on the latent scale
m6PhyloSig	treetip / (treetip + units + pi^2 / 3) per draw	applies the source binary-style residual convention
m7	multinomial2 model with diet_breadth	estimates intercept and slope probability effects
m10	repeated records with species + treetip random effects	separates species heterogeneity from phylogenetic structure
m10PhyloSig and m10spSig	draw-wise variance shares over species + treetip + units	partition repeated-observation variance

flowchart TB
    gaussian["m2<br/>Gaussian phylogenetic model"]
    signal["m2PhyloSig<br/>Gaussian signal"]
    ancestral["m3 → blupsm3<br/>ancestral effects"]
    poisson["m5<br/>Poisson counts"]
    infection["m6<br/>infection intercept"]
    binarySignal["m6PhyloSig<br/>latent-scale signal"]
    predictor["m7<br/>diet breadth"]
    repeated["m10<br/>species + phylogeny"]
    partition["m10PhyloSig / m10spSig<br/>variance shares"]

    gaussian --> signal
    gaussian --> ancestral
    ancestral --> poisson
    poisson --> infection
    infection --> binarySignal
    infection --> predictor
    predictor --> repeated
    repeated --> partition

The arrows describe the conceptual progression of the source analysis. They do not authorize inheritance of evidence: every derived quantity still needs its own governed inputs, observations, diagnostics, and verdict.

Evidence Decomposition

Bundle	Evaluated surface	Manifest verdict
evidence-001	workspace reload and repeated-measure data contract	not_comparable
evidence-002	posterior phylogenetic signal for the Gaussian model	not_comparable
evidence-003	ancestral-node posterior extraction and export	not_comparable
evidence-004	tree-linked ancestral plotting contract	not_comparable
evidence-005	Poisson phylogenetic parasite-count model	not_comparable
evidence-006	infection model and logit-scale summaries	not_comparable
evidence-007	intercept and slope transformation to probability scale	not_comparable
evidence-008	repeated-measure species + treetip random effects	not_comparable
evidence-009	phylogenetic, species, and residual variance partition	not_comparable

The nine claim records and bundle contracts are checked in, and all nine comparison verdicts are not_comparable. The repository's binary comparative utilities provide adjacent capability but do not reproduce the external generalized mixed-model posterior. No governed comparison execution or observation ledger currently closes that gap.

Preserve Dependencies Between Derived Claims

Several PCM5 quantities are derived from the same posterior draws. Their dependency is part of the evidence contract: an ancestral export, transformed probability, or variance share cannot be admitted when its source draws, parameter keys, or chain diagnostics are absent.

flowchart LR
    inputs["Admitted observations<br/>tree · model identity"]
    draws["Chain-indexed posterior<br/>draws and diagnostics"]
    nodes["Ancestral node<br/>draws"]
    probabilities["Draw-wise inverse-link<br/>probabilities"]
    variance["Draw-wise variance<br/>partitions"]
    figures["Tables and figures<br/>with stable identities"]

    inputs --> draws
    draws --> nodes --> figures
    draws --> probabilities --> figures
    draws --> variance --> figures

Derived claim	Material parent record	Required identity join
ancestral-node posterior	model draws and exact tree	chain, draw, node descendant set, tree revision and parameter scale
probability curve or contrast	paired intercept/slope draws	chain, draw, predictor value, link and inverse-link convention
phylogenetic signal	phylogenetic and residual variance draws	chain, draw, component names, covariance scaling and family-specific denominator
species-versus-phylogeny partition	species, phylogenetic and residual draws	chain, draw, random-effect levels and complete variance denominator
plotted node or curve	governed derived rows	source-row key, rendering transform, exclusions and artifact identity

Passing a parent identity check would not automatically match every child quantity; each transformation still needs its own observation and rule. Conversely, a visually plausible derived figure cannot compensate for absent draws or diagnostics upstream. The current contract-only state stops before the posterior-draw node in this graph.

Retained Record State

Record layer	Present in every bundle	What it establishes
claim and manifest	yes	stable claim ID, direct-parity intent, source basis, limitation, and not_comparable verdict
authored reference.R	yes	writes a contract naming PCM5_PGLMM.R and PCM5_extra_code.R; it does not execute their posterior models
authored analysis.py	yes	writes a bundle_contract_only record with BUILD_SCRIPT = None and no primary outputs
checks, provenance, and reviewer summaries	yes	source identity, required bundle structure, and readable current status
governed primary outputs	no	all nine results manifests record governed_primary_output_count: 0
claim-level observations	no	no aligned draws, coefficients, node summaries, transformed probabilities, diagnostics, or variance partitions are adjudicated

Contract wrappers make the missing work locatable; they are not implementations of the generalized posterior workflow. The results manifests describe possible local reference-contract.json and analysis-run.json rerun products, but those ephemeral files are not governed scientific outputs.

Keep The Nested Denominators Separate

PCM5 conclusions can depend on several populations at once. Each denominator must reconcile before a derived posterior comparison can be complete.

Population	Required accounting	Consequence of loss
source observations	declared, admitted, excluded, repeated, and missing records	response and repeated-measure population becomes ambiguous
represented species and tree tips	source species, matched tips, pruned tips, duplicates, and ordering	phylogenetic covariance and ancestral identities cannot be reconstructed
model executions and chains	initialized, completed, failed, excluded, and retained chains	convergence and independent-chain claims are not interpretable
posterior draws	generated, burn-in, thinned, non-finite, transformed, and retained draws per chain	intervals and variance shares acquire an unknown population
parameters, categories, nodes, and variance components	expected, present, absent, aliased, and comparable keys	successful rows hide missing scientific quantities
claim observations	selected, comparable, matched, mismatched, and not comparable rows	the bundle verdict lacks a complete eligible denominator

These counts are linked but not interchangeable. Ten thousand retained draws cannot compensate for one chain, missing species, an absent variance component, or an unidentifiable node. Report the denominator that owns each conclusion instead of citing the largest available count.

Current Verdict

All nine bundle manifests record not_comparable. The dossiers identify the source code, datasets, claim boundaries, and required outputs, but their governed primary-output count is zero and no governed lane currently reproduces the MCMCglmm Poisson, multinomial, ancestral, or multiple-random-effect posteriors under matched priors and chain policy.

This status also prevents an unsafe substitution: a phylogenetic logistic approximation, a Gaussian model, or a non-phylogenetic generalized model cannot inherit the PCM5 claim merely because it accepts similar columns.

Why The Logistic Surface Is Only Adjacent Capability

summarize_phylogenetic_logistic owns a reviewer-facing phylogenetic logistic fit. PCM5's infection claims instead come from MCMCglmm multinomial2 posteriors, latent-scale residual conventions, draw-wise probability transforms, and—in the repeated-observation model—both species and treetip random effects.

The shared binary response vocabulary does not make these estimands equal. A governed comparison would first need to decide whether any quantity is scientifically correspondent, state the mapping and limitations, and retain both executions. Until then the logistic surface may inform design and diagnostics, but it supplies no PCM5 primary output and closes no bundle.

Verdict And Follow-Through

Axis	Current state	Consequence
bundle verdict	all nine are not_comparable	no count, infection, ancestral, probability-scale, or variance-partition claim has posterior correspondence
source and input identity	governed	scripts, datasets, tree, workspace objects, and claim decomposition are reviewable; analytical rerun success is not implied
adjacent runtime capability	narrower binary and comparative methods exist	similar inputs or output columns do not make the model, link, priors, or random effects equivalent
resolution work	open	matched MCMCglmm execution, posterior observations, diagnostics, and claim adjudication remain required

Do not use open follow-through as the bundle verdict. Do not interpret not_comparable as a failed chain: the required comparable execution is absent, so a numerical pass/fail conclusion is not yet available.

Comparison Contract By Model Family

Family	Identity that must match before values are compared
Poisson count	link, exposure or offset, overdispersion treatment, priors, random effects, and count scale
binary-style infection	source family, latent or link scale, residual convention, threshold or category coding, and probability transform
repeated measures	observation unit, species identity, phylogenetic effect, residual structure, and covariance among repeated records
ancestral posterior	node identity, tree revision, state/parameter scale, retained draws, and summary rule
variance partition	every numerator and denominator component, covariance scaling, family-specific residual convention, and draw-wise calculation

Only after those identities align, both computational surfaces execute, and the eligible denominator is complete should a governed lane compare posterior draws or summaries and assign an Evidence Book verdict.

Inputs And Provenance

The dossier contains cichlid.csv, cichlidtree.nex, fishdata.RData, PCM5_PGLMM.R, and PCM5_extra_code.R. Identities and sources are recorded in the dataset catalog, provenance record, and runtime/reference mapping.

Derived Quantity Discipline

Probability-scale values must retain their link-scale source and transformation formula. Posterior phylogenetic signal must name the variance components and denominator; the convention for a binary-style model cannot be silently copied from the Gaussian model. Ancestral node effects must retain node-to-tree identity rather than relying on plotting order.

When a denominator contains phylogenetic, species, residual, or distribution-specific variance, record every term and the scale on which it is defined. Two quantities labelled “phylogenetic signal” are not comparable if their denominators or latent-scale conventions differ.

Transform Draws Before Summarizing

For m7, retain the paired intercept and slope draw identity, compute the linear predictor for each declared predictor value, apply the inverse-logit to each draw, and only then compute posterior modes, intervals, or contrasts. For m2PhyloSig, m6PhyloSig, m10PhyloSig, and m10spSig, calculate each ratio from variance components belonging to the same draw before summarizing.

This rule preserves posterior dependence and asymmetry. Applying a nonlinear transform to marginal modes, interval endpoints, or independently summarized variance components constructs a different quantity and cannot be treated as the source result.

Review Priorities

• Confirm response family, link, and observation unit for every model.
• Compare posterior values on matching scales and parameterizations.
• Preserve chain diagnostics before interpreting derived quantities.
• Keep species and phylogenetic variance components identifiable.
• Verify node identities in ancestral exports and figures.
• Retain the external MCMCglmm dependency in provenance and verdicts.

Current Scope

PCM5 provides nine explicit review units and the data/source context needed to close them. It does not claim a native PGLMM implementation or governed posterior parity. See the generalized mixed-model evidence boundary for closure requirements.