Likelihood, prediction, and uncertainty conventions

PyMixEF log likelihoods include data-dependent normalization constants unless a method document explicitly says that only an unnormalized research objective is available. AIC or likelihood comparison is refused when conventions differ.

Gaussian maximum likelihood uses

\[ \ell(\beta,\theta)=-\frac12\{n\log(2\pi)+\log|V_\theta| +(y-X\beta)^\mathsf{T} V_\theta^{-1}(y-X\beta)\}. \]

REML additionally integrates the fixed effects and records the exact constant convention in the result. Approximate likelihoods name the approximation (laplace, for example) in the result and report sensitivity controls.

Prediction modes are never implicit:

• population: fixed-effects/typical-subject prediction;

• conditional: includes empirical conditional random effects;

• new-subject: integrates or simulates new random effects as requested.

Every standard error identifies the covariance estimator, calculation scale, approximation, degrees-of-freedom method, and nuisance-parameter treatment in result metadata.

Objective comparison

pymixef.compare.compare() calculates objective differences only when the PyMixEF result and reference both declare:

• estimation method;

• objective convention;

• whether likelihood normalization constants are included.

Parameter estimates can still be aligned when objective conventions differ, but the objective difference remains unavailable. This prevents an apparently precise comparison of non-equivalent numbers.

ML, REML, and approximate likelihoods

• Gaussian ML objectives are comparable across compatible fixed-effect models under the same data and convention.

• REML integrates/adjusts for the fixed-effect space; REML objectives from different fixed-effect designs are not ordinary likelihood comparisons.

• Laplace GLMM objectives are conditional on the named approximation and controls. Report “first-order Laplace” rather than only “maximum likelihood.”

• Boundary variance tests can have nonstandard reference distributions; a naive chi-square LRT is not universal.

Conditional, population, marginal, and new-group

These terms are not interchangeable.

Output	Random-effect treatment	Answers
conditional	fitted modes for observed groups	fitted response for these groups
population/typical-group	random effect set to its reference value	fixed-effect curve at \(b=0\)
marginal	integrate over the random-effect distribution	population-average response under the model
new-group simulation	draw a new random effect	predictive distribution for an unseen group

With an identity link, the typical-group and random-effect-marginal mean can coincide. With nonlinear links, they generally do not.

For a GLMM with inverse link \(g^{-1}\), the distinction is explicit:

\[ \mu_\mathrm{typical}(x)=g^{-1}(x^\mathsf{T}\beta), \qquad \mu_\mathrm{marginal}(x) =\int g^{-1}(x^\mathsf{T}\beta+z^\mathsf{T}b)\, \phi(b;0,G)\,db. \]

Except for special link/distribution combinations, applying the inverse link before integrating is not the same as applying it after setting \(b=0\).

Link and response scales

Coefficients are estimated on the linear-predictor/link scale:

• identity: response-unit difference;

• logit: log odds; exponentiation gives a conditional odds ratio;

• log: log mean/rate; exponentiation gives a conditional ratio;

• other links require their own inverse transformation.

An odds ratio is not a risk ratio, probability ratio, or percentage-point change. Always state the link, conditional/marginal target, covariate reference values, and random-effect treatment.

Standard errors and intervals

Before reporting an interval, identify:

1. parameter/contrast and calculation scale;

2. covariance source (for example observed Hessian);

3. approximation and whether nuisance uncertainty is included;

4. reference distribution and degrees of freedom;

5. transformation to the reported scale;

6. multiplicity and prespecification where relevant.

If Hessian diagnostics are indefinite or unavailable, ordinary Hessian-based Wald intervals are not supported by the calculation.

Simulation uncertainty

Seeded simulation is reproducible, but finite replicates carry Monte Carlo error. Increase replicates until the simulation summary is stable for the decision scale and report the seed, replicate count, random-effect/residual components, and any parameter uncertainty.

Missing data

A likelihood calculated under missing at random does not verify that assumption. The data audit establishes which records were used; scientific missingness reasoning and sensitivity analysis establish how conclusions depend on unobserved outcomes.