Families and links

Family objects provide probability calculations, simulation, moments, and link metadata. They are useful both inside compatible model engines and as standalone distribution objects.

Shared family contract

Depending on the distribution, a pymixef.families.Family exposes log_prob/log_probability, logpdf, logpmf, cdf, logcdf, sf, logsf, rvs, random, mean, variance, and moments. Methods validate support and parameter domains.

import numpy as np
from pymixef.families import Bernoulli, Gaussian

normal = Gaussian()
log_density = normal.log_prob(np.array([0.0, 1.0]), mean=0.0, scale=1.0)

binary = Bernoulli()
draws = binary.rvs(mean=np.array([0.2, 0.8]), random_state=2026)

Consult the signature-level API because parameters differ by family.

Link functions

Built-ins are identity, log, logit, probit, cloglog, cauchit, inverse, and inverse_squared. Each pymixef.families.Link defines forward, inverse, and derivative calculations with domain checks.

from pymixef.families import get_link

logit = get_link("logit")
probability = logit.inverse(0.75)

The links namespace and uppercase constants such as LOGIT are convenience forms.

Distribution catalog

Area	Families
Continuous	Gaussian, StudentT, LogNormal, Gamma, InverseGaussian, Beta, Tweedie
Discrete	Bernoulli, Binomial, Poisson, NegativeBinomial1, NegativeBinomial2, GeneralizedPoisson, COMPoisson, Ordinal, Multinomial
Composition	ZeroInflated, Hurdle, Truncated, Censored
Survival	Exponential, LogNormalSurvival, Weibull, Gompertz, LogLogistic, PiecewiseExponential

Aliases such as Normal, NB1, NB2, NegativeBinomial, and WeibullSurvival are indexed in the alias reference.

Catalog support versus engine support

The broad catalog is a probability and representation layer. Current formula fit support is narrower:

Engine	Accepted family path
LMM	Gaussian with identity link
GLMM	Bernoulli, binomial, Poisson, and negative-binomial-2 with supported canonical-link parameterizations
MMRM	Gaussian continuous response with structured residual covariance

Wrappers, survival families, ordinal/multinomial families, and other catalog objects are not automatically executable by the current GLMM backend. Model validation checks the requested engine/family/link combination before optimization.

Conditional mean and link scale

In a GLMM,

\[ g\{\operatorname{E}(Y_{ij}\mid b_i)\}=x_{ij}^{T}\beta+z_{ij}^{T}b_i. \]

Coefficients live on the link scale. For a logit link, exponentiating a coefficient produces a conditional odds ratio—not a probability ratio or percentage-point difference. Setting \(b_i=0\) gives a typical-cluster conditional curve, not a random-effect-integrated marginal curve.

Composition objects

pymixef.families.ZeroInflated, pymixef.families.Hurdle, pymixef.families.Truncated, and pymixef.families.Censored wrap a base family while retaining the component definition. They support relevant standalone probability calculations and IR representation. Always confirm backend compatibility before attempting a fit.

API map

The complete signatures, parameters, aliases, and inherited methods are in pymixef.families.