GLMM families¶

All response-distribution families available for interlace.glmer(). Each family defines a link function, inverse link, variance function, and deviance residuals — mirroring R’s family() interface.

For the basic families ("binomial", "poisson", "gaussian", "negativebinomial"), pass a string to glmer(). For extended families, pass an instance directly.

GLMMFamily protocol¶

class GLMMFamily(*args, **kwargs)[source]¶

Protocol that all GLMM families must satisfy.

All concrete family classes (e.g. BinomialFamily, BetaFamily) must implement the five methods below. The protocol is runtime_checkable, so isinstance(obj, GLMMFamily) works at runtime.

name¶

Short string identifier for the family, e.g. "binomial", "beta".

Type:: str

link(mu)[source]¶: Apply the link function: map mean mu to the linear predictor eta.

linkinv(eta)[source]¶: Apply the inverse link: map linear predictor eta to mean mu.

mu_eta(eta)[source]¶: Derivative d(mu)/d(eta) of the inverse link.

variance(mu)[source]¶: Variance function V(mu) used by PIRLS.

dev_resids(y, mu, wt)[source]¶: Weighted unit deviance residuals.

Examples

>>> from interlace import BetaFamily
>>> from interlace.glmm_family import GLMMFamily
>>> fam = BetaFamily(phi=5.0)
>>> isinstance(fam, GLMMFamily)
True

Any object implementing these five methods can be passed as the family parameter to glmer().

Standard families (string shorthand)¶

These can be passed as strings to glmer(family=...):

String	Family	Link	Variance
`"binomial"`	Binomial	logit	mu(1-mu)
`"poisson"`	Poisson	log	mu
`"gaussian"`	Gaussian	identity	1
`"negativebinomial"`	NB2 (theta=1)	log	mu + mu^2/theta

Extended families (pass instances)¶

NegativeBinomial2Family¶

Quadratic mean-variance relationship: Var(Y) = mu + mu^2/theta.

from interlace import NegativeBinomial2Family

result = interlace.glmer(
    ..., family=NegativeBinomial2Family(theta=2.0), ...
)

Parameter	Default	Description
`theta`	1.0	Shape parameter. Smaller = more overdispersion.

NegativeBinomial1Family¶

Linear mean-variance relationship: Var(Y) = mu * (1 + alpha).

from interlace import NegativeBinomial1Family

result = interlace.glmer(
    ..., family=NegativeBinomial1Family(alpha=2.0), ...
)

Parameter	Default	Description
`alpha`	1.0	Overdispersion parameter. Must be positive.

BetaFamily¶

For continuous proportions (0, 1). Uses logit link with variance mu(1-mu)/(1+phi).

from interlace import BetaFamily

result = interlace.glmer(
    ..., family=BetaFamily(phi=5.0), ...
)

Parameter	Default	Description
`phi`	1.0	Precision parameter. Larger = less variance.

GammaFamily¶

For positive continuous data. Supports log and inverse links.

from interlace import GammaFamily

result = interlace.glmer(
    ..., family=GammaFamily(link="log"), ...
)

Parameter	Default	Description
`link`	`"log"`	Link function: `"log"` or `"inverse"`.

ZeroInflatedPoissonFamily¶

Mixture model: structural zeros with probability pi, Poisson counts otherwise.

from interlace import ZeroInflatedPoissonFamily

result = interlace.glmer(
    ..., family=ZeroInflatedPoissonFamily(pi=0.2), ...
)

Parameter	Default	Description
`pi`	0.1	Zero-inflation probability.

ZeroInflatedNB2Family¶

Mixture model: structural zeros with probability pi, NB2 counts otherwise.

from interlace import ZeroInflatedNB2Family

result = interlace.glmer(
    ..., family=ZeroInflatedNB2Family(pi=0.2, theta=2.0), ...
)

Parameter	Default	Description
`pi`	0.1	Zero-inflation probability.
`theta`	1.0	NB2 shape parameter.

HurdlePoissonFamily¶

Two-part model: y=0 observations carry no count information (score and Hessian are zero). The count component is Poisson.

from interlace import HurdlePoissonFamily

result = interlace.glmer(
    ..., family=HurdlePoissonFamily(), ...
)

ZeroOneInflatedBetaFamily¶

For proportions with exact 0s and/or 1s. Three-component mixture: point mass at 0, point mass at 1, and Beta distribution on (0, 1).

from interlace import ZeroOneInflatedBetaFamily

result = interlace.glmer(
    ..., family=ZeroOneInflatedBetaFamily(pi0=0.1, pi1=0.05, phi=5.0), ...
)

Parameter	Default	Description
`pi0`	0.1	Probability of structural zero.
`pi1`	0.05	Probability of structural one.
`phi`	1.0	Beta precision parameter.

Summary table¶

Family	Link	Variance	Use case
Binomial	logit	mu(1-mu)	Binary, proportions
Poisson	log	mu	Counts
Gaussian	identity	1	Continuous (same as `fit()`)
NB2	log	mu + mu^2/theta	Overdispersed counts (quadratic)
NB1	log	mu(1+alpha)	Overdispersed counts (linear)
Beta	logit	mu(1-mu)/(1+phi)	Continuous proportions in (0,1)
Gamma	log/inverse	mu^2	Positive continuous
ZIP	log	mu + pi*mu^2	Counts with excess zeros
ZINB2	log	NB2 + zero-inflation	Overdispersed counts with excess zeros
Hurdle Poisson	log	Poisson (y>0 only)	Counts with structural zeros
ZOIB	logit	Beta + point masses	Proportions with 0s and 1s