Family functions for glmmTMB
nbinom2(link = "log")nbinom1(link = "log")
nbinom12(link = "log")
compois(link = "log")
truncated_compois(link = "log")
genpois(link = "log")
truncated_genpois(link = "log")
truncated_poisson(link = "log")
truncated_nbinom2(link = "log")
truncated_nbinom1(link = "log")
beta_family(link = "logit")
betabinomial(link = "logit")
combinomial(link = "logit", allow_negative_nu = FALSE)
tweedie(link = "log")
skewnormal(link = "identity")
lognormal(link = "log")
ziGamma(link = "inverse")
t_family(link = "identity")
ordbeta(link = "logit")
ordinal(link = "logit")
bell(link = "log")
returns a list with (at least) components
length-1 character vector giving the family name
length-1 character vector specifying the link function
a function of either 1 (mean) or 2 (mean and dispersion
parameter) arguments giving a value proportional to the
predicted variance (scaled by sigma(.))
(character) link function for the conditional mean ("log", "logit", "probit", "inverse", "cloglog", "identity", or "sqrt")
(combinomial only) if TRUE, use an
identity link (rather than the default log link) for the dispersion
model, allowing the dispersion parameter \(\nu\) to go negative
(super-dispersion, i.e. over-dispersion beyond \(\nu \to 0\))
If specified, the dispersion model uses a log link; additional family parameters
(Student-t df, Tweedie power parameters, ordered beta cutpoints,
skew-normal skew parameters, etc.) use various link functions and
are accessible via family_params.
Denoting the variance as \(V\), the dispersion parameter
as \(\phi=\exp(\eta)\) (where \(\eta\) is the linear predictor from the dispersion model),
and the predicted mean as \(\mu\):
(from base R): constant \(V=\phi^2\)
(from base R) phi is the shape parameter. \(V=\mu\phi\)
a modified version of Gamma that skips checks for zero values, allowing it to be used to fit hurdle-Gamma models
Negative binomial distribution: quadratic parameterization (Hardin & Hilbe 2007). \(V=\mu(1+\mu/\phi) = \mu+\mu^2/\phi\).
Negative binomial distribution: linear parameterization (Hardin & Hilbe 2007). \(V=\mu(1+\phi)\). Note that the \(phi\) parameter has opposite meanings in the nbinom1 and nbinom2 families. In nbinom1 overdispersion increases with increasing phi (the Poisson limit is phi=0); in nbinom2 overdispersion decreases with increasing phi (the Poisson limit is reached as phi goes to infinity).
Negative binomial distribution: mixed linear/quadratic, as in the DESeq2 package or as described by Lindén and Mäntyniemi (2011). \(V=\mu(1+\phi+\mu/\psi)\). (In Lindén and Mäntyniemi's parameterization, \(\omega = \phi\) and \(\theta=1/\psi\).) If a dispersion model is specified, it applies only to the linear (phi) term.
Zero-truncated version of nbinom2: variance expression from Shonkwiler 2016. Simulation code (for this and the other truncated count distributions) is taken from C. Geyer's functions in the aster package; the algorithms are described in this vignette.
Conway-Maxwell Poisson distribution: parameterized with the exact mean (Huang 2017), which differs from the parameterization used in the COMPoissonReg package (Sellers & Shmueli 2010, Sellers & Lotze 2015). \(V=\mu\phi\).
Generalized Poisson distribution (Consul & Famoye 1992). \(V=\mu\exp(\eta)\). (Note that Consul & Famoye (1992) define \(\phi\) differently.) Our implementation is taken from the HMMpa package, based on Joe and Zhu (2005) and implemented by Vitali Witowski.
Beta distribution: parameterization of Ferrari and Cribari-Neto (2004) and the betareg package (Cribari-Neto and Zeileis 2010); \(V=\mu(1-\mu)/(\phi+1)\)
Beta-binomial distribution: parameterized according to Morris (1997). \(V=\mu(1-\mu)(n(\phi+n)/(\phi+1))\)
Conway-Maxwell-Binomial distribution: mean-parameterized as in Huang (2017) but for the binomial. The dispersion parameter \(\nu\) controls under- (\(\nu>1\)) and over- (\(\nu<1\)) dispersion relative to the binomial; \(\nu=1\) recovers the binomial. \(V\) has no closed form and is computed numerically.
Tweedie distribution: \(V=\phi\mu^{power}\). The power parameter is restricted to the interval \(1<power<2\), i.e. the compound Poisson-gamma distribution. Code taken from the tweedie package, written by Peter Dunn. The power parameter (designated psi in the list of parameters) uses the link function qlogis(psi-1.0); thus one can fix the power parameter to a specified value using start = list(psi = qlogis(fixed_power-1.0)), map = list(psi = factor(NA)).
Student-t distribution with adjustable scale and location parameters (also called a Pearson type VII distribution). The shape (degrees of freedom parameter) is fitted with a log link; it may be often be useful to fix the shape parameter using start = list(psi = log(fixed_df)), map = list(psi = factor(NA)).
Ordered beta regression from Kubinec (2022); fits continuous (e.g. proportion) data in the closed interval [0,1]. Unlike the implementation in the ordbeta package, this family will not automatically scale the data. If your response variable is defined on the closed interval [a,b], transform it to [0,1] via y_scaled <- (y-a)/(b-a).
Log-normal, parameterized by the mean and standard deviation on the data scale
Skew-normal, parameterized by the mean, standard deviation, and shape (Azzalini & Capitanio, 2014); constant \(V=\phi^2\)
Bell distribution (see Castellares et al 2018).
Cumulative-link (proportional odds) model for an ordinal response (Agresti 2010; e.g. MASS::polr, ordinal::clm). The response should be an ordered factor (or 1-based integer category codes). The K-1 ordered thresholds (cutpoints) are stored internally in the psi parameter vector via a softmax parameterization (cf. Koslik et al 2025): psi contains the log-weights of the K baseline (linear predictor = 0) category probabilities relative to the last category, and theta = qlogis(cumsum(softmax(c(psi, 0)))), which is increasing by construction; use family_params() to extract the thresholds on the threshold scale. As in other cumulative-link software, the linear predictor enters the model as \(\theta_j - \eta\) (so positive coefficients shift the response toward higher categories) and the fixed-effect intercept is fixed to zero (absorbed into the thresholds). Fitted/predicted mean values are reported as the expected category index \(E[Y] = \sum_j j P(Y=j)\). REML=TRUE integrates out the fixed effects but not the thresholds, so the appropriate REML correction for this family is unsettled; it is allowed but warns.
Agresti A (2010). "Analysis of Ordinal Categorical Data." 2nd ed. Hoboken, NJ: Wiley.
Koslik J-O, Dupont F, Auger-Méthé M, Marcoux M, Hussey N, & Heckman N (2025). "Flexible Unimodal Density Estimation in Hidden Markov Models." arXiv:2511.17071. tools:::Rd_expr_doi("10.48550/arXiv.2511.17071")
Azzalini A & Capitanio A (2014). "The skew-normal and related families." Cambridge: Cambridge University Press.
Castellares F, Ferrari SLP, & Lemonte AJ (2018) "On the Bell Distribution and Its Associated Regression Model for Count Data" Applied Mathematical Modelling 56: 172–85. tools:::Rd_expr_doi("10.1016/j.apm.2017.12.014")
Consul PC & Famoye F (1992). "Generalized Poisson regression model." Communications in Statistics: Theory and Methods 21:89–109.
Ferrari SLP, Cribari-Neto F (2004). "Beta Regression for Modelling Rates and Proportions." J. Appl. Stat. 31(7), 799-815.
Hardin JW & Hilbe JM (2007). "Generalized linear models and extensions." Stata Press.
Huang A (2017). "Mean-parametrized Conway–Maxwell–Poisson regression models for dispersed counts." Statistical Modelling 17(6), 1-22.
Joe H & Zhu R (2005). "Generalized Poisson Distribution: The Property of Mixture of Poisson and Comparison with Negative Binomial Distribution." Biometrical Journal 47(2): 219–29. tools:::Rd_expr_doi("10.1002/bimj.200410102").
Lindén, A & Mäntyniemi S. (2011). "Using the Negative Binomial Distribution to Model Overdispersion in Ecological Count Data." Ecology 92 (7): 1414–21. tools:::Rd_expr_doi("10.1890/10-1831.1").
Morris W (1997). "Disentangling Effects of Induced Plant Defenses and Food Quantity on Herbivores by Fitting Nonlinear Models." American Naturalist 150:299-327.
Kubinec R (2022). "Ordered Beta Regression: A Parsimonious, Well-Fitting Model for Continuous Data with Lower and Upper Bounds." Political Analysis. doi:10.1017/pan.2022.20.
Sellers K & Lotze T (2015). "COMPoissonReg: Conway-Maxwell Poisson (COM-Poisson) Regression". R package version 0.3.5. https://CRAN.R-project.org/package=COMPoissonReg
Sellers K & Shmueli G (2010) "A Flexible Regression Model for Count Data." Annals of Applied Statistics 4(2), 943–61. tools:::Rd_expr_doi("10.1214/09-AOAS306").
Shonkwiler, J. S. (2016). "Variance of the truncated negative binomial distribution." Journal of Econometrics 195(2), 209–210. tools:::Rd_expr_doi("10.1016/j.jeconom.2016.09.002").