Family functions for glmmTMB
nbinom2(link = "log")nbinom1(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")
tweedie(link = "log")
skewnormal(link = "identity")
lognormal(link = "log")
ziGamma(link = "inverse")
t_family(link = "identity")
ordbeta(link = "logit")
nbinom12(link = "log")
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")
If specified, the dispersion model uses a log link. 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). \(V=\mu(1-\mu)(n(\phi+n)/(\phi+1))\)
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).
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").