nbinom2 function

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")

tweedie(link = "log")

skewnormal(link = "identity")

lognormal(link = "log")

ziGamma(link = "inverse")

t_family(link = "identity")

ordbeta(link = "logit")

bell(link = "log")

Arguments

• link: (character) link function for the conditional mean ("log", "logit", "probit", "inverse", "cloglog", "identity", or "sqrt")

Returns

returns a list with (at least) components - family: length-1 character vector giving the family name

• link: length-1 character vector specifying the link function

• variance: a function of either 1 (mean) or 2 (mean and dispersion parameter) arguments giving a value proportional to the predicted variance (scaled by sigma(.))

Details

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 $mu$:

• gaussian: (from base R): constant $V=phi^2$
• Gamma: (from base R) phi is the shape parameter. $V=mu*phi$
• ziGamma: a modified version of Gamma that skips checks for zero values, allowing it to be used to fit hurdle-Gamma models
• nbinom2: Negative binomial distribution: quadratic parameterization (Hardin & Hilbe 2007). $V=mu*(1+mu/phi) = mu+mu^2/phi$.
• nbinom1: 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).
• nbinom12: 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.
• truncated_nbinom2: 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.
• compois: 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). $V=mu*phi$.
• genpois: 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: Beta distribution: parameterization of Ferrari and Cribari-Neto (2004) and the betareg package (Cribari-Neto and Zeileis 2010); $V=mu*(1-mu)/(phi+1)$
• betabinomial: Beta-binomial distribution: parameterized according to Morris (1997). $V=mu*(1-mu)*(n*(phi+n)/(phi+1))$
• tweedie: 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)).
• t_family: 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)).
• ordbeta: 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).
• lognormal: Log-normal, parameterized by the mean and standard deviation on the data scale
• skewnormal: Skew-normal, parameterized by the mean, standard deviation, and shape (Azzalini & Capitanio, 2014); constant $V=phi^2$
• bell: Bell distribution (see Castellares et al 2018).

References

• 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") .