Negative Binomial Regression

Log-linear count models with a mean-dependent dispersion

Statistics

Regression

Count Data

Negative binomial regression from its conditional count model through coefficients, likelihood, uncertainty, diagnostics, and bioinformatics study design.

Author

Jiahong Sun

Published

October 20, 2026

Modified

October 20, 2026

1 The Conditional Count Model

Negative binomial regression models a nonnegative count whose conditional variance may exceed its conditional mean. For independent experimental units $i=1,\ldots,n$, the commonly used NB2 model is

\[ Y_i\mid\mathbf x_i\sim\operatorname{NB}(\mu_i,\alpha), \qquad \log(\mu_i)=\mathbf x_i^\mathsf T\boldsymbol\beta, \]

with

\[ \boxed{ E(Y_i\mid\mathbf x_i)=\mu_i, \qquad \operatorname{Var}(Y_i\mid\mathbf x_i) =\mu_i+\alpha\mu_i^2 }, \qquad \alpha>0. \]

The mean model has the same log-linear form as Poisson regression, but the sampling distribution has an additional dispersion parameter. Conditional variance grows quadratically with the mean, so a single multiplicative variance factor cannot reproduce NB2 over the full mean range.

Quantity	Meaning
$Y_i$	Count observed on experimental unit $i$
$\mu_i$	Conditional expected count
$\boldsymbol\beta$	Log-mean regression coefficients
$\alpha$	NB2 dispersion; larger values imply more conditional variation
$1/\alpha$	Size or shape parameter in a common alternative convention
$\exp(\beta_j)$	Conditional mean ratio for a one-unit predictor contrast

Software conventions differ. Some packages call $1/\alpha$ the size, shape, or dispersion, so a larger reported value can mean less variation. A fitted model should always be documented by its variance function, not by a parameter name alone.

2 A Poisson–Gamma Construction

One derivation introduces a unit-specific latent multiplier $U_i$:

\[ Y_i\mid U_i,\mathbf x_i\sim\operatorname{Poisson}(U_i\mu_i), \qquad U_i\sim\operatorname{Gamma} \left(\frac{1}{\alpha},\text{rate}=\frac{1}{\alpha}\right). \]

Then $E(U_i)=1$ and $\operatorname{Var}(U_i)=\alpha$. The laws of total expectation and total variance give

\[ E(Y_i\mid\mathbf x_i)=E(U_i\mu_i)=\mu_i \]

and

\[ \begin{aligned} \operatorname{Var}(Y_i\mid\mathbf x_i) &=E\{\operatorname{Var}(Y_i\mid U_i,\mathbf x_i)\} +\operatorname{Var}\{E(Y_i\mid U_i,\mathbf x_i)\}\\ &=\mu_i+\alpha\mu_i^2. \end{aligned} \]

Marginalizing $U_i$ yields the NB2 probability mass function

\[ P(Y_i=y\mid\mu_i,\alpha) =\frac{\Gamma(y+1/\alpha)}{\Gamma(1/\alpha)y!} \left(\frac{1}{1+\alpha\mu_i}\right)^{1/\alpha} \left(\frac{\alpha\mu_i}{1+\alpha\mu_i}\right)^y. \]

This construction explains one source of overdispersion: unobserved rates vary across units. It does not prove that gamma-distributed heterogeneity is the true biological mechanism. Dependence, omitted predictors, excess zeros, outliers, or measurement artifacts can create similar residual patterns.

As $\alpha\downarrow0$, the latent multiplier concentrates at one and the model approaches Poisson regression. The Poisson model is therefore a boundary case; standard likelihood-ratio reference distributions need care when testing $H_0:\alpha=0$.

3 Coefficients, Contrasts, and Exposure

For a binary condition $A_i$ and covariates $\mathbf z_i$,

\[ \log(\mu_i)=\beta_0+\beta_AA_i +\boldsymbol\gamma^\mathsf T\mathbf z_i. \]

At common $\mathbf z$,

\[ \frac{E(Y\mid A=1,\mathbf z)}{E(Y\mid A=0,\mathbf z)} =\exp(\beta_A). \]

Dispersion changes the likelihood weights and uncertainty, not the meaning of this mean ratio. The contrast is conditional on the modeled covariates and is not automatically causal. Causal interpretation additionally needs a well-defined intervention, exchangeability, positivity, consistency, and adequate handling of selection, missingness, measurement, and interference.

When unit $i$ has exposure $t_i>0$, write

\[ \log(\mu_i)=\log(t_i)+\mathbf x_i^\mathsf T\boldsymbol\beta. \]

The fixed-coefficient term $\log(t_i)$ is an offset, and exponentiated slopes are conditional rate ratios. The offset still defines the estimand under a negative binomial observation model; dispersion cannot repair an inappropriate or poorly measured denominator.

4 Likelihood, Estimation, and Uncertainty

For NB2, the log-likelihood contribution, omitting no parameter-dependent terms, is

\[ \begin{aligned} \ell_i(\boldsymbol\beta,\alpha) ={}&\log\Gamma(y_i+1/\alpha)-\log\Gamma(1/\alpha)-\log(y_i!)\\ &-\frac{1}{\alpha}\log(1+\alpha\mu_i) +y_i\log\left(\frac{\alpha\mu_i}{1+\alpha\mu_i}\right), \end{aligned} \]

where $\mu_i=\exp(\mathbf x_i^\mathsf T\boldsymbol\beta)$. Numerical maximum likelihood estimates $\boldsymbol\beta$ and $\alpha$ jointly or by alternating updates. At a regular interior fit, the inverse observed or expected information provides model-based covariance estimates. Wald, score, likelihood-ratio, and profile-likelihood procedures answer different approximations to the same coefficient or dispersion question.

The estimand might be a population conditional log mean ratio $\beta_A$; the maximum-likelihood rule $\hat\beta_A(Y_1,\ldots,Y_n)$ is its estimator; and the number obtained from the observed dataset is its estimate. Its standard error describes repeated-sampling uncertainty under the fitted data-generating model. It is not the spread of individual counts and does not measure biological importance.

Sparse data and small numbers of independent units can make $\hat\alpha$ unstable. In high-dimensional sequencing, gene-wise dispersions are therefore commonly stabilized by borrowing information across genes. This shrinkage estimates a nuisance mean–variance trend more reliably, but genes are parallel outcomes, not additional biological replicates for a treatment contrast.

5 Worked Interpretation

Suppose a fitted NB2 model for mutation counts is

\[ \log(\mu_i)=\log(t_i)-2.1+0.50A_i, \qquad \hat\alpha=0.15, \]

where $t_i$ is callable genome exposure in a common unit. The estimated rate ratio is $\exp(0.50)\approx1.65$: at the same modeled covariates, condition 1 has an estimated mutation rate 65% higher than condition 0.

If a unit has fitted mean $\hat\mu=20$, its fitted conditional variance is

\[ 20+0.15(20)^2=80, \]

not 20 as under a Poisson model. That extra variation widens predictive and coefficient uncertainty. It does not mean that 15% of counts are erroneous or that every unit’s latent rate differs by 15%.

6 Assumptions, Diagnostics, and Sensitivity

Assumption or issue	Why it matters	Assessment	Response or sensitivity analysis
Correct event and experimental unit	A count’s component events are not independent study replicates	Trace sampling, aggregation, and preprocessing	Fit at the experimental-unit level; model repeated or clustered units explicitly
Correct log-mean structure	Dispersion can absorb omitted curvature, interactions, batch, or exposure	Residuals against predictors and fitted means; partial and stratified checks	Revise the mean model before interpreting $\alpha$
NB2 variance shape	One $\alpha$ imposes $\mu+\alpha\mu^2$	Compare residual spread across mean ranges; simulate fitted counts	Consider NB1, quasi-likelihood, covariate-dependent dispersion, or another count family
Conditional independence	Correlation invalidates naive likelihood information	Map pairing, repeated measures, families, sites, and batches	GEE, mixed models, cluster-aware covariance, or justified aggregation
No separate zero mechanism	NB2 may still misrepresent structural or censored zeros	Compare fitted and observed zeros conditionally; investigate assay mechanisms	Hurdle or zero-inflated model only when a distinct process is defensible
Stable dispersion information	Boundary or noisy estimates distort inference	Profile $\alpha$; examine sensitivity to influential units and shrinkage	Report weak identification; use validated shrinkage or bootstrap checks

Useful diagnostics include randomized quantile or simulation-based residuals, observed-versus-replicated zero and upper-tail counts, residual spread across the fitted-mean range, leverage and deletion checks, and predictive calibration. A good global deviance does not exclude local lack of fit.

Compare scientifically credible alternatives: Poisson with model-based and sandwich uncertainty, quasi-Poisson, NB1 and NB2, altered mean structures, cluster-aware models, and mechanism-based zero models. Stability of the target contrast is more informative than selecting whichever distribution has the smallest criterion in one dataset.

7 Bioinformatics Design and Interpretation

For bulk RNA-seq, let $K_{gj}$ be the raw count for gene $g$ in biological sample $j$. A common model is

\[ K_{gj}\sim\operatorname{NB}(\mu_{gj},\alpha_g), \qquad \log(\mu_{gj})=\log(s_j)+\mathbf x_j^\mathsf T\boldsymbol\beta_g, \]

where $s_j$ is an effective library-size factor. Independently sampled organisms, patients, or cultures are experimental units. Reads and technical lanes improve measurement of a sample; genes provide parallel tests and information for nuisance-trend estimation. Neither substitutes for biological replication.

Design variables should represent condition, blocking, batch, pairing, and other prespecified structure. Normalization factors address library scale and composition; they do not remove arbitrary batch effects. Dispersion represents residual replicate variability conditional on the design, combining biological heterogeneity and technical variation that remains after preprocessing.

Statistically, $\exp(\beta_{gA})$ is a model-based expression mean ratio and its interval quantifies sampling uncertainty under the analysis. Biologically, a credible mechanism also depends on absolute abundance, effect magnitude, replicate consistency, tissue or cell composition, annotation, technical bias, and independent validation. A small adjusted $p$-value is not itself evidence of large or causal biological change.

8 Connections and Take-home Messages

Negative binomial distribution and overdispersion derives the probability law and its Poisson–gamma mixture.
Diagnosing and modeling overdispersion separates evidence of extra variation from its possible mechanisms and remedies.
Poisson regression is the equidispersed boundary model.
Offsets and modeling event rates explains rate denominators and exposure uncertainty.
Robust standard errors and robust regression distinguishes covariance robustness from distributional modeling.
Likelihood-ratio, Wald, and score tests compares coefficient-testing procedures.

NB2 regression combines a log-linear conditional mean with variance $\mu+\alpha\mu^2$.
Exponentiated slopes remain conditional count or rate ratios; dispersion changes weighting and uncertainty, not coefficient scale.
The Poisson–gamma mixture is an interpretation, not proof of a biological latent-rate mechanism.
Mean structure, exposure, dependence, zeros, and influential units must be checked before attributing lack of fit to negative binomial dispersion.
Bioinformatics inference is supported by independent biological samples, not reads, genes, cells, or technical aliquots counted as replicates.

9 References

Lawless, J. F. (1987). Negative binomial and mixed Poisson regression. The Canadian Journal of Statistics, 15(3), 209–225. https://doi.org/10.2307/3314912
McCullagh, P., and Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). Chapman and Hall/CRC. https://doi.org/10.1201/9780203753736
Cameron, A. C., and Trivedi, P. K. (2013). Regression Analysis of Count Data (2nd ed.). Cambridge University Press. https://doi.org/10.1017/CBO9781139013567
Robinson, M. D., McCarthy, D. J., and Smyth, G. K. (2010). edgeR: a Bioconductor package for differential expression analysis of digital gene expression data. Bioinformatics, 26(1), 139–140. https://doi.org/10.1093/bioinformatics/btp616
Love, M. I., Huber, W., and Anders, S. (2014). Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2. Genome Biology, 15, 550. https://doi.org/10.1186/s13059-014-0550-8
Venables, W. N., and Ripley, B. D. glm.nb: fit a negative binomial generalized linear model. MASS package documentation. https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/glm.nb.html

--- title: "Negative Binomial Regression" subtitle: "Log-linear count models with a mean-dependent dispersion" description: "Negative binomial regression from its conditional count model through coefficients, likelihood, uncertainty, diagnostics, and bioinformatics study design." author: "Jiahong Sun" date: "2026-10-20" date-modified: "2026-10-20" categories: [Statistics, Regression, Negative Binomial Regression, Count Data] toc: true number-sections: true code-fold: show draft: false --- # The Conditional Count Model Negative binomial regression models a nonnegative count whose conditional variance may exceed its conditional mean. For independent experimental units $i=1,\ldots,n$, the commonly used NB2 model is $$ Y_i\mid\mathbf x_i\sim\operatorname{NB}(\mu_i,\alpha), \qquad \log(\mu_i)=\mathbf x_i^\mathsf T\boldsymbol\beta, $$ with $$ \boxed{ E(Y_i\mid\mathbf x_i)=\mu_i, \qquad \operatorname{Var}(Y_i\mid\mathbf x_i) =\mu_i+\alpha\mu_i^2 }, \qquad \alpha>0. $$ The mean model has the same log-linear form as [Poisson regression](poisson-regression.qmd), but the sampling distribution has an additional dispersion parameter. Conditional variance grows quadratically with the mean, so a single multiplicative variance factor cannot reproduce NB2 over the full mean range. | Quantity | Meaning | |---|---| | $Y_i$ | Count observed on experimental unit $i$ | | $\mu_i$ | Conditional expected count | | $\boldsymbol\beta$ | Log-mean regression coefficients | | $\alpha$ | NB2 dispersion; larger values imply more conditional variation | | $1/\alpha$ | Size or shape parameter in a common alternative convention | | $\exp(\beta_j)$ | Conditional mean ratio for a one-unit predictor contrast | Software conventions differ. Some packages call $1/\alpha$ the size, shape, or dispersion, so a larger reported value can mean *less* variation. A fitted model should always be documented by its variance function, not by a parameter name alone. # A Poisson--Gamma Construction One derivation introduces a unit-specific latent multiplier $U_i$: $$ Y_i\mid U_i,\mathbf x_i\sim\operatorname{Poisson}(U_i\mu_i), \qquad U_i\sim\operatorname{Gamma} \left(\frac{1}{\alpha},\text{rate}=\frac{1}{\alpha}\right). $$ Then $E(U_i)=1$ and $\operatorname{Var}(U_i)=\alpha$. The laws of total expectation and total variance give $$ E(Y_i\mid\mathbf x_i)=E(U_i\mu_i)=\mu_i $$ and $$ \begin{aligned} \operatorname{Var}(Y_i\mid\mathbf x_i) &=E\{\operatorname{Var}(Y_i\mid U_i,\mathbf x_i)\} +\operatorname{Var}\{E(Y_i\mid U_i,\mathbf x_i)\}\\ &=\mu_i+\alpha\mu_i^2. \end{aligned} $$ Marginalizing $U_i$ yields the NB2 probability mass function $$ P(Y_i=y\mid\mu_i,\alpha) =\frac{\Gamma(y+1/\alpha)}{\Gamma(1/\alpha)y!} \left(\frac{1}{1+\alpha\mu_i}\right)^{1/\alpha} \left(\frac{\alpha\mu_i}{1+\alpha\mu_i}\right)^y. $$ This construction explains one source of overdispersion: unobserved rates vary across units. It does not prove that gamma-distributed heterogeneity is the true biological mechanism. Dependence, omitted predictors, excess zeros, outliers, or measurement artifacts can create similar residual patterns. As $\alpha\downarrow0$, the latent multiplier concentrates at one and the model approaches Poisson regression. The Poisson model is therefore a boundary case; standard likelihood-ratio reference distributions need care when testing $H_0:\alpha=0$. # Coefficients, Contrasts, and Exposure For a binary condition $A_i$ and covariates $\mathbf z_i$, $$ \log(\mu_i)=\beta_0+\beta_AA_i +\boldsymbol\gamma^\mathsf T\mathbf z_i. $$ At common $\mathbf z$, $$ \frac{E(Y\mid A=1,\mathbf z)}{E(Y\mid A=0,\mathbf z)} =\exp(\beta_A). $$ Dispersion changes the likelihood weights and uncertainty, not the meaning of this mean ratio. The contrast is conditional on the modeled covariates and is not automatically causal. Causal interpretation additionally needs a well-defined intervention, exchangeability, positivity, consistency, and adequate handling of selection, missingness, measurement, and interference. When unit $i$ has exposure $t_i>0$, write $$ \log(\mu_i)=\log(t_i)+\mathbf x_i^\mathsf T\boldsymbol\beta. $$ The fixed-coefficient term $\log(t_i)$ is an offset, and exponentiated slopes are conditional rate ratios. The [offset](offsets-and-modeling-event-rates.qmd) still defines the estimand under a negative binomial observation model; dispersion cannot repair an inappropriate or poorly measured denominator. # Likelihood, Estimation, and Uncertainty For NB2, the log-likelihood contribution, omitting no parameter-dependent terms, is $$ \begin{aligned} \ell_i(\boldsymbol\beta,\alpha) ={}&\log\Gamma(y_i+1/\alpha)-\log\Gamma(1/\alpha)-\log(y_i!)\\ &-\frac{1}{\alpha}\log(1+\alpha\mu_i) +y_i\log\left(\frac{\alpha\mu_i}{1+\alpha\mu_i}\right), \end{aligned} $$ where $\mu_i=\exp(\mathbf x_i^\mathsf T\boldsymbol\beta)$. Numerical maximum likelihood estimates $\boldsymbol\beta$ and $\alpha$ jointly or by alternating updates. At a regular interior fit, the inverse observed or expected information provides model-based covariance estimates. Wald, score, likelihood-ratio, and profile-likelihood procedures answer different approximations to the same coefficient or dispersion question. The estimand might be a population conditional log mean ratio $\beta_A$; the maximum-likelihood rule $\hat\beta_A(Y_1,\ldots,Y_n)$ is its estimator; and the number obtained from the observed dataset is its estimate. Its standard error describes repeated-sampling uncertainty under the fitted data-generating model. It is not the spread of individual counts and does not measure biological importance. Sparse data and small numbers of independent units can make $\hat\alpha$ unstable. In high-dimensional sequencing, gene-wise dispersions are therefore commonly stabilized by borrowing information across genes. This shrinkage estimates a nuisance mean--variance trend more reliably, but genes are parallel outcomes, not additional biological replicates for a treatment contrast. # Worked Interpretation Suppose a fitted NB2 model for mutation counts is $$ \log(\mu_i)=\log(t_i)-2.1+0.50A_i, \qquad \hat\alpha=0.15, $$ where $t_i$ is callable genome exposure in a common unit. The estimated rate ratio is $\exp(0.50)\approx1.65$: at the same modeled covariates, condition 1 has an estimated mutation rate 65% higher than condition 0. If a unit has fitted mean $\hat\mu=20$, its fitted conditional variance is $$ 20+0.15(20)^2=80, $$ not 20 as under a Poisson model. That extra variation widens predictive and coefficient uncertainty. It does not mean that 15% of counts are erroneous or that every unit's latent rate differs by 15%. # Assumptions, Diagnostics, and Sensitivity | Assumption or issue | Why it matters | Assessment | Response or sensitivity analysis | |---|---|---|---| | Correct event and experimental unit | A count's component events are not independent study replicates | Trace sampling, aggregation, and preprocessing | Fit at the experimental-unit level; model repeated or clustered units explicitly | | Correct log-mean structure | Dispersion can absorb omitted curvature, interactions, batch, or exposure | Residuals against predictors and fitted means; partial and stratified checks | Revise the mean model before interpreting $\alpha$ | | NB2 variance shape | One $\alpha$ imposes $\mu+\alpha\mu^2$ | Compare residual spread across mean ranges; simulate fitted counts | Consider NB1, quasi-likelihood, covariate-dependent dispersion, or another count family | | Conditional independence | Correlation invalidates naive likelihood information | Map pairing, repeated measures, families, sites, and batches | GEE, mixed models, cluster-aware covariance, or justified aggregation | | No separate zero mechanism | NB2 may still misrepresent structural or censored zeros | Compare fitted and observed zeros conditionally; investigate assay mechanisms | Hurdle or zero-inflated model only when a distinct process is defensible | | Stable dispersion information | Boundary or noisy estimates distort inference | Profile $\alpha$; examine sensitivity to influential units and shrinkage | Report weak identification; use validated shrinkage or bootstrap checks | Useful diagnostics include randomized quantile or simulation-based residuals, observed-versus-replicated zero and upper-tail counts, residual spread across the fitted-mean range, leverage and deletion checks, and predictive calibration. A good global deviance does not exclude local lack of fit. Compare scientifically credible alternatives: Poisson with model-based and sandwich uncertainty, quasi-Poisson, NB1 and NB2, altered mean structures, cluster-aware models, and mechanism-based zero models. Stability of the target contrast is more informative than selecting whichever distribution has the smallest criterion in one dataset. # Bioinformatics Design and Interpretation For bulk RNA-seq, let $K_{gj}$ be the raw count for gene $g$ in biological sample $j$. A common model is $$ K_{gj}\sim\operatorname{NB}(\mu_{gj},\alpha_g), \qquad \log(\mu_{gj})=\log(s_j)+\mathbf x_j^\mathsf T\boldsymbol\beta_g, $$ where $s_j$ is an effective library-size factor. Independently sampled organisms, patients, or cultures are experimental units. Reads and technical lanes improve measurement of a sample; genes provide parallel tests and information for nuisance-trend estimation. Neither substitutes for biological replication. Design variables should represent condition, blocking, batch, pairing, and other prespecified structure. Normalization factors address library scale and composition; they do not remove arbitrary batch effects. Dispersion represents residual replicate variability conditional on the design, combining biological heterogeneity and technical variation that remains after preprocessing. Statistically, $\exp(\beta_{gA})$ is a model-based expression mean ratio and its interval quantifies sampling uncertainty under the analysis. Biologically, a credible mechanism also depends on absolute abundance, effect magnitude, replicate consistency, tissue or cell composition, annotation, technical bias, and independent validation. A small adjusted $p$-value is not itself evidence of large or causal biological change. # Connections and Take-home Messages - [Negative binomial distribution and overdispersion](../probability-randomness/negative-binomial-distribution-and-overdispersion.qmd) derives the probability law and its Poisson--gamma mixture. - [Diagnosing and modeling overdispersion](diagnosing-and-modeling-overdispersion.qmd) separates evidence of extra variation from its possible mechanisms and remedies. - [Poisson regression](poisson-regression.qmd) is the equidispersed boundary model. - [Offsets and modeling event rates](offsets-and-modeling-event-rates.qmd) explains rate denominators and exposure uncertainty. - [Robust standard errors and robust regression](robust-standard-errors-and-robust-regression.qmd) distinguishes covariance robustness from distributional modeling. - [Likelihood-ratio, Wald, and score tests](../statistical-inference/likelihood-ratio-wald-and-score-tests.qmd) compares coefficient-testing procedures. 1. NB2 regression combines a log-linear conditional mean with variance $\mu+\alpha\mu^2$. 2. Exponentiated slopes remain conditional count or rate ratios; dispersion changes weighting and uncertainty, not coefficient scale. 3. The Poisson--gamma mixture is an interpretation, not proof of a biological latent-rate mechanism. 4. Mean structure, exposure, dependence, zeros, and influential units must be checked before attributing lack of fit to negative binomial dispersion. 5. Bioinformatics inference is supported by independent biological samples, not reads, genes, cells, or technical aliquots counted as replicates. # References - Lawless, J. F. (1987). Negative binomial and mixed Poisson regression. *The Canadian Journal of Statistics*, 15(3), 209--225. <https://doi.org/10.2307/3314912> - McCullagh, P., and Nelder, J. A. (1989). *Generalized Linear Models* (2nd ed.). Chapman and Hall/CRC. <https://doi.org/10.1201/9780203753736> - Cameron, A. C., and Trivedi, P. K. (2013). *Regression Analysis of Count Data* (2nd ed.). Cambridge University Press. <https://doi.org/10.1017/CBO9781139013567> - Robinson, M. D., McCarthy, D. J., and Smyth, G. K. (2010). edgeR: a Bioconductor package for differential expression analysis of digital gene expression data. *Bioinformatics*, 26(1), 139--140. <https://doi.org/10.1093/bioinformatics/btp616> - Love, M. I., Huber, W., and Anders, S. (2014). Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2. *Genome Biology*, 15, 550. <https://doi.org/10.1186/s13059-014-0550-8> - Venables, W. N., and Ripley, B. D. `glm.nb`: fit a negative binomial generalized linear model. *MASS package documentation*. <https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/glm.nb.html>