Beta-binomial Distribution

Binomial counts with heterogeneous success probabilities

Statistics

Probability

Distributions

Proportion Data

The beta-binomial distribution, including its mixture derivation, moments, intracluster correlation, overdispersion, and biological interpretation.

Author

Jiahong Sun

Published

September 14, 2026

Modified

September 14, 2026

1 Topic at a Glance

One-sentence summary: The beta-binomial distribution models a bounded success count when the success probability varies across clusters or experimental units.

Item	Description
Domain	Probability & Randomness; discrete mixture distributions
Support	$Y\in\{0,1,\ldots,n\}$ for fixed $n$
Parameters	Trial count $n$ and beta shapes $a,b>0$, or mean probability $\pi$ and intracluster correlation $\rho$
Key variance	$n\pi(1-\pi)\{1+(n-1)\rho\}$
Typical applications	Methylated-read counts, allele counts, clustered binary outcomes, spectral-count proportions
Prerequisites	Bernoulli and binomial distributions; gamma and beta distributions

2 A Binomial Probability That Varies Across Units

The ordinary binomial model assumes one fixed success probability $p$ across $n$ conditionally independent trials. The beta-binomial instead introduces a latent probability:

\[ P\sim\operatorname{Beta}(a,b), \qquad Y\mid P=p\sim\operatorname{Binomial}(n,p), \]

where $a,b>0$. Conditional on $P$, the trials are independent and share one probability. Marginally, they are exchangeable and positively associated because they share the same random $P$.

Integrating out $P$ gives

\[ \boxed{ P(Y=y) ={n\choose y}\frac{B(y+a,n-y+b)}{B(a,b)}, \qquad y=0,1,\ldots,n. } \]

Here $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$ is the beta function. Unlike a continuous beta variable, $Y$ is a discrete count and can place positive probability on both endpoints 0 and $n$.

3 Deriving the Marginal Mass Function

The beta-binomial probability follows directly from the multiplication rule and marginalization:

\[ \begin{aligned} P(Y=y) &=\int_0^1 P(Y=y\mid p)f_P(p)\,dp\\ &={n\choose y}\frac{1}{B(a,b)} \int_0^1 p^{y+a-1}(1-p)^{n-y+b-1}\,dp\\ &={n\choose y}\frac{B(y+a,n-y+b)}{B(a,b)}. \end{aligned} \]

The integral is the normalizing beta function with updated shapes. This same algebra has two interpretations that should not be conflated:

Sampling heterogeneity: each cluster has its own latent probability drawn from a population distribution, giving a beta-binomial sampling law across clusters.
Bayesian prediction: a beta prior updated by binomial data produces a beta posterior; integrating a future binomial count over that posterior gives a beta-binomial posterior predictive law.

The formula is shared, but the random quantity and scientific claim differ.

4 Mean, Variance, and Intracluster Correlation

Define

\[ \pi=E(P)=\frac{a}{a+b}, \qquad \rho=\frac{1}{a+b+1}. \]

The parameter $\pi$ is the mean success probability. The concentration $a+b$ controls heterogeneity: larger concentration means smaller $\rho$ and a tighter distribution of $P$ around $\pi$.

Using iterated expectation,

\[ \boxed{E(Y)=E\{E(Y\mid P)\}=n\pi.} \]

The law of total variance gives

\[ \begin{aligned} \operatorname{Var}(Y) &=E\{nP(1-P)\}+\operatorname{Var}(nP)\\ &=n\pi(1-\pi)\{1+(n-1)\rho\}. \end{aligned} \]

Therefore

\[ \boxed{ \operatorname{Var}(Y) =n\pi(1-\pi)\{1+(n-1)\rho\}. } \]

The binomial variance is multiplied by the design effect $1+(n-1)\rho$. Even a modest within-cluster correlation can matter when $n$ is large.

To see where $\rho$ comes from, write $Y=\sum_{j=1}^nX_j$ with $X_j\mid P\sim\operatorname{Bernoulli}(P)$. For distinct trials $j\neq k$,

\[ \operatorname{Cov}(X_j,X_k) =\operatorname{Var}(P), \]

and because each marginal $X_j$ is Bernoulli with mean $\pi$,

\[ \operatorname{Corr}(X_j,X_k) =\frac{\operatorname{Var}(P)}{\pi(1-\pi)} =\frac{1}{a+b+1} =\rho. \]

Thus the standard beta-binomial imposes one positive correlation for every pair of trials in a cluster. This exchangeable dependence is useful but restrictive.

5 Limiting and Special Cases

Hold $\pi=a/(a+b)$ fixed while $a+b\to\infty$. Then $P$ concentrates at $\pi$, $\rho\to0$, and

\[ Y\Rightarrow\operatorname{Binomial}(n,\pi). \]

If $a=b=1$, then $P$ is uniform on $(0,1)$ and

\[ P(Y=y)=\frac{1}{n+1} \]

for every $y=0,\ldots,n$: the count itself is discrete uniform. If $a$ and $b$ are both below one, the latent probability favors values near 0 and 1, often producing a U-shaped count distribution with many endpoint counts.

The beta-binomial only adds positive association and overdispersion relative to a binomial with probability $\pi$. Sampling without replacement can instead produce negative association and underdispersion; the hypergeometric model is then the natural finite-population law.

6 A Predictive View of Reinforcement

After observing $s$ successes and $f$ failures under a beta–Bernoulli model,

\[ P\mid\text{data}\sim\operatorname{Beta}(a+s,b+f), \]

so the predictive probability that the next trial succeeds is

\[ P(X_{s+f+1}=1\mid\text{data}) =\frac{a+s}{a+b+s+f}. \]

Earlier successes increase the predictive chance of another success, and earlier failures decrease it. This is the Pólya-urn representation of the beta-binomial: reinforcement produces exchangeable, positively correlated trials. It is an illuminating probability construction, not a claim that literal reinforcement occurs in a biological assay.

7 Worked Example: Methylated Reads

Suppose each biological sample yields $n=20$ usable reads at a CpG site. Across specimens, let the latent methylation probability follow

\[ P_i\sim\operatorname{Beta}(8,2), \]

and, conditionally on $P_i$, let methylated-read counts satisfy

\[ Y_i\mid P_i\sim\operatorname{Binomial}(20,P_i). \]

Then

\[ \pi=\frac{8}{10}=0.8, \qquad \rho=\frac{1}{11}\approx0.091, \]

and

\[ E(Y_i)=20(0.8)=16. \]

The beta-binomial variance is

\[ \operatorname{Var}(Y_i) =20(0.8)(0.2)\left\{1+19\left(\frac{1}{11}\right)\right\} \approx8.73. \]

A binomial model fixed at $p=0.8$ would have variance $3.2$. The extra variation represents between-sample heterogeneity in the latent methylation probability under this model.

Reads within one library are technical subsamples. The biological sample $i$ is the experimental unit, and treatment replication comes from independently sampled specimens assigned or observed under each condition. A beta-binomial observation model can represent read-level sampling plus between-sample heterogeneity, but it cannot rescue a design with no independent biological replication.

Statistically, a group difference concerns a contrast between distributions or mean probabilities after accounting for dispersion and design. Biologically, it concerns methylation in the target tissue, cell mixture, developmental stage, and population represented by the specimens; those interpretations require design knowledge beyond the count model.

8 Assumptions and Diagnostic Questions

Assumption or implication	Why it matters	Diagnostic question	Response or sensitivity check
Denominator $n$ is fixed or conditioned upon	the support is $0,\ldots,n$	is total coverage informative about biology or detection?	jointly model totals or test sensitivity to conditioning
One latent $P$ per cluster	it defines which trials share dependence	are reads grouped by specimen, locus, batch, or another unit?	align the hierarchy with the data-generating process
Beta-distributed heterogeneity	mean and dispersion determine a specific shape	are latent proportions multimodal or concentrated at boundaries?	use richer random-effects, mixture, or inflated models
Exchangeable positive correlation	every within-cluster pair has correlation $\rho$	does dependence vary by position, time, molecule, or read family?	model the dependence structure explicitly
Conditional binomial sampling	trials share $P$ and are independent given it	are there PCR duplicates, mapping bias, or allele-specific errors?	deduplicate, model measurement error, or aggregate appropriately
Independent biological units	between-unit inference needs genuine replication	are repeated reads or cells being counted as samples?	analyze at the specimen level or use a hierarchical model

Diagnostics should be conditional on denominators and predictors. Compare fitted and observed variance, endpoint frequencies, tails, and replicate-level proportions. Simulated replicated datasets are especially useful because residual overdispersion can coexist with zero inflation, omitted covariates, or a wrong grouping level.

9 Estimation and Interpretation

For independent observations $(y_i,n_i)$, the likelihood is

\[ L(a,b) =\prod_i {n_i\choose y_i} \frac{B(y_i+a,n_i-y_i+b)}{B(a,b)}. \]

Maximum-likelihood estimates usually require numerical optimization. Moment estimates can be unstable, particularly with few clusters, similar $n_i$, or probabilities near a boundary. In high-throughput studies, dispersion estimates are commonly stabilized by borrowing information across loci or features.

That borrowing estimates a nuisance variation pattern; it does not change the experimental unit. Uncertainty for a treatment contrast must still reflect the number and structure of independent biological samples. Report the mean-probability contrast or another scientifically defined effect, its uncertainty, the dispersion convention, and the scale on which the effect is interpreted.

10 Common Pitfalls

Calling an observed proportion beta-binomial: the beta-binomial models its numerator conditional on a known denominator; the latent $P$ is beta distributed.
Pooling biological samples: combining counts erases between-sample variation, the feature the mixture was introduced to represent.
Treating all read pairs as independent after marginalization: they are conditionally independent given $P$, but positively associated marginally.
Interpreting $\rho$ as a causal biological interaction: it is the correlation induced by a shared random probability under the model.
Assuming overdispersion uniquely selects the beta-binomial: omitted predictors, contamination, zero/one inflation, and dependence can produce similar symptoms.
Ignoring denominator variation: equal proportions based on 5 and 500 trials have different sampling information.
Confusing prior prediction with population heterogeneity: identical algebra can support different inferential stories.

11 Connections to the Knowledge Map

Bernoulli and binomial distributions provides the conditional sampling model and binomial reference variance.
Gamma and beta distributions develops beta shapes, moments, and beta–binomial conjugacy.
Joint, marginal, and conditional distributions explains how integrating over $P$ creates a marginal mixture distribution.
Independence and conditional independence distinguishes conditional independence from marginal dependence.
Covariance and correlation supplies the variance-of-a-sum and intracluster-correlation calculations.
Counting rules and combinatorial probability contrasts binomial overdispersion with sampling-without-replacement underdispersion.

12 Take-home Messages

A beta-binomial count is binomial conditional on a beta-distributed success probability.
Marginalization produces positive within-cluster correlation and variance inflated by $1+(n-1)\rho$.
Mean probability and concentration separate location from heterogeneity, with the binomial recovered as concentration grows.
The same beta-binomial formula can describe population heterogeneity or Bayesian prediction; the scientific interpretation must be stated.
The standard model assumes one exchangeable positive dependence pattern and may miss boundary inflation, multimodality, or structured correlation.
In sequencing applications, specimens are biological units while reads supply within-sample measurement information.

13 References

Skellam, J. G. “A Probability Distribution Derived from the Binomial Distribution by Regarding the Probability of Success as Variable between the Sets of Trials.” Journal of the Royal Statistical Society: Series B 10(2), 1948, 257–261. https://doi.org/10.1111/j.2517-6161.1948.tb00014.x.
Johnson, N. L., Kemp, A. W., and Kotz, S. Univariate Discrete Distributions, 3rd ed. Wiley, 2005. https://doi.org/10.1002/0471715816.
Crowder, M. J. “Beta-binomial ANOVA for Proportions.” Applied Statistics 27(1), 1978, 34–37. https://doi.org/10.2307/2346223.
Feng, H., Conneely, K. N., and Wu, H. “A Bayesian Hierarchical Model to Detect Differentially Methylated Loci from Single Nucleotide Resolution Sequencing Data.” Nucleic Acids Research 42(8), 2014, e69. https://doi.org/10.1093/nar/gku154.
Pham, T. V., Piersma, S. R., Warmoes, M., and Jimenez, C. R. “On the Beta-binomial Model for Analysis of Spectral Count Data in Label-free Tandem Mass Spectrometry-based Proteomics.” Bioinformatics 26(3), 2010, 363–369. https://doi.org/10.1093/bioinformatics/btp677.
Yee, T. W. “The Beta-binomial Distribution.” VGAM Documentation. https://search.r-project.org/CRAN/refmans/VGAM/html/betabinomUC.html.

--- title: "Beta-binomial Distribution" subtitle: "Binomial counts with heterogeneous success probabilities" description: "The beta-binomial distribution, including its mixture derivation, moments, intracluster correlation, overdispersion, and biological interpretation." author: "Jiahong Sun" date: "2026-09-14" date-modified: "2026-09-14" categories: [Statistics, Probability, Distributions, Proportion Data] toc: true number-sections: true draft: false --- # Topic at a Glance > **One-sentence summary:** The beta-binomial distribution models a bounded success count when the success probability varies across clusters or experimental units. | Item | Description | |---|---| | Domain | Probability & Randomness; discrete mixture distributions | | Support | $Y\in\{0,1,\ldots,n\}$ for fixed $n$ | | Parameters | Trial count $n$ and beta shapes $a,b>0$, or mean probability $\pi$ and intracluster correlation $\rho$ | | Key variance | $n\pi(1-\pi)\{1+(n-1)\rho\}$ | | Typical applications | Methylated-read counts, allele counts, clustered binary outcomes, spectral-count proportions | | Prerequisites | [Bernoulli and binomial distributions](bernoulli-and-binomial-distributions.qmd); [gamma and beta distributions](gamma-and-beta-distributions.qmd) | # A Binomial Probability That Varies Across Units The ordinary binomial model assumes one fixed success probability $p$ across $n$ conditionally independent trials. The beta-binomial instead introduces a latent probability: $$ P\sim\operatorname{Beta}(a,b), \qquad Y\mid P=p\sim\operatorname{Binomial}(n,p), $$ where $a,b>0$. Conditional on $P$, the trials are independent and share one probability. Marginally, they are exchangeable and positively associated because they share the same random $P$. Integrating out $P$ gives $$ \boxed{ P(Y=y) ={n\choose y}\frac{B(y+a,n-y+b)}{B(a,b)}, \qquad y=0,1,\ldots,n. } $$ Here $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$ is the beta function. Unlike a continuous beta variable, $Y$ is a discrete count and can place positive probability on both endpoints 0 and $n$. # Deriving the Marginal Mass Function The beta-binomial probability follows directly from the multiplication rule and marginalization: $$ \begin{aligned} P(Y=y) &=\int_0^1 P(Y=y\mid p)f_P(p)\,dp\\ &={n\choose y}\frac{1}{B(a,b)} \int_0^1 p^{y+a-1}(1-p)^{n-y+b-1}\,dp\\ &={n\choose y}\frac{B(y+a,n-y+b)}{B(a,b)}. \end{aligned} $$ The integral is the normalizing beta function with updated shapes. This same algebra has two interpretations that should not be conflated: - **Sampling heterogeneity:** each cluster has its own latent probability drawn from a population distribution, giving a beta-binomial sampling law across clusters. - **Bayesian prediction:** a beta prior updated by binomial data produces a beta posterior; integrating a future binomial count over that posterior gives a beta-binomial posterior predictive law. The formula is shared, but the random quantity and scientific claim differ. # Mean, Variance, and Intracluster Correlation Define $$ \pi=E(P)=\frac{a}{a+b}, \qquad \rho=\frac{1}{a+b+1}. $$ The parameter $\pi$ is the mean success probability. The concentration $a+b$ controls heterogeneity: larger concentration means smaller $\rho$ and a tighter distribution of $P$ around $\pi$. Using iterated expectation, $$ \boxed{E(Y)=E\{E(Y\mid P)\}=n\pi.} $$ The law of total variance gives $$ \begin{aligned} \operatorname{Var}(Y) &=E\{nP(1-P)\}+\operatorname{Var}(nP)\\ &=n\pi(1-\pi)\{1+(n-1)\rho\}. \end{aligned} $$ Therefore $$ \boxed{ \operatorname{Var}(Y) =n\pi(1-\pi)\{1+(n-1)\rho\}. } $$ The binomial variance is multiplied by the **design effect** $1+(n-1)\rho$. Even a modest within-cluster correlation can matter when $n$ is large. To see where $\rho$ comes from, write $Y=\sum_{j=1}^nX_j$ with $X_j\mid P\sim\operatorname{Bernoulli}(P)$. For distinct trials $j\neq k$, $$ \operatorname{Cov}(X_j,X_k) =\operatorname{Var}(P), $$ and because each marginal $X_j$ is Bernoulli with mean $\pi$, $$ \operatorname{Corr}(X_j,X_k) =\frac{\operatorname{Var}(P)}{\pi(1-\pi)} =\frac{1}{a+b+1} =\rho. $$ Thus the standard beta-binomial imposes one positive correlation for every pair of trials in a cluster. This exchangeable dependence is useful but restrictive. # Limiting and Special Cases Hold $\pi=a/(a+b)$ fixed while $a+b\to\infty$. Then $P$ concentrates at $\pi$, $\rho\to0$, and $$ Y\Rightarrow\operatorname{Binomial}(n,\pi). $$ If $a=b=1$, then $P$ is uniform on $(0,1)$ and $$ P(Y=y)=\frac{1}{n+1} $$ for every $y=0,\ldots,n$: the count itself is discrete uniform. If $a$ and $b$ are both below one, the latent probability favors values near 0 and 1, often producing a U-shaped count distribution with many endpoint counts. The beta-binomial only adds positive association and overdispersion relative to a binomial with probability $\pi$. Sampling without replacement can instead produce negative association and underdispersion; the [hypergeometric model](counting-rules-and-combinatorial-probability.qmd) is then the natural finite-population law. # A Predictive View of Reinforcement After observing $s$ successes and $f$ failures under a beta--Bernoulli model, $$ P\mid\text{data}\sim\operatorname{Beta}(a+s,b+f), $$ so the predictive probability that the next trial succeeds is $$ P(X_{s+f+1}=1\mid\text{data}) =\frac{a+s}{a+b+s+f}. $$ Earlier successes increase the predictive chance of another success, and earlier failures decrease it. This is the Pólya-urn representation of the beta-binomial: reinforcement produces exchangeable, positively correlated trials. It is an illuminating probability construction, not a claim that literal reinforcement occurs in a biological assay. # Worked Example: Methylated Reads Suppose each biological sample yields $n=20$ usable reads at a CpG site. Across specimens, let the latent methylation probability follow $$ P_i\sim\operatorname{Beta}(8,2), $$ and, conditionally on $P_i$, let methylated-read counts satisfy $$ Y_i\mid P_i\sim\operatorname{Binomial}(20,P_i). $$ Then $$ \pi=\frac{8}{10}=0.8, \qquad \rho=\frac{1}{11}\approx0.091, $$ and $$ E(Y_i)=20(0.8)=16. $$ The beta-binomial variance is $$ \operatorname{Var}(Y_i) =20(0.8)(0.2)\left\{1+19\left(\frac{1}{11}\right)\right\} \approx8.73. $$ A binomial model fixed at $p=0.8$ would have variance $3.2$. The extra variation represents between-sample heterogeneity in the latent methylation probability under this model. Reads within one library are technical subsamples. The biological sample $i$ is the experimental unit, and treatment replication comes from independently sampled specimens assigned or observed under each condition. A beta-binomial observation model can represent read-level sampling plus between-sample heterogeneity, but it cannot rescue a design with no independent biological replication. Statistically, a group difference concerns a contrast between distributions or mean probabilities after accounting for dispersion and design. Biologically, it concerns methylation in the target tissue, cell mixture, developmental stage, and population represented by the specimens; those interpretations require design knowledge beyond the count model. # Assumptions and Diagnostic Questions | Assumption or implication | Why it matters | Diagnostic question | Response or sensitivity check | |---|---|---|---| | Denominator $n$ is fixed or conditioned upon | the support is $0,\ldots,n$ | is total coverage informative about biology or detection? | jointly model totals or test sensitivity to conditioning | | One latent $P$ per cluster | it defines which trials share dependence | are reads grouped by specimen, locus, batch, or another unit? | align the hierarchy with the data-generating process | | Beta-distributed heterogeneity | mean and dispersion determine a specific shape | are latent proportions multimodal or concentrated at boundaries? | use richer random-effects, mixture, or inflated models | | Exchangeable positive correlation | every within-cluster pair has correlation $\rho$ | does dependence vary by position, time, molecule, or read family? | model the dependence structure explicitly | | Conditional binomial sampling | trials share $P$ and are independent given it | are there PCR duplicates, mapping bias, or allele-specific errors? | deduplicate, model measurement error, or aggregate appropriately | | Independent biological units | between-unit inference needs genuine replication | are repeated reads or cells being counted as samples? | analyze at the specimen level or use a hierarchical model | Diagnostics should be conditional on denominators and predictors. Compare fitted and observed variance, endpoint frequencies, tails, and replicate-level proportions. Simulated replicated datasets are especially useful because residual overdispersion can coexist with zero inflation, omitted covariates, or a wrong grouping level. # Estimation and Interpretation For independent observations $(y_i,n_i)$, the likelihood is $$ L(a,b) =\prod_i {n_i\choose y_i} \frac{B(y_i+a,n_i-y_i+b)}{B(a,b)}. $$ Maximum-likelihood estimates usually require numerical optimization. Moment estimates can be unstable, particularly with few clusters, similar $n_i$, or probabilities near a boundary. In high-throughput studies, dispersion estimates are commonly stabilized by borrowing information across loci or features. That borrowing estimates a nuisance variation pattern; it does not change the experimental unit. Uncertainty for a treatment contrast must still reflect the number and structure of independent biological samples. Report the mean-probability contrast or another scientifically defined effect, its uncertainty, the dispersion convention, and the scale on which the effect is interpreted. # Common Pitfalls 1. **Calling an observed proportion beta-binomial:** the beta-binomial models its numerator conditional on a known denominator; the latent $P$ is beta distributed. 2. **Pooling biological samples:** combining counts erases between-sample variation, the feature the mixture was introduced to represent. 3. **Treating all read pairs as independent after marginalization:** they are conditionally independent given $P$, but positively associated marginally. 4. **Interpreting $\rho$ as a causal biological interaction:** it is the correlation induced by a shared random probability under the model. 5. **Assuming overdispersion uniquely selects the beta-binomial:** omitted predictors, contamination, zero/one inflation, and dependence can produce similar symptoms. 6. **Ignoring denominator variation:** equal proportions based on 5 and 500 trials have different sampling information. 7. **Confusing prior prediction with population heterogeneity:** identical algebra can support different inferential stories. # Connections to the Knowledge Map - [Bernoulli and binomial distributions](bernoulli-and-binomial-distributions.qmd) provides the conditional sampling model and binomial reference variance. - [Gamma and beta distributions](gamma-and-beta-distributions.qmd) develops beta shapes, moments, and beta--binomial conjugacy. - [Joint, marginal, and conditional distributions](joint-marginal-and-conditional-distributions.qmd) explains how integrating over $P$ creates a marginal mixture distribution. - [Independence and conditional independence](independence-and-conditional-independence.qmd) distinguishes conditional independence from marginal dependence. - [Covariance and correlation](covariance-and-correlation.qmd) supplies the variance-of-a-sum and intracluster-correlation calculations. - [Counting rules and combinatorial probability](counting-rules-and-combinatorial-probability.qmd) contrasts binomial overdispersion with sampling-without-replacement underdispersion. # Take-home Messages 1. A beta-binomial count is binomial conditional on a beta-distributed success probability. 2. Marginalization produces positive within-cluster correlation and variance inflated by $1+(n-1)\rho$. 3. Mean probability and concentration separate location from heterogeneity, with the binomial recovered as concentration grows. 4. The same beta-binomial formula can describe population heterogeneity or Bayesian prediction; the scientific interpretation must be stated. 5. The standard model assumes one exchangeable positive dependence pattern and may miss boundary inflation, multimodality, or structured correlation. 6. In sequencing applications, specimens are biological units while reads supply within-sample measurement information. # References - Skellam, J. G. “A Probability Distribution Derived from the Binomial Distribution by Regarding the Probability of Success as Variable between the Sets of Trials.” *Journal of the Royal Statistical Society: Series B* 10(2), 1948, 257–261. [https://doi.org/10.1111/j.2517-6161.1948.tb00014.x](https://doi.org/10.1111/j.2517-6161.1948.tb00014.x). - Johnson, N. L., Kemp, A. W., and Kotz, S. *Univariate Discrete Distributions*, 3rd ed. Wiley, 2005. [https://doi.org/10.1002/0471715816](https://doi.org/10.1002/0471715816). - Crowder, M. J. “Beta-binomial ANOVA for Proportions.” *Applied Statistics* 27(1), 1978, 34–37. [https://doi.org/10.2307/2346223](https://doi.org/10.2307/2346223). - Feng, H., Conneely, K. N., and Wu, H. “A Bayesian Hierarchical Model to Detect Differentially Methylated Loci from Single Nucleotide Resolution Sequencing Data.” *Nucleic Acids Research* 42(8), 2014, e69. [https://doi.org/10.1093/nar/gku154](https://doi.org/10.1093/nar/gku154). - Pham, T. V., Piersma, S. R., Warmoes, M., and Jimenez, C. R. “On the Beta-binomial Model for Analysis of Spectral Count Data in Label-free Tandem Mass Spectrometry-based Proteomics.” *Bioinformatics* 26(3), 2010, 363–369. [https://doi.org/10.1093/bioinformatics/btp677](https://doi.org/10.1093/bioinformatics/btp677). - Yee, T. W. “The Beta-binomial Distribution.” *VGAM Documentation*. [https://search.r-project.org/CRAN/refmans/VGAM/html/betabinomUC.html](https://search.r-project.org/CRAN/refmans/VGAM/html/betabinomUC.html).