Nothing
Chapter 09: Models for the Binomial family
In glmbayes: Bayesian Generalized Linear Models (IID Samples)
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
library(glmbayes)
1. Introductory Discussion
Binomial generalized linear models (GLMs) are used when the response represents binary outcomes (success/failure) or proportions (successes out of trials). They are among the most widely used GLMs in applied statistics, powering models for:
- clinical trial response rates
- yes/no survey outcomes
- conversion rates in marketing
- event occurrence indicators
- grouped binomial data (successes out of n)
Binomial regression is a standard generalized linear model [@NelderWedderburn1972; @McCullagh1989; @Agresti2015].
In classical statistics, these models are fit using:
glm(..., family = binomial(link = ...))
In glmbayes, the Bayesian analogue is:
glmb(..., family = binomial(link = ...), pfamily = dNormal(mu, Sigma))
This chapter introduces:
- the structure of binomial GLMs
- the available link functions (logit, probit, cloglog)
- how to specify these models in glmbayes
- worked examples for each link function
We build on the foundations from Chapters 07 and 08, especially the role of link functions, log‑concavity, and prior specification.
2. Binomial Likelihood and Weighted Formulation
Binomial data arise in several equivalent representations:
- Binary outcomes: (y_i \in {0,1})
- Proportions: (y_i = \text{successes}_i / n_i) with weights (w_i = n_i)
- Two‑column matrix: (\text{cbind(successes, failures)})
In all cases, the underlying sampling model is
[
Y_i \sim \text{Binomial}(n_i, \mu_i), \qquad 0 < \mu_i < 1,
]
where:
- (n_i) is the number of trials,
- (\mu_i = \Pr(Y_i = 1)) is the success probability.
2.1 Linear predictor and mean structure
A binomial GLM links the mean (\mu_i) to a linear predictor through
[
\eta_i = x_i^\top \beta, \qquad \mu_i = g^{-1}(\eta_i),
]
where (g(\cdot)) is the chosen link function (logit, probit, cloglog, etc.).
2.2 Weighted binomial log‑likelihood
Using weights (w_i = n_i), the log‑likelihood (up to constants) becomes
[
\ell(\beta)
=
\sum_{i=1}^n
w_i\Big[
y_i \log(\mu_i)
+
(1-y_i)\log(1-\mu_i)
\Big].
]
This form is used by both glm() and the Bayesian functions glmb() and rglmb().
2.3 Exponential‑family representation
The binomial likelihood belongs to the exponential family [@McCullagh1989; @Agresti2015].
For a model with linear predictor
[
\eta_i = x_i^\top \beta,
]
and mean
[
\mu_i = g^{-1}(\eta_i),
]
the contribution of observation (i) to the log‑likelihood can be written as
[
\ell_i(\beta)
=
w_i\Big[
y_i \log(\mu_i)
+
(1-y_i)\log(1-\mu_i)
\Big],
]
where (w_i) is the number of trials (or a user‑supplied weight).
This representation does not require the link to be canonical.
The variance of a binomial observation is always
[
\mathrm{Var}(Y_i) = \mu_i(1-\mu_i),
]
regardless of the link function.
2.4 Link functions
The link function determines how the mean response relates to the linear predictor:
[
g(\mu_i) = \eta_i.
]
The binomial() family in R supports several link functions:
| Link Function | Formula | Notes |
|---------------|---------|-------|
| logit (canonical) | ( \eta = \log(\mu/(1-\mu)) ) | Most common; canonical link; induces log‑concavity in (\eta) |
| probit | ( \eta = \Phi^{-1}(\mu) ) | Based on the standard normal CDF; induces log‑concavity in (\eta) |
| cloglog | ( \eta = \log[-\log(1-\mu)] ) | Asymmetric; useful for rare events; induces log‑concavity in (\eta) |
| cauchit | ( \eta = \tan[\pi(\mu - 1/2)] ) | Heavy‑tailed; does not preserve log‑concavity in general |
| identity | ( \eta = \mu ) | Must ensure (0 < \mu < 1); does not preserve log‑concavity in general |
In this chapter we focus on the three most commonly used links:
- logit (Section 4)
- probit (Section 5)
- cloglog (Section 6)
Each link produces a different transformation (g^{-1}(\eta)) and therefore a different expression for the log‑likelihood and its derivatives. For the logit, probit, and cloglog links, the resulting log‑likelihood is known to be log‑concave in the linear predictor (\eta), which is crucial for stable envelope construction and accept–reject sampling in glmbayes.
The explicit formulas for each link are provided at the beginning of their respective sections.
3. Specifying Binomial Models in glmbayes
The general Bayesian call is:
glmb(
formula,
family = binomial(link = "logit" | "probit" | "cloglog"),
pfamily = dNormal(mu = mu, Sigma = V),
data = ...
)
3.1 Prior Specification
As in earlier chapters, the recommended workflow is:
ps <- Prior_Setup(formula, family = binomial(link = "logit"), data = ...)
mu <- ps$mu
V <- ps$Sigma
This produces:
- a Zellner‑type g‑prior [@GriffinBrown2010]
- prior means aligned with the null model
- prior variances scaled relative to the likelihood curvature
You may override these defaults for more informative priors (see Chapter 14).
4. Logit Link: Likelihood, Prior, Posterior
For the logit link:
[
\eta_i = \log!\left(\frac{\mu_i}{1-\mu_i}\right),
\qquad
\mu_i = \frac{1}{1+e^{-\eta_i}}.
]
Weighted Log-Likelihood
[
\ell_{\text{logit}}(\beta)
=
\sum_{i=1}^n
w_i\Big[
y_i\,\eta_i - \log(1+e^{\eta_i})
\Big].
]
Normal Prior
[
\log p(\beta)
=
-\tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0)
+ \text{const}.
]
Log-Posterior
[
\log p(\beta \mid y)
=
\sum_{i=1}^n
w_i\Big[
y_i\,\eta_i - \log(1+e^{\eta_i})
\Big]
-
\tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0)
+ \text{const}.
]
The logit link is the canonical choice for binomial GLMs [@McCullagh1989; @Agresti2015]:
[
\eta = \log\left(\frac{\mu}{1-\mu}\right)
]
It is symmetric and interpretable in terms of log‑odds.
4.1 Example Data
We use the Menarche dataset introduced in Chapter 08:
data(menarche,package="MASS")
Age2 <- menarche$Age - 13
Menarche_Model_Data <- data.frame(
Age = menarche$Age,
Total = menarche$Total,
Menarche = menarche$Menarche,
Age2 = Age2
)
Menarche_Model_Data
4.2 Prior Setup
ps <- Prior_Setup(
cbind(Menarche, Total - Menarche) ~ Age2,
family = binomial(link = "logit"),
data = Menarche_Model_Data
)
mu <- ps$mu
V <- ps$Sigma
4.3 Fit the Model
glmb.logit <- glmb(
cbind(Menarche, Total - Menarche) ~ Age2,
family = binomial(link = "logit"),
pfamily = dNormal(mu = mu, Sigma = V),
data = Menarche_Model_Data,
n = 1000
)
4.4 Summary
summary(glmb.logit)
This produces:
- posterior means and modes
- posterior standard deviations
- tail probabilities
- credible intervals
- DIC and effective parameter count [@Spiegelhalter2002]
The slope parameter typically shows strong evidence of increasing probability of menarche with age.
5. Probit Link: Likelihood, Prior, Posterior
The probit link is a common alternative to the logit when a latent normal formulation is convenient [@McCullagh1989; @Agresti2015].
For the probit link:
[
\eta_i = \Phi^{-1}(\mu_i),
\qquad
\mu_i = \Phi(\eta_i),
]
where (\Phi) is the standard normal CDF.
Weighted Log-Likelihood
[
\ell_{\text{probit}}(\beta)
=
\sum_{i=1}^n
w_i\Big[
y_i \log\Phi(\eta_i)
+
(1-y_i)\log\big(1-\Phi(\eta_i)\big)
\Big].
]
Normal Prior
[
\log p(\beta)
=
-\tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0)
+ \text{const}.
]
Log-Posterior
[
\log p(\beta \mid y)
=
\sum_{i=1}^n
w_i\Big[
y_i \log\Phi(\eta_i)
+
(1-y_i)\log\big(1-\Phi(\eta_i)\big)
\Big]
-
\tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0)
+ \text{const}.
]
It is similar to the logit link but has slightly thinner tails and a more Gaussian interpretation.
5.1 Prior Setup
ps2 <- Prior_Setup(
cbind(Menarche, Total - Menarche) ~ Age2,
family = binomial(link = "probit"),
data = Menarche_Model_Data
)
mu2 <- ps2$mu
V2 <- ps2$Sigma
5.2 Fit the Model
glmb.probit <- glmb(
cbind(Menarche, Total - Menarche) ~ Age2,
family = binomial(link = "probit"),
pfamily = dNormal(mu = mu2, Sigma = V2),
data = Menarche_Model_Data,
n = 1000
)
5.3 Summary
summary(glmb.probit)
The probit model typically yields:
- slightly smaller slope estimates
- slightly smaller posterior standard deviations
- similar fitted probabilities
The DIC often favors probit slightly for smooth S‑shaped curves (as seen in Chapter 05).
6. Complementary Log-Log (cloglog) Link: Likelihood, Prior, Posterior
The complementary log–log link is often used for asymmetric response curves and hazard‑type interpretations [@McCullagh1989; @Agresti2015].
For the cloglog link:
[
\eta_i = \log!\big[-\log(1-\mu_i)\big],
\qquad
\mu_i = 1 - \exp!\big(-e^{\eta_i}\big).
]
Weighted Log-Likelihood
[
\ell_{\text{cloglog}}(\beta)
=
\sum_{i=1}^n
w_i\Big[
y_i \log!\big(1 - e^{-e^{\eta_i}}\big)
+
(1-y_i)\big(-e^{\eta_i}\big)
\Big].
]
Normal Prior
[
\log p(\beta)
=
-\tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0)
+ \text{const}.
]
Log-Posterior
[
\log p(\beta \mid y)
=
\sum_{i=1}^n
w_i\Big[
y_i \log!\big(1 - e^{-e^{\eta_i}}\big)
+
(1-y_i)\big(-e^{\eta_i}\big)
\Big]
-
\tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0)
+ \text{const}.
]
The cloglog link is asymmetric:
It is useful when:
- the event probability increases rapidly near 0
- the hazard interpretation is meaningful
- the response curve is skewed
6.1 Prior Setup
ps3 <- Prior_Setup(
cbind(Menarche, Total - Menarche) ~ Age2,
family = binomial(link = "cloglog"),
data = Menarche_Model_Data
)
mu3 <- ps3$mu
V3 <- ps3$Sigma
6.2 Fit the Model
glmb.cloglog <- glmb(
cbind(Menarche, Total - Menarche) ~ Age2,
family = binomial(link = "cloglog"),
pfamily = dNormal(mu = mu3, Sigma = V3),
data = Menarche_Model_Data,
n = 1000
)
6.3 Summary
summary(glmb.cloglog)
The cloglog model often fits poorly for symmetric S‑shaped curves (as shown in Chapter 05), but it is valuable for:
- rare events
- survival‑type interpretations
- asymmetric growth curves
7. Comparing Link Functions
The Deviance Information Criterion (DIC) provides a Bayesian analogue to AIC [@Spiegelhalter2002]:
DIC_comp<-rbind(
extractAIC(glmb.logit),
extractAIC(glmb.probit),
extractAIC(glmb.cloglog))
rownames(DIC_comp)<-c("logit","probit","cloglog")
DIC_comp
Typical patterns:
- logit and probit often perform similarly
- probit may slightly outperform logit for smooth curves
- cloglog is best when the response is asymmetric or hazard‑like
The effective number of parameters (pD) is part of the same framework [@Spiegelhalter2002] and helps diagnose model complexity.
8. Concluding Discussion
Binomial GLMs are a core component of the glmbayes package. Their log‑concave likelihoods make them ideal for the envelope‑based accept‑reject sampler, and the familiar link functions allow analysts to choose models that match the scientific context [@McCullagh1989; @Gelman2013].
This chapter demonstrated:
- how binomial GLMs are structured
- how link functions differ
- how to specify priors using Prior_Setup
- how to fit logit, probit, and cloglog models
- how to compare models using DIC
In the next chapter, we extend these ideas to Poisson models, which share many structural similarities but introduce new considerations for count data.
Appendix A. Bayes Rules! companion — Perth rain (weather_perth)
Book: [@JohnsonLauEtAl2022], Chapter 13 — same model and informative priors as Chapter 08, Appendix A.
Model: raintomorrow ~ humidity9am (logit link)
Requires bayesrules.
library(bayesrules)
weather <- bayesrules::weather_perth
weather$raintomorrow <- as.integer(weather$raintomorrow == "Yes")
mu_w <- matrix(c(-1.4, 0.07), nrow = 1)
colnames(mu_w) <- c("(Intercept)", "humidity9am")
Sigma_w <- diag(c(0.7^2, 0.035^2))
dimnames(Sigma_w) <- list(colnames(mu_w), colnames(mu_w))
book_br09 <- data.frame(
parameter = c("(Intercept)", "humidity9am"),
book_lo = c(-5.08785, 0.04147),
book_hi = c(-4.13450, 0.05487),
book_mid = c(-4.611175, 0.04817),
check.names = FALSE
)
set.seed(2026)
glmb_rain <- glmb(
raintomorrow ~ humidity9am,
family = binomial(),
pfamily = dNormal(mu = mu_w, Sigma = Sigma_w),
data = weather,
n = 2000
)
print(glmb_rain)
br09_compare <- data.frame(
parameter = book_br09$parameter,
`Book 80% lo` = book_br09$book_lo,
`Book 80% hi` = book_br09$book_hi,
`Book midpoint` = book_br09$book_mid,
`glmb Post.Mean` = as.numeric(glmb_rain$coef.means[book_br09$parameter]),
`glmb Post.Sd` = sapply(book_br09$parameter, function(p)
sd(glmb_rain$coefficients[, p, drop = TRUE])),
check.names = FALSE
)
knitr::kable(br09_compare, digits = 4,
caption = "Bayes Rules! Ch. 13 vs. glmb() (informative priors)")
Multivariable extensions (humidity3pm, raintoday) use Prior_Setup() defaults on extra coefficients unless you add further informative book priors (Chapter 14).
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
library(glmbayes)
1. Introductory Discussion
Binomial generalized linear models (GLMs) are used when the response represents binary outcomes (success/failure) or proportions (successes out of trials). They are among the most widely used GLMs in applied statistics, powering models for:
- clinical trial response rates
- yes/no survey outcomes
- conversion rates in marketing
- event occurrence indicators
- grouped binomial data (successes out of n)
Binomial regression is a standard generalized linear model [@NelderWedderburn1972; @McCullagh1989; @Agresti2015].
In classical statistics, these models are fit using:
glm(..., family = binomial(link = ...))
In glmbayes, the Bayesian analogue is:
glmb(..., family = binomial(link = ...), pfamily = dNormal(mu, Sigma))
This chapter introduces:
- the structure of binomial GLMs
- the available link functions (logit, probit, cloglog)
- how to specify these models in glmbayes
- worked examples for each link function
We build on the foundations from Chapters 07 and 08, especially the role of link functions, log‑concavity, and prior specification.
2. Binomial Likelihood and Weighted Formulation
Binomial data arise in several equivalent representations:
- Binary outcomes: (y_i \in {0,1})
- Proportions: (y_i = \text{successes}_i / n_i) with weights (w_i = n_i)
- Two‑column matrix: (\text{cbind(successes, failures)})
In all cases, the underlying sampling model is
[ Y_i \sim \text{Binomial}(n_i, \mu_i), \qquad 0 < \mu_i < 1, ]
where: - (n_i) is the number of trials, - (\mu_i = \Pr(Y_i = 1)) is the success probability.
2.1 Linear predictor and mean structure
A binomial GLM links the mean (\mu_i) to a linear predictor through
[ \eta_i = x_i^\top \beta, \qquad \mu_i = g^{-1}(\eta_i), ]
where (g(\cdot)) is the chosen link function (logit, probit, cloglog, etc.).
2.2 Weighted binomial log‑likelihood
Using weights (w_i = n_i), the log‑likelihood (up to constants) becomes
[ \ell(\beta) = \sum_{i=1}^n w_i\Big[ y_i \log(\mu_i) + (1-y_i)\log(1-\mu_i) \Big]. ]
This form is used by both glm() and the Bayesian functions glmb() and rglmb().
2.3 Exponential‑family representation
The binomial likelihood belongs to the exponential family [@McCullagh1989; @Agresti2015].
For a model with linear predictor
[
\eta_i = x_i^\top \beta,
]
and mean
[
\mu_i = g^{-1}(\eta_i),
]
the contribution of observation (i) to the log‑likelihood can be written as
[
\ell_i(\beta)
=
w_i\Big[
y_i \log(\mu_i)
+
(1-y_i)\log(1-\mu_i)
\Big],
]
where (w_i) is the number of trials (or a user‑supplied weight).
This representation does not require the link to be canonical.
The variance of a binomial observation is always
[
\mathrm{Var}(Y_i) = \mu_i(1-\mu_i),
]
regardless of the link function.
2.4 Link functions
The link function determines how the mean response relates to the linear predictor: [ g(\mu_i) = \eta_i. ]
The binomial() family in R supports several link functions:
| Link Function | Formula | Notes | |---------------|---------|-------| | logit (canonical) | ( \eta = \log(\mu/(1-\mu)) ) | Most common; canonical link; induces log‑concavity in (\eta) | | probit | ( \eta = \Phi^{-1}(\mu) ) | Based on the standard normal CDF; induces log‑concavity in (\eta) | | cloglog | ( \eta = \log[-\log(1-\mu)] ) | Asymmetric; useful for rare events; induces log‑concavity in (\eta) | | cauchit | ( \eta = \tan[\pi(\mu - 1/2)] ) | Heavy‑tailed; does not preserve log‑concavity in general | | identity | ( \eta = \mu ) | Must ensure (0 < \mu < 1); does not preserve log‑concavity in general |
In this chapter we focus on the three most commonly used links:
- logit (Section 4)
- probit (Section 5)
- cloglog (Section 6)
Each link produces a different transformation (g^{-1}(\eta)) and therefore a different expression for the log‑likelihood and its derivatives. For the logit, probit, and cloglog links, the resulting log‑likelihood is known to be log‑concave in the linear predictor (\eta), which is crucial for stable envelope construction and accept–reject sampling in glmbayes.
The explicit formulas for each link are provided at the beginning of their respective sections.
3. Specifying Binomial Models in glmbayes
The general Bayesian call is:
glmb( formula, family = binomial(link = "logit" | "probit" | "cloglog"), pfamily = dNormal(mu = mu, Sigma = V), data = ... )
3.1 Prior Specification
As in earlier chapters, the recommended workflow is:
ps <- Prior_Setup(formula, family = binomial(link = "logit"), data = ...) mu <- ps$mu V <- ps$Sigma
This produces:
- a Zellner‑type g‑prior [@GriffinBrown2010]
- prior means aligned with the null model
- prior variances scaled relative to the likelihood curvature
You may override these defaults for more informative priors (see Chapter 14).
4. Logit Link: Likelihood, Prior, Posterior
For the logit link: [ \eta_i = \log!\left(\frac{\mu_i}{1-\mu_i}\right), \qquad \mu_i = \frac{1}{1+e^{-\eta_i}}. ]
Weighted Log-Likelihood
[ \ell_{\text{logit}}(\beta) = \sum_{i=1}^n w_i\Big[ y_i\,\eta_i - \log(1+e^{\eta_i}) \Big]. ]
Normal Prior
[ \log p(\beta) = -\tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0) + \text{const}. ]
Log-Posterior
[ \log p(\beta \mid y) = \sum_{i=1}^n w_i\Big[ y_i\,\eta_i - \log(1+e^{\eta_i}) \Big] - \tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0) + \text{const}. ]
The logit link is the canonical choice for binomial GLMs [@McCullagh1989; @Agresti2015]:
[ \eta = \log\left(\frac{\mu}{1-\mu}\right) ]
It is symmetric and interpretable in terms of log‑odds.
4.1 Example Data
We use the Menarche dataset introduced in Chapter 08:
data(menarche,package="MASS") Age2 <- menarche$Age - 13 Menarche_Model_Data <- data.frame( Age = menarche$Age, Total = menarche$Total, Menarche = menarche$Menarche, Age2 = Age2 ) Menarche_Model_Data
4.2 Prior Setup
ps <- Prior_Setup( cbind(Menarche, Total - Menarche) ~ Age2, family = binomial(link = "logit"), data = Menarche_Model_Data ) mu <- ps$mu V <- ps$Sigma
4.3 Fit the Model
glmb.logit <- glmb( cbind(Menarche, Total - Menarche) ~ Age2, family = binomial(link = "logit"), pfamily = dNormal(mu = mu, Sigma = V), data = Menarche_Model_Data, n = 1000 )
4.4 Summary
summary(glmb.logit)
This produces:
- posterior means and modes
- posterior standard deviations
- tail probabilities
- credible intervals
- DIC and effective parameter count [@Spiegelhalter2002]
The slope parameter typically shows strong evidence of increasing probability of menarche with age.
5. Probit Link: Likelihood, Prior, Posterior
The probit link is a common alternative to the logit when a latent normal formulation is convenient [@McCullagh1989; @Agresti2015].
For the probit link: [ \eta_i = \Phi^{-1}(\mu_i), \qquad \mu_i = \Phi(\eta_i), ] where (\Phi) is the standard normal CDF.
Weighted Log-Likelihood
[ \ell_{\text{probit}}(\beta) = \sum_{i=1}^n w_i\Big[ y_i \log\Phi(\eta_i) + (1-y_i)\log\big(1-\Phi(\eta_i)\big) \Big]. ]
Normal Prior
[ \log p(\beta) = -\tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0) + \text{const}. ]
Log-Posterior
[ \log p(\beta \mid y) = \sum_{i=1}^n w_i\Big[ y_i \log\Phi(\eta_i) + (1-y_i)\log\big(1-\Phi(\eta_i)\big) \Big] - \tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0) + \text{const}. ]
It is similar to the logit link but has slightly thinner tails and a more Gaussian interpretation.
5.1 Prior Setup
ps2 <- Prior_Setup( cbind(Menarche, Total - Menarche) ~ Age2, family = binomial(link = "probit"), data = Menarche_Model_Data ) mu2 <- ps2$mu V2 <- ps2$Sigma
5.2 Fit the Model
glmb.probit <- glmb( cbind(Menarche, Total - Menarche) ~ Age2, family = binomial(link = "probit"), pfamily = dNormal(mu = mu2, Sigma = V2), data = Menarche_Model_Data, n = 1000 )
5.3 Summary
summary(glmb.probit)
The probit model typically yields:
- slightly smaller slope estimates
- slightly smaller posterior standard deviations
- similar fitted probabilities
The DIC often favors probit slightly for smooth S‑shaped curves (as seen in Chapter 05).
6. Complementary Log-Log (cloglog) Link: Likelihood, Prior, Posterior
The complementary log–log link is often used for asymmetric response curves and hazard‑type interpretations [@McCullagh1989; @Agresti2015].
For the cloglog link: [ \eta_i = \log!\big[-\log(1-\mu_i)\big], \qquad \mu_i = 1 - \exp!\big(-e^{\eta_i}\big). ]
Weighted Log-Likelihood
[ \ell_{\text{cloglog}}(\beta) = \sum_{i=1}^n w_i\Big[ y_i \log!\big(1 - e^{-e^{\eta_i}}\big) + (1-y_i)\big(-e^{\eta_i}\big) \Big]. ]
Normal Prior
[ \log p(\beta) = -\tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0) + \text{const}. ]
Log-Posterior
[ \log p(\beta \mid y) = \sum_{i=1}^n w_i\Big[ y_i \log!\big(1 - e^{-e^{\eta_i}}\big) + (1-y_i)\big(-e^{\eta_i}\big) \Big] - \tfrac12(\beta-\mu_0)^\top \Sigma_0^{-1}(\beta-\mu_0) + \text{const}. ]
The cloglog link is asymmetric:
It is useful when:
- the event probability increases rapidly near 0
- the hazard interpretation is meaningful
- the response curve is skewed
6.1 Prior Setup
ps3 <- Prior_Setup( cbind(Menarche, Total - Menarche) ~ Age2, family = binomial(link = "cloglog"), data = Menarche_Model_Data ) mu3 <- ps3$mu V3 <- ps3$Sigma
6.2 Fit the Model
glmb.cloglog <- glmb( cbind(Menarche, Total - Menarche) ~ Age2, family = binomial(link = "cloglog"), pfamily = dNormal(mu = mu3, Sigma = V3), data = Menarche_Model_Data, n = 1000 )
6.3 Summary
summary(glmb.cloglog)
The cloglog model often fits poorly for symmetric S‑shaped curves (as shown in Chapter 05), but it is valuable for:
- rare events
- survival‑type interpretations
- asymmetric growth curves
7. Comparing Link Functions
The Deviance Information Criterion (DIC) provides a Bayesian analogue to AIC [@Spiegelhalter2002]:
DIC_comp<-rbind( extractAIC(glmb.logit), extractAIC(glmb.probit), extractAIC(glmb.cloglog)) rownames(DIC_comp)<-c("logit","probit","cloglog") DIC_comp
Typical patterns:
- logit and probit often perform similarly
- probit may slightly outperform logit for smooth curves
- cloglog is best when the response is asymmetric or hazard‑like
The effective number of parameters (pD) is part of the same framework [@Spiegelhalter2002] and helps diagnose model complexity.
8. Concluding Discussion
Binomial GLMs are a core component of the glmbayes package. Their log‑concave likelihoods make them ideal for the envelope‑based accept‑reject sampler, and the familiar link functions allow analysts to choose models that match the scientific context [@McCullagh1989; @Gelman2013].
This chapter demonstrated:
- how binomial GLMs are structured
- how link functions differ
- how to specify priors using Prior_Setup
- how to fit logit, probit, and cloglog models
- how to compare models using DIC
In the next chapter, we extend these ideas to Poisson models, which share many structural similarities but introduce new considerations for count data.
Appendix A. Bayes Rules! companion — Perth rain (weather_perth)
Book: [@JohnsonLauEtAl2022], Chapter 13 — same model and informative priors as Chapter 08, Appendix A.
Model: raintomorrow ~ humidity9am (logit link)
Requires
bayesrules.
library(bayesrules) weather <- bayesrules::weather_perth weather$raintomorrow <- as.integer(weather$raintomorrow == "Yes") mu_w <- matrix(c(-1.4, 0.07), nrow = 1) colnames(mu_w) <- c("(Intercept)", "humidity9am") Sigma_w <- diag(c(0.7^2, 0.035^2)) dimnames(Sigma_w) <- list(colnames(mu_w), colnames(mu_w)) book_br09 <- data.frame( parameter = c("(Intercept)", "humidity9am"), book_lo = c(-5.08785, 0.04147), book_hi = c(-4.13450, 0.05487), book_mid = c(-4.611175, 0.04817), check.names = FALSE )
set.seed(2026) glmb_rain <- glmb( raintomorrow ~ humidity9am, family = binomial(), pfamily = dNormal(mu = mu_w, Sigma = Sigma_w), data = weather, n = 2000 ) print(glmb_rain)
br09_compare <- data.frame( parameter = book_br09$parameter, `Book 80% lo` = book_br09$book_lo, `Book 80% hi` = book_br09$book_hi, `Book midpoint` = book_br09$book_mid, `glmb Post.Mean` = as.numeric(glmb_rain$coef.means[book_br09$parameter]), `glmb Post.Sd` = sapply(book_br09$parameter, function(p) sd(glmb_rain$coefficients[, p, drop = TRUE])), check.names = FALSE ) knitr::kable(br09_compare, digits = 4, caption = "Bayes Rules! Ch. 13 vs. glmb() (informative priors)")
Multivariable extensions (humidity3pm, raintoday) use Prior_Setup() defaults on extra coefficients unless you add further informative book priors (Chapter 14).
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