Package 'bang' reference manual

Title:	Bayesian Analysis, No Gibbs
Description:	Provides functions for the Bayesian analysis of some simple commonly-used models, without using Markov Chain Monte Carlo (MCMC) methods such as Gibbs sampling. The 'rust' package <https://cran.r-project.org/package=rust> is used to simulate a random sample from the required posterior distribution, using the generalized ratio-of-uniforms method. See Wakefield, Gelfand and Smith (1991) <DOI:10.1007/BF01889987> for details. At the moment three conjugate hierarchical models are available: beta-binomial, gamma-Poisson and a 1-way analysis of variance (ANOVA).
Authors:	Paul J. Northrop [aut, cre, cph], Benjamin D. Hall [aut, cph]
Maintainer:	Paul J. Northrop <[email protected]>
License:	GPL (>= 2)
Version:	1.0.4
Built:	2024-10-16 06:21:30 UTC
Source:	CRAN

bang: Bayesian Analysis, No Gibbs

Description

Performs Bayesian analyses using some simple commonly-used models. The multivariate generalized ratio-of-uniforms method is used to simulate random samples from the required posterior distribution. The user can either choose hyperparameter values of a default prior distribution or specify their own prior distribution.

Details

Currently three conjugate hierarchical models are available: beta-binomial, gamma-Poisson and 1-way Analysis of Variance (ANOVA). The function hef produces random posterior samples from for the beta-binomial and gamma-Poisson models. The function hanova1 does this for the 1-way Analysis of Variance (ANOVA). The rust package is used to produce these samples.

See vignette("bang-a-vignette", package = "bang") for a brief introduction to the package and vignette("bang-b-hef-vignette", package = "bang") and vignette("bang-c-anova-vignette", package = "bang") for illustrations of the use of the hef and hanova1 functions.

Author(s)

Maintainer: Paul J. Northrop [email protected] [copyright holder]

Authors:

Benjamin D. Hall [email protected] [copyright holder]

References

Northrop, P. J. (2017). rust: Ratio-of-Uniforms Simulation with Transformation. R package version 1.2.3. https://cran.r-project.org/package=rust.

Coagulation time data

Description

Coagulation time in seconds for blood drawn from 24 animals randomly allocated to four different diets from Box, Hunter, and Hunter (1978). The data frame coagulation has 24 rows and 2 columns. Each row relates to a different animal. Column 1 contains the coagulation times. Column 2 contains a label for the type of diet: one of A, B, C or D.

Usage

coagulation
coagulation

Format

A data frame with 24 rows and 2 columns.

Source

Table 11.2 of Gelman, A., Carlin, J. B., Stern, H. S. Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014) Bayesian Data Analysis. Chapman & Hall / CRC. http://www.stat.columbia.edu/~gelman/book/

References

Box, G. E. P., Hunter, W. G., and Hunter, J. S. (1978). Statistics for Experimenters. New York: Wiley.

Posterior sampling for a 1-way hierarchical ANOVA

Description

Produces random samples from the posterior distribution of the parameters of a 1-way hierarchical ANOVA model.

Usage

hanova1(
  n = 1000,
  resp,
  fac,
  ...,
  prior = "default",
  hpars = NULL,
  param = c("trans", "original"),
  init = NULL,
  mu0 = 0,
  sigma0 = Inf,
  nrep = NULL
)
hanova1(
  n = 1000,
  resp,
  fac,
  ...,
  prior = "default",
  hpars = NULL,
  param = c("trans", "original"),
  init = NULL,
  mu0 = 0,
  sigma0 = Inf,
  nrep = NULL
)

Arguments

`n`	A numeric scalar. The size of posterior sample required.
`resp`	A numeric vector. Response values.
`fac`	A vector of class `factor` indicating the group from which the corresponding element of `resp` originates. Must have the same length as `resp`.
`...`	Optional further arguments to be passed to `ru`.
`prior`	The log-prior for the parameters of the hyperprior distribution. If the user wishes to specify their own prior then `prior` must be an object returned from a call to `set_user_prior`. Otherwise, `prior` is a character scalar giving the name of the required in-built prior. If `prior` is not supplied then a default prior is used. See Details.
`hpars`	A numeric vector. Used to set parameters (if any) in an in-built prior. If `prior = cauchy` then `hpars` is a numeric vector of length 2 giving the respective scale parameters of the half-Cauchy priors for $\sigma_\alpha$ and $\sigma$ .
`param`	A character scalar. If `param = "trans"` (the default) then the marginal posterior of hyperparameter vector $\phi$ is reparameterized in terms of $log \sigma_\alpha, log \sigma$ . If `param = "original"` the original parameterization, i.e. $\sigma_\alpha, \sigma$ is used. The former tends to make the optimizations involved in the ratio-of-uniforms algorithm more stable and to increase the probability of acceptance, but at the expense of slower function evaluations.
`init`	A numeric vector. Optional initial estimates sent to `ru` in the search for the mode of the posterior density of (perhaps a subset of) the hyperparameter vector $\phi$ . If an in-built prior is used then `ru` is used to sample from the marginal posterior density of $(\sigma_\alpha, \sigma)$ , so `init` must have length 2. Otherwise, `init` has length equal to the argument `anova_d` supplied to `set_user_prior`.
`mu0`, `sigma0`	A numeric scalar. Mean and standard deviation of a normal prior for $\mu$ . Only used if an in-built prior is used or if `anova_d = 2` is supplied in a call to `set_user_prior` to set a user-defined prior. The default, `sigma0 = Inf`, sets an improper uniform prior for $\mu$ .
`nrep`	A numeric scalar. If `nrep` is not `NULL` then `nrep` gives the number of replications of the original dataset simulated from the posterior predictive distribution. Each replication is based on one of the samples from the posterior distribution. Therefore, `nrep` must not be greater than `n`. In that event `nrep` is set equal to `n`.

Details

Consider $I$ independent experiments in which the $ni$ responses $y$ $i$ from experiment/group $i$ are normally distributed with mean $\theta i$ and standard deviation $\sigma$ . The population parameters $\theta$ 1, ..., $\theta$ $I$ are modelled as random samples from a normal distribution with mean $\mu$ and standard deviation $\sigma_\alpha$ . Let $\phi = (\mu, \sigma_\alpha, \sigma)$ . Conditionally on $\theta$ 1, ..., $\theta$ $I$ , $y$ 1, ..., $y$ $I$ are independent of each other and are independent of $\phi$ . A hyperprior is placed on $\phi$ . The user can either choose parameter values of a default hyperprior or specify their own hyperprior using set_user_prior.

The ru function in the rust package is used to draw a random sample from the marginal posterior of the hyperparameter vector $\phi$ . Then, conditional on these values, population parameters are sampled directly from the conditional posterior density of $\theta$ 1, ..., $\theta$ $I$ given $\phi$ and the data. See the vignette("bang-c-anova-vignette", package = "bang") for details.

The following priors are specified up to proportionality.

Priors:

prior = "bda" (the default): $\pi(\mu, \sigma_\alpha, \sigma) = 1/\sigma,$ that is, a uniform prior for $(\mu, \sigma_\alpha, log \sigma)$ , for $\sigma_\alpha > 0$ and $\sigma > 0$ . The data must contain at least 3 groups, that is, fac must have at least 3 levels, for a proper posterior density to be obtained. [See Sections 5.7 and 11.6 of Gelman et al. (2014).]

prior = "unif": $\pi(\mu, \sigma_\alpha, \sigma) = 1,$ that is, a uniform prior for $(\mu, \sigma_\alpha, \sigma)$ , for $\sigma_\alpha > 0$ and $\sigma > 0$ . [See Section 11.6 of Gelman et al. (2014).]

prior = "cauchy": independent half-Cauchy priors for $\sigma_\alpha$ and $\sigma$ with respective scale parameters $A_\alpha$ and $A$ , that is, $\pi(\sigma_\alpha, \sigma) = 1 / [(1 + \sigma_\alpha^2 / A_\alpha^2) (1 + \sigma^2 / A^2)].$ [See Gelman (2006).] The scale parameters ( $A_\alpha$ , $A$ ) are specified using hpars = ( $A_\alpha$ , $A$ ). The default setting is hpars = c(10, 10).

Parameterizations for sampling:

param = "original" is ( $\mu, \sigma_\alpha, \sigma$ ), param = "trans" (the default) is $\phi1 = \mu, \phi2 = log \sigma_\alpha, \phi3 = log \sigma$ .

Value

An object (list) of class "hef", which has the same structure as an object of class "ru" returned from ru. In particular, the columns of the n-row matrix sim_vals contain the simulated values of $\phi$ . In addition this list contains the arguments model, resp, fac and prior detailed above and an n by $I$ matrix theta_sim_vals: column $i$ contains the simulated values of $\theta$ $i$ . Also included are data = cbind(resp, fac) and summary_stats a list containing: the number of groups I; the numbers of responses each group ni; the total number of observations; the sample mean response in each group; the sum of squared deviations from the group means s; the arguments to hanova1 mu0 and sigma0; call: the matched call to hanova1.

References

Gelman, A., Carlin, J. B., Stern, H. S. Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014) Bayesian Data Analysis. Chapman & Hall / CRC.

Gelman, A. (2006) Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 1(3), 515-533. doi:10.1214/06-BA117A.

Examples

# ======= Late 21st Century Global Temperature Data =======

# Extract data for RCP2.6
RCP26_2 <- temp2[temp2$RCP == "rcp26", ]

# Sample from the posterior under the default `noninformative' flat prior
# for (mu, sigma_alpha, log(sigma)).  Ratio-of-uniforms is used to sample
# from the marginal posterior for (log(sigma_alpha), log(sigma)).
temp_res <- hanova1(resp = RCP26_2[, 1], fac = RCP26_2[, 2])

# Plot of sampled values of (sigma_alpha, sigma)
plot(temp_res, params = "ru")

# Plot of sampled values of (log(sigma_alpha), log(sigma))
# (centred at (0,0))
plot(temp_res, ru_scale = TRUE)

# Plot of sampled values of (mu, sigma_alpha, sigma)
plot(temp_res)

# Estimated marginal posterior densities of the mean for each GCM
plot(temp_res, params = "pop", which_pop = "all", one_plot = TRUE)

# Posterior sample quantiles
probs <- c(2.5, 25, 50, 75, 97.5) / 100
round(t(apply(temp_res$sim_vals, 2, quantile, probs = probs)), 2)

# Ratio-of-uniforms information and posterior sample summaries
summary(temp_res)

# ======= Coagulation time data, from Table 11.2 Gelman et al (2014) =======

# With only 4 groups the posterior for sigma_alpha has a heavy right tail if
# the default `noninformative' flat prior for (mu, sigma_alpha, log(sigma))
# is used.  If we try to sample from the marginal posterior for
# (sigma_alpha, sigma) using the default generalized ratio-of-uniforms
# runing parameter value r = 1/2 then the acceptance region is not bounded.

# Two remedies: reparameterize the posterior and/or increase the value of r.

# (log(sigma_alpha), log(sigma)) parameterization, ru parameter r = 1/2
coag1 <- hanova1(resp = coagulation[, 1], fac = coagulation[, 2])

# (sigma_alpha, sigma) parameterization, ru parameter r = 1
coag2 <- hanova1(resp = coagulation[, 1], fac = coagulation[, 2],
               param = "original", r = 1)

# Values to compare to those in Table 11.3 of Gelman et al (2014)
all1 <- cbind(coag1$theta_sim_vals, coag1$sim_vals)
all2 <- cbind(coag2$theta_sim_vals, coag2$sim_vals)
round(t(apply(all1, 2, quantile, probs = probs)), 1)
round(t(apply(all2, 2, quantile, probs = probs)), 1)

# Pairwise plots of posterior samples from the group means
plot(coag1, which_pop = "all", plot_type = "pairs")

# Independent half-Cauchy priors for sigma_alpha and sigma
coag3 <- hanova1(resp = coagulation[, 1], fac = coagulation[, 2],
                 param = "original", prior = "cauchy", hpars = c(10, 1e6))

# ======= Late 21st Century Global Temperature Data =======

# Extract data for RCP2.6
RCP26_2 <- temp2[temp2$RCP == "rcp26", ]

# Sample from the posterior under the default `noninformative' flat prior
# for (mu, sigma_alpha, log(sigma)).  Ratio-of-uniforms is used to sample
# from the marginal posterior for (log(sigma_alpha), log(sigma)).
temp_res <- hanova1(resp = RCP26_2[, 1], fac = RCP26_2[, 2])

# Plot of sampled values of (sigma_alpha, sigma)
plot(temp_res, params = "ru")

# Plot of sampled values of (log(sigma_alpha), log(sigma))
# (centred at (0,0))
plot(temp_res, ru_scale = TRUE)

# Plot of sampled values of (mu, sigma_alpha, sigma)
plot(temp_res)

# Estimated marginal posterior densities of the mean for each GCM
plot(temp_res, params = "pop", which_pop = "all", one_plot = TRUE)

# Posterior sample quantiles
probs <- c(2.5, 25, 50, 75, 97.5) / 100
round(t(apply(temp_res$sim_vals, 2, quantile, probs = probs)), 2)

# Ratio-of-uniforms information and posterior sample summaries
summary(temp_res)

# ======= Coagulation time data, from Table 11.2 Gelman et al (2014) =======

# With only 4 groups the posterior for sigma_alpha has a heavy right tail if
# the default `noninformative' flat prior for (mu, sigma_alpha, log(sigma))
# is used.  If we try to sample from the marginal posterior for
# (sigma_alpha, sigma) using the default generalized ratio-of-uniforms
# runing parameter value r = 1/2 then the acceptance region is not bounded.

# Two remedies: reparameterize the posterior and/or increase the value of r.

# (log(sigma_alpha), log(sigma)) parameterization, ru parameter r = 1/2
coag1 <- hanova1(resp = coagulation[, 1], fac = coagulation[, 2])

# (sigma_alpha, sigma) parameterization, ru parameter r = 1
coag2 <- hanova1(resp = coagulation[, 1], fac = coagulation[, 2],
               param = "original", r = 1)

# Values to compare to those in Table 11.3 of Gelman et al (2014)
all1 <- cbind(coag1$theta_sim_vals, coag1$sim_vals)
all2 <- cbind(coag2$theta_sim_vals, coag2$sim_vals)
round(t(apply(all1, 2, quantile, probs = probs)), 1)
round(t(apply(all2, 2, quantile, probs = probs)), 1)

# Pairwise plots of posterior samples from the group means
plot(coag1, which_pop = "all", plot_type = "pairs")

# Independent half-Cauchy priors for sigma_alpha and sigma
coag3 <- hanova1(resp = coagulation[, 1], fac = coagulation[, 2],
                 param = "original", prior = "cauchy", hpars = c(10, 1e6))

Hierarchical Exponential Family Model

Description

Produces random samples from the posterior distribution of the parameters of certain hierarchical exponential family models.

Usage

hef(
  n = 1000,
  model = c("beta_binom", "gamma_pois"),
  data,
  ...,
  prior = "default",
  hpars = NULL,
  param = c("trans", "original"),
  init = NULL,
  nrep = NULL
)
hef(
  n = 1000,
  model = c("beta_binom", "gamma_pois"),
  data,
  ...,
  prior = "default",
  hpars = NULL,
  param = c("trans", "original"),
  init = NULL,
  nrep = NULL
)

Arguments

`n`	An integer scalar. The size of the posterior sample required.
`model`	A character string. Abbreviated name for the response-population distribution combination. For a hierarchical normal model see `hanova1` (hierarchical one-way analysis of variance (ANOVA)).
`data`	A numeric matrix. The format depends on `model`. See Details.
`...`	Optional further arguments to be passed to `ru`.
`prior`	The log-prior for the parameters of the hyperprior distribution. If the user wishes to specify their own prior then `prior` must be an object returned from a call to `set_user_prior`. Otherwise, `prior` is a character scalar giving the name of the required in-built prior. If `prior` is not supplied then a default prior is used. See Details.
`hpars`	A numeric vector. Used to set parameters (if any) in an in-built prior.
`param`	A character scalar. If `param = "trans"` (the default) then the marginal posterior of hyperparameter vector $\phi$ is reparameterized in a way designed to improve the efficiency of sampling from this posterior. If `param = "original"` the original parameterization is used. The former tends to make the optimizations involved in the ratio-of-uniforms algorithm more stable and to increase the probability of acceptance, but at the expense of slower function evaluations.
`init`	A numeric vector of length 2. Optional initial estimates for the search for the mode of the posterior density of the hyperparameter vector $\phi$ .
`nrep`	A numeric scalar. If `nrep` is not `NULL` then `nrep` gives the number of replications of the original dataset simulated from the posterior predictive distribution. Each replication is based on one of the samples from the posterior distribution. Therefore, `nrep` must not be greater than `n`. In that event `nrep` is set equal to `n`.

Details

Conditional on population-specific parameter vectors $\theta$ 1, ..., $\theta$ $J$ the observed response data $y$ 1, ..., $y$ J within each population are modelled as random samples from a distribution in an exponential family. The population parameters $\theta$ 1, ..., $\theta$ $J$ are modelled as random samples from a common population distribution, chosen to be conditionally conjugate to the response distribution, with hyperparameter vector $\phi$ . Conditionally on $\theta$ 1, ..., $\theta$ $J$ , $y$ 1, ..., $y$ $J$ are independent of each other and are independent of $\phi$ . A hyperprior is placed on $\phi$ . The user can either choose parameter values of a default hyperprior or specify their own hyperprior using set_user_prior.

We outline each model, specify the format of the data, give the default (log-)priors (up to an additive constant) and detail the choices of ratio-of-uniforms parameterization param.

Beta-binomial: For $j = 1, ..., J$ , $Yj | pj$ are i.i.d binomial $(nj, pj)$ , where $pj$ is the probability of success in group $j$ and $nj$ is the number of trials in group $j$ . $pj$ are i.i.d. beta $(\alpha, \beta)$ , so and $\phi = (\alpha, \beta)$ . data is a 2-column matrix: the numbers of successes in column 1 and the corresponding numbers of trials in column 2.

Priors:

prior = "bda" (the default): $log \pi(\alpha, \beta) = - 2.5 log(\alpha + \beta), \alpha > 0, \beta > 0.$ [See Section 5.3 of Gelman et al. (2014).]

prior = "gamma": independent gamma priors on $\alpha$ and $\beta$ , i.e. $log \pi(\alpha, \beta) = (s1 - 1)log\alpha - r1 \alpha + (s2 - 1)log\beta - r2 \beta, \alpha > 0, \beta > 0.$ where the respective shape ( $s1$ , $s2$ ) and rate ( $r1$ , $r2$ ) parameters are specified using hpars = $(s1, r1, s2, r2)$ . The default setting is hpars = c(1, 0.01, 1, 0.01).

Parameterizations for sampling:

param = "original" is ( $\alpha, \beta$ ), param = "trans" (the default) is $\phi1 = logit(\alpha/(\alpha+\beta)) = log(\alpha/\beta), \phi2 = log(\alpha+\beta)$ . See Section 5.3 of Gelman et al. (2014).

Gamma-Poisson: For $j = 1, ..., J$ , $Yj | \lambda$ j are i.i.d Poisson( $e$ j $\lambda$ j), where $ej$ is the exposure in group $j$ , based on the total length of observation time and/or size of the population at risk of the event of interest and $\lambda$ j is the mean number of events per unit of exposure. $\lambda$ j are i.i.d. gamma $(\alpha, \beta)$ , so $\phi = (\alpha, \beta)$ . data is a 2-column matrix: the counts $yj$ of the numbers of events in column 1 and the corresponding exposures $ej$ in column 2.

Priors:

prior = "gamma" (the default): independent gamma priors on $\alpha$ and $\beta$ , i.e. $log \pi(\alpha, \beta) = (s1 - 1)log\alpha - r1 \alpha + (s2 - 1)log\beta - r2 \beta, \alpha > 0, \beta > 0.$ where the respective shape ( $s1$ , $s2$ ) and rate ( $r1$ , $r2$ ) parameters are specified using hpars = $(s1, r1, s2, r2)$ . The default setting is hpars = c(1, 0.01, 1, 0.01).

Parameterizations for sampling:

param = "original" is ( $\alpha, \beta$ ), param = "trans" (the default) is $\phi1 = log(\alpha/\beta), \phi2 = log(\beta).$

Value

If nrep is not NULL then this list also contains data_rep, a numerical matrix with nrep columns. Each column contains a replication of the first column of the original data data[, 1], simulated from the posterior predictive distribution.

References

Gelman, A., Carlin, J. B., Stern, H. S. Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014) Bayesian Data Analysis. Chapman & Hall / CRC. http://www.stat.columbia.edu/~gelman/book/

Examples

############################ Beta-binomial #################################

# ------------------------- Rat tumor data ------------------------------- #

# Default prior, sampling on (rotated) (log(mean), log(alpha + beta)) scale
rat_res <- hef(model = "beta_binom", data = rat)

# Hyperparameters alpha and beta
plot(rat_res)
# Parameterization used for sampling
plot(rat_res, ru_scale = TRUE)

summary(rat_res)

# Choose rats with extreme sample probabilities
pops <- c(which.min(rat[, 1] / rat[, 2]), which.max(rat[, 1] / rat[, 2]))
# Population-specific posterior samples: separate plots
plot(rat_res, params = "pop", plot_type = "both", which_pop = pops)
# Population-specific posterior samples: one plot
plot(rat_res, params = "pop", plot_type = "dens", which_pop = pops,
     one_plot = TRUE, add_legend = TRUE)

# Default prior, sampling on (rotated) (alpha, beta) scale
rat_res <- hef(model = "beta_binom", data = rat, param = "original")

plot(rat_res)
plot(rat_res, ru_scale = TRUE)

summary(rat_res)

# To produce a plot akin to Figure 5.3 of Gelman et al. (2014) we
# (a) Use the same prior for (alpha, beta)
# (b) Don't use axis rotation (rotate = FALSE)
# (c) Plot on the scale used for ratio-of-uniforms sampling (ru_scale = TRUE)
# (d) Note that the mode is relocated to (0, 0) in the plot
rat_res <- hef(model = "beta_binom", data = rat, rotate = FALSE)

plot(rat_res, ru_scale = TRUE)

# This is the estimated location of the posterior mode
rat_res$f_mode

# User-defined prior, passing parameters
# (equivalent to prior = "gamma" with hpars = c(1, 0.01, 1, 0.01))
user_prior <- function(x, hpars) {
  return(dexp(x[1], hpars[1], log = TRUE) + dexp(x[2], hpars[2], log = TRUE))
}
user_prior_fn <- set_user_prior(user_prior, hpars = c(0.01, 0.01))
rat_res <- hef(model = "beta_binom", data = rat, prior = user_prior_fn)

plot(rat_res)

summary(rat_res)

############################ Gamma-Poisson #################################

# ------------------------ Pump failure data ------------------------------ #

pump_res <- hef(model = "gamma_pois", data = pump)
# Hyperparameters alpha and beta

plot(pump_res)

# Parameterization used for sampling
plot(pump_res, ru_scale = TRUE)
summary(pump_res)

# Choose pumps with extreme sample rates
pops <- c(which.min(pump[, 1] / pump[, 2]), which.max(pump[, 1] / pump[, 2]))
plot(pump_res, params = "pop", plot_type = "dens", which_pop = pops)
############################ Beta-binomial #################################

# ------------------------- Rat tumor data ------------------------------- #

# Default prior, sampling on (rotated) (log(mean), log(alpha + beta)) scale
rat_res <- hef(model = "beta_binom", data = rat)

# Hyperparameters alpha and beta
plot(rat_res)
# Parameterization used for sampling
plot(rat_res, ru_scale = TRUE)

summary(rat_res)

# Choose rats with extreme sample probabilities
pops <- c(which.min(rat[, 1] / rat[, 2]), which.max(rat[, 1] / rat[, 2]))
# Population-specific posterior samples: separate plots
plot(rat_res, params = "pop", plot_type = "both", which_pop = pops)
# Population-specific posterior samples: one plot
plot(rat_res, params = "pop", plot_type = "dens", which_pop = pops,
     one_plot = TRUE, add_legend = TRUE)

# Default prior, sampling on (rotated) (alpha, beta) scale
rat_res <- hef(model = "beta_binom", data = rat, param = "original")

plot(rat_res)
plot(rat_res, ru_scale = TRUE)

summary(rat_res)

# To produce a plot akin to Figure 5.3 of Gelman et al. (2014) we
# (a) Use the same prior for (alpha, beta)
# (b) Don't use axis rotation (rotate = FALSE)
# (c) Plot on the scale used for ratio-of-uniforms sampling (ru_scale = TRUE)
# (d) Note that the mode is relocated to (0, 0) in the plot
rat_res <- hef(model = "beta_binom", data = rat, rotate = FALSE)

plot(rat_res, ru_scale = TRUE)

# This is the estimated location of the posterior mode
rat_res$f_mode

# User-defined prior, passing parameters
# (equivalent to prior = "gamma" with hpars = c(1, 0.01, 1, 0.01))
user_prior <- function(x, hpars) {
  return(dexp(x[1], hpars[1], log = TRUE) + dexp(x[2], hpars[2], log = TRUE))
}
user_prior_fn <- set_user_prior(user_prior, hpars = c(0.01, 0.01))
rat_res <- hef(model = "beta_binom", data = rat, prior = user_prior_fn)

plot(rat_res)

summary(rat_res)

############################ Gamma-Poisson #################################

# ------------------------ Pump failure data ------------------------------ #

pump_res <- hef(model = "gamma_pois", data = pump)
# Hyperparameters alpha and beta

plot(pump_res)

# Parameterization used for sampling
plot(pump_res, ru_scale = TRUE)
summary(pump_res)

# Choose pumps with extreme sample rates
pops <- c(which.min(pump[, 1] / pump[, 2]), which.max(pump[, 1] / pump[, 2]))
plot(pump_res, params = "pop", plot_type = "dens", which_pop = pops)

Plot diagnostics for a hef object

Description

plot method for class "hef".

Usage

## S3 method for class 'hef'
plot(
  x,
  y,
  ...,
  params = c("hyper", "ru", "pop"),
  which_pop = NULL,
  plot_type = NULL,
  one_plot = FALSE,
  add_legend = FALSE,
  legend_position = "topright",
  legend_text = NULL,
  num = 100
)
## S3 method for class 'hef'
plot(
  x,
  y,
  ...,
  params = c("hyper", "ru", "pop"),
  which_pop = NULL,
  plot_type = NULL,
  one_plot = FALSE,
  add_legend = FALSE,
  legend_position = "topright",
  legend_text = NULL,
  num = 100
)

Arguments

`x`	an object of class "hef", a result of a call to `ru`.
`y`	Not used.
`...`	Additional arguments passed to `plot.ru`, `hist` or `pairs`. In particular, `ru_scale = TRUE` produces a plot using the parameterization used for ratio-of-uniforms sampling.
`params`	A character scalar that determines to which parameters the plots relate. `"hyper"`: the posterior sample of all hyperparameter values in $\phi$ is plotted using `plot.ru`. `"ru"`: only the posterior sample generated using `ru` is plotted using `plot.ru`. This produces a different plot to `params = "hyper"` if `ru` is used only on a subset of $\phi$ . For example, this may be the case if `x` is the result of a call to `hanova1`. See vignette("bang-c-anova-vignette", package = "bang") for information. `"pop"`: posterior samples and/or densities of the population-specific parameter $\theta$ are plotted. The population(s) included are determined by `which_pop` and the type of plot is determined by `plot_type`. If `plot_type` is not supplied then it is set to `"dens"`.
`which_pop`	An integer vector or character scalar. If `params = "pop"` then `which_pop` indicates which populations to include in the plot. If `which_pop` is supplied then `params` is set to "pop". If `which_pop = "all"` then all populations are included. If there are many populations then this may fail if `plot_type = "pairs"` and/or `one_plot = FALSE`.
`plot_type`	A character scalar that determines the type of plot produced when `params = "pop"`. If `plot_type` is supplied then `params` is set automatically to `"pop"`. `"sim"`: histograms of the posterior samples of $\theta$ for the populations in `which_pop`. `"dens"`: estimates of the marginal posterior densities of $\theta$ for the populations in `which_pop`. `"both"`: both the histograms and estimated posterior densities. `"pairs"`: pairwise scatter plots of the posterior samples of $\theta$ for the populations in `which_pop`, which must have length greater than one.
`one_plot`, `add_legend`, `legend_position`, `legend_text`	Only relevant if `plot_type = "dens"`. If `one_plot = TRUE` then the estimated marginal posterior densities are plotted in the same graph and if `add_legend = TRUE` then a legend is added to this plot using `legend` in the position indicated by the character scalar `legend_position`. A character vector `legend_text` may be used to override the default legend text.
`num`	A numeric scalar. If `plot_type == "dens"` or `plot_type == "both"` then `num` gives the number of points at which the marginal densities are evaluated to produce plots.

Examples

See the examples in hef and hanova1.

Posterior predictive checks for a hef object

Description

pp_check method for class "hef". This provides an interface to the functions that perform posterior predictive checks in the bayesplot package. See PPC-overview for details of these functions.

Usage

## S3 method for class 'hef'
pp_check(object, fun = NULL, raw = FALSE, nrep = NULL, ...)
## S3 method for class 'hef'
pp_check(object, fun = NULL, raw = FALSE, nrep = NULL, ...)

Arguments

`object`	An object of class "hef", a result of a call to `hef` or `hanova1`.
`fun`	The plotting function to call. Can be any of the functions detailed at PPC-overview. The "ppc_" prefix can optionally be dropped if fun is specified as a string.
`raw`	Only relevant if `object$model = "beta_binom"` or `object$model = "gamma_pois"`. If `raw = TRUE` then the raw responses are used in the plots. Otherwise, the proportions of successes are used in the `beta_binom` case and the exposure-adjusted rate in the `gamma_pois` case. In both cases the values used are `object$data[, 1] / object$data[, 2]` and the equivalent in `object$data_rep`.
`nrep`	The number of predictive replicates to use. If `nrep` is supplied then the first `nrep` rows of `object$data_rep` are used. Otherwise, or if `nrep` is greater than `nrow(object$data_rep)`, then all rows are used.
`...`	Additional arguments passed on to bayesplot functions. See Examples below.

Details

For details of these functions see PPC-overview. See also the vignettes Conjugate Hierarchical Models, Hierarchical 1-way Analysis of Variance and the bayesplot vignette Graphical posterior predictive checks.

The general idea is to compare the observed data object$data with a matrix object$data_rep in which each row is a replication of the observed data simulated from the posterior predictive distribution. For greater detail see Chapter 6 of Gelman et al. (2013).

Value

A ggplot object that can be further customized using the ggplot2 package.

References

Jonah Gabry (2016). bayesplot: Plotting for Bayesian Models. R package version 1.1.0. https://CRAN.R-project.org/package=bayesplot

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Analysis. Chapman & Hall/CRC Press, London, third edition. (Chapter 6). http://www.stat.columbia.edu/~gelman/book/

Examples

############################ Beta-binomial #################################

# ------------------------- Rat tumor data ------------------------------- #

rat_res <- hef(model = "beta_binom", data = rat, nrep = 50)

# Overlaid density estimates
pp_check(rat_res)

# Overlaid distribution function estimates
pp_check(rat_res, fun = "ecdf_overlay")

# Multiple histograms
pp_check(rat_res, fun = "hist", nrep = 8)

# Multiple boxplots
pp_check(rat_res, fun = "boxplot")
# Predictive medians vs observed median
pp_check(rat_res, fun = "stat", stat = "median")

# Predictive (mean, sd) vs observed (mean, sd)
pp_check(rat_res, fun = "stat_2d", stat = c("mean", "sd"))

############################ Gamma-Poisson #################################

# ------------------------ Pump failure data ------------------------------ #

pump_res <- hef(model = "gamma_pois", data = pump, nrep = 50)


# Overlaid density estimates
pp_check(pump_res)
# Predictive (mean, sd) vs observed (mean, sd)
pp_check(pump_res, fun = "stat_2d", stat = c("mean", "sd"))


###################### One-way Hierarchical ANOVA ##########################

#----------------- Late 21st Century Global Temperature Data ------------- #

RCP26_2 <- temp2[temp2$RCP == "rcp26", ]
temp_res <- hanova1(resp = RCP26_2[, 1], fac = RCP26_2[, 2], nrep = 50)

# Overlaid density estimates
pp_check(temp_res)
# Predictive (mean, sd) vs observed (mean, sd)
pp_check(temp_res, fun = "stat_2d", stat = c("mean", "sd"))

############################ Beta-binomial #################################

# ------------------------- Rat tumor data ------------------------------- #

rat_res <- hef(model = "beta_binom", data = rat, nrep = 50)

# Overlaid density estimates
pp_check(rat_res)

# Overlaid distribution function estimates
pp_check(rat_res, fun = "ecdf_overlay")

# Multiple histograms
pp_check(rat_res, fun = "hist", nrep = 8)

# Multiple boxplots
pp_check(rat_res, fun = "boxplot")
# Predictive medians vs observed median
pp_check(rat_res, fun = "stat", stat = "median")

# Predictive (mean, sd) vs observed (mean, sd)
pp_check(rat_res, fun = "stat_2d", stat = c("mean", "sd"))

############################ Gamma-Poisson #################################

# ------------------------ Pump failure data ------------------------------ #

pump_res <- hef(model = "gamma_pois", data = pump, nrep = 50)


# Overlaid density estimates
pp_check(pump_res)
# Predictive (mean, sd) vs observed (mean, sd)
pp_check(pump_res, fun = "stat_2d", stat = c("mean", "sd"))


###################### One-way Hierarchical ANOVA ##########################

#----------------- Late 21st Century Global Temperature Data ------------- #

RCP26_2 <- temp2[temp2$RCP == "rcp26", ]
temp_res <- hanova1(resp = RCP26_2[, 1], fac = RCP26_2[, 2], nrep = 50)

# Overlaid density estimates
pp_check(temp_res)
# Predictive (mean, sd) vs observed (mean, sd)
pp_check(temp_res, fun = "stat_2d", stat = c("mean", "sd"))

Print method for objects of class "hef"

Description

print method for class "hef".

Usage

## S3 method for class 'hef'
print(x, ...)
## S3 method for class 'hef'
print(x, ...)

Arguments

`x`	an object of class "hef", a result of a call to `hef` or `hanova1`.
`...`	Additional optional arguments. At present no optional arguments are used.

Details

Prints the original call to hef or hanova1, the name of the model and the number of populations in the hierarchical model.

Value

The argument x, invisibly, as for all print methods.

Print method for objects of class "summary.hef"

Description

print method for class "summary.hef".

Usage

## S3 method for class 'summary.hef'
print(x, ...)
## S3 method for class 'summary.hef'
print(x, ...)

Arguments

`x`	an object of class "summary.hef", a result of a call to `summary.hef`.
`...`	Additional optional arguments to be passed to `print`.

Details

What is printed depends on the argument params supplied to summary.hef.

Value

The argument x, invisibly, as for all print methods.

Pump-failure data

Description

Data on pump failures from Gaver, D. P. and O'Muircheartaigh, I. G. (1987). The matrix pump has 10 rows and 2 columns. Each row relates to a different pump system. The first column contains the number of pump failures. The second column contains the length of operating time, in thousands of hours.

Usage

pump
pump

Format

A matrix 10 rows and 2 columns.

Source

Table 3 of Gaver, D. P. and O'Muircheartaigh, I. G. (1987). See also Gelfand, A. E. and Smith, A. F. M. (1990).

References

Gaver, D. P. and O'Muircheartaigh, I. G. (1987) Robust Empirical Bayes Analyses of Event Rates. Technometrics, 29, 1-15. doi:10.1080/00401706.1987.10488178

Gelfand, A. E. and Smith, A. F. M. (1990) Sampling-Based Approaches to Calculating Marginal Densities. Journal of the American Statistical Association, 85(410), 398-409. doi:10.1080/01621459.1990.10476213

Rat tumor data

Description

Tumor incidence in 71 groups of rate from Tarone (1982). The matrix rat has 71 rows and 2 columns. Each row relates to a different group of rats. The first column (y) contains the number of rats with tumors. The second column (n) contains the total number of rats.

Usage

rat
rat

Format

A matrix with 71 rows and 2 columns.

Source

Table 5.1 of Gelman, A., Carlin, J. B., Stern, H. S. Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014) Bayesian Data Analysis, Chapman & Hall / CRC. http://www.stat.columbia.edu/~gelman/book/data/rats.asc

References

Tarone, R. E. (1982) The use of historical information in testing for a trend in proportions. Biometrics, 38, 215-220. doi:10.2307/2530304

Set a user-defined prior

Description

Constructs a user-defined prior distribution for use as the argument prior in hef or hanova1.

Usage

set_user_prior(
  prior,
  ...,
  model = c("beta_binom", "gamma_pois", "anova1"),
  anova_d = 2
)
set_user_prior(
  prior,
  ...,
  model = c("beta_binom", "gamma_pois", "anova1"),
  anova_d = 2
)

Arguments

`prior`	An R function returning the log of the prior density for of (perhaps a subset of) the hyperparameter vector $\phi$ .
`...`	Further arguments giving the names and values of any parameters involved in the function `prior`.
`model`	A character string. Abbreviated name of the model: "beta_binom" for beta-binomial and "gamma_pois" for gamma-Poisson (see `hef`), "anova1" for 1-way ANOVA (see `hanova1`).
`anova_d`	An integer scalar. Only relevant if `model = anova1`. If `anova_d = 2` then `prior` must return the log-prior density for the standard deviations $(\sigma_\alpha, \sigma)$ and a normal prior with mean `mu0` and standard deviation `sigma0` is used for $\mu$ . The values of `mu0 = 0` and `sigma0 = Inf` are set in the call to `hanova1`, with default values `mu0 = 0` and `sigma0 = Inf`. If `anova_d = 3` then `prior` must return the log-prior density for $(\mu, \sigma_\alpha, \sigma)$ .

Details

For details of the hyperparameters in $\phi$ see the Details section of hef for the models beta_binom and gamma_pois and of hanova1 for the model anova1.

Value

A list of class "bang_prior". Will contain the component prior, the user-supplied function to evaluate the log of the prior, and any arguments supplied in ....

Examples

# User-defined prior, passing parameters
# (equivalent to prior = "gamma" with hpars = c(1, 0.01, 1, 0.01))
user_prior <- function(x, hpars) {
  return(dexp(x[1], hpars[1], log = TRUE) + dexp(x[2], hpars[2], log = TRUE))
}
user_prior_fn <- set_user_prior(user_prior, hpars = c(0.01, 0.01))
# User-defined prior, passing parameters
# (equivalent to prior = "gamma" with hpars = c(1, 0.01, 1, 0.01))
user_prior <- function(x, hpars) {
  return(dexp(x[1], hpars[1], log = TRUE) + dexp(x[2], hpars[2], log = TRUE))
}
user_prior_fn <- set_user_prior(user_prior, hpars = c(0.01, 0.01))

Simulate from a beta-binomial posterior predictive distribution

Description

Simulates nrep draws from the posterior predictive distribution of the beta-binomial model described in hef. This function is called within hef when the argument nrep is supplied.

Usage

sim_pred_beta_binom(theta_sim_vals, data, nrep)
sim_pred_beta_binom(theta_sim_vals, data, nrep)

Arguments

`theta_sim_vals`	A numeric matrix with `nrow(data)` columns. Each row of `theta_sim_vals` contains binomial success probabilities simulated from their posterior distribution.
`data`	A 2-column numeric matrix: the numbers of successes in column 1 and the corresponding numbers of trials in column 2.
`nrep`	A numeric scalar. The number of replications of the original dataset simulated from the posterior predictive distribution. If `nrep` is greater than `nrow(theta_sim_vals)` then `nrep` is set equal to `nrow(theta_sim_vals)`.

Value

A numeric matrix with nrep columns. Each column contains a draw from the posterior predictive distribution of the number of successes.

Examples

rat_res <- hef(model = "beta_binom", data = rat)
rat_sim_pred <- sim_pred_beta_binom(rat_res$theta_sim_vals, rat, 50)
rat_res <- hef(model = "beta_binom", data = rat)
rat_sim_pred <- sim_pred_beta_binom(rat_res$theta_sim_vals, rat, 50)

Simulate from a gamma-Poisson posterior predictive distribution

Description

Simulates nrep draws from the posterior predictive distribution of the beta-binomial model described in hef. This function is called within hef when the argument nrep is supplied.

Usage

sim_pred_gamma_pois(theta_sim_vals, data, nrep)
sim_pred_gamma_pois(theta_sim_vals, data, nrep)

Arguments

`theta_sim_vals`	A numeric matrix with `nrow(data)` columns. Each row of `theta_sim_vals` contains binomial success probabilities simulated from their posterior distribution.
`data`	A 2-column numeric matrix: the numbers of successes in column 1 and the corresponding numbers of trials in column 2.
`nrep`	A numeric scalar. The number of replications of the original dataset simulated from the posterior predictive distribution. If `nrep` is greater than `nrow(theta_sim_vals)` then `nrep` is set equal to `nrow(theta_sim_vals)`.

Value

A numeric matrix with nrep columns. Each column contains a draw from the posterior predictive distribution of the number of successes.

Examples

pump_res <- hef(model = "gamma_pois", data = pump)
pump_sim_pred <- sim_pred_gamma_pois(pump_res$theta_sim_vals, pump, 50)
pump_res <- hef(model = "gamma_pois", data = pump)
pump_sim_pred <- sim_pred_gamma_pois(pump_res$theta_sim_vals, pump, 50)

Simulate from a one-way hierarchical ANOVA posterior predictive distribution

Description

Simulates nrep draws from the posterior predictive distribution of the one-way hierarchical ANOVA model described in hanova1. This function is called within hanova1 when the argument nrep is supplied.

Usage

sim_pred_hanova1(theta_sim_vals, sim_vals, fac, nrep)
sim_pred_hanova1(theta_sim_vals, sim_vals, fac, nrep)

Arguments

`theta_sim_vals`	A numeric matrix with `length(fac)` columns. Each row of `theta_sim_vals` contains normal means simulated from their posterior distribution.
`sim_vals`	A numeric matrix with `length(fac)` columns. Each row of `sim_vals` contains normal standard deviations $\sigma$ simulated from their posterior distribution.
`fac`	The argument `fac` to `hanova1`, that is, a vector of class factor indicating group membership.
`nrep`	A numeric scalar. The number of replications of the original dataset simulated from the posterior predictive distribution. If `nrep` is greater than `nrow(theta_sim_vals)` then `nrep` is set equal to `nrow(theta_sim_vals)`.

Value

A numeric matrix with nrep columns. Each column contains a draw from the posterior predictive distribution of the number of successes.

Examples

RCP26_2 <- temp2[temp2$RCP == "rcp26", ]
temp_res <- hanova1(resp = RCP26_2[, 1], fac = RCP26_2[, 2])
sim_pred <- sim_pred_hanova1(temp_res$theta_sim_vals, temp_res$sim_vals,
                             RCP26_2[, 2], 50)
RCP26_2 <- temp2[temp2$RCP == "rcp26", ]
temp_res <- hanova1(resp = RCP26_2[, 1], fac = RCP26_2[, 2])
sim_pred <- sim_pred_hanova1(temp_res$theta_sim_vals, temp_res$sim_vals,
                             RCP26_2[, 2], 50)

Summarizing hef objects

Description

summary method for class "hef".

Usage

## S3 method for class 'hef'
summary(
  object,
  ...,
  params = c("hyper", "pop"),
  which_pop = 1:ncol(object$theta_sim_vals)
)
## S3 method for class 'hef'
summary(
  object,
  ...,
  params = c("hyper", "pop"),
  which_pop = 1:ncol(object$theta_sim_vals)
)

Arguments

`object`	an object of class "hef", a result of a call to `hef`.
`...`	Additional arguments passed on to `summary.ru`.
`params`	A character scalar. If `params = "hyper"` then the posterior samples of all hyperparameter values in $\phi$ are summarized using `summary.ru`. If `params = "pop"` then only posterior samples of the populations specified in `which_pop` are summarized.
`which_pop`	An integer vector. If `params = "pop"` then `which_pop` indicates which populations, i.e. which columns of `object$theta_sim_vals` to summarize, using `summary`. The default is all populations.

Examples

# Beta-binomial model, rat data
rat_res <- hef(model = "beta_binom", data = rat)

# Posterior summaries of the hyperparameters alpha and beta
summary(rat_res)

# Posterior summaries of the binomial probability for rats 1 to 3
summary(rat_res, params = "pop", which_pop = 1:3)
# Beta-binomial model, rat data
rat_res <- hef(model = "beta_binom", data = rat)

# Posterior summaries of the hyperparameters alpha and beta
summary(rat_res)

# Posterior summaries of the binomial probability for rats 1 to 3
summary(rat_res, params = "pop", which_pop = 1:3)

Mid 21st Century Global Temperature Projection Data

Description

Indices of global temperature change from late 20th century (1970-1999) to mid 21st century (2020-2049) based on data produced by the Fifth Coupled Model Intercomparison Project (CMIP5).

Usage

temp1
temp1

Format

A data frame with 270 rows and 4 columns.

Column 1, index: anomaly of 2020-2049 mean relative to the 1970-1999 mean.
Column 2, GCM: Abbreviated name of General Circulation Model.
Column 3, RCP: Representative Concentration Pathway. One of rcp26, rcp45, rcp60, rcp85.
Column 4, run: Simulation run number.

Details

The data frame temp1 data frame has 270 rows and 4 columns. Each row relates to a climate projection run from one of 38 different General Circulation Models (GCMs) under a particular Representative Concentration Pathway (RCP). Use table(temp1[, c("GCM", "RCP")]) to see the numbers of runs under each RCP for each GCM. See Van Vuuren et al (2011) for an overview of RCPs and Northrop and Chandler (2014) for analyses of a similar older dataset (CMIP3). Column 1 contains the anomaly of the mean global temperature over the time period 2020-2049 relative to the mean global temperature over 1970-1999, i.e. the latter subtracted from the former. Column 2 contains an abbreviation for the name of the climate modelling research group and the GCM. Column 3 contains the RCP in the format rcpxx where xx is a radiative forcing level resulting from an anticipated future greenhouse gas emissions. Column 4 is the simulation run number.

Source

The raw data from which the indices are calculated are monthly CMIP5 scenario runs for global surface air temperature (tas) downloaded from the KNMI Climate Explorer (https://climexp.knmi.nl/) on 4/3/2015.

References

Northrop, P.J. and R.E. Chandler (2014). Quantifying Sources of Uncertainty in Projections of Future Climate. Journal of Climate, 27, 8793-8808. doi:10.1175/JCLI-D-14-00265.1

Van Vuuren, D. P., Edmonds, J., Kainuma, M., Riahi, K. Thomson, A., Hibbard, K., Hurtt, G. C., Kram, T., Krey, V., Lamarque, J.-F. (2011). The representative concentration pathways: an overview. Climatic change, 109, 5-31. doi:10.1007/s10584-011-0148-z

Late 21st Century Global Temperature Projection Data

Description

Indices of global temperature change from late 20th century (1970-1999) to late 21st century (2069-2098) based on data produced by the Fifth Coupled Model Intercomparison Project (CMIP5).

Usage

temp2
temp2

Format

A data frame with 270 rows and 4 columns.

Column 1, index: anomaly of 2069-2098 mean relative to the 1970-1999 mean.
Column 2, GCM: Abbreviated name of General Circulation Model.
Column 3, RCP: Representative Concentration Pathway. One of rcp26, rcp45, rcp60, rcp85.
Column 4, run: Simulation run number.

Details

The data frame temp2 data frame has 270 rows and 4 columns. Each row relates to a climate projection run from one of 38 different General Circulation Models (GCMs) under a particular Representative Concentration Pathway (RCP). Use table(temp2[, c("GCM", "RCP")]) to see the numbers of runs under each RCP for each GCM. See Van Vuuren et al (2011) for an overview of RCPs and Northrop and Chandler (2014) for analyses of a similar older dataset (CMIP3). Column 1 contains the anomaly of the mean global temperature over the time period 2069-2098 relative to the mean global temperature over 1970-1999, i.e. the latter subtracted from the former. Column 2 contains an abbreviation for the name of the climate modelling research group and the GCM. Column 3 contains the RCP in the format rcpxx where xx is a radiative forcing level resulting from an anticipated future greenhouse gas emissions. Column 4 is the simulation run number.

Source

References

Northrop, P.J. and R.E. Chandler (2014). Quantifying Sources of Uncertainty in Projections of Future Climate. Journal of Climate, 27, 8793-8808. doi:10.1175/JCLI-D-14-00265.1

Weight Gained by Rats

Description

Data from an experiment to study weight gained by 10 rats fed on four different diets, defined by a combination of the amount of protein (low and high) and by the source of protein (beef and cereal).

Usage

weight_gain
weight_gain

Format

A data frame with 40 rows and 3 columns.

Column 1, source: source of protein, a factor with levels Beef and Cereal.
Column 2, type: amount of protein, a factor with levels High and Low.
Column 3, weightgain: weight gained, in grams.

Source

D. J. Hand, A. D. Lunn, K. J. McConway, and E. Ostrowski (1994). A Handbook of Small Datasets, Chapman and Hall/CRC, London.

Package 'bang'

Help Index

bang: Bayesian Analysis, No Gibbs

Description

Details

Author(s)

References

See Also

Coagulation time data

Description

Usage

Format

Source

References

Posterior sampling for a 1-way hierarchical ANOVA

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Hierarchical Exponential Family Model

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Plot diagnostics for a hef object

Description

Usage

Arguments

Examples

See Also

Posterior predictive checks for a hef object

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Print method for objects of class "hef"

Description

Usage

Arguments

Details

Value

See Also

Print method for objects of class "summary.hef"

Description

Usage

Arguments

Details

Value

See Also

Pump-failure data

Description

Usage

Format

Source

References

Rat tumor data

Description

Usage

Format

Source

References

Set a user-defined prior

Description

Usage

Arguments

Details

Value

See Also