Posterior sampling for a 1-way hierarchical ANOVA

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

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.

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.

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 \(\phi1 = \mu, \phi2 = log \sigma_\alpha, \phi3 = log \sigma\).

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)
#>              2.5%  25%  50%  75% 97.5%
#> mu           1.07 1.19 1.24 1.30  1.40
#> sigma[alpha] 0.34 0.40 0.43 0.49  0.60
#> sigma        0.04 0.05 0.06 0.06  0.07

# Ratio-of-uniforms information and posterior sample summaries
summary(temp_res)
#> ru bounding box:  
#>                box         vals1        vals2 conv
#> a        1.0000000  0.0000000000  0.000000000    0
#> b1minus -0.1466810 -0.2326509989  0.001997344    0
#> b2minus -0.1475368 -0.0004397794 -0.234538810    0
#> b1plus   0.1755008  0.3048045191  0.001126515    0
#> b2plus   0.1739269 -0.0010104521  0.300416382    0
#> 
#> estimated probability of acceptance:  
#> [1] 0.5473454
#> 
#> sample summary 
#>        mu          sigma[alpha]        sigma        
#>  Min.   :0.9404   Min.   :0.2808   Min.   :0.03864  
#>  1st Qu.:1.1881   1st Qu.:0.3983   1st Qu.:0.05182  
#>  Median :1.2392   Median :0.4350   Median :0.05650  
#>  Mean   :1.2415   Mean   :0.4464   Mean   :0.05721  
#>  3rd Qu.:1.2994   3rd Qu.:0.4871   3rd Qu.:0.06195  
#>  Max.   :1.5743   Max.   :0.6859   Max.   :0.09005  

# ======= 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)
#>              2.5%  25%  50%  75% 97.5%
#> theta[1]     58.6 60.4 61.2 62.0  63.8
#> theta[2]     63.8 65.2 65.8 66.5  68.0
#> theta[3]     65.8 67.1 67.7 68.4  69.8
#> theta[4]     59.3 60.6 61.1 61.7  63.0
#> mu           55.5 62.3 63.9 65.6  72.8
#> sigma[alpha]  2.0  3.6  5.1  8.1  25.6
#> sigma         1.8  2.2  2.4  2.7   3.5
round(t(apply(all2, 2, quantile, probs = probs)), 1)
#>              2.5%  25%  50%  75% 97.5%
#> theta[1]     58.7 60.5 61.3 62.1  63.8
#> theta[2]     63.9 65.2 65.9 66.5  67.8
#> theta[3]     65.8 67.1 67.8 68.4  69.8
#> theta[4]     59.4 60.6 61.1 61.7  62.9
#> mu           55.4 62.1 63.9 65.8  73.8
#> sigma[alpha]  1.9  3.4  4.9  7.5  23.0
#> sigma         1.8  2.2  2.4  2.7   3.4

# 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))