| gbp {Rgbp} | R Documentation |
Fitting Gaussian, Poisson, and Binomial Hierarchical Models
Description
gbp fits Bayesian hierarchical models using the Uniform distribution on the second level variance component (variance of the prior distribution), which enables good frequentist repeated sampling properties.
Usage
gbp(y, se.or.n, covariates, mean.PriorDist, model, intercept,
confidence.lvl, n.AR, n.AR.factor, trial.scale, save.result,
normal.CI, t, u)Arguments
y |
a (k by 1) vector of k groups' sample means for Gaussian or of each group's number of successful trials for Binomial and Poisson data, where k is the number of groups (or units) in a dataset. |
se.or.n |
a (k by 1) vector composed of the standard errors of all groups for Gaussian or of each group's total number of trials for Binomial and Poisson data. |
covariates |
(optional) a matrix of covariate(s) each column of which corresponds to one covariate. |
mean.PriorDist |
(optional) a numeric value for the second-level mean parameter, i.e. the mean of prior distribution, if you know this value a priori. |
model |
a character string indicating which hierarchical model to fit. "gaussian" for Gaussian data, "poisson" for Poisson, and "binomial" for Binomial. Default is "gaussian" |
intercept |
|
confidence.lvl |
a |
n.AR |
Only for Binomial model. If |
n.AR.factor |
Only for Binomial model. If |
trial.scale |
A scale used in the trial distribution of |
save.result |
Only for Binomial model with the acceptance-rejection sampling. |
normal.CI |
Only applicable for Gaussian data. If |
t |
Non-negative constant to determine the hyper-prior distribution of r for the Binomial model with the acceptance-rejection method. If t is positive, then the hyper-prior distribution of r is proper, otherwise improper. |
u |
A positive constant to determine the hyper-prior distribution of r for the Binomial model with the acceptance-rejection method. |
Details
gbp fits a hierarchical model whose first-level hierarchy is a distribution of observed data and second-level is a conjugate prior distribution on the first-level parameter. To be specific, for Normal data, gbp constructs a two-level Normal-Normal multilevel model. V_{j} (=\sigma^{2}/n_{j}) is assumed to be known or to be accurately estimated, and subscript j indicates j-th group
(or unit) in a dataset.
(y_{j} | \theta_{j}) \stackrel{ind}{\sim} N(\theta_{j}, \sigma^{2}_{j})
(\theta_{j} |\mu_{0j} , A) \stackrel{ind}{\sim} N(\mu_{0j}, A)
\mu_{0j} = x_{j}'\beta
for j = 1, \ldots, k, where k is the number of groups (units) in a dataset.
For Poisson data, gbp builds a two-level Poisson-Gamma multilevel model. A square bracket below indicates [mean, variance] of distribution, a constant multiplied to the notation representing Gamma distribution (Gam) is a scale, and y_{j}=\frac{z_{j}}{n_{j}}.
(z_{j} | \theta_{j}) \stackrel{ind}{\sim} Pois(n_{j}\theta_{j})
(\theta_{j} | r, \mu_{0j}) \stackrel{ind}{\sim} \frac{1}{r}Gam(r\mu_{0j})\stackrel{ind}{\sim}Gam[\mu_{0j}, \mu_{0j} / r]
log(\mu_{0j}) = x_{j}'\beta
for j = 1, \ldots, k, where k is the number of groups (units) in a dataset.
For Binomial data, gbp sets a two-level Binomial-Beta multilevel model. A square bracket below indicates [mean, variance] of distribution and y_{j} = \frac{z_{j}}{n_{j}}.
(z_{j} | \theta_{j}) \stackrel{ind}{\sim} Bin(n_{j}, \theta_{j})
(\theta_{j} | r, \mu_{0j}) \stackrel{ind}{\sim} Beta(r\mu_{0j}, r(1-\mu_{0j})) \stackrel{ind}{\sim} Beta[\mu_{0j}, \mu_{0j}(1 - \mu_{0j}) / (r + 1)]
logit(\mu_{0j}) = x_{j}'\beta
for j = 1, \ldots, k, where k is the number of groups (units) in a dataset.
For reference, based on the above notations, the Uniform prior distribution on the second level variance component (variance of the prior distribution) is dA for Gaussian and d(\frac{1}{r})
(= \frac{dr}{r^{2}}) for Binomial and Poisson data. The second level variance component can be interpreted as variation among the first-level parameters (\theta_{j}) or variance of ensemble information.
Under this setting, the argument y in gbp is a (k by 1) vector of k groups' sample means (y_{j}'s in the description of Normal-Normal model above) for Gaussian or of each group's number of successful trials (z_{j}'s) for Binomial and Poisson data, where k is the number of groups (or units) in a dataset.
The argument se.or.n is a (k by 1) vector composed of the standard errors (V_{j}'s) of all groups for Gaussian or of each group's total number of trials (n_{j}'s) for Binomial and Poisson data.
As for two optional arguments, covariates and mean.PriorDist, there are three feasible
combinations of them to run gbp. The first situation is when we do not have any covariate and do not
know a mean of the prior distribution (\mu_{0}) a priori. In this case, assigning none of two
optional arguments, such as "gbp(z, n, model = "binomial")", will lead to a correct model. gbp
will automatically fit a regression with only an intercept term to estimate a common mean of the prior
distribution (exchangeability).
The second situation is when we have covariate(s) and do not know a mean of the prior distribution (\mu_{0}) a priori. In this case, assigning a matrix, X, each column of which corresponds to one covariate, such as "gbp(z, n, X, model = "poisson")", will lead to a correct model. Default of gbp is to fit a regression including an intercept term to estimate a mean of the prior distribution. Double exchangeability will hold in this case.
The last case is when we know a mean of the prior distribution (\mu_{0}) a priori. Now, we do
not need to estimate regression coefficients at all because we know a true value of \mu_{0} (strong assumption).
Designating this value into the argument of gbp like
"gbp(y, se, mean.PriorDist = 3)" is enough to account for it. For reference,
mean.PriorDist has a stronger priority than covariates, which means that when both
arguments are designated, gbp will fit a hierarchical model using the known mean of prior distribution, mean.PriorDist.
gbp returns an object of class "gbp" which provides three relevant functions plot, print, and summary.
Value
An object of class gbp comprises of:
sample.mean |
sample mean of each group (or unit) |
se |
if Gaussian data, standard error of sample mean in each group (or unit) |
n |
if Binomial and Poisson data, total number of trials of each group (or unit) |
prior.mean |
numeric if entered, NA if not entered |
prior.mean.hat |
estimate of prior mean by a regression if prior mean is not assigned a priori |
shrinkage |
shrinkage estimate of each group (adjusted posterior mean) |
sd.shrinkage |
posterior standard deviation of shrinkage |
post.mean |
posterior mean of each group |
post.sd |
posterior standard deviation of each group |
post.intv.low |
lower bound of 100*confidence.lvl% posterior interval (quantile of posterior distribution) |
post.intv.upp |
upper bound of 100*confidence.lvl% posterior interval (quantile of posterior distribution) |
model |
"gaussian" for Gaussian, "poisson" for Poisson, and "binomial" for Binomial data |
X |
a covariate vector or matrix if designated. NA if not |
beta.new |
regression coefficient estimates |
beta.var |
estimated variance matrix of regression coefficient |
intercept |
whether TRUE or FALSE |
a.new |
a posterior mode of |
a.var |
posterior variance of |
confidence.lvl |
confidence level based on which confidence interval is constructed |
weight |
(only for Binomial model) weights for acceptance-rejection method |
p |
(only for Binomial model) posterior samples of p based on the acceptance-rejection method, if this method was used. This is a (k by nsim) matrix. Each row contains posteiror samples of each random effect. |
alpha |
(only for Binomial model) posterior samples of alpha based on the acceptance-rejection method, if this method was used |
beta |
(only for Binomial model) posterior samples of beta based on the acceptance-rejection method, if this method was used |
accept.rate |
(only for Binomial model) the acceptance rate of the acceptance-rejection method, if this method was used |
n.AR |
(Only for Binomial model) If |
n.AR.factor |
(only for Binomial model) If |
Author(s)
Hyungsuk Tak, Joseph Kelly, and Carl Morris
References
Tak, H., Kelly, J., and Morris, C. (2017) Rgbp: An R Package for Gaussian, Poisson, and Binomial Random Effects Models with Frequency Coverage Evaluations. Journal of Statistical Software. 78, 5, 1–33.
Morris, C. and Lysy, M. (2012). Shrinkage Estimation in Multilevel Normal Models. Statistical Science. 27, 1, 115–134.
Examples
# Loading datasets
data(schools)
y <- schools$y
se <- schools$se
# Arbitrary covariate for schools data
x2 <- rep(c(-1, 0, 1, 2), 2)
# baseball data where z is Hits and n is AtBats
z <- c(18, 17, 16, 15, 14, 14, 13, 12, 11, 11, 10, 10, 10, 10, 10, 9, 8, 7)
n <- c(45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45)
# One covariate: 1 if a player is an outfielder and 0 otherwise
x1 <- c(1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0)
################################################################
# Gaussian Regression Interactive Multilevel Modeling (GRIMM) #
################################################################
###################################################################################
# If we do not have any covariate and do not know a mean of the prior distribution#
###################################################################################
g <- gbp(y, se, model = "gaussian")
g
print(g, sort = FALSE)
summary(g)
plot(g)
plot(g, sort = FALSE)
### when we want to simulate pseudo datasets considering the estimated values
### as true ones.
gcv <- coverage(g, nsim = 10)
### gcv$coverageRB, gcv$coverage10, gcv$average.coverageRB, gcv$average.coverage10,
### gcv$minimum.coverageRB, gcv$minimum.coverage10, gcv$raw.resultRB,
### gcv$raw.result10.
### when we want to simulate pseudo datasets based on different values of A
### and a regression coefficient (intercept),
### not using estimated values as true ones.
gcv <- coverage(g, A.or.r = 9, reg.coef = 10, nsim = 10)
##################################################################################
# If we have one covariate and do not know a mean of the prior distribution yet, #
##################################################################################
g <- gbp(y, se, x2, model = "gaussian")
g
print(g, sort = FALSE)
summary(g)
plot(g)
plot(g, sort = FALSE)
### when we want to simulate pseudo datasets considering the estimated values
### as true ones.
gcv <- coverage(g, nsim = 10)
### gcv$coverageRB, gcv$coverage10, gcv$average.coverageRB, gcv$average.coverage10,
### gcv$minimum.coverageRB, gcv$minimum.coverage10, gcv$raw.resultRB,
### gcv$raw.result10.
### when we want to simulate pseudo datasets based on different values of A
### and regression coefficients, not using estimated values
### as true ones. Two values of reg.coef are for beta0 and beta1
gcv <- coverage(g, A.or.r = 9, reg.coef = c(10, 1), nsim = 10)
################################################
# If we know a mean of the prior distribution, #
################################################
g <- gbp(y, se, mean.PriorDist = 8, model = "gaussian")
g
print(g, sort = FALSE)
summary(g)
plot(g)
plot(g, sort = FALSE)
### when we want to simulate pseudo datasets considering the estimated values
### as true ones.
gcv <- coverage(g, nsim = 10)
### gcv$coverageRB, gcv$coverage10, gcv$average.coverageRB, gcv$average.coverage10,
### gcv$minimum.coverageRB, gcv$minimum.coverage10, gcv$raw.resultRB,
### gcv$raw.result10.
### when we want to simulate pseudo datasets based on different values of A and
### 2nd level mean as true ones, not using estimated values as true ones.
coverage(g, A.or.r = 9, mean.PriorDist = 5, nsim = 10)
###############################################################
# Binomial Regression Interactive Multilevel Modeling (BRIMM) #
###############################################################
###################################################################################
# If we do not have any covariate and do not know a mean of the prior distribution#
###################################################################################
b <- gbp(z, n, model = "binomial")
b
print(b, sort = FALSE)
summary(b)
plot(b)
plot(b, sort = FALSE)
### when we want to simulate pseudo datasets considering the estimated values
### as true ones.
bcv <- coverage(b, nsim = 10)
### bcv$coverageRB, bcv$coverage10, bcv$average.coverageRB, bcv$average.coverage10,
### bcv$minimum.coverageRB, bcv$minimum.coverage10, bcv$raw.resultRB,
### bcv$raw.result10.
### when we want to simulate pseudo datasets based on different values of r
### and a regression coefficient (intercept),
### not using estimated values as true ones.
bcv <- coverage(b, A.or.r = 60, reg.coef = -1, nsim = 10)
##################################################################################
# If we have one covariate and do not know a mean of the prior distribution yet, #
##################################################################################
b <- gbp(z, n, x1, model = "binomial")
b
print(b, sort = FALSE)
summary(b)
plot(b)
plot(b, sort = FALSE)
### when we want to simulate pseudo datasets considering the estimated values
### as true ones.
bcv <- coverage(b, nsim = 10)
### bcv$coverageRB, bcv$coverage10, bcv$average.coverageRB, bcv$average.coverage10,
### bcv$minimum.coverageRB, bcv$minimum.coverage10, bcv$raw.resultRB,
### bcv$raw.result10.
### when we want to simulate pseudo datasets based on different values of r
### and regression coefficients, not using estimated values
### as true ones. Two values of reg.coef are for beta0 and beta1
bcv <- coverage(b, A.or.r = 60, reg.coef = c(-1, 0), nsim = 10)
################################################
# If we know a mean of the prior distribution, #
################################################
b <- gbp(z, n, mean.PriorDist = 0.265, model = "binomial")
b
print(b, sort = FALSE)
summary(b)
plot(b)
plot(b, sort = FALSE)
### when we want to simulate pseudo datasets considering the estimated values
### as true ones.
bcv <- coverage(b, nsim = 10)
### bcv$coverageRB, bcv$coverage10, bcv$average.coverageRB, bcv$average.coverage10,
### bcv$minimum.coverageRB, bcv$minimum.coverage10, bcv$raw.resultRB,
### bcv$raw.result10.
### when we want to simulate pseudo datasets based on different values of r and
### 2nd level mean as true ones, not using estimated values as true ones.
bcv <- coverage(b, A.or.r = 60, mean.PriorDist = 0.3, nsim = 10)
##############################################################
# Poisson Regression Interactive Multilevel Modeling (PRIMM) #
##############################################################
################################################
# If we know a mean of the prior distribution, #
################################################
p <- gbp(z, n, mean.PriorDist = 0.265, model = "poisson")
p
print(p, sort = FALSE)
summary(p)
plot(p)
plot(p, sort = FALSE)
### when we want to simulate pseudo datasets considering the estimated values
### as true ones.
pcv <- coverage(p, nsim = 10)
### pcv$coverageRB, pcv$coverage10, pcv$average.coverageRB, pcv$average.coverage10,
### pcv$minimum.coverageRB, pcv$minimum.coverage10, pcv$raw.resultRB,
### pcv$raw.result10.
### when we want to simulate pseudo datasets based on different values of r and
### 2nd level mean as true ones, not using estimated values as true ones.
pcv <- coverage(p, A.or.r = 60, mean.PriorDist = 0.3, nsim = 10)