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Last data update: 2014.03.03

R: Fitting Gaussian, Poisson, and Binomial Hierarchical Models

gbp	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`	`TRUE` or `FALSE` flag indicating whether an intercept should be included in the regression. Default is `TRUE`.
`confidence.lvl`	a `float` between 0 and 1 to estimate 100*confidence.lvl% intervals. Default is 0.95.
`n.AR`	Only for Binomial model. If `n.AR = 1000`, all the results will be based on 1000 posterior samples using the acceptance-rejection sampling. Default is 0.
`n.AR.factor`	Only for Binomial model. If `n.AR = 1000` and `n.AR.factor = 4`, then the acceptance-rejection method will sample 4 * 1000 trial samples, and accept or reject them until the method gets n.AR posterior samples. Default is 4.
`trial.scale`	A scale used in the trial distribution of α. If resultant weight has too mamy 0's, scale should be smaller than before. If resultant weight has too few 0's, scale should be larger than before. If there are relatively huge weights, scale should be larger than before. Default is NA.
`save.result`	Only for Binomial model with the acceptance-rejection sampling. If `save.result` is TRUE, all the results of weights and posterior samples will be saved in the gbp object. Default is TRUE.
`normal.CI`	Only applicable for Gaussian data. If `TRUE` then a Normal approximation is used to construct intervals for the level 1 group means. If `FALSE` (default) then a Skew-Normal distribution is used. Setting the value to `TRUE` may result in speed improvements but may lead to intervals that under cover.
`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. dr/(t + r)^{u+1}.
`u`	A positive constant to determine the hyper-prior distribution of r for the Binomial model with the acceptance-rejection method. dr/(t + r)^{u+1}.

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 (=σ^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 | θ_j) ~ indep N(θ_j, σ_j^2)

(θ_j | μ0_j, A) ~ indep N(μ0_j, A)

μ0_j = x_j'β

for j = 1, …, 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 = z_j / n_j.

(z_j | θ_j) ~ indep Pois(n_jθ_j)

(θ_j | r, μ0_j) ~ indep Gam(rμ0_j) / r ~ indep Gam[μ0_j, μ0_j / r]

log(μ0_j) = x_j'β

for j = 1, …, 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 = z_j / n_j.

(z_j | θ_j) ~ indep Bin(n_j, θ_j)

(θ_j | r, μ0_j) ~ indep Beta(rμ0_j, r(1 - μ0_j)) ~ indep Beta[μ0_j, μ0_j(1 - μ0_j) / (r + 1)]

logit(μ0_j) = x_j'β

for j = 1, …, 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(1 / r) (= dr / r^2) for Binomial and Poisson data. The second level variance component can be interpreted as variation among the first-level parameters (θ_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 (μ0) 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 (μ0) 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 (μ0) a priori. Now, we do not need to estimate regression coefficients at all because we know a true value of μ0 (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 α defined as log(A) for Gaussian or log(1 / r) for Binomial and Poisson data. Practical meaning (variation of ensemble information) of estimating α will appear in `summary(gbp.object)`.
`a.var`	posterior variance of α
`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 = 1000`, all the results will be based on 1000 posterior samples using the acceptance-rejection sampling. Default is 0.
`n.AR.factor`	(only for Binomial model) If `n.AR = 1000` and `n.AR.factor = 4`, then the acceptance-rejection method will sample 4 * 1000 trial posteior samples, and accept or reject them. Default is 4.

Author(s)

Hyungsuk Tak, Joseph Kelly, and Carl Morris

References

Morris, C. and Lysy, M. (2012). Shrinkage Estimation in Multilevel Normal Models. Statistical Science. 27. 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)

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(Rgbp)
Loading required package: sn
Loading required package: stats4

Attaching package: 'sn'

The following object is masked from 'package:stats':

    sd

Loading required package: mnormt
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/Rgbp/gbp.Rd_%03d_medium.png", width=480, height=480)
> ### Name: gbp
> ### Title: Fitting Gaussian, Poisson, and Binomial Hierarchical Models
> ### Aliases: gbp
> ### Keywords: methods
> 
> ### ** 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
Summary for each group (sorted by the descending order of se): 

     obs.mean   se prior.mean shrinkage low.intv post.mean upp.intv post.sd
8       12.00 18.0       8.17     0.734   -10.21      9.19     29.9   10.23
3       -3.00 16.0       8.17     0.685   -17.13      4.65     22.5   10.10
1       28.00 15.0       8.17     0.657    -2.32     14.98     38.8   10.56
4        7.00 11.0       8.17     0.507    -8.78      7.59     23.6    8.26
6        1.00 11.0       8.17     0.507   -13.03      4.63     20.1    8.44
2        8.00 10.0       8.17     0.459    -7.25      8.08     23.4    7.81
7       18.00 10.0       8.17     0.459    -1.29     13.48     30.8    8.18
5       -1.00  9.0       8.17     0.408   -13.30      2.74     16.7    7.63
Mean          12.5       8.17     0.552    -9.16      8.17     25.7    8.90
>     print(g, sort = FALSE)
Summary for each group: 

     obs.mean   se prior.mean shrinkage low.intv post.mean upp.intv post.sd
1       28.00 15.0       8.17     0.657    -2.32     14.98     38.8   10.56
2        8.00 10.0       8.17     0.459    -7.25      8.08     23.4    7.81
3       -3.00 16.0       8.17     0.685   -17.13      4.65     22.5   10.10
4        7.00 11.0       8.17     0.507    -8.78      7.59     23.6    8.26
5       -1.00  9.0       8.17     0.408   -13.30      2.74     16.7    7.63
6        1.00 11.0       8.17     0.507   -13.03      4.63     20.1    8.44
7       18.00 10.0       8.17     0.459    -1.29     13.48     30.8    8.18
8       12.00 18.0       8.17     0.734   -10.21      9.19     29.9   10.23
Mean          12.5       8.17     0.552    -9.16      8.17     25.7    8.90
>     summary(g)
Main summary:

                       obs.mean   se prior.mean shrinkage low.intv post.mean
Group with min(se)        -1.00  9.0       8.17     0.408   -13.30      2.74
Group with median(se)1     1.00 11.0       8.17     0.507   -13.03      4.63
Group with median(se)2     7.00 11.0       8.17     0.507    -8.78      7.59
Group with max(se)        12.00 18.0       8.17     0.734   -10.21      9.19
Overall Mean                    12.5       8.17     0.552    -9.16      8.17
                       upp.intv post.sd
Group with min(se)         16.7    7.63
Group with median(se)1     20.1    8.44
Group with median(se)2     23.6    8.26
Group with max(se)         29.9   10.23
Overall Mean               25.7    8.90


Estimation summary for the second-level variance component:
alpha = log(A) for Gaussian or alpha =  log(1/r) for Binomial and Poisson data:

 post.mode.alpha post.sd.alpha post.mode.A
            4.77          1.14         118


Estimation summary for the regression coefficient :

      estimate   se z.val p.val
beta1    8.168 5.73 1.425 0.154
>     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
Summary for each group (sorted by the descending order of se): 

     obs.mean   se   X1 prior.mean shrinkage low.intv post.mean upp.intv
8       12.00 18.0  2.0       8.45     0.666   -14.83      9.64     35.2
3       -3.00 16.0  1.0       8.32     0.612   -20.05      3.93     23.9
1       28.00 15.0 -1.0       8.07     0.581    -5.14     16.43     43.4
4        7.00 11.0  2.0       8.45     0.427   -11.30      7.62     26.2
6        1.00 11.0  0.0       8.20     0.427   -14.51      4.07     20.8
2        8.00 10.0  0.0       8.20     0.381    -8.15      8.08     24.3
7       18.00 10.0  1.0       8.32     0.381    -1.59     14.31     32.3
5       -1.00  9.0 -1.0       8.07     0.333   -15.14      2.02     17.7
Mean          12.5  0.5       8.26     0.476   -11.34      8.26     28.0
     post.sd
8      12.74
3      11.20
1      12.39
4       9.56
6       8.98
2       8.27
7       8.63
5       8.35
Mean   10.02
>     print(g, sort = FALSE)
Summary for each group: 

     obs.mean   se   X1 prior.mean shrinkage low.intv post.mean upp.intv
1       28.00 15.0 -1.0       8.07     0.581    -5.14     16.43     43.4
2        8.00 10.0  0.0       8.20     0.381    -8.15      8.08     24.3
3       -3.00 16.0  1.0       8.32     0.612   -20.05      3.93     23.9
4        7.00 11.0  2.0       8.45     0.427   -11.30      7.62     26.2
5       -1.00  9.0 -1.0       8.07     0.333   -15.14      2.02     17.7
6        1.00 11.0  0.0       8.20     0.427   -14.51      4.07     20.8
7       18.00 10.0  1.0       8.32     0.381    -1.59     14.31     32.3
8       12.00 18.0  2.0       8.45     0.666   -14.83      9.64     35.2
Mean          12.5  0.5       8.26     0.476   -11.34      8.26     28.0
     post.sd
1      12.39
2       8.27
3      11.20
4       9.56
5       8.35
6       8.98
7       8.63
8      12.74
Mean   10.02
>     summary(g)
Main summary:

                       obs.mean   se   X1 prior.mean shrinkage low.intv
Group with min(se)        -1.00  9.0 -1.0       8.07     0.333    -15.1
Group with median(se)1     1.00 11.0  0.0       8.20     0.427    -14.5
Group with median(se)2     7.00 11.0  2.0       8.45     0.427    -11.3
Group with max(se)        12.00 18.0  2.0       8.45     0.666    -14.8
Overall Mean                    12.5  0.5       8.26     0.476    -11.3
                       post.mean upp.intv post.sd
Group with min(se)          2.02     17.7    8.35
Group with median(se)1      4.07     20.8    8.98
Group with median(se)2      7.62     26.2    9.56
Group with max(se)          9.64     35.2   12.74
Overall Mean                8.26     28.0   10.02


Estimation summary for the second-level variance component:
alpha = log(A) for Gaussian or alpha =  log(1/r) for Binomial and Poisson data:

 post.mode.alpha post.sd.alpha post.mode.A
            5.09          1.17         162


Estimation summary for the regression coefficient :

      estimate    se z.val p.val
beta1    8.198 6.656 1.232 0.218
beta2    0.126 5.698 0.022 0.982
>     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
Summary for each group (sorted by the descending order of se): 

     obs.mean   se prior.mean shrinkage low.intv post.mean upp.intv post.sd
8       12.00 18.0          8     0.786   -6.767      8.86     26.1    8.36
3       -3.00 16.0          8     0.743  -13.448      5.18     19.3    8.38
1       28.00 15.0          8     0.718    0.595     13.64     34.6    8.95
4        7.00 11.0          8     0.578   -6.642      7.58     21.4    7.15
6        1.00 11.0          8     0.578  -10.701      5.04     18.1    7.34
2        8.00 10.0          8     0.531   -5.426      8.00     21.4    6.85
7       18.00 10.0          8     0.531    0.123     12.69     28.6    7.27
5       -1.00  9.0          8     0.478  -11.516      3.30     15.4    6.87
Mean          12.5          8     0.618   -6.723      8.04     23.1    7.65
>     print(g, sort = FALSE)
Summary for each group: 

     obs.mean   se prior.mean shrinkage low.intv post.mean upp.intv post.sd
1       28.00 15.0          8     0.718    0.595     13.64     34.6    8.95
2        8.00 10.0          8     0.531   -5.426      8.00     21.4    6.85
3       -3.00 16.0          8     0.743  -13.448      5.18     19.3    8.38
4        7.00 11.0          8     0.578   -6.642      7.58     21.4    7.15
5       -1.00  9.0          8     0.478  -11.516      3.30     15.4    6.87
6        1.00 11.0          8     0.578  -10.701      5.04     18.1    7.34
7       18.00 10.0          8     0.531    0.123     12.69     28.6    7.27
8       12.00 18.0          8     0.786   -6.767      8.86     26.1    8.36
Mean          12.5          8     0.618   -6.723      8.04     23.1    7.65
>     summary(g)
Main summary:

                       obs.mean   se prior.mean shrinkage low.intv post.mean
Group with min(se)        -1.00  9.0          8     0.478   -11.52      3.30
Group with median(se)1     1.00 11.0          8     0.578   -10.70      5.04
Group with median(se)2     7.00 11.0          8     0.578    -6.64      7.58
Group with max(se)        12.00 18.0          8     0.786    -6.77      8.86
Overall Mean                    12.5          8     0.618    -6.72      8.04
                       upp.intv post.sd
Group with min(se)         15.4    6.87
Group with median(se)1     18.1    7.34
Group with median(se)2     21.4    7.15
Group with max(se)         26.1    8.36
Overall Mean               23.1    7.65


Estimation summary for the second-level variance component:
alpha = log(A) for Gaussian or alpha = log(1/r) for Binomial and Poisson data:

 post.mode.alpha post.sd.alpha post.mode.A
            4.48          1.13        88.4
>     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)  
$coverageRB
[1] 0.998 0.999 1.000 0.999 0.999 0.995 0.991 0.999

$coverageS
[1] 1.0 1.0 1.0 1.0 1.0 0.9 1.0 1.0

$average.coverageRB
[1] 0.998

$average.coverageS
[1] 0.988

$overall.coverageRB
[1] 0.998

$se.overal.coverageRB
[1] 8e-04

$se.coverageRB
[1] 0.0015 0.0008 0.0001 0.0004 0.0002 0.0024 0.0055 0.0006

$se.coverageS
[1] 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0

$raw.resultRB
          [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
[1,] 0.9847640 0.9999998 1.0000000 0.9997908 1.0000000 0.9993732 0.9999259
[2,] 0.9997135 0.9999210 0.9999861 0.9919308 0.9999628 0.9999285 0.9997725
[3,] 1.0000000 1.0000000 0.9999972 1.0000000 0.9999992 0.9998540 0.9999694
[4,] 0.9999383 0.9988361 0.9999987 0.9999946 1.0000000 0.9997252 0.9992767
[5,] 0.9999885 0.9999871 0.9999962 0.9974014 0.9995615 0.9990831 0.9989867
[6,] 0.9999989 0.9999988 0.9824083 0.9999999 0.9807068 0.9984266 0.9998216
[7,] 0.9999048 0.9990768 0.9950878 0.9989887 0.9820918 0.9999989 0.9990731
[8,] 1.0000000 0.9950041 1.0000000 0.9999996 1.0000000 0.9999995 1.0000000
          [,8]      [,9]     [,10]
[1,] 1.0000000 0.9991794 1.0000000
[2,] 0.9997985 0.9998286 0.9962021
[3,] 0.9999999 0.9999223 0.9984980
[4,] 0.9959410 0.9999973 0.9990709
[5,] 0.9992161 0.9998059 0.9997078
[6,] 0.9928254 0.9999976 0.9999999
[7,] 0.9443528 0.9959980 0.9996104
[8,] 0.9952561 0.9999686 1.0000000

$raw.resultS
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    1    1    1    1    1    1    1    1    1     1
[2,]    1    1    1    1    1    1    1    1    1     1
[3,]    1    1    1    1    1    1    1    1    1     1
[4,]    1    1    1    1    1    1    1    1    1     1
[5,]    1    1    1    1    1    1    1    1    1     1
[6,]    1    1    1    1    1    1    1    0    1     1
[7,]    1    1    1    1    1    1    1    1    1     1
[8,]    1    1    1    1    1    1    1    1    1     1

$confidence.lvl
[1] 0.95

$effective.n
[1] 10

$model
[1] "gr"

$case
[1] 5

$betas
[1] NA

$A.r
[1] 9

$priormeanused
[1] 5

> 
>   ###############################################################
>   # 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
Summary for each group (sorted by  the ascending order of n): 

     obs.mean  n prior.mean shrinkage low.intv post.mean upp.intv post.sd
1       0.400 45      0.267     0.622    0.222     0.317    0.421  0.0507
2       0.378 45      0.267     0.622    0.218     0.309    0.408  0.0487
3       0.356 45      0.267     0.622    0.213     0.300    0.396  0.0469
4       0.333 45      0.267     0.622    0.207     0.292    0.385  0.0453
5       0.311 45      0.267     0.622    0.202     0.284    0.374  0.0440
6       0.311 45      0.267     0.622    0.202     0.284    0.374  0.0440
7       0.289 45      0.267     0.622    0.195     0.275    0.363  0.0431
8       0.267 45      0.267     0.622    0.188     0.267    0.354  0.0424
9       0.244 45      0.267     0.622    0.180     0.258    0.345  0.0421
10      0.244 45      0.267     0.622    0.180     0.258    0.345  0.0421
11      0.222 45      0.267     0.622    0.172     0.250    0.337  0.0422
12      0.222 45      0.267     0.622    0.172     0.250    0.337  0.0422
13      0.222 45      0.267     0.622    0.172     0.250    0.337  0.0422
14      0.222 45      0.267     0.622    0.172     0.250    0.337  0.0422
15      0.222 45      0.267     0.622    0.172     0.250    0.337  0.0422
16      0.200 45      0.267     0.622    0.163     0.242    0.330  0.0426
17      0.178 45      0.267     0.622    0.154     0.233    0.323  0.0433
18      0.156 45      0.267     0.622    0.144     0.225    0.318  0.0444
Mean          45      0.267     0.622    0.185     0.266    0.357  0.0439
>     print(b, sort = FALSE)
Summary for each group: 

     obs.mean  n prior.mean shrinkage low.intv post.mean upp.intv post.sd
1       0.400 45      0.267     0.622    0.222     0.317    0.421  0.0507
2       0.378 45      0.267     0.622    0.218     0.309    0.408  0.0487
3       0.356 45      0.267     0.622    0.213     0.300    0.396  0.0469
4       0.333 45      0.267     0.622    0.207     0.292    0.385  0.0453
5       0.311 45      0.267     0.622    0.202     0.284    0.374  0.0440
6       0.311 45      0.267     0.622    0.202     0.284    0.374  0.0440
7       0.289 45      0.267     0.622    0.195     0.275    0.363  0.0431
8       0.267 45      0.267     0.622    0.188     0.267    0.354  0.0424
9       0.244 45      0.267     0.622    0.180     0.258    0.345  0.0421
10      0.244 45      0.267     0.622    0.180     0.258    0.345  0.0421
11      0.222 45      0.267     0.622    0.172     0.250    0.337  0.0422
12      0.222 45      0.267     0.622    0.172     0.250    0.337  0.0422
13      0.222 45      0.267     0.622    0.172     0.250    0.337  0.0422
14      0.222 45      0.267     0.622    0.172     0.250    0.337  0.0422
15      0.222 45      0.267     0.622    0.172     0.250    0.337  0.0422
16      0.200 45      0.267     0.622    0.163     0.242    0.330  0.0426
17      0.178 45      0.267     0.622    0.154     0.233    0.323  0.0433
18      0.156 45      0.267     0.622    0.144     0.225    0.318  0.0444
Mean          45      0.267     0.622    0.185     0.266    0.357  0.0439
>     summary(b)
Main summary:

                             obs.mean  n prior.mean shrinkage low.intv
Group with min(obs.mean)        0.156 45      0.267     0.622    0.144
Group with median(obs.mean)1    0.244 45      0.267     0.622    0.180
Group with median(obs.mean)2    0.244 45      0.267     0.622    0.180
Group with max(obs.mean)        0.400 45      0.267     0.622    0.222
Overall Mean                          45      0.267     0.622    0.185
                             post.mean upp.intv post.sd
Group with min(obs.mean)         0.225    0.318  0.0444
Group with median(obs.mean)1     0.258    0.345  0.0421
Group with median(obs.mean)2     0.258    0.345  0.0421
Group with max(obs.mean)         0.317    0.421  0.0507
Overall Mean                     0.266    0.357  0.0439


Estimation summary for the second-level variance component:
alpha = log(A) for Gaussian or alpha =  log(1/r) for Binomial and Poisson data:

 post.mode.alpha post.sd.alpha post.mode.r
           -4.31          0.82        74.1


Estimation summary for the regression coefficient :

      estimate  se   z.val p.val
beta1   -1.012 0.1 -10.153     0
>     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
Summary for each group (sorted by  the ascending order of n): 

     obs.mean  n   X1 prior.mean shrinkage low.intv post.mean upp.intv post.sd
1       0.400 45 1.00      0.310     0.715    0.248     0.335    0.429  0.0462
2       0.378 45 1.00      0.310     0.715    0.244     0.329    0.420  0.0448
3       0.356 45 1.00      0.310     0.715    0.240     0.323    0.411  0.0437
4       0.333 45 1.00      0.310     0.715    0.236     0.316    0.403  0.0429
5       0.311 45 1.00      0.310     0.715    0.230     0.310    0.396  0.0424
6       0.311 45 0.00      0.233     0.715    0.179     0.256    0.341  0.0415
7       0.289 45 0.00      0.233     0.715    0.175     0.249    0.331  0.0400
8       0.267 45 0.00      0.233     0.715    0.171     0.243    0.323  0.0388
9       0.244 45 0.00      0.233     0.715    0.166     0.237    0.315  0.0380
10      0.244 45 1.00      0.310     0.715    0.210     0.291    0.379  0.0432
11      0.222 45 0.00      0.233     0.715    0.161     0.230    0.308  0.0377
12      0.222 45 0.00      0.233     0.715    0.161     0.230    0.308  0.0377
13      0.222 45 0.00      0.233     0.715    0.161     0.230    0.308  0.0377
14      0.222 45 1.00      0.310     0.715    0.202     0.285    0.375  0.0441
15      0.222 45 1.00      0.310     0.715    0.202     0.285    0.375  0.0441
16      0.200 45 0.00      0.233     0.715    0.155     0.224    0.302  0.0377
17      0.178 45 0.00      0.233     0.715    0.148     0.218    0.297  0.0381
18      0.156 45 0.00      0.233     0.715    0.140     0.211    0.292  0.0389
Mean          45 0.44      0.267     0.715    0.191     0.267    0.351  0.0410
>     print(b, sort = FALSE)
Summary for each group: 

     obs.mean  n   X1 prior.mean shrinkage low.intv post.mean upp.intv post.sd
1       0.400 45 1.00      0.310     0.715    0.248     0.335    0.429  0.0462
2       0.378 45 1.00      0.310     0.715    0.244     0.329    0.420  0.0448
3       0.356 45 1.00      0.310     0.715    0.240     0.323    0.411  0.0437
4       0.333 45 1.00      0.310     0.715    0.236     0.316    0.403  0.0429
5       0.311 45 1.00      0.310     0.715    0.230     0.310    0.396  0.0424
6       0.311 45 0.00      0.233     0.715    0.179     0.256    0.341  0.0415
7       0.289 45 0.00      0.233     0.715    0.175     0.249    0.331  0.0400
8       0.267 45 0.00      0.233     0.715    0.171     0.243    0.323  0.0388
9       0.244 45 0.00      0.233     0.715    0.166     0.237    0.315  0.0380
10      0.244 45 1.00      0.310     0.715    0.210     0.291    0.379  0.0432
11      0.222 45 0.00      0.233     0.715    0.161     0.230    0.308  0.0377
12      0.222 45 0.00      0.233     0.715    0.161     0.230    0.308  0.0377
13      0.222 45 0.00      0.233     0.715    0.161     0.230    0.308  0.0377
14      0.222 45 1.00      0.310     0.715    0.202     0.285    0.375  0.0441
15      0.222 45 1.00      0.310     0.715    0.202     0.285    0.375  0.0441
16      0.200 45 0.00      0.233     0.715    0.155     0.224    0.302  0.0377
17      0.178 45 0.00      0.233     0.715    0.148     0.218    0.297  0.0381
18      0.156 45 0.00      0.233     0.715    0.140     0.211    0.292  0.0389
Mean          45 0.44      0.267     0.715    0.191     0.267    0.351  0.0410
>     summary(b)
Main summary:

                             obs.mean  n    X1 prior.mean shrinkage low.intv
Group with min(obs.mean)        0.156 45 0.000      0.233     0.715    0.140
Group with median(obs.mean)1    0.244 45 0.000      0.233     0.715    0.166
Group with median(obs.mean)2    0.244 45 1.000      0.310     0.715    0.210
Group with max(obs.mean)        0.400 45 1.000      0.310     0.715    0.248
Overall Mean                          45 0.444      0.267     0.715    0.191
                             post.mean upp.intv post.sd
Group with min(obs.mean)         0.211    0.292  0.0389
Group with median(obs.mean)1     0.237    0.315  0.0380
Group with median(obs.mean)2     0.291    0.379  0.0432
Group with max(obs.mean)         0.335    0.429  0.0462
Overall Mean                     0.267    0.351  0.0410


Estimation summary for the second-level variance component:
alpha = log(A) for Gaussian or alpha =  log(1/r) for Binomial and Poisson data:

 post.mode.alpha post.sd.alpha post.mode.r
           -4.73         0.957         113


Estimation summary for the regression coefficient :

      estimate    se  z.val p.val
beta1   -1.194 0.131 -9.129 0.000
beta2    0.389 0.187  2.074 0.038
>     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
Summary for each group (sorted by  the ascending order of n): 

     obs.mean  n prior.mean shrinkage low.intv post.mean upp.intv post.sd
1       0.400 45      0.265      0.65    0.230     0.312    0.401  0.0439
2       0.378 45      0.265      0.65    0.224     0.304    0.391  0.0428
3       0.356 45      0.265      0.65    0.218     0.297    0.382  0.0418
4       0.333 45      0.265      0.65    0.212     0.289    0.372  0.0409
5       0.311 45      0.265      0.65    0.206     0.281    0.363  0.0401
6       0.311 45      0.265      0.65    0.206     0.281    0.363  0.0401
7       0.289 45      0.265      0.65    0.200     0.273    0.354  0.0395
8       0.267 45      0.265      0.65    0.193     0.266    0.345  0.0390
9       0.244 45      0.265      0.65    0.186     0.258    0.337  0.0386
10      0.244 45      0.265      0.65    0.186     0.258    0.337  0.0386
11      0.222 45      0.265      0.65    0.179     0.250    0.329  0.0383
12      0.222 45      0.265      0.65    0.179     0.250    0.329  0.0383
13      0.222 45      0.265      0.65    0.179     0.250    0.329  0.0383
14      0.222 45      0.265      0.65    0.179     0.250    0.329  0.0383
15      0.222 45      0.265      0.65    0.179     0.250    0.329  0.0383
16      0.200 45      0.265      0.65    0.172     0.242    0.321  0.0381
17      0.178 45      0.265      0.65    0.164     0.234    0.313  0.0381
18      0.156 45      0.265      0.65    0.156     0.227    0.306  0.0383
Mean          45      0.265      0.65    0.191     0.265    0.346  0.0395
>     print(b, sort = FALSE)
Summary for each group: 

     obs.mean  n prior.mean shrinkage low.intv post.mean upp.intv post.sd
1       0.400 45      0.265      0.65    0.230     0.312    0.401  0.0439
2       0.378 45      0.265      0.65    0.224     0.304    0.391  0.0428
3       0.356 45      0.265      0.65    0.218     0.297    0.382  0.0418
4       0.333 45      0.265      0.65    0.212     0.289    0.372  0.0409
5       0.311 45      0.265      0.65    0.206     0.281    0.363  0.0401
6       0.311 45      0.265      0.65    0.206     0.281    0.363  0.0401
7       0.289 45      0.265      0.65    0.200     0.273    0.354  0.0395
8       0.267 45      0.265      0.65    0.193     0.266    0.345  0.0390
9       0.244 45      0.265      0.65    0.186     0.258    0.337  0.0386
10      0.244 45      0.265      0.65    0.186     0.258    0.337  0.0386
11      0.222 45      0.265      0.65    0.179     0.250    0.329  0.0383
12      0.222 45      0.265      0.65    0.179     0.250    0.329  0.0383
13      0.222 45      0.265      0.65    0.179     0.250    0.329  0.0383
14      0.222 45      0.265      0.65    0.179     0.250    0.329  0.0383
15      0.222 45      0.265      0.65    0.179     0.250    0.329  0.0383
16      0.200 45      0.265      0.65    0.172     0.242    0.321  0.0381
17      0.178 45      0.265      0.65    0.164     0.234    0.313  0.0381
18      0.156 45      0.265      0.65    0.156     0.227    0.306  0.0383
Mean          45      0.265      0.65    0.191     0.265    0.346  0.0395
>     summary(b)
Main summary:

                             obs.mean  n prior.mean shrinkage low.intv
Group with min(obs.mean)        0.156 45      0.265      0.65    0.156
Group with median(obs.mean)1    0.244 45      0.265      0.65    0.186
Group with median(obs.mean)2    0.244 45      0.265      0.65    0.186
Group with max(obs.mean)        0.400 45      0.265      0.65    0.230
Overall Mean                          45      0.265      0.65    0.191
                             post.mean upp.intv post.sd
Group with min(obs.mean)         0.227    0.306  0.0383
Group with median(obs.mean)1     0.258    0.337  0.0386
Group with median(obs.mean)2     0.258    0.337  0.0386
Group with max(obs.mean)         0.312    0.401  0.0439
Overall Mean                     0.265    0.346  0.0395


Estimation su