Estimating Coverage Probability

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Description

coverage estimates Rao-Blackwellized and simple unbiased coverage probabilities.

Usage

1
coverage(gbp.object, A.or.r, reg.coef, mean.PriorDist, nsim = 100)

Arguments

gbp.object

a resultant object of gbp function.

A.or.r

(optional) a given true numeric value of A for Gaussian data or of r for Binomial and Poisson data. If not designated, the estimated value in the gbp.object object will be considered as a true value.

reg.coef

(optional) a given true (m by 1) vector for regression coefficients, β, where m is the number of regression coefficients including an intercept. If not designated, the estimated value in the gbp.object object will be considered as a true value.

mean.PriorDist

(optional) a given true numeric value for the mean of (second-level) prior distribution. If not designated, the previously known value in the gbp.object object will be considered as a known prior mean.

nsim

number of datasets to be generated. Default is 100.

Details

As for the argument gbp.object, if the result of gbp is designated to b, for example
"b <- gbp(z, n, model = "binomial")", the argument gbp.object indicates this b.

Data generating process is based on a second-level hierarchical model. The first-level hierarchy is a distribution of observed data and the 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 2-level model. σ_j^2 below is assumed to be known or to be accurately estimated (s^2) and subscript j indicates j-th group 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 multi-level model. A square bracket below indicates [mean, variance] of distribution and a constant multiplied to the notation representing Gamma distribution (Gam) is a scale. Also, for consistent notation, y_j = z_j / n_j and n_j can be interpreted as j-th group's exposure only in this Poisson-Gamma hierarchical model.

(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 multi-level model. For reference, 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.

From now on, the subscript (i) means i-th simulation and the subscript j indicates j-th group. So, notations with a subscript (i) are (k by 1) vectors, for example θ_i' = (\thate_(i)1, θ_(i)2, ..., θ_(i)k).

Pseudo-data generating process starts from the second-level hierarchy to the first-level. coverage first generates true parameters (θ_(i)) for k groups at the second-level and then moves onto the first-level to simulate pseudo-data sets, y_(i) for Gaussian or z_(i) for Binomial and Poisson data, given previously generated true parameters (θ_(i)).

So, in order to generate pseudo-datasets, coverage needs parameters of prior distribution, (A (or r) and β (reg.coef)) or (A (or r) and μ0). From here, we have four options to run coverage.

First, if any values related to the prior distribution are not designated like coverage(b, nsim = 10), then coverage will regard estimated values (or known prior mean, μ0) in b (gbp.object) as given true values when it generates lots of pseudo-datasets. After sampling θ_(i) from the prior distribution determined by these estimated values (or known prior mean) in b (gbp.object), coverage creates an i-th pseudo-dataset based on θ_(i) just sampled.

Second, coverage allows us to try different true values in generating datasets. Suppose gbp.object is based on the model with a known prior mean, μ0. Then, we can try either different A.or.r or mean.PriorDist. For example, coverage(b, A.or.r = 20, nsim = 10),
coverage(b, mean.PriorDist = 0.5, nsim = 10), or
coverage(b, A.or.r = 20, mean.PriorDist = 0.5, nsim = 10). Note that we cannot set reg.coef because the second-level mean (prior mean) is known in gbp.object to begin with.

Suppose gbp.object is based on the model with an unknown prior mean. In this case, gbp.object has the estimation result of regression model, linear regression for Normal-Normal, log-linear regression for Poisson-Gamma, or logistic regression for Binomial-Beta, (only intercept term if there is no covariate) to estimate the unknown prior mean. Then, we can try some options: one or two of (A.or.r, mean.PriorDist, reg.coef). For example, coverage(b, A.or.r = 20, nsim = 10), coverage(b, mean.PriorDist = 0.5, nsim = 10), or coverage(b, reg.coef = 0.1, nsim = 10) with no covariate where reg.coef is a designated intercept term. Estimates in gbp.object will be used for undesignated values. Also, we can try appropriate combinations of two arguments. For example, coverage(b, A.or.r = 20, mean.PriorDist = 0.5, nsim = 10) and
coverage(b, A.or.r = 20, reg.coef = 0.1, nsim = 10). If we have one covariate, a 2 by 1 vector should be designated for reg.coef, one for an intercept term and the other for a regression coefficient of the covariate. Note that the two arguments, mean.PriorDist and reg.coef, cannot be assigned together because we do not need reg.coef given mean.PriorDist.

The simple unbiased estimator of coverage probability in j-th group is a sample mean of indicators over all simulated datasets. The j-th indicator in i-th simulation is 1 if the estimated interval of the j-th group on i-th simulated dataset contains a true parameter θ_(i)j that generated the observed value of the j-th group in the i-th dataset.

Rao-Blackwellized unbiased estimator for group j is a conditional expectation of the simple unbiased estimator given a sufficient statistic, y_j for Gaussian or z_j for Binomial and Poisson data.

Value

coverageRB

Rao-Blackwellized unbiased coverage estimate for each group averaged over all simulations.

coverageS

Simple unbiased coverage estimate for each group averaged over all simulations.

average.coverageRB

Overall Rao-Blackwellized unbiased coverage estimate across all the groups and simulations.

overall.coverageRB

Overall Rao-Blackwellized unbiased coverage estimate across all the groups and simulations.

average.coverageS

Overall simple unbiased coverage estimate across all the groups and simulations.

se.coverageRB

Standard error of Rao-Blackwellized unbiased coverage estimate for each group.

se.overall.coverageRB

Standard error of the overall Rao-Blackwellized unbiased coverage estimate.

se.coverageS

Standard error of simple unbiased coverage estimate for each group.

raw.resultRB

All the Rao-Blackwellized unbiased coverage estimates for every group and for every simulation.

raw.resultS

All the simple unbiased coverage estimates for every group and for every simulation.

confidence.lvl

Nominal confidence level

effective.n

The number of simulated data sets used to calculate the coverage estimates. The data sets may cause some errors in fitting models. For example, the data set may be against the conditions for the posteiror propriety in Binomial data.

model

The model being used, "br", "pr", or "gr".

case

One of the cases used to re-draw the coverage plot by coverage.plot.

betas

The regression coefficient used to generate simulated data sets.

A.r

The hyper-parameter value (A for Gaussian model, and r for both Binomial and Poisson models) used to generate simulated data sets.

priormeanused

The value of the prior mean(s) used to generate simulated data sets.

Author(s)

Hyungsuk Tak, Joseph Kelly, and Carl Morris

References

Christiansen, C. and Morris, C. (1997). Hierarchical Poisson Regression Modeling. Journal of the American Statistical Association. 92. 618-632.

Examples

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  # 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 Multi-level 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")

    ### when we want to simulate pseudo datasets considering the estimated values 
    ### as true ones.
    gcv <- coverage(g, nsim = 10)  

    ### gcv$coverageRB, gcv$coverageS, gcv$average.coverageRB, gcv$average.coverageS,
    ### gcv$minimum.coverageRB, gcv$raw.resultRB, gcv$raw.resultS

    ### gcv <- coverage(g, mean.PriorDist = 3, nsim = 100)
    ### gcv <- coverage(g, A.or.r = 150, nsim = 100)
    ### gcv <- coverage(g, reg.coef = 10, nsim = 100)
    ### gcv <- coverage(g, A.or.r = 150, mean.PriorDist = 3, nsim = 100)
    ### gcv <- coverage(g, A.or.r = 150, reg.coef = 10, nsim = 100)

    ##################################################################################
    # If we have one covariate and do not know a mean of the prior distribution yet, #
    ##################################################################################

    g <- gbp(y, se, x2, model = "gaussian")
 
    ### when we want to simulate pseudo datasets considering the estimated values 
    ### as true ones.
    gcv <- coverage(g, nsim = 10)  
 
    ### gcv$coverageRB, gcv$coverageS, gcv$average.coverageRB, gcv$average.coverageS,
    ### gcv$minimum.coverageRB, gcv$raw.resultRB, gcv$raw.resultS

    ### gcv <- coverage(g, mean.PriorDist = 3, nsim = 100)
    ### gcv <- coverage(g, A.or.r = 200, nsim = 100)
    ### gcv <- coverage(g, reg.coef = c(10, 2), nsim = 100)
    ### gcv <- coverage(g, A.or.r = 200, mean.PriorDist = 3, nsim = 100)
    ### gcv <- coverage(g, A.or.r = 200, reg.coef = c(10, 2), nsim = 100)

    ################################################
    # If we know a mean of the prior distribution, #
    ################################################

    g <- gbp(y, se, mean.PriorDist = 8, model = "gaussian")

    ### when we want to simulate pseudo datasets considering the estimated values 
    ### as true ones.
    gcv <- coverage(g, nsim = 10)  

    ### gcv$coverageRB, gcv$coverageS, gcv$average.coverageRB, gcv$average.coverageS,
    ### gcv$minimum.coverageRB, gcv$raw.resultRB, gcv$raw.resultS

    ### gcv <- coverage(g, mean.PriorDist = 3, nsim = 100)
    ### gcv <- coverage(g, A.or.r = 150, nsim = 100)
    ### gcv <- coverage(g, A.or.r = 150, mean.PriorDist = 3, nsim = 100)

  ################################################################
  # Binomial Regression Interactive Multi-level 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")

    ### when we want to simulate pseudo datasets considering the estimated values 
    ### as true ones.
    bcv <- coverage(b, nsim = 10)  

    ### bcv$coverageRB, bcv$coverageS, bcv$average.coverageRB, bcv$average.coverageS,
    ### bcv$minimum.coverageRB, bcv$raw.resultRB, bcv$raw.resultS

    ### bcv <- coverage(b, mean.PriorDist = 0.2, nsim = 100)
    ### bcv <- coverage(b, A.or.r = 50, nsim = 100)
    ### bcv <- coverage(b, reg.coef = -1.5, nsim = 100)
    ### bcv <- coverage(b, A.or.r = 50, mean.PriorDist = 0.2, nsim = 100)
    ### bcv <- coverage(b, A.or.r = 50, reg.coef = -1.5, nsim = 100)

    ##################################################################################
    # If we have one covariate and do not know a mean of the prior distribution yet, #
    ##################################################################################

    b <- gbp(z, n, x1, model = "binomial")

    ### when we want to simulate pseudo datasets considering the estimated values 
    ### as true ones.
    bcv <- coverage(b, nsim = 10)  

    ### bcv$coverageRB, bcv$coverageS, bcv$average.coverageRB, bcv$average.coverageS,
    ### bcv$minimum.coverageRB, bcv$raw.resultRB, bcv$raw.resultS

    ### bcv <- coverage(b, mean.PriorDist = 0.2, nsim = 100)
    ### bcv <- coverage(b, A.or.r = 50, nsim = 100)
    ### bcv <- coverage(b, reg.coef = c(-1.5, 0), nsim = 100)
    ### bcv <- coverage(b, A.or.r = 40, mean.PriorDist = 0.2, nsim = 100)
    ### bcv <- coverage(b, A.or.r = 40, reg.coef = c(-1.5, 0), nsim = 100)

    ################################################
    # If we know a mean of the prior distribution, #
    ################################################

    b <- gbp(z, n, mean.PriorDist = 0.265, model = "binomial")

    ### when we want to simulate pseudo datasets considering the estimated values 
    ### as true ones.
    bcv <- coverage(b, nsim = 10)  

    ### bcv$coverageRB, bcv$coverageS, bcv$average.coverageRB, bcv$average.coverageS,
    ### bcv$minimum.coverageRB, bcv$raw.resultRB, bcv$raw.resultS

    ### bcv <- coverage(b, mean.PriorDist = 0.2, nsim = 100)
    ### bcv <- coverage(b, A.or.r = 50, nsim = 100)
    ### bcv <- coverage(b, A.or.r = 40, mean.PriorDist = 0.2, nsim = 100)

  ###############################################################
  # Poisson Regression Interactive Multi-level Modeling (PRIMM) #
  ###############################################################

    ################################################
    # If we know a mean of the prior distribution, #
    ################################################

    p <- gbp(z, n, mean.PriorDist = 0.265, model = "poisson")

    ### when we want to simulate pseudo datasets considering the estimated values 
    ### as true ones.
    pcv <- coverage(p, nsim = 10)  

    ### pcv$coverageRB, pcv$coverageS, pcv$average.coverageRB, pcv$average.coverageS,
    ### pcv$minimum.coverageRB, pcv$raw.resultRB, pcv$raw.resultS

    ### pcv <- coverage(p, mean.PriorDist = 0.265, nsim = 100)
    ### pcv <- coverage(p, A.or.r = 150, nsim = 100)
    ### pcv <- coverage(p, A.or.r = 150, mean.PriorDist = 0.265, nsim = 100)