Description Usage Arguments Details Value Warnings References See Also Examples
Computes a table of Hellinger distances between marginal posterior distributions
for different parameters in the NNHM
induced by the actual heterogeneity priors specified in tau.prior
and
posterior benchmarks proposed in the Supplementary Material of Ott et al. (2021).
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df 
data frame with one column "y" containing the (transformed) effect estimates for the individual studies and one column "sigma" containing the standard errors of these estimates. 
tau.prior 
list of prior specifications, which are either functions returning the probability densities of the actual priors of interest for the heterogeneity parameter tau or character strings specifying priors implemented in the 
H.dist.method 
method for computation of Hellinger distances between marginal posterior densities. Either 
m_inf 
parameter value m=m_{inf} of the SGC(m) prior,
such that the median relative latent model complexity (RLMC) is close to 0.
If set to 
M_inf 
parameter value M=M_{inf} of the SIGC(M) prior,
such that the median relative latent model complexity (RLMC) is close to 1.
If set to 
rlmc0 
RLMC target value for the SGC(m_{inf}) benchmark prior (typically close to 0).
Is required only if 
rlmc1 
RLMC target value for the SIGC(M_{inf}) benchmark prior (typically close to 1).
Is required only if 
mu.mean 
mean of the normal prior for the effect mu. 
mu.sd 
standard deviation of the normal prior for the effect mu. 
The three posterior benchmarks used are introduced in the Supplementary Material of Ott et al. (2021, Sections 2.2.1 and 2.5, see also Section 3.4 in Ott at al. (2021) for Jeffreys reference prior), where they are denoted by po_{m_{inf}}(Ψ), po_{J}(Ψ) and po_{M_{inf}}(Ψ). Here, Ψ \in \{ μ, τ, θ_1, ..., θ_k, θ_{new} \} denotes the parameter of interest in the NNHM, where θ_{i} is the random effect in the ith study and θ_{new} the predicted effect for a new study.
Note that Jeffreys reference posterior po_{J} is proper if there are at least two studies in the metaanalysis data set. It is based on the improper Jeffreys reference prior, which is minimally informative given the data.
If integralbased computation (H.dist.method = "integral"
) of Hellinger distances is selected (the default), numerical integration is applied to obtain the Hellinger distance between the two marginal posterior densities (by using the function H
).
If momentbased computation (H.dist.method = "moment"
) is selected, the marginal densities are first approximated by normal densities with the same means and standard deviations and then the Hellinger distance between these normal densities can be obtained by an analytical formula (implemented in the function H_normal
).
The default values for mu.mean
and mu.sd
are suitable for effects mu on the log odds (ratio) scale.
A list with two elements:
The first element named "table" is a matrix containing the Hellinger distance estimates and the
second element called "par" is a named vector giving the parameter values
of the benchmark priors.
The vector "par" has the following three components:
m_inf
, M_inf
and C
.
The matrix "table" contains the Hellinger distance estimates between marginal posteriors
and has 3 columns and n*(k+3) rows,
where n=length(tau.prior
) is the number of actual heterogeneity priors specified
and k is the number of studies in the metaanalysis data set
(so that there are k+3 parameters of interest).
The columns of the matrix give the following Hellinger distance estimates between two marginal posteriors (for the parameter of interest Ψ varying with rows) induced by the following two heterogeneity priors (from left to right):
H(po_m_inf, po_act) 
benchmark prior SGC(m_inf) and actual prior 
H(po_J, po_act) 
Jeffreys reference prior π_J and actual prior 
H(po_M_inf, po_act) 
benchmark prior SIGC(M_inf) and actual prior 
The actual heterogenity prior and the parameter of interest Ψ vary with the rows in the following order:
mu, pri_act_1 
Ψ=μ and first actual prior in 
mu, pri_act_2 
Ψ=μ and second actual prior in 
... 

mu, pri_act_n 
Ψ=μ and nth actual prior in 
tau, pri_act_1 
Ψ=τ and first actual prior in 
... 

tau, pri_act_n 
Ψ=τ and nth actual prior 
theta_1, pri_act_1 
Ψ=θ_1 and first actual prior 
... 

theta_k, pri_act_n 
Ψ=θ_k and nth actual prior 
theta_new, pri_act_1 
Ψ=θ_{new} and first actual prior 
... 

theta_new, pri_act_n 
Ψ=θ_{new} and nth actual prior 
A warning message will be displayed if one of the parameters m_inf
or M_inf
has a value larger than 5*10^6, since this may lead to numerical problems
in the bayesmeta
function
used for computation of the marginal posteriors.
If the integralbased method is used to compute Hellinger distances (H.dist.method = "integral"
),
numerical problems may occur in some cases, which may lead to implausible outputs.
Therefore, we generally recommend to doublecheck the results of the integralbased method using the momentbased method (H.dist.method = "moment"
)  especially if the former results are implausibe. If large differences between the two methods are observed, we recommend to rely on the momentbased method unless a normal approximation of the involved densities is inappropriate.
Ott, M., Plummer, M., Roos, M. Supplementary Material: How vague is vague? How informative is informative? Reference analysis for Bayesian metaanalysis. Revised for Statistics in Medicine. 2021.
Ott, M., Plummer, M., Roos, M. How vague is vague? How informative is informative? Reference analysis for Bayesian metaanalysis. Manuscript revised for Statistics in Medicine. 2021.
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