sensitivity_learning_table_flexible: Posterior, sensitivity and learning estimates for Bayesian...

In sl4bayesmeta: Sensitivity and learning for Bayesian meta-analysis

Description

Generates a table containing posterior, sensitivity and learning estimates and effective RLMC values for a meta-analysis data set with effect expressed as log-odds ratios or log-odds. Is a generalized version of the sensitivity_learning_table() function, which enables a flexible choice of multiple parameters that are fixed in the latter function. This function supports computation of the reference within-study standard deviation based on both the geometric mean (the default) and a weighted harmonic mean. Assumes a Bayesian normal-normal hierarchical model with different priors (half-normal, exponential, half-Cauchy and Lomax) for the between-study standard deviation. These priors are either tail_alpha_static-tail adjusted or adjusted to a target RLMC value (with respect to a tail probability tail_alpha_dynamic).

Usage

sensitivity_learning_table_flexible(df, tail_alpha_static=0.04550026, 
                                    U1=1, U2=2, 
                                    tail_alpha_dynamic=0.5, 
                                    rlmc1=0.25, rlmc2=0.5,
                                    scale_effect="OR",
                                    type_sigma_ref="geometric", 
                                    mu_mean=0, mu_sd=4, 
                                    grid_epsilon=0.00354)

Arguments

df	a data frame containing a column "y" with the estimates of the log-odds (ratios) for the individual studies, a column "sigma" with the corresponding standard errors and a column "labels" with labels for the studies
tail_alpha_static	tail probability for the static heterogeneity prior tail-adjustment
U1	first reference threshold for the static tail-adjustment
U2	second reference threshold for the static tail-adjustment
tail_alpha_dynamic	tail probability for the dynamic heterogeneity prior adjustment based on RLMC. The default 0.5 corresponds to a median-based adjustment.
rlmc1	first target value for RLMC
rlmc2	second target value for RLMC
scale_effect	either "OR" or "logOR". Defaults to "OR". Specifies if the effect estimates are given on the odds (ratio) or log-odds (ratio) scale.
type_sigma_ref	either "geometric" or "harmonic". Defaults to "geometric". Specifies if the geometric mean or a weighted harmonic mean is used to compute the reference standard deviation. See sigma_ref for details.
mu_mean	mean of the normal prior for the effect mu on the log-odds (ratio) scale. Defaults to 0.
mu_sd	standard deviation of the normal prior for the effect mu on the log-odds (ratio) scale. Defaults to 4.
grid_epsilon	step size of the grid for the epsilon-local sensitivity computation. See pri_par_epsilon_grid for details.

Details

The sensitivity measure used is epsilon-local sensitivity (Roos et al. 2015), which is based on the Hellinger distance. Here, learning refers to the ability of the data to modify the prior. It is quantified by the Hellinger distance between the prior and its marginal posterior. See Ott et al. (2020) for a description of the implemented methodology and some examples. Since the sensitivity and learning measures are invariant with respect to monotone transformations of the parameter, the sensitvity and learning estimates do not change if we switch from the odds (ratio) scale to the log-odds (ratio) scale or vice versa.

The shape parameter of the Lomax prior is fixed at 1. The posterior estimates are obtained from the bayesmeta() function in the package bayesmeta.

The default values of the parameters correspond to the values used in the basic
sensitivity_learning_table() function. These values have been chosen as follows: for tail_alpha_static, the default value is based on the half-normal prior with scale parameter 0.5 and the reference threshold U1=1: we have P[HN(0.5)>1]=0.04550026. For the mean and the standard deviation of the normal prior for the effect mu, we use the default values recommended by Roever (2018) for outcomes on the log-odds ratio scale, which are also suitable for the log-odds scale. For grid_epsilon, the default value 0.00354 corresponds to the Hellinger distance between two unit-variance normal distributions with means 0 and 0.01 (Roos et al., 2015).

Value

A matrix with the different estimates in the columns and one row per heterogeneity prior (i.e. the prior for the between-study standard deviation). If scale_effect="OR", the quantities given in the columns are as follows, where OR=exp(mu) denotes the effect on the odds ratio or odds scale and tau the heterogeneity standard deviation:

U	reference threshold for prior adjustment
tail_prob	tail probability for prior adjustment
par_val	scale parameter value of the heterogeneity prior
MRLMC	median relative latent model complexity estimated from MC sampling
median_post_OR	posterior median for the effect OR-exp(mu)
95CrI_post_OR_low	lower end point of the 95 % shortest credible interval (CrI) for the effect OR
95CrI_post_OR_up	upper end point of the 95 % shortest CrI for the effect OR
length_95CrI_post_OR	length of the 95 % shortest CrI for the effect OR
median_post_tau	posterior median for tau
95CrI_post_tau_low	lower end point of the 95 % shortest CrI for tau
95CrI_post_tau_up	upper end point of the 95 % shortest CrI for tau
length_95CrI_post_tau	length of the 95 % shortest CrI for tau
L_mu	learning estimate for the effect mu
L_tau	learning estimate for tau
S_mu	sensitivity estimate for the effect mu
S_tau	sensitivity estimate for tau

If scale_effect="logOR", then the estimates in columns 5-8 and the corresponding column names are adapted as follows (the remaining columns remain unchanged), where mu denotes the effect on the log-odds ratio or log-odds scale:

median_post_mu	posterior median for the effect mu
95CrI_post_mu_low	lower end point of the 95 % shortest credible interval (CrI) for the effect mu
95CrI_post_mu_up	upper end point of the 95 % shortest CrI for the effect mu
length_95CrI_post_mu	length of the 95 % shortest CrI for the effect mu

The following heterogeneity priors are given in the rows (from top to bottom):

HN_U1tail, EXP_U1tail, HC_U1tail, LMX_U1tail	half-normal, exponential, half-Caucy and Lomax prior, all tail_alpha_static-tail adjusted with threshold U=U1 (static)
HN_rlmc1, EXP_rlmc1, HC_rlmc1, LMX_rlmc1	dynamically adjusted with target RLMC=rlmc1 and tail probability tail_alpha_dynamic
HN_U2tail, EXP_U2tail5, HC_U2tail5, LMX_U2tail	tail_alpha_static-tail adjusted with threshold U=U2 (static)
HN_rlmc2, EXP_rlmc2, HC_rlmc2, LMX_rlmc2	dynamically adjusted with target RLMC=rlmc2 and tail probability tail_alpha_dynamic

Warning

This function takes ca. 5-10 minutes to run on the acute graft rejection data set given in the example below.

Note

For effects which are not on the log-odds (ratio) scale, the prior for the effect needs to be adjusted, but otherwise the same code can be used if a Bayesian normal-normal hierarchical model is appropriate.

References

Ott, M., Hunanyan, S., Held, L., Roos, M. Sensitivity quantification in Bayesian meta-analysis. Manuscript revised for Research Synthesis Methods. 2020.

Roever C. Bayesian random-effects meta-analysis using the bayesmeta R package. Journal of Statistical Software (accepted), 2018.

Roos, M., Martins, T., Held, L., Rue, H. (2015). Sensitivity analysis for Bayesian hierarchical models. Bayesian Analysis 10(2), 321–349. https://projecteuclid.org/euclid.ba/1422884977

See Also

bayesmeta in package bayesmeta, pri_par_adjust_static,
pri_par_adjust_dynamic, effective_rlmc, pri_par_epsilon_grid

Examples

# Acute Graft rejection (AGR) data analyzed in Friede et al. (2017),  
# Sect. 3.2, URL: https://doi.org/10.1002/bimj.201500236
# First study: experimental group: 14 cases out of 61; 
# control group: 15 cases out of 20 
# Second study: experimental group: 4 cases out of 36; 
# control group: 11 cases out of 36 
rT <- c(14,4)
nT <- c(61,36)
rC <- c(15,11)
nC <- c(20,36)
df <- data.frame(y = log((rT*(nC-rC))/(rC*(nT-rT))), # log-OR
                 sigma = sqrt(1/rT+1/(nT-rT)+1/rC+1/(nC-rC)), # SE(log-OR)
                 labels = c(1:2))
  
# compute the table for the AGR data with RLMC target values 0.4 and 0.7
# and a weighted harmonic mean to compute the reference sd
# warning: it takes ca. 5-10 minutes to run this function
# on the above data set!
sensitivity_learning_table_flexible(df, rlmc1=0.4, rlmc2=0.7, 
                                    type_sigma_ref="harmonic")

sl4bayesmeta documentation built on Feb. 18, 2020, 3:02 p.m.