BFI: Bayesian Federated Inference
In BFI: Bayesian Federated Inference
bfi R Documentation
Bayesian Federated Inference
Description
bfi function can be used (on the central server) to combine inference results from separate datasets (without combining the data) to approximate what would have been inferred had the datasets been merged. This function can handle linear, logistic and survival regression models.
Usage
bfi(theta_hats = NULL,
A_hats,
Lambda,
family = c("gaussian", "binomial", "survival"),
basehaz = c("weibul", "exp", "gomp", "poly", "pwexp", "unspecified"),
stratified = FALSE,
strat_par = NULL,
center_spec = NULL,
theta_A_polys = NULL,
treat_round = NULL,
for_ATE = NULL,
p,
q_ls,
center_zero_sample = FALSE,
which_cent_zeros,
zero_sample_covs,
refer_cats,
zero_cats,
lev_no_ref_zeros)
Arguments
theta_hats
a list of L vectors of the maximum a posteriori (MAP) estimates of the model parameters in the L centers. These vectors must have equal dimensions. See ‘Details’.
A_hats
a list of L minus curvature matrices for L centers. These matrices must have equal dimensions. See ‘Details’.
Lambda
a list of L+1 matrices. The k^{\th} matrix is the chosen inverse variance-covariance matrix of the Gaussian distribution that is used as prior distribution in center k, where k=1,2,\ldots,L. The last matrix is the chosen variance-covariance matrix for the Gaussian prior of the (fictive) combined data set.
If stratified = FALSE, all L+1 matrices must have equal dimensions. While, if stratified = TRUE, the first L matrices must have equal dimensions and the last matrix should have a different (greater) dimention than the others. See ‘Details’.
family
a character string representing the family name used for the local centers. Can be abbreviated.
basehaz
a character string representing one of the available baseline hazard functions; exponential ("exp"), Weibull ("weibul", the default), Gompertz ("gomp"), exponentiated polynomial ("poly"), piecewise exponential ("pwexp"), and unspecified baseline hazard ("unspecified"). It is only used when family = "survival". Can be abbreviated. If basehaz = "unspecified", it means that a (semi-parametric) Cox model is considered, and the parameters (regression coefficients) are estimated using the partial log-likelihood.
stratified
logical flag for performing the stratified analysis. If stratified = TRUE, the parameter(s) selected in the strat_par argument are allowed to be different across centers (to deal with heterogeneity across centers), except when the argument center_spec is not NULL. Default is stratified = FALSE. See ‘Details’ and ‘Examples’.
strat_par
an integer vector for indicating the stratification parameter(s). It can be used to deal with heterogeneity due to center-specific parameters.
For the "binomial" and "gaussian" families it is a one- or two-element integer vector so that the values 1 and/or 2 are/is used to indicate that the “intercept” and/or “sigma2” are allowed to vary, respectively. For the "binomial" family the length of the vector should be at most one which refers to “intercept”, and the value of this element should be 1 (to handel heterogeneity across outcome means). For "gaussian" this vector can be 1 for indicating the “intercept” only (handeling heterogeneity across outcome means), 2 for indicating the “sigma2” only (handeling heterogeneity due to nuisance parameter), and c(1, 2) for both “intercept” and “sigma2”. When family = "survival", this vector can contain any combination of values ranging from 1 to the maximum number of parameters of the baseline hazard function, i.e., 1 for "exp", 2 for "weibul" and "gomp", max_order + 1 for "poly", and n_intervals for "pwexp".
For example, for "weibul", strat_par could be 1, 2 or c(1, 2), where 1 represents \omega_1 and 2 represents \omega_2.
This argument is used only when stratified = TRUE and center_spec = NULL. Default is strat_par = NULL. See ‘Details’ and ‘Examples’.
center_spec
a vector of L elements to account for the heterogeneity across centers due to clustering. This argument is used only when stratified = TRUE and strat_par = NULL. Each element represents a specific feature of the corresponding center. There must be only one specific value or attribute for each center. This vector could be a numeric, characteristic or factor vector. Note that, the order of the centers in the vector center_spec must be the same as in the list of the argument theta_hats.
The used data type in the argument center_spec must be categorical. Default is center_spec = NULL. See also ‘Details’ and ‘Examples’.
theta_A_polys
a list with L elements so that each element is the array theta_A_ploy (the output of the MAP.estimation function, MAP.estimation()$theta_A_ploy) for the corresponding center. This argument, theta_A_polys, is only used if family = "survival" and basehaz = "poly". See ‘Details’ and ‘Examples’.
treat_round
a character string representing the "first" and "second" rounds of estimating the treatment effect.
for_ATE
a list of L vectors of 9 elements to calculate the average treatment effects (ATEs) only for the binomial and gaussian families. These vectors must have equal dimensions. If treat_round = "first", then for_ATE must be NULL. If treat_round = "second", then for_ATE must be a list for binomial and gaussian, while for survival, for_ATE must be NULL. It should be defined using the output of MAP.estimation()$for_ATE obtained from the first round. See ‘Details’ and ‘Examples’.
p
an integer representing the number of covariates/coefficients. It can be found from the output of the MAP.estimation function, MAP.estimation()$np). This argument, p, is only used if stratified = TRUE and family = "survival".
q_ls
a vector with L elements in which each element is the order (minus 1) of the exponentiated polynomial baseline hazard function for the corresponding center, i.e., each element is the value of q_l (the output of the MAP.estimation function, MAP.estimation()$q_l). This argument, q_ls, is only used if family = "survival", family = "survival" and basehaz = "poly". It can also be a scalar which represents the maximum value of the q_l's across the centers.
center_zero_sample
logical flag indicating whether the center has a categorical covariate with no observations/individuals in one of the categories. It is used to address heterogeneity across centers due to center-specific covariates. Default is center_zero_sample = FALSE. For more detailes see ‘References’.
which_cent_zeros
an integer vector representing the center(s) which has one categorical covariate with no individuals in one of the categories. It is used if center_zero_sample = TRUE.
zero_sample_covs
a vector in which each element is a character string representing the categorical covariate that has no samples/observations in one of its categories for the corresponding center. Each element of the vector can be obtained from the output of the MAP.estimation function for the corresponding center, MAP.estimation()$zero_sample_cov. It is used when center_zero_sample = TRUE.
refer_cats
a vector in which each element is a character string representing the reference category for the corresponding center. Each element of the vector can be obtained from the output of the MAP.estimation function for the corresponding center, MAP.estimation()$refer_cat. This vector is used when center_zero_sample = TRUE.
zero_cats
a vector in which each element is a character string representing the category with no samples/observations for the corresponding center. Each element of the vector can be obtained from the output of the MAP.estimation function for the corresponding center, i.e., MAP.estimation()$zero_cat. It is used when center_zero_sample = TRUE.
lev_no_ref_zeros
a list in which the number of elements equals the length of the which_cent_zeros argument. Each element of the list is a vector containing the names of the levels of the categorical covariate that has no samples/observations in one of its categories for the corresponding center. However, the name of the category with no samples and the name of the reference category are excluded from this vector. Each element of the list can be obtained from the output of the MAP.estimation function, i.e., MAP.estimation()$lev_no_ref_zero.
This argument is used if center_zero_sample = TRUE.
Details
bfi function implements the BFI approach described in the papers Jonker et. al. (2024a), Pazira et. al. (2024) and Jonker et. al. (2024b) given in the references.
The inference results gathered from different (L) centers are combined, and the BFI estimates of the model parameters and curvature matrix evaluated at that point are returned.
The inference result from each center must be obtained using the MAP.estimation function separately, and then all of these results (coming from different centers) should be compiled into a list to be used as an input of bfi().
The models in the different centers should be defined in exactly the same way; among others, exactly the same covariates should be included in the models. The parameter vectors should be defined exactly the same, so that the L vectors and matrices in the input lists theta_hat's and A_hat's are defined in the same way (e.g., the covariates need to be included in the models in the same order).
Note that the order of the elements in the lists theta_hats, A_hats and Lambda, must be the same with respect to the centers, so that in every list the element at the \ell^{\th} position is from the center \ell. This should also be the case for the vector center_spec.
If for the locations intercept = FALSE, the stratified analysis is not possible anymore for the binomial family.
If stratified = FALSE, both strat_par and center_spec must be NULL (the defaults), while if stratified = TRUE only one of the two must be NULL.
If stratified = FALSE and all the L+1 matrices in Lambda are equal, it is sufficient to give a (list of) one matrix only.
In both cases of the stratified argument (TRUE or FALSE), if only the first L matrices are equal, the argument Lambda can be a list of two matrices, so that the fist matrix represents the chosen variance-covariance matrix for local centers and the second one is the chosen matrix for the combined data set.
The last matrix of the list in the argument Lambda can be built by the function inv.prior.cov().
If the data type used in the argument center_spec is continuous or categorical with the number of categories equal to the number of centers, one can use stratified = TRUE and center_spec = NULL, and set strat_par not to NULL (i.e., to 1, 2 or both (1, 2)). Indeed, in this case, the stratification parameter(s) given in the argument strat_par are assumed to be different across the centers.
When family = 'survival' and basehaz = 'poly', the arguments theta_hats and A_hats should not be provided. Instead, the theta_A_polys and q_ls arguments should be defined using the local information, specifically MAP.estimation()$theta_A_poly and MAP.estimation()$q_l, respectively. See Example 3 in ‘Examples’.
For estimating the treatment effect, in the first round (treat_round = "first"), the argument for_ATE must be NULL (the default) and the family must be set to binomial (family is handled automatically.)
Value
bfi returns a list containing the following components:
theta_hat
the vector of estimates obtained by combining the inference results from the L centers with the 'BFI' methodology. If an intercept was fitted in every center and stratified = FALSE, there is only one general “intercept” in this vector, while if stratified = TRUE and strat_par = 1, there are L different intercepts in the model, for each center one. If treatment is not 'NULL', when treat_round = 'first', theta_hat gives \hat{\boldsymbol{\gamma}}_{BFI}, and when treat_round = 'second', theta_hat is the treatment effect \hat \zeta_{BFI};
A_hat
minus the curvature (or Hessian) matrix obtained by the 'BFI' method for the combined model. If stratified = TRUE, the dimension of the matrix is always greater than when stratified = FALSE;
sd
the vector of (posterior) standard deviation of the estimates in theta_hat obtained from the matrix in A_hat, i.e., the vector equals sqrt(diag(solve(A_hat))) which equals the square root of the elements at the diagonal of the inverse of the A_hat matrix.
family
the family object used;
basehaz
the baseline hazard function used;
stratified
whether a stratified analysis was done or not;
strat_par
the stratification parameter(s) used;
Ave_Treat
the estimates of the average treatment effect. Two diffterent estimations (IPTW and wIPTW) if the family is gaussian or binomial, and for the survival family it is 'NULL'. For more detailes see ‘References’.
Author(s)
Hassan Pazira and Marianne Jonker
Maintainer: Hassan Pazira hassan.pazira@radboudumc.nl
References
Jonker M.A., Pazira H. and Coolen A.C.C. (2024a). Bayesian federated inference for estimating statistical models based on non-shared multicenter data sets, Statistics in Medicine, 43(12): 2421-2438. <https://doi.org/10.1002/sim.10072>
Pazira H., Massa E., Weijers J.A.M., Coolen A.C.C. and Jonker M.A. (2025b). Bayesian Federated Inference for Survival Models, Journal of Applied Statistics (Accepted). <https://arxiv.org/abs/2404.17464>
Jonker M.A., Pazira H. and Coolen A.C.C. (2025a). Bayesian Federated Inference for regression models based on non-shared medical center data, Research Synthesis Methods, 1-41. <https://doi.org/10.1017/rsm.2025.6>
See Also
MAP.estimation and inv.prior.cov
Examples
#################################################
## Example 1: y ~ Binomial (L = 2 centers) ##
#################################################
# Setting a seed for reproducibility
set.seed(112358)
#------------------------------------#
# Data Simulation for Local Center 1 #
#------------------------------------#
n1 <- 30 # sample size of center 1
X1 <- data.frame(x1=rnorm(n1), # continuous variable
x2=sample(0:2, n1, replace=TRUE)) # categorical variable
# make dummy variables
X1x2_1 <- ifelse(X1$x2 == '1', 1, 0)
X1x2_2 <- ifelse(X1$x2 == '2', 1, 0)
X1$x2 <- as.factor(X1$x2)
# regression coefficients
beta <- 1:4 # beta[1] is the intercept
# linear predictor:
eta1 <- beta[1] + X1$x1 * beta[2] + X1x2_1 * beta[3] + X1x2_2 * beta[4]
# inverse of the link function ( g^{-1}(\eta) = \mu ):
mu1 <- binomial()$linkinv(eta1)
y1 <- rbinom(n1, 1, mu1)
#------------------------------------#
# Data Simulation for Local Center 2 #
#------------------------------------#
n2 <- 50 # sample size of center 2
X2 <- data.frame(x1=rnorm(n2), # continuous variable
x2=sample(0:2, n2, replace=TRUE)) # categorical variable
# make dummy variables:
X2x2_1 <- ifelse(X2$x2 == '1', 1, 0)
X2x2_2 <- ifelse(X2$x2 == '2', 1, 0)
X2$x2 <- as.factor(X2$x2)
# linear predictor:
eta2 <- beta[1] + X2$x1 * beta[2] + X2x2_1 * beta[3] + X2x2_2 * beta[4]
# inverse of the link function:
mu2 <- binomial()$linkinv(eta2)
y2 <- rbinom(n2, 1, mu2)
#---------------------------#
# MAP Estimates at Center 1 #
#---------------------------#
# Assume the same inverse covariance matrix (Lambda) for both centers:
Lambda <- inv.prior.cov(X1, lambda = 0.01, family = 'binomial')
fit1 <- MAP.estimation(y1, X1, family = 'binomial', Lambda)
theta_hat1 <- fit1$theta_hat # intercept and coefficient estimates
A_hat1 <- fit1$A_hat # minus the curvature matrix
#---------------------------#
# MAP Estimates at Center 2 #
#---------------------------#
fit2 <- MAP.estimation(y2, X2, family='binomial', Lambda)
theta_hat2 <- fit2$theta_hat
A_hat2 <- fit2$A_hat
#-----------------------#
# BFI at Central Server #
#-----------------------#
theta_hats <- list(theta_hat1, theta_hat2)
A_hats <- list(A_hat1, A_hat2)
bfi <- bfi(theta_hats, A_hats, Lambda, family='binomial')
class(bfi)
summary(bfi, cur_mat=TRUE)
###---------------------###
### Stratified Analysis ###
###---------------------###
# By running the following line an error appears because
# when stratified = TRUE, both 'strat_par' and 'center_spec' can not be NULL:
Just4check1 <- try(bfi(theta_hats, A_hats, Lambda, family = 'binomial',
stratified = TRUE), TRUE)
class(Just4check1) # By default, both 'strat_par' and 'center_spec' are NULL!
# By running the following line an error appears because when stratified = TRUE,
# last matrix in 'Lambda' should not have the same dim. as the other local matrices:
Just4check2 <- try(bfi(theta_hats, A_hats, Lambda, stratified = TRUE,
strat_par = 1), TRUE)
class(Just4check2) # All matices in Lambda have the same dimension!
# Stratified analysis when 'intercept' varies across two centers:
newLam <- inv.prior.cov(X1, lambda=c(0.1, 0.3), family = 'binomial',
stratified = TRUE, strat_par = 1)
bfi <- bfi(theta_hats, A_hats, list(Lambda, newLam), family = 'binomial',
stratified=TRUE, strat_par=1)
summary(bfi, cur_mat=TRUE)
###---------------------###
### Treatment Effect ###
###---------------------###
set.seed(112358)
#------------------------------------#
# Data Simulation for Local Center 1 #
#------------------------------------#
n1 <- 30 # sample size of center 1
X1 <- data.frame(x1=rnorm(n1), # continuous variable
treatment=sample(1:2, n1, replace=TRUE)) # categorical variable
X1$treatment <- as.factor(X1$treatment)
# regression coefficients
beta <- 1:3 # beta[1] is the intercept
# make dummy variable
X1x2_2 <- ifelse(X1$treatment == '2', 1, 0)
# linear predictor:
eta1 <- beta[1] + X1$x1 * beta[2] + X1x2_2 * beta[3]
# inverse of the link function ( g^{-1}(\eta) = \mu ):
mu1 <- binomial()$linkinv(eta1)
y1 <- rbinom(n1, 1, mu1)
#------------------------------------#
# Data Simulation for Local Center 2 #
#------------------------------------#
n2 <- 50 # sample size of center 2
X2 <- data.frame(x1=rnorm(n2), # continuous variable
treatment=sample(1:2, n2, replace=TRUE)) # categorical variable
X2$treatment <- as.factor(X2$treatment)
# make dummy variables:
X2x2_2 <- ifelse(X2$treatment == '2', 1, 0)
# linear predictor:
eta2 <- beta[1] + X2$x1 * beta[2] + X2x2_2 * beta[3]
# inverse of the link function:
mu2 <- binomial()$linkinv(eta2)
y2 <- rbinom(n2, 1, mu2)
# The algorithm works even if the order of the covariates are not
# the same across centers
X2 <- X2[,c("treatment","x1")]
#-----------------------#
# Observational data #
#-----------------------#
# For observational data (RWD), we need two rounds for estimating treatment effect:
#-------------#
# First Round #
#-------------#
## Center 1:
Lambda1 <- inv.prior.cov(X1, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="first")
fit1_r1 <- MAP.estimation(y1, X1, family = 'binomial', Lambda = Lambda1,
treatment = "treatment", treat_round = "first")
# In the first round, the output is without the treatment!
summary(fit1_r1)
## Center 2:
Lambda2 <- inv.prior.cov(X2, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="first")
fit2_r1 <- MAP.estimation(y2, X2, family = 'binomial', Lambda = Lambda2,
treatment = "treatment", treat_round = "first")
fit2_r1
## Centeral Server:
theta_hats_r1 <- list(fit1_r1$theta_hat, fit2_r1$theta_hat)
A_hats_r1 <- list(fit1_r1$A_hat, fit2_r1$A_hat)
fitbfi_r1 <- bfi(theta_hats_r1, A_hats_r1, Lambda1, family = 'binomial',
treat_round = "first")
summary(fitbfi_r1, cur_mat = TRUE)
#--------------#
# Second Round #
#--------------#
## Center 1:
Lambda11 <- inv.prior.cov(X1, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="second")
fit1_r2 <- MAP.estimation(y1, X1, family = 'binomial', Lambda = Lambda11,
treatment = "treatment", treat_round = "second",
gamma_bfi = fitbfi_r1$theta_hat)
# In the second round, the output is only with the treatment!
summary(fit1_r2)
## Center 2:
Lambda22 <- inv.prior.cov(X2, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="second")
fit2_r2 <- MAP.estimation(y2, X2, family = 'binomial', Lambda = Lambda22,
treatment = "treatment", treat_round = "second",
gamma_bfi = fitbfi_r1$theta_hat)
fit2_r2$propensity # Propensity Score
fit2_r2$for_ATE # will be used in central server
fit2_r2
## Centeral Server:
theta_hats_r2 <- list(fit1_r2$theta_hat, fit2_r2$theta_hat)
A_hats_r2 <- list(fit1_r2$A_hat, fit2_r2$A_hat)
for_ATEs <- list(fit1_r2$for_ATE, fit2_r2$for_ATE)
fitbfi_r2 <- bfi(theta_hats_r2, A_hats_r2, Lambda11, family = 'binomial',
treat_round = "second", for_ATE = for_ATEs)
fitbfi_r2$S_var
fitbfi_r2$Ave_Treat
summary(fitbfi_r2)
#--------------------#
# Randomized Trial #
#--------------------#
# For Randomized Control Trial (RCT), we need only one round (the second round) for
# estimating treatment effect. Because we do not need to estimate propensity score.
# For example, in a 1:1 randomized trial, the propensity scores are, by definition,
# equal to 0.5. Here we use 'RCT_propens', instead of 'gamma_bfi':
## Center 1:
Lambda11 <- inv.prior.cov(X1, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="second")
fit1_r2 <- MAP.estimation(y1, X1, family = 'binomial', Lambda = Lambda11,
treatment = "treatment", treat_round = "second",
RCT_propens = rep(0.5, n1)) # gamma_bfi = NULL
summary(fit1_r2)
## Center 2:
Lambda22 <- inv.prior.cov(X2, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="second")
fit2_r2 <- MAP.estimation(y2, X2, family = 'binomial', Lambda = Lambda22,
treatment = "treatment", treat_round = "second",
RCT_propens = rep(0.5, n2)) # gamma_bfi = NULL
fit2_r2$for_ATE # will be used in central server
fit2_r2
## Centeral Server:
theta_hats_r2 <- list(fit1_r2$theta_hat, fit2_r2$theta_hat)
A_hats_r2 <- list(fit1_r2$A_hat, fit2_r2$A_hat)
for_ATEs <- list(fit1_r2$for_ATE, fit2_r2$for_ATE)
fitbfi_r2 <- bfi(theta_hats_r2, A_hats_r2, Lambda11, family = 'binomial',
treat_round = "second", for_ATE = for_ATEs)
fitbfi_r2$S_var
fitbfi_r2$Ave_Treat
summary(fitbfi_r2)
#################################################
## Example 2: y ~ Gaussian (L = 3 centers) ##
#################################################
# Setting a seed for reproducibility
set.seed(112358)
p <- 3 # number of coefficients without 'intercept'
theta <- c(1, rep(2, p), 1.5) # reg. coef.s ('intercept' is 1) & 'sigma2' = 1.5
#------------------------------------#
# Data Simulation for Local Center 1 #
#------------------------------------#
n1 <- 30 # sample size of center 1
X1 <- data.frame(matrix(rnorm(n1 * p), n1, p)) # continuous variables
# linear predictor:
eta1 <- theta[1] + as.matrix(X1)
# inverse of the link function ( g^{-1}(\eta) = \mu ):
mu1 <- gaussian()$linkinv(eta1)
y1 <- rnorm(n1, mu1, sd = sqrt(theta[5]))
#------------------------------------#
# Data Simulation for Local Center 2 #
#------------------------------------#
n2 <- 40 # sample size of center 2
X2 <- data.frame(matrix(rnorm(n2 * p), n2, p)) # continuous variables
# linear predictor:
eta2 <- theta[1] + as.matrix(X2)
# inverse of the link function:
mu2 <- gaussian()$linkinv(eta2)
y2 <- rnorm(n2, mu2, sd = sqrt(theta[5]))
#------------------------------------#
# Data Simulation for Local Center 3 #
#------------------------------------#
n3 <- 50 # sample size of center 3
X3 <- data.frame(matrix(rnorm(n3 * p), n3, p)) # continuous variables
# linear predictor:
eta3 <- theta[1] + as.matrix(X3)
# inverse of the link function:
mu3 <- gaussian()$linkinv(eta3)
y3 <- rnorm(n3, mu3, sd = sqrt(theta[5]))
#---------------------------#
# Inverse Covariance Matrix #
#---------------------------#
# Creating the inverse covariance matrix for the Gaussian prior distribution:
# the same for both centers
Lambda <- inv.prior.cov(X1, lambda = 0.05, family='gaussian')
#---------------------------#
# MAP Estimates at Center 1 #
#---------------------------#
fit1 <- MAP.estimation(y1, X1, family = 'gaussian', Lambda)
theta_hat1 <- fit1$theta_hat # intercept and coefficient estimates
A_hat1 <- fit1$A_hat # minus the curvature matrix
#---------------------------#
# MAP Estimates at Center 2 #
#---------------------------#
fit2 <- MAP.estimation(y2, X2, family = 'gaussian', Lambda)
theta_hat2 <- fit2$theta_hat
A_hat2 <- fit2$A_hat
#---------------------------#
# MAP Estimates at Center 3 #
#---------------------------#
fit3 <- MAP.estimation(y3, X3, family = 'gaussian', Lambda)
theta_hat3 <- fit3$theta_hat
A_hat3 <- fit3$A_hat
#-----------------------#
# BFI at Central Server #
#-----------------------#
A_hats <- list(A_hat1, A_hat2, A_hat3)
theta_hats <- list(theta_hat1, theta_hat2, theta_hat3)
bfi <- bfi(theta_hats, A_hats, Lambda, family = 'gaussian')
summary(bfi, cur_mat=TRUE)
###---------------------###
### Stratified Analysis ###
###---------------------###
# Stratified analysis when 'intercept' varies across two centers:
newLam1 <- inv.prior.cov(X1, lambda = c(0.1,0.3), family = 'gaussian',
stratified = TRUE, strat_par = 1, L = 3)
# 'newLam1' is used as the prior for combined data and
# 'Lambda' is used as the prior for locals
list_newLam1 <- list(Lambda, newLam1)
bfi1 <- bfi(theta_hats, A_hats, list_newLam1, family = 'gaussian',
stratified = TRUE, strat_par = 1)
summary(bfi1, cur_mat = TRUE)
# Stratified analysis when 'sigma2' varies across two centers:
newLam2 <- inv.prior.cov(X1, lambda = c(0.1,0.3), family = 'gaussian',
stratified = TRUE, strat_par = 2, L = 3)
# 'newLam2' is used as the prior for combined data and 'Lambda' is used as
# the prior for locals
list_newLam2 <- list(Lambda, newLam2)
bfi2 <- bfi(theta_hats, A_hats, list_newLam2, family = 'gaussian',
stratified = TRUE, strat_par=2)
summary(bfi2, cur_mat = TRUE)
# Stratified analysis when 'intercept' and 'sigma2' vary across 2 centers:
newLam3 <- inv.prior.cov(X1, lambda = c(0.1,0.2,0.3), family = 'gaussian',
stratified = TRUE, strat_par = c(1, 2), L = 3)
# 'newLam3' is used as the prior for combined data and 'Lambda' is used as
# the prior for locals
list_newLam3 <- list(Lambda, newLam3)
bfi3 <- bfi(theta_hats, A_hats, list_newLam3, family = 'gaussian',
stratified = TRUE, strat_par = 1:2)
summary(bfi3, cur_mat = TRUE)
###----------------------------###
### Center Specific Covariates ###
###----------------------------###
# Assume the first and third centers have the same center-specific covariate value
# of 'High', while this value for the second center is 'Low', i.e.,
# center_spec = c('High','Low','High')
newLam4 <- inv.prior.cov(X1, lambda=c(0.1, 0.2, 0.3), family='gaussian',
stratified = TRUE, center_spec = c('High','Low','High'),
L = 3)
# 'newLam4' is used as the prior for combined data and 'Lambda' is used as
# the prior for locals
l_newLam4 <- list(Lambda, newLam4)
bfi4 <- bfi(theta_hats, A_hats, l_newLam4, family = 'gaussian',
stratified = TRUE, center_spec = c('High','Low','High'))
summary(bfi4, cur_mat = TRUE)
###---------------------###
### Treatment Effect ###
###---------------------###
set.seed(112358)
#-----------------------------#
# New Data for Local Center 1 #
#-----------------------------#
# Generating new data with 'treatment' variable
# We cansider the first variable (X1$X1) to be the treatment:
X1$X1 <- sample(0:1, n1, replace=TRUE) # categorical variable
eta1 <- theta[1] + as.matrix(X1)
mu1 <- gaussian()$linkinv(eta1)
y1 <- rnorm(n1, mu1, sd = sqrt(theta[5]))
#-----------------------------#
# New Data for Local Center 2 #
#-----------------------------#
# We cansider the first variable (X2$X1) to be the treatment:
X2$X1 <- sample(0:1, n2, replace=TRUE) # categorical variable
eta2 <- theta[1] + as.matrix(X2)
mu2 <- gaussian()$linkinv(eta2)
y2 <- rnorm(n2, mu2, sd = sqrt(theta[5]))
#-----------------------------#
# New Data for Local Center 3 #
#-----------------------------#
# We cansider the first variable (X3$X1) to be the treatment:
X3$X1 <- sample(0:1, n3, replace=TRUE) # categorical variable
# linear predictor:
eta3 <- theta[1] + as.matrix(X3)
# inverse of the link function:
mu3 <- gaussian()$linkinv(eta3)
y3 <- rnorm(n3, mu3, sd = sqrt(theta[5]))
#-----------------------#
# Observational data #
#-----------------------#
# For observational data (RWD), we need two rounds for estimating treatment effect:
#-------------#
# First Round #
#-------------#
## Center 1:
Lambda1 <- inv.prior.cov(X1, lambda = 0.01, family = 'binomial',
treatment = "X1", treat_round="first")
# When treat_round = "first", the family will automatically set to 'binomial',
# even if family = 'gaussian' or family = 'survival'.
fit1_r1 <- MAP.estimation(y1, X1, family = 'gaussian', Lambda = Lambda1,
treatment = "X1", treat_round = "first")
# Althghou family = 'gaussian', the output is based on 'binomial'!
# The output without the treatment (X1) in the first round!
summary(fit1_r1)
## Center 2:
Lambda2 <- inv.prior.cov(X2, lambda = 0.01, family = 'gaussian',
treatment = "X1", treat_round="first")
fit2_r1 <- MAP.estimation(y2, X2, family = 'gaussian', Lambda = Lambda2,
treatment = "X1", treat_round = "first")
fit2_r1
## Center 3:
Lambda3 <- inv.prior.cov(X3, lambda = 0.01, family = 'gaussian',
treatment = "X1", treat_round="first")
fit3_r1 <- MAP.estimation(y3, X3, family = 'gaussian', Lambda = Lambda3,
treatment = "X1", treat_round = "first")
## Centeral Server:
theta_hats_r1 <- list(fit1_r1$theta_hat, fit2_r1$theta_hat, fit3_r1$theta_hat)
A_hats_r1 <- list(fit1_r1$A_hat, fit2_r1$A_hat, fit3_r1$A_hat)
fitbfi_r1 <- bfi(theta_hats_r1, A_hats_r1, Lambda1, family = 'gaussian',
treat_round = "first") # same results with 'binomial'
# The output without the treatment (X1) in the first round!
summary(fitbfi_r1, cur_mat = TRUE)
#--------------#
# Second Round #
#--------------#
## Center 1:
Lambda11 <- inv.prior.cov(X1, lambda = 0.01, family = 'gaussian',
treatment = "X1", treat_round="second")
fit1_r2 <- MAP.estimation(y1, X1, family = 'gaussian', Lambda = Lambda11,
treatment = "X1", treat_round = "second",
gamma_bfi = fitbfi_r1$theta_hat)
# The output with only the treatment (X1) in the second round!
summary(fit1_r2)
## Center 2:
Lambda22 <- inv.prior.cov(X2, lambda = 0.01, family = 'gaussian', treatment = "X1",
treat_round="second")
fit2_r2 <- MAP.estimation(y2, X2, family = 'gaussian', Lambda = Lambda22,
treatment = "X1", treat_round = "second",
gamma_bfi = fitbfi_r1$theta_hat)
## Center 3:
Lambda33 <- inv.prior.cov(X3, lambda = 0.01, family = 'gaussian', treatment = "X1",
treat_round="second")
fit3_r2 <- MAP.estimation(y3, X3, family = 'gaussian', Lambda = Lambda33,
treatment = "X1", treat_round = "second",
gamma_bfi = fitbfi_r1$theta_hat)
## Centeral Server:
theta_hats_r2 <- list(fit1_r2$theta_hat, fit2_r2$theta_hat, fit3_r2$theta_hat)
A_hats_r2 <- list(fit1_r2$A_hat, fit2_r2$A_hat, fit3_r2$A_hat)
for_ATEs <- list(fit1_r2$for_ATE, fit2_r2$for_ATE, fit3_r2$for_ATE)
fitbfi_r2 <- bfi(theta_hats_r2, A_hats_r2, Lambda11, family = 'gaussian',
treat_round = "second", for_ATE = for_ATEs)
fitbfi_r2$Ave_Treat
fitbfi_r2$S_var
summary(fitbfi_r2)
####################################################
## Example 3: Survival family (L = 2 centers) ##
####################################################
# Setting a seed for reproducibility
set.seed(112358)
p <- 3
theta <- c(1:4, 5, 6) # regression coefficients (1:4) & omega's (5:6)
#---------------------------------------------#
# Simulating Survival data for Local Center 1 #
#---------------------------------------------#
n1 <- 30
X1 <- data.frame(matrix(rnorm(n1 * p), n1, p)) # continuous (normal) variables
# Simulating survival data ('time' and 'status') from 'Weibull' with
# a predefined censoring rate of 0.3:
y1 <- surv.simulate(Z = list(X1), beta = theta[1:p], a = theta[5],
b = theta[6], u1 = 0.1, cen_rate = 0.3,
gen_data_from = "weibul")$D[[1]][, 1:2]
## MAP Estimates at Center 1
Lambda <- inv.prior.cov(X1, lambda = c(0.1, 1), family = "survival",
basehaz = "poly")
fit1 <- MAP.estimation(y1, X1, family = 'survival', Lambda = Lambda,
basehaz = "poly")
theta_hat1 <- fit1$theta_hat # coefficient estimates
A_hat1 <- fit1$A_hat # minus the curvature matrix
summary(fit1, cur_mat=TRUE)
fit1$theta_A_poly # Only when family = "survival" and basehaz ="poly"
#---------------------------------------------#
# Simulating Survival data for Local Center 2 #
#---------------------------------------------#
n2 <- 30
X2 <- data.frame(matrix(rnorm(n2 * p), n2, p)) # continuous (normal) variables
# Survival simulated data from 'Weibull' with a predefined censoring rate of 0.3:
y2 <- surv.simulate(Z = list(X2), beta = theta[1:p], a = theta[5],
b = theta[6],u1 = 0.1, cen_rate = 0.3,
gen_data_from = "weibul")$D[[1]][, 1:2]
## MAP Estimates at Center 2
fit2 <- MAP.estimation(y2, X2, family = 'survival', Lambda = Lambda,
basehaz = "poly")
theta_hat2 <- fit2$theta_hat
A_hat2 <- fit2$A_hat
summary(fit2, cur_mat=TRUE)
#-----------------------#
# BFI at Central Server #
#-----------------------#
# When family = 'survival' and basehaz = "poly", only 'theta_A_polys'
# should be defined instead of 'theta_hats' and 'A_hats':
theta_A_hats <- list(fit1$theta_A_poly, fit2$theta_A_poly)
qls <- c(fit1$q_l, fit2$q_l)
bfi <- bfi(Lambda = Lambda, family = 'survival', theta_A_polys = theta_A_hats,
basehaz = "poly", q_ls = qls)
summary(bfi, cur_mat=TRUE)
###---------------------###
### Stratified Analysis ###
###---------------------###
# Stratified analysis when first parameter ('omega_0') varies across two centers:
(newLam0 <- inv.prior.cov(X1, lambda = c(rep(1, 3), 0.3, 0.7, rep(2,2)),
family = 'survival', stratified = TRUE,
basehaz = c("poly"), strat_par = 1, L = 2))
# 'newLam0' is used as the prior for combined data and 'Lambda' is used as for locals:
list_newLam0 <- list(Lambda, newLam0)
bfi0 <- bfi(Lambda = list_newLam0, family = 'survival', theta_A_polys = theta_A_hats,
stratified = TRUE, basehaz = c("poly"), p = 3, q_ls = qls, strat_par = 1)
summary(bfi0, cur_mat = TRUE)
# Stratified analysis when the first and second parameters ('omega_0' and 'omega_1')
# vary across two centers:
newLam1 <- inv.prior.cov(X1, lambda = c(rep(1, 3), 0.3, 0.7, 0.5, 0.8, 2),
family = 'survival', stratified = TRUE, basehaz = c("poly"),
strat_par = c(1, 2), L = 2)
# 'newLam1' is used as the prior for combined data:
list_newLam1 <- list(Lambda, newLam1)
bfi1 <- bfi(Lambda = list_newLam1, family = 'survival', theta_A_polys = theta_A_hats,
stratified = TRUE, basehaz = c("poly"), p = 3, q_ls = qls,
strat_par = c(1, 2))
summary(bfi1, cur_mat = TRUE)
###---------------------###
### Treatment Effect ###
###---------------------###
set.seed(112358)
#-----------------------------#
# New Data for Local Center 1 #
#-----------------------------#
# Generating new data with 'treatment' variable
# We cansider the first variable (X1$X1) to be the treatment
X1$X1 <- sample(0:1, n1, replace=TRUE) # categorical variable
y1 <- surv.simulate(Z = list(X1), beta = theta[1:p], a = theta[5], b = theta[6],
u1 = 0.1, cen_rate = 0.3, gen_data_from = "weibul")$D[[1]][, 1:2]
#-----------------------------#
# New Data for Local Center 2 #
#-----------------------------#
# We cansider the first variable (X2$X1) to be the treatment!
X2$X1 <- sample(0:1, n2, replace=TRUE) # categorical variable
y2 <- surv.simulate(Z = list(X2), beta = theta[1:p], a = theta[5], b = theta[6],
u1 = 0.1, cen_rate = 0.3, gen_data_from = "weibul")$D[[1]][, 1:2]
#-------------#
# First Round #
#-------------#
## Center 1:
Lambda1 <- inv.prior.cov(X1, lambda = 0.01, family = 'survival',
treatment = "X1", treat_round="first")
# When treat_round = "first", the family will automatically set to 'binomial',
# even if family = 'gaussian' or family = 'survival'.
fit1_r1 <- MAP.estimation(y1, X1, family = 'survival', # 'basehaz' is not needed!
Lambda = Lambda1, treatment = "X1", treat_round = "first")
# While family = 'survival', the output is based on 'binomial' with no 'Intercept'!
# The output without the treatment (X1) in the first round!
summary(fit1_r1)
## Center 2:
Lambda2 <- inv.prior.cov(X2, lambda = 0.01, family = 'survival',
treatment = "X1", treat_round="first")
fit2_r1 <- MAP.estimation(y2, X2, family = 'survival', Lambda = Lambda2,
treatment = "X1", treat_round = "first")
fit2_r1
## Centeral Server:
theta_hats_r1 <- list(fit1_r1$theta_hat, fit2_r1$theta_hat)
A_hats_r1 <- list(fit1_r1$A_hat, fit2_r1$A_hat)
fitbfi_r1 <- bfi(theta_hats_r1, A_hats_r1, Lambda1, family = 'survival',
treat_round = "first")
# In the first round output is based on 'binomial', and without
# the intercept and treatment (X1):
summary(fitbfi_r1, cur_mat = TRUE)
#--------------#
# Second Round #
#--------------#
## Center 1:
Lambda11 <- inv.prior.cov(X1, lambda = 0.01, family = 'survival',
basehaz = "unspecified", treatment = "X1",
treat_round="second")
fit1_r2 <- MAP.estimation(y1, X1, family = 'survival', Lambda = Lambda11,
basehaz = "unspecified", treatment = "X1",
treat_round = "second", gamma_bfi = fitbfi_r1$theta_hat)
# The output with only the treatment (X1) in the second round!
summary(fit1_r2)
## Center 2:
Lambda22 <- inv.prior.cov(X2, lambda = 0.01, family = 'survival',
basehaz = "unspecified", treatment = "X1",
treat_round="second")
fit2_r2 <- MAP.estimation(y2, X2, family = 'survival', basehaz = "unspecified",
Lambda = Lambda22, treatment = "X1",
treat_round = "second", gamma_bfi = fitbfi_r1$theta_hat)
fit2_r2
## Centeral Server:
theta_hats_r2 <- list(fit1_r2$theta_hat, fit2_r2$theta_hat)
A_hats_r2 <- list(fit1_r2$A_hat, fit2_r2$A_hat)
fitbfi_r2 <- bfi(theta_hats_r2, A_hats_r2, Lambda11, family = 'survival',
basehaz = "unspecified", treat_round = "second")
# When family = 'survival', 'for_ATE' is not calculated.
summary(fitbfi_r2)
BFI documentation built on June 8, 2025, 12:41 p.m.
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| bfi | R Documentation |
Bayesian Federated Inference
Description
bfi function can be used (on the central server) to combine inference results from separate datasets (without combining the data) to approximate what would have been inferred had the datasets been merged. This function can handle linear, logistic and survival regression models.
Usage
bfi(theta_hats = NULL,
A_hats,
Lambda,
family = c("gaussian", "binomial", "survival"),
basehaz = c("weibul", "exp", "gomp", "poly", "pwexp", "unspecified"),
stratified = FALSE,
strat_par = NULL,
center_spec = NULL,
theta_A_polys = NULL,
treat_round = NULL,
for_ATE = NULL,
p,
q_ls,
center_zero_sample = FALSE,
which_cent_zeros,
zero_sample_covs,
refer_cats,
zero_cats,
lev_no_ref_zeros)
Arguments
theta_hats |
a list of |
A_hats |
a list of |
Lambda |
a list of |
family |
a character string representing the family name used for the local centers. Can be abbreviated. |
basehaz |
a character string representing one of the available baseline hazard functions; |
stratified |
logical flag for performing the stratified analysis. If |
strat_par |
an integer vector for indicating the stratification parameter(s). It can be used to deal with heterogeneity due to center-specific parameters.
For the |
center_spec |
a vector of |
theta_A_polys |
a list with |
treat_round |
a character string representing the |
for_ATE |
a list of |
p |
an integer representing the number of covariates/coefficients. It can be found from the output of the |
q_ls |
a vector with |
center_zero_sample |
logical flag indicating whether the center has a categorical covariate with no observations/individuals in one of the categories. It is used to address heterogeneity across centers due to center-specific covariates. Default is |
which_cent_zeros |
an integer vector representing the center(s) which has one categorical covariate with no individuals in one of the categories. It is used if |
zero_sample_covs |
a vector in which each element is a character string representing the categorical covariate that has no samples/observations in one of its categories for the corresponding center. Each element of the vector can be obtained from the output of the |
refer_cats |
a vector in which each element is a character string representing the reference category for the corresponding center. Each element of the vector can be obtained from the output of the |
zero_cats |
a vector in which each element is a character string representing the category with no samples/observations for the corresponding center. Each element of the vector can be obtained from the output of the |
lev_no_ref_zeros |
a list in which the number of elements equals the length of the |
Details
bfi function implements the BFI approach described in the papers Jonker et. al. (2024a), Pazira et. al. (2024) and Jonker et. al. (2024b) given in the references.
The inference results gathered from different (L) centers are combined, and the BFI estimates of the model parameters and curvature matrix evaluated at that point are returned.
The inference result from each center must be obtained using the MAP.estimation function separately, and then all of these results (coming from different centers) should be compiled into a list to be used as an input of bfi().
The models in the different centers should be defined in exactly the same way; among others, exactly the same covariates should be included in the models. The parameter vectors should be defined exactly the same, so that the L vectors and matrices in the input lists theta_hat's and A_hat's are defined in the same way (e.g., the covariates need to be included in the models in the same order).
Note that the order of the elements in the lists theta_hats, A_hats and Lambda, must be the same with respect to the centers, so that in every list the element at the \ell^{\th} position is from the center \ell. This should also be the case for the vector center_spec.
If for the locations intercept = FALSE, the stratified analysis is not possible anymore for the binomial family.
If stratified = FALSE, both strat_par and center_spec must be NULL (the defaults), while if stratified = TRUE only one of the two must be NULL.
If stratified = FALSE and all the L+1 matrices in Lambda are equal, it is sufficient to give a (list of) one matrix only.
In both cases of the stratified argument (TRUE or FALSE), if only the first L matrices are equal, the argument Lambda can be a list of two matrices, so that the fist matrix represents the chosen variance-covariance matrix for local centers and the second one is the chosen matrix for the combined data set.
The last matrix of the list in the argument Lambda can be built by the function inv.prior.cov().
If the data type used in the argument center_spec is continuous or categorical with the number of categories equal to the number of centers, one can use stratified = TRUE and center_spec = NULL, and set strat_par not to NULL (i.e., to 1, 2 or both (1, 2)). Indeed, in this case, the stratification parameter(s) given in the argument strat_par are assumed to be different across the centers.
When family = 'survival' and basehaz = 'poly', the arguments theta_hats and A_hats should not be provided. Instead, the theta_A_polys and q_ls arguments should be defined using the local information, specifically MAP.estimation()$theta_A_poly and MAP.estimation()$q_l, respectively. See Example 3 in ‘Examples’.
For estimating the treatment effect, in the first round (treat_round = "first"), the argument for_ATE must be NULL (the default) and the family must be set to binomial (family is handled automatically.)
Value
bfi returns a list containing the following components:
theta_hat |
the vector of estimates obtained by combining the inference results from the |
A_hat |
minus the curvature (or Hessian) matrix obtained by the |
sd |
the vector of (posterior) standard deviation of the estimates in |
family |
the |
basehaz |
the baseline hazard function used; |
stratified |
whether a stratified analysis was done or not; |
strat_par |
the stratification parameter(s) used; |
Ave_Treat |
the estimates of the average treatment effect. Two diffterent estimations (IPTW and wIPTW) if the family is |
Author(s)
Hassan Pazira and Marianne Jonker
Maintainer: Hassan Pazira hassan.pazira@radboudumc.nl
References
Jonker M.A., Pazira H. and Coolen A.C.C. (2024a). Bayesian federated inference for estimating statistical models based on non-shared multicenter data sets, Statistics in Medicine, 43(12): 2421-2438. <https://doi.org/10.1002/sim.10072>
Pazira H., Massa E., Weijers J.A.M., Coolen A.C.C. and Jonker M.A. (2025b). Bayesian Federated Inference for Survival Models, Journal of Applied Statistics (Accepted). <https://arxiv.org/abs/2404.17464>
Jonker M.A., Pazira H. and Coolen A.C.C. (2025a). Bayesian Federated Inference for regression models based on non-shared medical center data, Research Synthesis Methods, 1-41. <https://doi.org/10.1017/rsm.2025.6>
See Also
MAP.estimation and inv.prior.cov
Examples
#################################################
## Example 1: y ~ Binomial (L = 2 centers) ##
#################################################
# Setting a seed for reproducibility
set.seed(112358)
#------------------------------------#
# Data Simulation for Local Center 1 #
#------------------------------------#
n1 <- 30 # sample size of center 1
X1 <- data.frame(x1=rnorm(n1), # continuous variable
x2=sample(0:2, n1, replace=TRUE)) # categorical variable
# make dummy variables
X1x2_1 <- ifelse(X1$x2 == '1', 1, 0)
X1x2_2 <- ifelse(X1$x2 == '2', 1, 0)
X1$x2 <- as.factor(X1$x2)
# regression coefficients
beta <- 1:4 # beta[1] is the intercept
# linear predictor:
eta1 <- beta[1] + X1$x1 * beta[2] + X1x2_1 * beta[3] + X1x2_2 * beta[4]
# inverse of the link function ( g^{-1}(\eta) = \mu ):
mu1 <- binomial()$linkinv(eta1)
y1 <- rbinom(n1, 1, mu1)
#------------------------------------#
# Data Simulation for Local Center 2 #
#------------------------------------#
n2 <- 50 # sample size of center 2
X2 <- data.frame(x1=rnorm(n2), # continuous variable
x2=sample(0:2, n2, replace=TRUE)) # categorical variable
# make dummy variables:
X2x2_1 <- ifelse(X2$x2 == '1', 1, 0)
X2x2_2 <- ifelse(X2$x2 == '2', 1, 0)
X2$x2 <- as.factor(X2$x2)
# linear predictor:
eta2 <- beta[1] + X2$x1 * beta[2] + X2x2_1 * beta[3] + X2x2_2 * beta[4]
# inverse of the link function:
mu2 <- binomial()$linkinv(eta2)
y2 <- rbinom(n2, 1, mu2)
#---------------------------#
# MAP Estimates at Center 1 #
#---------------------------#
# Assume the same inverse covariance matrix (Lambda) for both centers:
Lambda <- inv.prior.cov(X1, lambda = 0.01, family = 'binomial')
fit1 <- MAP.estimation(y1, X1, family = 'binomial', Lambda)
theta_hat1 <- fit1$theta_hat # intercept and coefficient estimates
A_hat1 <- fit1$A_hat # minus the curvature matrix
#---------------------------#
# MAP Estimates at Center 2 #
#---------------------------#
fit2 <- MAP.estimation(y2, X2, family='binomial', Lambda)
theta_hat2 <- fit2$theta_hat
A_hat2 <- fit2$A_hat
#-----------------------#
# BFI at Central Server #
#-----------------------#
theta_hats <- list(theta_hat1, theta_hat2)
A_hats <- list(A_hat1, A_hat2)
bfi <- bfi(theta_hats, A_hats, Lambda, family='binomial')
class(bfi)
summary(bfi, cur_mat=TRUE)
###---------------------###
### Stratified Analysis ###
###---------------------###
# By running the following line an error appears because
# when stratified = TRUE, both 'strat_par' and 'center_spec' can not be NULL:
Just4check1 <- try(bfi(theta_hats, A_hats, Lambda, family = 'binomial',
stratified = TRUE), TRUE)
class(Just4check1) # By default, both 'strat_par' and 'center_spec' are NULL!
# By running the following line an error appears because when stratified = TRUE,
# last matrix in 'Lambda' should not have the same dim. as the other local matrices:
Just4check2 <- try(bfi(theta_hats, A_hats, Lambda, stratified = TRUE,
strat_par = 1), TRUE)
class(Just4check2) # All matices in Lambda have the same dimension!
# Stratified analysis when 'intercept' varies across two centers:
newLam <- inv.prior.cov(X1, lambda=c(0.1, 0.3), family = 'binomial',
stratified = TRUE, strat_par = 1)
bfi <- bfi(theta_hats, A_hats, list(Lambda, newLam), family = 'binomial',
stratified=TRUE, strat_par=1)
summary(bfi, cur_mat=TRUE)
###---------------------###
### Treatment Effect ###
###---------------------###
set.seed(112358)
#------------------------------------#
# Data Simulation for Local Center 1 #
#------------------------------------#
n1 <- 30 # sample size of center 1
X1 <- data.frame(x1=rnorm(n1), # continuous variable
treatment=sample(1:2, n1, replace=TRUE)) # categorical variable
X1$treatment <- as.factor(X1$treatment)
# regression coefficients
beta <- 1:3 # beta[1] is the intercept
# make dummy variable
X1x2_2 <- ifelse(X1$treatment == '2', 1, 0)
# linear predictor:
eta1 <- beta[1] + X1$x1 * beta[2] + X1x2_2 * beta[3]
# inverse of the link function ( g^{-1}(\eta) = \mu ):
mu1 <- binomial()$linkinv(eta1)
y1 <- rbinom(n1, 1, mu1)
#------------------------------------#
# Data Simulation for Local Center 2 #
#------------------------------------#
n2 <- 50 # sample size of center 2
X2 <- data.frame(x1=rnorm(n2), # continuous variable
treatment=sample(1:2, n2, replace=TRUE)) # categorical variable
X2$treatment <- as.factor(X2$treatment)
# make dummy variables:
X2x2_2 <- ifelse(X2$treatment == '2', 1, 0)
# linear predictor:
eta2 <- beta[1] + X2$x1 * beta[2] + X2x2_2 * beta[3]
# inverse of the link function:
mu2 <- binomial()$linkinv(eta2)
y2 <- rbinom(n2, 1, mu2)
# The algorithm works even if the order of the covariates are not
# the same across centers
X2 <- X2[,c("treatment","x1")]
#-----------------------#
# Observational data #
#-----------------------#
# For observational data (RWD), we need two rounds for estimating treatment effect:
#-------------#
# First Round #
#-------------#
## Center 1:
Lambda1 <- inv.prior.cov(X1, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="first")
fit1_r1 <- MAP.estimation(y1, X1, family = 'binomial', Lambda = Lambda1,
treatment = "treatment", treat_round = "first")
# In the first round, the output is without the treatment!
summary(fit1_r1)
## Center 2:
Lambda2 <- inv.prior.cov(X2, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="first")
fit2_r1 <- MAP.estimation(y2, X2, family = 'binomial', Lambda = Lambda2,
treatment = "treatment", treat_round = "first")
fit2_r1
## Centeral Server:
theta_hats_r1 <- list(fit1_r1$theta_hat, fit2_r1$theta_hat)
A_hats_r1 <- list(fit1_r1$A_hat, fit2_r1$A_hat)
fitbfi_r1 <- bfi(theta_hats_r1, A_hats_r1, Lambda1, family = 'binomial',
treat_round = "first")
summary(fitbfi_r1, cur_mat = TRUE)
#--------------#
# Second Round #
#--------------#
## Center 1:
Lambda11 <- inv.prior.cov(X1, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="second")
fit1_r2 <- MAP.estimation(y1, X1, family = 'binomial', Lambda = Lambda11,
treatment = "treatment", treat_round = "second",
gamma_bfi = fitbfi_r1$theta_hat)
# In the second round, the output is only with the treatment!
summary(fit1_r2)
## Center 2:
Lambda22 <- inv.prior.cov(X2, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="second")
fit2_r2 <- MAP.estimation(y2, X2, family = 'binomial', Lambda = Lambda22,
treatment = "treatment", treat_round = "second",
gamma_bfi = fitbfi_r1$theta_hat)
fit2_r2$propensity # Propensity Score
fit2_r2$for_ATE # will be used in central server
fit2_r2
## Centeral Server:
theta_hats_r2 <- list(fit1_r2$theta_hat, fit2_r2$theta_hat)
A_hats_r2 <- list(fit1_r2$A_hat, fit2_r2$A_hat)
for_ATEs <- list(fit1_r2$for_ATE, fit2_r2$for_ATE)
fitbfi_r2 <- bfi(theta_hats_r2, A_hats_r2, Lambda11, family = 'binomial',
treat_round = "second", for_ATE = for_ATEs)
fitbfi_r2$S_var
fitbfi_r2$Ave_Treat
summary(fitbfi_r2)
#--------------------#
# Randomized Trial #
#--------------------#
# For Randomized Control Trial (RCT), we need only one round (the second round) for
# estimating treatment effect. Because we do not need to estimate propensity score.
# For example, in a 1:1 randomized trial, the propensity scores are, by definition,
# equal to 0.5. Here we use 'RCT_propens', instead of 'gamma_bfi':
## Center 1:
Lambda11 <- inv.prior.cov(X1, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="second")
fit1_r2 <- MAP.estimation(y1, X1, family = 'binomial', Lambda = Lambda11,
treatment = "treatment", treat_round = "second",
RCT_propens = rep(0.5, n1)) # gamma_bfi = NULL
summary(fit1_r2)
## Center 2:
Lambda22 <- inv.prior.cov(X2, lambda = 0.01, family = 'binomial',
treatment = "treatment", treat_round="second")
fit2_r2 <- MAP.estimation(y2, X2, family = 'binomial', Lambda = Lambda22,
treatment = "treatment", treat_round = "second",
RCT_propens = rep(0.5, n2)) # gamma_bfi = NULL
fit2_r2$for_ATE # will be used in central server
fit2_r2
## Centeral Server:
theta_hats_r2 <- list(fit1_r2$theta_hat, fit2_r2$theta_hat)
A_hats_r2 <- list(fit1_r2$A_hat, fit2_r2$A_hat)
for_ATEs <- list(fit1_r2$for_ATE, fit2_r2$for_ATE)
fitbfi_r2 <- bfi(theta_hats_r2, A_hats_r2, Lambda11, family = 'binomial',
treat_round = "second", for_ATE = for_ATEs)
fitbfi_r2$S_var
fitbfi_r2$Ave_Treat
summary(fitbfi_r2)
#################################################
## Example 2: y ~ Gaussian (L = 3 centers) ##
#################################################
# Setting a seed for reproducibility
set.seed(112358)
p <- 3 # number of coefficients without 'intercept'
theta <- c(1, rep(2, p), 1.5) # reg. coef.s ('intercept' is 1) & 'sigma2' = 1.5
#------------------------------------#
# Data Simulation for Local Center 1 #
#------------------------------------#
n1 <- 30 # sample size of center 1
X1 <- data.frame(matrix(rnorm(n1 * p), n1, p)) # continuous variables
# linear predictor:
eta1 <- theta[1] + as.matrix(X1)
# inverse of the link function ( g^{-1}(\eta) = \mu ):
mu1 <- gaussian()$linkinv(eta1)
y1 <- rnorm(n1, mu1, sd = sqrt(theta[5]))
#------------------------------------#
# Data Simulation for Local Center 2 #
#------------------------------------#
n2 <- 40 # sample size of center 2
X2 <- data.frame(matrix(rnorm(n2 * p), n2, p)) # continuous variables
# linear predictor:
eta2 <- theta[1] + as.matrix(X2)
# inverse of the link function:
mu2 <- gaussian()$linkinv(eta2)
y2 <- rnorm(n2, mu2, sd = sqrt(theta[5]))
#------------------------------------#
# Data Simulation for Local Center 3 #
#------------------------------------#
n3 <- 50 # sample size of center 3
X3 <- data.frame(matrix(rnorm(n3 * p), n3, p)) # continuous variables
# linear predictor:
eta3 <- theta[1] + as.matrix(X3)
# inverse of the link function:
mu3 <- gaussian()$linkinv(eta3)
y3 <- rnorm(n3, mu3, sd = sqrt(theta[5]))
#---------------------------#
# Inverse Covariance Matrix #
#---------------------------#
# Creating the inverse covariance matrix for the Gaussian prior distribution:
# the same for both centers
Lambda <- inv.prior.cov(X1, lambda = 0.05, family='gaussian')
#---------------------------#
# MAP Estimates at Center 1 #
#---------------------------#
fit1 <- MAP.estimation(y1, X1, family = 'gaussian', Lambda)
theta_hat1 <- fit1$theta_hat # intercept and coefficient estimates
A_hat1 <- fit1$A_hat # minus the curvature matrix
#---------------------------#
# MAP Estimates at Center 2 #
#---------------------------#
fit2 <- MAP.estimation(y2, X2, family = 'gaussian', Lambda)
theta_hat2 <- fit2$theta_hat
A_hat2 <- fit2$A_hat
#---------------------------#
# MAP Estimates at Center 3 #
#---------------------------#
fit3 <- MAP.estimation(y3, X3, family = 'gaussian', Lambda)
theta_hat3 <- fit3$theta_hat
A_hat3 <- fit3$A_hat
#-----------------------#
# BFI at Central Server #
#-----------------------#
A_hats <- list(A_hat1, A_hat2, A_hat3)
theta_hats <- list(theta_hat1, theta_hat2, theta_hat3)
bfi <- bfi(theta_hats, A_hats, Lambda, family = 'gaussian')
summary(bfi, cur_mat=TRUE)
###---------------------###
### Stratified Analysis ###
###---------------------###
# Stratified analysis when 'intercept' varies across two centers:
newLam1 <- inv.prior.cov(X1, lambda = c(0.1,0.3), family = 'gaussian',
stratified = TRUE, strat_par = 1, L = 3)
# 'newLam1' is used as the prior for combined data and
# 'Lambda' is used as the prior for locals
list_newLam1 <- list(Lambda, newLam1)
bfi1 <- bfi(theta_hats, A_hats, list_newLam1, family = 'gaussian',
stratified = TRUE, strat_par = 1)
summary(bfi1, cur_mat = TRUE)
# Stratified analysis when 'sigma2' varies across two centers:
newLam2 <- inv.prior.cov(X1, lambda = c(0.1,0.3), family = 'gaussian',
stratified = TRUE, strat_par = 2, L = 3)
# 'newLam2' is used as the prior for combined data and 'Lambda' is used as
# the prior for locals
list_newLam2 <- list(Lambda, newLam2)
bfi2 <- bfi(theta_hats, A_hats, list_newLam2, family = 'gaussian',
stratified = TRUE, strat_par=2)
summary(bfi2, cur_mat = TRUE)
# Stratified analysis when 'intercept' and 'sigma2' vary across 2 centers:
newLam3 <- inv.prior.cov(X1, lambda = c(0.1,0.2,0.3), family = 'gaussian',
stratified = TRUE, strat_par = c(1, 2), L = 3)
# 'newLam3' is used as the prior for combined data and 'Lambda' is used as
# the prior for locals
list_newLam3 <- list(Lambda, newLam3)
bfi3 <- bfi(theta_hats, A_hats, list_newLam3, family = 'gaussian',
stratified = TRUE, strat_par = 1:2)
summary(bfi3, cur_mat = TRUE)
###----------------------------###
### Center Specific Covariates ###
###----------------------------###
# Assume the first and third centers have the same center-specific covariate value
# of 'High', while this value for the second center is 'Low', i.e.,
# center_spec = c('High','Low','High')
newLam4 <- inv.prior.cov(X1, lambda=c(0.1, 0.2, 0.3), family='gaussian',
stratified = TRUE, center_spec = c('High','Low','High'),
L = 3)
# 'newLam4' is used as the prior for combined data and 'Lambda' is used as
# the prior for locals
l_newLam4 <- list(Lambda, newLam4)
bfi4 <- bfi(theta_hats, A_hats, l_newLam4, family = 'gaussian',
stratified = TRUE, center_spec = c('High','Low','High'))
summary(bfi4, cur_mat = TRUE)
###---------------------###
### Treatment Effect ###
###---------------------###
set.seed(112358)
#-----------------------------#
# New Data for Local Center 1 #
#-----------------------------#
# Generating new data with 'treatment' variable
# We cansider the first variable (X1$X1) to be the treatment:
X1$X1 <- sample(0:1, n1, replace=TRUE) # categorical variable
eta1 <- theta[1] + as.matrix(X1)
mu1 <- gaussian()$linkinv(eta1)
y1 <- rnorm(n1, mu1, sd = sqrt(theta[5]))
#-----------------------------#
# New Data for Local Center 2 #
#-----------------------------#
# We cansider the first variable (X2$X1) to be the treatment:
X2$X1 <- sample(0:1, n2, replace=TRUE) # categorical variable
eta2 <- theta[1] + as.matrix(X2)
mu2 <- gaussian()$linkinv(eta2)
y2 <- rnorm(n2, mu2, sd = sqrt(theta[5]))
#-----------------------------#
# New Data for Local Center 3 #
#-----------------------------#
# We cansider the first variable (X3$X1) to be the treatment:
X3$X1 <- sample(0:1, n3, replace=TRUE) # categorical variable
# linear predictor:
eta3 <- theta[1] + as.matrix(X3)
# inverse of the link function:
mu3 <- gaussian()$linkinv(eta3)
y3 <- rnorm(n3, mu3, sd = sqrt(theta[5]))
#-----------------------#
# Observational data #
#-----------------------#
# For observational data (RWD), we need two rounds for estimating treatment effect:
#-------------#
# First Round #
#-------------#
## Center 1:
Lambda1 <- inv.prior.cov(X1, lambda = 0.01, family = 'binomial',
treatment = "X1", treat_round="first")
# When treat_round = "first", the family will automatically set to 'binomial',
# even if family = 'gaussian' or family = 'survival'.
fit1_r1 <- MAP.estimation(y1, X1, family = 'gaussian', Lambda = Lambda1,
treatment = "X1", treat_round = "first")
# Althghou family = 'gaussian', the output is based on 'binomial'!
# The output without the treatment (X1) in the first round!
summary(fit1_r1)
## Center 2:
Lambda2 <- inv.prior.cov(X2, lambda = 0.01, family = 'gaussian',
treatment = "X1", treat_round="first")
fit2_r1 <- MAP.estimation(y2, X2, family = 'gaussian', Lambda = Lambda2,
treatment = "X1", treat_round = "first")
fit2_r1
## Center 3:
Lambda3 <- inv.prior.cov(X3, lambda = 0.01, family = 'gaussian',
treatment = "X1", treat_round="first")
fit3_r1 <- MAP.estimation(y3, X3, family = 'gaussian', Lambda = Lambda3,
treatment = "X1", treat_round = "first")
## Centeral Server:
theta_hats_r1 <- list(fit1_r1$theta_hat, fit2_r1$theta_hat, fit3_r1$theta_hat)
A_hats_r1 <- list(fit1_r1$A_hat, fit2_r1$A_hat, fit3_r1$A_hat)
fitbfi_r1 <- bfi(theta_hats_r1, A_hats_r1, Lambda1, family = 'gaussian',
treat_round = "first") # same results with 'binomial'
# The output without the treatment (X1) in the first round!
summary(fitbfi_r1, cur_mat = TRUE)
#--------------#
# Second Round #
#--------------#
## Center 1:
Lambda11 <- inv.prior.cov(X1, lambda = 0.01, family = 'gaussian',
treatment = "X1", treat_round="second")
fit1_r2 <- MAP.estimation(y1, X1, family = 'gaussian', Lambda = Lambda11,
treatment = "X1", treat_round = "second",
gamma_bfi = fitbfi_r1$theta_hat)
# The output with only the treatment (X1) in the second round!
summary(fit1_r2)
## Center 2:
Lambda22 <- inv.prior.cov(X2, lambda = 0.01, family = 'gaussian', treatment = "X1",
treat_round="second")
fit2_r2 <- MAP.estimation(y2, X2, family = 'gaussian', Lambda = Lambda22,
treatment = "X1", treat_round = "second",
gamma_bfi = fitbfi_r1$theta_hat)
## Center 3:
Lambda33 <- inv.prior.cov(X3, lambda = 0.01, family = 'gaussian', treatment = "X1",
treat_round="second")
fit3_r2 <- MAP.estimation(y3, X3, family = 'gaussian', Lambda = Lambda33,
treatment = "X1", treat_round = "second",
gamma_bfi = fitbfi_r1$theta_hat)
## Centeral Server:
theta_hats_r2 <- list(fit1_r2$theta_hat, fit2_r2$theta_hat, fit3_r2$theta_hat)
A_hats_r2 <- list(fit1_r2$A_hat, fit2_r2$A_hat, fit3_r2$A_hat)
for_ATEs <- list(fit1_r2$for_ATE, fit2_r2$for_ATE, fit3_r2$for_ATE)
fitbfi_r2 <- bfi(theta_hats_r2, A_hats_r2, Lambda11, family = 'gaussian',
treat_round = "second", for_ATE = for_ATEs)
fitbfi_r2$Ave_Treat
fitbfi_r2$S_var
summary(fitbfi_r2)
####################################################
## Example 3: Survival family (L = 2 centers) ##
####################################################
# Setting a seed for reproducibility
set.seed(112358)
p <- 3
theta <- c(1:4, 5, 6) # regression coefficients (1:4) & omega's (5:6)
#---------------------------------------------#
# Simulating Survival data for Local Center 1 #
#---------------------------------------------#
n1 <- 30
X1 <- data.frame(matrix(rnorm(n1 * p), n1, p)) # continuous (normal) variables
# Simulating survival data ('time' and 'status') from 'Weibull' with
# a predefined censoring rate of 0.3:
y1 <- surv.simulate(Z = list(X1), beta = theta[1:p], a = theta[5],
b = theta[6], u1 = 0.1, cen_rate = 0.3,
gen_data_from = "weibul")$D[[1]][, 1:2]
## MAP Estimates at Center 1
Lambda <- inv.prior.cov(X1, lambda = c(0.1, 1), family = "survival",
basehaz = "poly")
fit1 <- MAP.estimation(y1, X1, family = 'survival', Lambda = Lambda,
basehaz = "poly")
theta_hat1 <- fit1$theta_hat # coefficient estimates
A_hat1 <- fit1$A_hat # minus the curvature matrix
summary(fit1, cur_mat=TRUE)
fit1$theta_A_poly # Only when family = "survival" and basehaz ="poly"
#---------------------------------------------#
# Simulating Survival data for Local Center 2 #
#---------------------------------------------#
n2 <- 30
X2 <- data.frame(matrix(rnorm(n2 * p), n2, p)) # continuous (normal) variables
# Survival simulated data from 'Weibull' with a predefined censoring rate of 0.3:
y2 <- surv.simulate(Z = list(X2), beta = theta[1:p], a = theta[5],
b = theta[6],u1 = 0.1, cen_rate = 0.3,
gen_data_from = "weibul")$D[[1]][, 1:2]
## MAP Estimates at Center 2
fit2 <- MAP.estimation(y2, X2, family = 'survival', Lambda = Lambda,
basehaz = "poly")
theta_hat2 <- fit2$theta_hat
A_hat2 <- fit2$A_hat
summary(fit2, cur_mat=TRUE)
#-----------------------#
# BFI at Central Server #
#-----------------------#
# When family = 'survival' and basehaz = "poly", only 'theta_A_polys'
# should be defined instead of 'theta_hats' and 'A_hats':
theta_A_hats <- list(fit1$theta_A_poly, fit2$theta_A_poly)
qls <- c(fit1$q_l, fit2$q_l)
bfi <- bfi(Lambda = Lambda, family = 'survival', theta_A_polys = theta_A_hats,
basehaz = "poly", q_ls = qls)
summary(bfi, cur_mat=TRUE)
###---------------------###
### Stratified Analysis ###
###---------------------###
# Stratified analysis when first parameter ('omega_0') varies across two centers:
(newLam0 <- inv.prior.cov(X1, lambda = c(rep(1, 3), 0.3, 0.7, rep(2,2)),
family = 'survival', stratified = TRUE,
basehaz = c("poly"), strat_par = 1, L = 2))
# 'newLam0' is used as the prior for combined data and 'Lambda' is used as for locals:
list_newLam0 <- list(Lambda, newLam0)
bfi0 <- bfi(Lambda = list_newLam0, family = 'survival', theta_A_polys = theta_A_hats,
stratified = TRUE, basehaz = c("poly"), p = 3, q_ls = qls, strat_par = 1)
summary(bfi0, cur_mat = TRUE)
# Stratified analysis when the first and second parameters ('omega_0' and 'omega_1')
# vary across two centers:
newLam1 <- inv.prior.cov(X1, lambda = c(rep(1, 3), 0.3, 0.7, 0.5, 0.8, 2),
family = 'survival', stratified = TRUE, basehaz = c("poly"),
strat_par = c(1, 2), L = 2)
# 'newLam1' is used as the prior for combined data:
list_newLam1 <- list(Lambda, newLam1)
bfi1 <- bfi(Lambda = list_newLam1, family = 'survival', theta_A_polys = theta_A_hats,
stratified = TRUE, basehaz = c("poly"), p = 3, q_ls = qls,
strat_par = c(1, 2))
summary(bfi1, cur_mat = TRUE)
###---------------------###
### Treatment Effect ###
###---------------------###
set.seed(112358)
#-----------------------------#
# New Data for Local Center 1 #
#-----------------------------#
# Generating new data with 'treatment' variable
# We cansider the first variable (X1$X1) to be the treatment
X1$X1 <- sample(0:1, n1, replace=TRUE) # categorical variable
y1 <- surv.simulate(Z = list(X1), beta = theta[1:p], a = theta[5], b = theta[6],
u1 = 0.1, cen_rate = 0.3, gen_data_from = "weibul")$D[[1]][, 1:2]
#-----------------------------#
# New Data for Local Center 2 #
#-----------------------------#
# We cansider the first variable (X2$X1) to be the treatment!
X2$X1 <- sample(0:1, n2, replace=TRUE) # categorical variable
y2 <- surv.simulate(Z = list(X2), beta = theta[1:p], a = theta[5], b = theta[6],
u1 = 0.1, cen_rate = 0.3, gen_data_from = "weibul")$D[[1]][, 1:2]
#-------------#
# First Round #
#-------------#
## Center 1:
Lambda1 <- inv.prior.cov(X1, lambda = 0.01, family = 'survival',
treatment = "X1", treat_round="first")
# When treat_round = "first", the family will automatically set to 'binomial',
# even if family = 'gaussian' or family = 'survival'.
fit1_r1 <- MAP.estimation(y1, X1, family = 'survival', # 'basehaz' is not needed!
Lambda = Lambda1, treatment = "X1", treat_round = "first")
# While family = 'survival', the output is based on 'binomial' with no 'Intercept'!
# The output without the treatment (X1) in the first round!
summary(fit1_r1)
## Center 2:
Lambda2 <- inv.prior.cov(X2, lambda = 0.01, family = 'survival',
treatment = "X1", treat_round="first")
fit2_r1 <- MAP.estimation(y2, X2, family = 'survival', Lambda = Lambda2,
treatment = "X1", treat_round = "first")
fit2_r1
## Centeral Server:
theta_hats_r1 <- list(fit1_r1$theta_hat, fit2_r1$theta_hat)
A_hats_r1 <- list(fit1_r1$A_hat, fit2_r1$A_hat)
fitbfi_r1 <- bfi(theta_hats_r1, A_hats_r1, Lambda1, family = 'survival',
treat_round = "first")
# In the first round output is based on 'binomial', and without
# the intercept and treatment (X1):
summary(fitbfi_r1, cur_mat = TRUE)
#--------------#
# Second Round #
#--------------#
## Center 1:
Lambda11 <- inv.prior.cov(X1, lambda = 0.01, family = 'survival',
basehaz = "unspecified", treatment = "X1",
treat_round="second")
fit1_r2 <- MAP.estimation(y1, X1, family = 'survival', Lambda = Lambda11,
basehaz = "unspecified", treatment = "X1",
treat_round = "second", gamma_bfi = fitbfi_r1$theta_hat)
# The output with only the treatment (X1) in the second round!
summary(fit1_r2)
## Center 2:
Lambda22 <- inv.prior.cov(X2, lambda = 0.01, family = 'survival',
basehaz = "unspecified", treatment = "X1",
treat_round="second")
fit2_r2 <- MAP.estimation(y2, X2, family = 'survival', basehaz = "unspecified",
Lambda = Lambda22, treatment = "X1",
treat_round = "second", gamma_bfi = fitbfi_r1$theta_hat)
fit2_r2
## Centeral Server:
theta_hats_r2 <- list(fit1_r2$theta_hat, fit2_r2$theta_hat)
A_hats_r2 <- list(fit1_r2$A_hat, fit2_r2$A_hat)
fitbfi_r2 <- bfi(theta_hats_r2, A_hats_r2, Lambda11, family = 'survival',
basehaz = "unspecified", treat_round = "second")
# When family = 'survival', 'for_ATE' is not calculated.
summary(fitbfi_r2)
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