Author: Hassan Pazira
Date: 2025-03-22

# An Introduction to BFI in R

#### Hassan Pazira

#### Marianne Jonker

#### 2025-03-22

## Overview

The R package `BFI` (Bayesian Federated Inference) provides several functions to perform Bayesian Federated Inference for two types of models (`GLM` and `Survival`) using multicenter data without the need to combine or share them. This tutorial focuses on `GLM` models. Two commonly used families for `GLM` models, `"binomial"` and `"gaussian"`, are available for this version of the package. The most commonly used functions include `bfi()`, `MAP.estimation()`, and `inv.prior.cov()`. In the following, we will see how the `BFI` package can be applied to real data sets included in the package.

## How to use it?

Before we go on, we first install and load the `BFI` package:

    # Install and load the BFI package from CRAN:
    install.packages("BFI")
    library(BFI)

    ## Warning: package 'BFI' was built under R version 4.3.3

By using the following code, we can see that there are two available data sets in the package: `trauma` and `Nurses`.

    data(package = "BFI")

The `trauma` data set can be utilized for the `"binomial"` family and `Nurses` data set can be used for `"gaussian"` family. To avoid repetition, we will only use the `trauma` data set. By loading the package, the data sets included will be loaded and can be inspected as follows:

    # Get the number of rows and columns
    dim(trauma)

    ## [1] 371   6

    # To get an idea of the data set, print the first 7 rows
    head(trauma, 7)

    ##   sex age hospital ISS GCS mortality
    ## 1   1  20        3  24  15         0
    ## 2   0  38        3  34  13         0
    ## 3   0  37        3  50  15         0
    ## 4   0  17        3  43   4         1
    ## 5   0  49        3  29  15         0
    ## 6   0  30        3  22  15         0
    ## 7   1  84        2  66   3         1

This data set consists of data of 371 trauma patients from three hospitals (peripheral hospital without a neuro-surgical unit, `'status=1'`, peripheral hospital with a neuro-surgical unit, `status=2`, and academic medical center, `status=3`).

As we can see, the data set has six columns. The covariates `sex` (dichotomous), `age` (continuous), `ISS` (Injury Severity Score, continuous), and `GCS` (Glasgow Coma Scale, continuous), which serve as the predictors. `mortality` is the response variable, while `hospital` is a categorical variable which indicates the hospitals involved in the study. For more information about this data set use

    # Get some info about the data set from the help file
    ?trauma

We will analyze the data with a `logistic` regression model. First we standardize the covariates. This is not necessary for the analysis, but is done for the interpretability of the accuracy of the estimates.

    trauma$age <- scale(trauma$age)
    trauma$ISS <- scale(trauma$ISS)
    trauma$GCS <- scale(trauma$GCS)
    trauma$hospital <- as.factor(trauma$hospital)

By using the following code we can see there are three hospitals involved in the study:

    length(levels(trauma$hospital))

    ## [1] 3

### MAP estimations

Therefore, the `MAP.estimation` function should be applied to these 3 local data sets separately to obtain the MAP estimations. Note that, in practice, we do not have access to the combined data, and each center should perform the analysis independently and send the output to the central server, as follows:

    # Center 1:
    # X1 <- data.frame(sex=trauma$sex[trauma$hospital==1],
    #                  age=trauma$age[trauma$hospital==1],
    #                  ISS=trauma$ISS[trauma$hospital==1],
    #                  GCS=trauma$GCS[trauma$hospital==1])
    X1 <- subset(trauma, hospital == 1, select = c(sex, age, ISS, GCS))
    Lambda1 <- inv.prior.cov(X1, lambda=0.01, L=3, family="binomial")
    fit1 <- MAP.estimation(y=trauma$mortality[trauma$hospital==1], X=X1, family="binomial", Lambda=Lambda1)
    summary(fit1)

    ##
    ## Summary of the local model:
    ##
    ##    Formula: y ~ sex + age + ISS + GCS
    ##     Family: 'binomial'
    ##       Link: 'Logit'
    ##
    ## Coefficients:
    ##
    ##             Estimate Std.Dev  CI 2.5% CI 97.5%
    ## (Intercept)  -2.1226  0.8074  -3.7050  -0.5402
    ## sex          -5.2795  2.6305 -10.4351  -0.1239
    ## age           1.8864  0.7210   0.4732   3.2996
    ## ISS           2.3741  0.9250   0.5611   4.1872
    ## GCS          -2.4522  0.8295  -4.0781  -0.8264
    ##
    ## Dispersion parameter (sigma2):  1
    ##             log Lik Posterior:  -10.74
    ##                   Convergence:  0

    # Center 2:
    # X2 <- data.frame(sex=trauma$sex[trauma$hospital==2],
    #                  age=trauma$age[trauma$hospital==2],
    #                  ISS=trauma$ISS[trauma$hospital==2],
    #                  GCS=trauma$GCS[trauma$hospital==2])
    X2 <- subset(trauma, hospital == 2, select = c(sex, age, ISS, GCS))
    Lambda2 <- inv.prior.cov(X2, lambda=0.01, L=3, family="binomial")
    fit2 <- MAP.estimation(y=trauma$mortality[trauma$hospital==2], X=X2, family="binomial", Lambda=Lambda2)
    summary(fit2)

    ##
    ## Summary of the local model:
    ##
    ##    Formula: y ~ sex + age + ISS + GCS
    ##     Family: 'binomial'
    ##       Link: 'Logit'
    ##
    ## Coefficients:
    ##
    ##             Estimate Std.Dev CI 2.5% CI 97.5%
    ## (Intercept)  -0.7854  0.3789 -1.5280  -0.0427
    ## sex          -0.7850  0.6999 -2.1568   0.5868
    ## age           1.9601  0.4662  1.0465   2.8738
    ## ISS           0.5216  0.3673 -0.1983   1.2415
    ## GCS          -1.8737  0.4115 -2.6803  -1.0672
    ##
    ## Dispersion parameter (sigma2):  1
    ##             log Lik Posterior:  -36.87
    ##                   Convergence:  0

    # Center 3:
    # X3 <- data.frame(sex=trauma$sex[trauma$hospital==3],
    #                  age=trauma$age[trauma$hospital==3],
    #                  ISS=trauma$ISS[trauma$hospital==3],
    #                  GCS=trauma$GCS[trauma$hospital==3])
    X3 <- subset(trauma, hospital == 3, select = c(sex, age, ISS, GCS))
    Lambda3 <- inv.prior.cov(X3, lambda=0.01, L=3, family="binomial")
    fit3 <- MAP.estimation(y=trauma$mortality[trauma$hospital==3], X=X3, family="binomial", Lambda=Lambda3)
    summary(fit3)

    ##
    ## Summary of the local model:
    ##
    ##    Formula: y ~ sex + age + ISS + GCS
    ##     Family: 'binomial'
    ##       Link: 'Logit'
    ##
    ## Coefficients:
    ##
    ##             Estimate Std.Dev CI 2.5% CI 97.5%
    ## (Intercept)  -2.3144  0.4020 -3.1024  -1.5264
    ## sex           0.1580  0.5714 -0.9619   1.2780
    ## age           1.3031  0.2955  0.7239   1.8824
    ## ISS           0.2967  0.2638 -0.2203   0.8138
    ## GCS          -2.4221  0.4130 -3.2316  -1.6125
    ##
    ## Dispersion parameter (sigma2):  1
    ##             log Lik Posterior:  -50.63
    ##                   Convergence:  0

It can be seen that all algorithms have converged (`Convergence: 0`). If `Convergence: 1` occurs, you can increase the number of iteration in the optimization process of `optim()` by adding `control = list(maxit=500)` to the function `MAP.estimation`, as shown below:

    # Example for Center 3:
    fit3 <- MAP.estimation(y=trauma$mortality[trauma$hospital==3], X=X3, family="binomial", Lambda=Lambda3, control = list(maxit=500))

To see more information about the data set, such as the number of observations and parameters, we can use the output of the `MAP.estimation` function as follows:

    # number of samples in center 1
    fit1$n

    ## [1] 49

    # number of parameters in center 1
    fit1$np

    ## [1] 5

    # number of samples in center 2
    fit2$n

    ## [1] 106

    # number of samples in center 3
    fit3$n

    ## [1] 216

Additionally, before conducting the analysis, we can use the `n.par` function to retrieve this information.

The outputs `fit1`,`fit2`, and `fit3` from the local centers should be sent to the central server for further analysis. To send these lists from R to the central server (which also uses R), you can save them in a format that R can easily read, such as an RDS file.

    # Save fit1 as an RDS file
    saveRDS(fit1, file="fit1.rds")

    # Save fit2 as an RDS file
    saveRDS(fit2, file="fit2.rds")

    # Save fit3 as an RDS file
    saveRDS(fit3, file="fit3.rds")

### BFI estimations

Now, the received files can be loaded in R using the following lines:

    # Load the RDS files
    fit1 <- readRDS("fit1.rds") # use the relative path to the file
    fit2 <- readRDS("fit2.rds") # use the relative path to the file
    fit3 <- readRDS("fit3.rds") # use the relative path to the file

On the central server, the `bfi()` function can be used to obtain the BFI estimations:

    theta_hats <- list(fit1$theta_hat, fit2$theta_hat, fit3$theta_hat)
    A_hats     <- list(fit1$A_hat, fit2$A_hat, fit3$A_hat)
    Lambda_com <- inv.prior.cov(X1, lambda=0.01, L=3, family="binomial")
    Lambdas    <- list(Lambda1, Lambda2, Lambda3, Lambda_com)
    BFI_fits   <- bfi(theta_hats, A_hats, Lambdas, family="binomial")
    summary(BFI_fits, cur_mat=TRUE)

    ##
    ## Summary of the BFI model:
    ##
    ##     Family: 'binomial'
    ##       Link: 'Logit'
    ##
    ## Coefficients:
    ##
    ##             Estimate Std.Dev CI 2.5% CI 97.5%
    ## (Intercept)  -1.4434  0.2465 -1.9265  -0.9602
    ## sex          -0.2473  0.4187 -1.0680   0.5733
    ## age           1.2189  0.2190  0.7897   1.6481
    ## ISS           0.4939  0.1945  0.1127   0.8751
    ## GCS          -1.7375  0.2491 -2.2258  -1.2492
    ##
    ## Dispersion parameter (sigma2):  1
    ##
    ## Minus the Curvature Matrix:
    ##
    ##             (Intercept)     sex     age      ISS      GCS
    ## (Intercept)     30.4330  7.7926  0.7578   8.3797 -16.6608
    ## sex              7.7926  7.8026  1.3061   1.8414  -4.6925
    ## age              0.7578  1.3061 31.5230  -4.8356  15.7494
    ## ISS              8.3797  1.8414 -4.8356  30.7881 -11.7189
    ## GCS            -16.6608 -4.6925 15.7494 -11.7189  34.4508

### Comparison

To compare the performance of the BFI methodology, we can combine the data sets and obtain the MAP estimations based on the combined data:

    # MAP estimates of the combined data:
    X_combined  <- data.frame(sex=trauma$sex,
                              age=trauma$age,
                              ISS=trauma$ISS,
                              GCS=trauma$GCS)
    Lambda   <- inv.prior.cov(X=X_combined, lambda=0.01, L=3, family="binomial")
    fit_comb  <- MAP.estimation(y=trauma$mortality, X=X_combined, family="binomial", Lambda=Lambda)
    summary(fit_comb, cur_mat=TRUE)

    ##
    ## Summary of the local model:
    ##
    ##    Formula: y ~ sex + age + ISS + GCS
    ##     Family: 'binomial'
    ##       Link: 'Logit'
    ##
    ## Coefficients:
    ##
    ##             Estimate Std.Dev CI 2.5% CI 97.5%
    ## (Intercept)  -1.6255  0.2398 -2.0955  -1.1556
    ## sex          -0.3357  0.3990 -1.1178   0.4463
    ## age           1.3703  0.2056  0.9673   1.7733
    ## ISS           0.5498  0.1862  0.1848   0.9149
    ## GCS          -1.9981  0.2379 -2.4645  -1.5318
    ##
    ## Dispersion parameter (sigma2):  1
    ##             log Lik Posterior:  -109.4
    ##                   Convergence:  0
    ##
    ## Minus the Curvature Matrix:
    ##
    ##             (Intercept)     sex     age      ISS      GCS
    ## (Intercept)     33.5675  8.4942  2.3840   8.3535 -18.2832
    ## sex              8.4942  8.5042  1.8643   1.4055  -4.5326
    ## age              2.3840  1.8643 37.1516  -6.1748  18.0223
    ## ISS              8.3535  1.4055 -6.1748  33.4408 -12.8543
    ## GCS            -18.2832 -4.5326 18.0223 -12.8543  38.5342

Now, we can see the difference between the BFI and combined estimates:

    # Squared Errors:
    (fit_comb$theta_hat - BFI_fits$theta_hat)^2

    ##      (Intercept)        sex        age         ISS        GCS
    ## [1,]  0.03319286 0.00781985 0.02292217 0.003132086 0.06791077

which are close to zero, as expected!

For more details see the following references.

## References

Jonker M.A., Pazira H. and Coolen A.C.C. (2024). 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

## Contact

If you find any errors, have any suggestions, or would like to request that something be added, please file an issue at issue report or send an email to: hassan.pazira@radboudumc.nl or Marianne.Jonker@radboudumc.nl.