# Fit a general-gamma, GPS, K-gamma, KM or efron model for a contingency table. This function fits a non-parametric empirical Bayes model to an AE-drug contingency table using one of several empirical Bayes approaches with specified hyperparameter, if is required. Supported models include the "general-gamma", "GPS", "K-gamma", "KM", and "efron". ```r pvEBayes( contin_table, model = "general-gamma", alpha = NULL, K = NULL, p = NULL, c0 = NULL, maxi = NULL, tol_ecm = 1e-04, rtol_efron = 1e-10, rtol_KM = 1e-06, n_posterior_draws = 1000, E = NULL ) ``` ## Arguments - `contin_table`: an IxJ contingency table showing pairwise counts of adverse events for I AEs (along the rows) and J drugs (along the columns). - `model`: the model to fit. Available models are "general-gamma", "K-gamma", "GPS", "KM" and "efron". Default to "general-gamma". Note that the input for model is case-sensitive. - `alpha`: numeric between 0 and 1. The hyperparameter of "general-gamma" model. It is needed if "general-gamma" model is used. Small 'alpha' encourages shrinkage on mixture weights of the estimated prior distribution. See Tan et al. (2025) for further details. - `K`: a integer greater than or equal to 2 indicating the number of mixture components in the prior distribution. It is needed if "K-gamma" model is used. See Tan et al. (2025) for further details. - `p`: a integer greater than or equal to 2. It is needed if "efron" mode is used. Larger p leads to smoother estimated prior distribution. See Narasimhan and Efron (2020) for detail. - `c0`: numeric and greater than 0. It is needed if "efron" mode is used. Large c0 encourage estimated prior distribution shrink toward discrete uniform. See Narasimhan and Efron (2020) for detail. - `maxi`: a upper limit of iteration for the ECM algorithm. - `tol_ecm`: a tolerance parameter used for the ECM stopping rule, defined as the absolute change in the joint marginal likelihood between two consecutive iterations. It is used when 'GPS', 'K-gamma' or 'general-gamma' model is fitted. Default to be 1e-4. - `rtol_efron`: a tolerance parameter used when 'efron' model is fitted. Default to 1e-10. See 'stats::nlminb' for detail. - `rtol_KM`: a tolerance parameter used when 'KM' model is fitted. Default to be 1e-6. See 'CVXR::solve' for detail. - `n_posterior_draws`: a number of posterior draws for each AE-drug combination. - `E`: A matrix of expected counts under the null model for the SRS frequency table. If `NULL` (default), the expected counts are estimated from the SRS data using 'estimate_null_expected_count()'. ## Returns The function returns an S3 object of class `pvEBayes` containing the estimated model parameters as well as posterior draws for each AE-drug combination if the number of posterior draws is specified. ## Details This function implements the ECM algorithm proposed by Tan et al. (2025), providing a stable and efficient implementation of Gamma-Poisson Shrinker(GPS), K-gamma and "general-gamma" methods for signal estimation and signal detection in Spontaneous Reporting System (SRS) data table. Method "GPS" is proposed by DuMouchel (1999) and it is a parametric empirical Bayes model with a two gamma mixture prior distribution. Methods "K-gamma" and "general-gamma" are non-parametric empirical Bayes models, introduced by Tan et al. (2025). The number of mixture components "K" needs to be prespecified when fitting a "K-gamma" model. For "general-gamma", the mixture weights are regularized by a Dirichlet hyper prior with hyperparameter $0 \leq \alpha < 1$ that controls the shrinkage strength. As "alpha" approaches 0, less non-empty mixture components exist in the fitted model. When $\alpha = 0$, the Dirichlet distribution is an improper prior still offering a reasonable posterior inference that represents the strongest shrinkage of the "general-gamma" model. Parameter estimation for the "KM" model is formulated as a convex optimization problem. The objective function and constraints follow the same construction as in `REBayes`. Parameter estimation is performed using the open-source convex optimization package `CVXR`. The implementation of the "efron" model in this package is adapted from the `deconvolveR` package, developed by Bradley Efron and Balasubramanian Narasimhan. The original implementation in `deconvolveR` does not support an exposure or offset parameter in the Poisson model, which corresponds to the expected null value ($E_{ij}$) for each AE-drug combination. To address this, we modified the relevant code to allow for the inclusion of $E_{ij}$ in the Poisson likelihood. In addition, we implemented a method for estimating the degrees of freedom, enabling AIC- or BIC-based hyperparameter selection for the "efron" model (Tan et al. 2025). See `pvEBayes_tune` for details. ## Examples ```r set.seed(1) ref_table <- statin2025_44 # set up signal matrix with one signal at entry (1,1) sig_mat <- matrix(1, nrow(ref_table), ncol(ref_table)) sig_mat[c(1, 5), 1] <- 2 simu_table <- generate_contin_table( ref_table = ref_table, signal_mat = sig_mat, n_table = 1 )[[1]] # fit general-gamma model with a specified alpha fit <- pvEBayes( contin_table = simu_table, model = "general-gamma", alpha = 0.3, n_posterior_draws = 1000, E = NULL, maxi = NULL ) # fit K-gamma model with K = 3 fit_Kgamma <- pvEBayes( contin_table = simu_table, model = "K-gamma", K = 3, n_posterior_draws = 1000 ) # fit Efron model with specified hyperparameters # p = 40, c0 = 0.05 ## Not run: fit_efron <- pvEBayes( contin_table = simu_table, model = "efron", p = 40, c0 = 0.05, n_posterior_draws = 1000 ) ## End(Not run) # fit GPS model and comapre with 'openEBGM' fit_gps <- pvEBayes(simu_table, model = "GPS") ## Not run: ## Optional comparison with openEBGM (only if installed) ## tol_ecm is the absolute tolerance for ECM stopping rule. ## It is set to ensure comparability to `openEBGM`. fit_gps <- pvEBayes(simu_table, model = "GPS", tol_ecm = 1e-2) if (requireNamespace("openEBGM", quietly = TRUE)) { E <- estimate_null_expected_count(simu_table) simu_table_stacked <- as.data.frame(as.table(simu_table)) simu_table_stacked$E <- as.vector(E) colnames(simu_table_stacked) <- c("var1", "var2", "N", "E") simu_table_stacked_squash <- openEBGM::autoSquash(simu_table_stacked) hyper_estimates <- openEBGM::hyperEM(simu_table_stacked_squash, theta_init = c(2, 1, 2, 2, 0.2), method = "nlminb", N_star = NULL, zeroes = TRUE, param_upper = Inf, LL_tol = 1e-2, max_iter = 10000 ) } theta_hat <- hyper_estimates$estimates qn <- openEBGM::Qn(theta_hat, N = simu_table_stacked$N, E = simu_table_stacked$E ) simu_table_stacked$q05 <- openEBGM::quantBisect(5, theta_hat = theta_hat, N = simu_table_stacked$N, E = simu_table_stacked$E, qn = qn ) ## obtain the detected signal provided by openEBGM simu_table_stacked %>% subset(q05 > 1.001) ## detected signal from pvEBayes presented in the same way as openEBGM fit_gps %>% summary(return = "posterior draws") %>% apply(c(2, 3), quantile, prob = 0.05) %>% as.table() %>% as.data.frame() %>% subset(Freq > 1.001) ## End(Not run) ``` ## References DuMouchel W. Bayesian data mining in large frequency tables, with an application to the FDA spontaneous reporting system. **The American Statistician.** 1999; 1;53(3):177-90. Tan Y, Markatou M and Chakraborty S. Flexible Empirical Bayesian Approaches to Pharmacovigilance for Simultaneous Signal Detection and Signal Strength Estimation in Spontaneous Reporting Systems Data. **Statistics in Medicine.** 2025; 44: 18-19, https://doi.org/10.1002/sim.70195. Narasimhan B, Efron B. deconvolveR: A G-modeling program for deconvolution and empirical Bayes estimation. **Journal of Statistical Software**. 2020; 2;94:1-20. Koenker R, Gu J. REBayes: an R package for empirical Bayes mixture methods. **Journal of Statistical Software**. 2017; 4;82:1-26. Fu, A, Narasimhan, B, Boyd, S. CVXR: An R Package for Disciplined Convex Optimization. **Journal of Statistical Software**. 2020; 94;14:1-34.

pvEBayes function