# Fits the second stage model in the two-stage federated data analysis approach. The function 'FederatedApproachStage2()' fits the second stage model of the two-stage federated data analysis approach to assess surrogacy. ```r FederatedApproachStage2( Dataset, Intercept.S, alpha, Intercept.T, beta, sigma.SS, sigma.ST, sigma.TT, Obs.per.trial, Trial.ID, R.list, Alpha = 0.05 ) ``` ## Arguments - `Dataset`: A data frame with the correct columns (See Data Format). - `Intercept.S`: Estimated intercepts for the surrogate endpoint. - `alpha`: Estimated treatment effects for the surrogate endpoint. - `Intercept.T`: Estimated intercepts for the true endpoint. - `beta`: Estimated treatment effects for the true endpoint. - `sigma.SS`: Estimated variance of the error terms for the surrogate endpoint. - `sigma.ST`: Estimated covariance between the error terms of the surrogate and true endpoint. - `sigma.TT`: Estimated variance of the error terms for the true endpoint. - `Obs.per.trial`: Number of subjects in the trial. - `Trial.ID`: Trial indicator. - `R.list`: List of the variance-covariance matrices of the fixed effects. - `Alpha`: The $\alpha$-level that is used to determine the confidence intervals around $R^2_{trial}$ and $R^2_{indiv}$. Default 0.05. ## Returns Returns an object of class "FederatedApproachStage2()" that can be used to evaluate surrogacy. * Indiv.R2: a data frame that contains the $R^2_{indiv}$ and 95% confidence interval to evaluate surrogacy at the trial level. * Trial.R2: a data frame that contains the $R^2_{trial}$ and 95% confidence interval to evaluate surrogacy at the trial level. * Fixed.Effects: a data frame that contains the average of the estimated fixed effects. * D: estimated $D$ matrix. * Obs.Per.Trial: number of observations in each trial. ## Model The two-stage federated data analysis approach that can be used to assess surrogacy in the meta-analytic multiple-trial setting (Continuous-continuous case), but without the need of sharing data. Instead, each organization conducts separate analyses on their data set using the so-called "first stage" model. The results of these analyses are then aggregated at a central analysis hub, where the aggregated results are analyzed using a "second stage" model and the necessary metrics ($R^2_{trial}$ and $R^2_{indiv}$) for the validation of the surrogate endpoint are obtained. This function fits the second stage model, where a method-of-moments estimator is used to obtain the variance-covariance matrix $D$ from which the $R^2_{trial}$ can be derived. The $R^2_{indiv}$ is obtained with a weighted average of the elements in $\Sigma_i$. ## Data Format A data frame that combines the results of the first stage models and contains: * a column with the trial indicator * a column with the number of subjects in the trial * a column with the estimated intercepts for the surrogate * a column with the estimated treatment effects for the surrogate * a column with the estimated intercepts for the true endpoint * a column with the estimated treatment effects for the true endpoint * a column with the variances of the error term for the surrogate endpoint * a column with the covariances between the error terms of the surrogate and true endpoint * a column with the variances of the error term for the true endpoint A list that combines all the variance-covariance matrices of the fixed effects obtained using the first stage model ## Examples ```r ## Not run: #As an example, the federated data analysis approach can be applied to the Schizo data set data(Schizo) Schizo <- Schizo[order(Schizo$InvestId, Schizo$Id),] #Create separate datasets for each investigator Schizo_datasets <- list() for (invest_id in 1:198) { Schizo_datasets[[invest_id]] <- Schizo[Schizo$InvestId == invest_id, ] assign(paste0("Schizo", invest_id), Schizo_datasets[[invest_id]]) } #Fit the first stage model for each dataset separately results_stage1 <- list() invest_ids <- list() i <- 1 for (invest_id in 1:198) { dataset <- Schizo_datasets[[invest_id]] skip_to_next <- FALSE tryCatch(FederatedApproachStage1(dataset, Surr=CGI, True=PANSS, Treat=Treat, Trial.ID = InvestId, Min.Treat.Size = 5, Alpha = 0.05), error = function(e) { skip_to_next <<- TRUE}) #if the trial does not have the minimum required number, skip to the next if(skip_to_next) { next } results_stage1[[invest_id]] <- FederatedApproachStage1(dataset, Surr=CGI, True=PANSS, Treat=Treat, Trial.ID = InvestId, Min.Treat.Size = 5, Alpha = 0.05) assign(paste0("stage1_invest", invest_id), results_stage1[[invest_id]]) invest_ids[[i]] <- invest_id #keep a list of ids with datasets with required number of patients i <- i+1 } invest_ids <- unlist(invest_ids) invest_ids #Combine the results of the first stage models for (invest_id in invest_ids) { dataset <- results_stage1[[invest_id]]$Results.Stage.1 if (invest_id == invest_ids[1]) { all_results_stage1<- dataset } else { all_results_stage1 <- rbind(all_results_stage1,dataset) } } all_results_stage1 #that combines the results of the first stage models R.list <- list() i <- 1 for (invest_id in invest_ids) { R <- results_stage1[[invest_id]]$R.i R.list[[i]] <- as.matrix(R[1:4,1:4]) i <- i+1 } R.list #list that combines all the variance-covariance matrices of the fixed effects fit <- FederatedApproachStage2(Dataset = all_results_stage1, Intercept.S = Intercept.S, alpha = alpha, Intercept.T = Intercept.T, beta = beta, sigma.SS = sigma.SS, sigma.ST = sigma.ST, sigma.TT = sigma.TT, Obs.per.trial = n, Trial.ID = Trial.ID, R.list = R.list) summary(fit) ## End(Not run) ``` ## References Florez, A. J., Molenberghs G, Verbeke G, Alonso, A. (2019). A closed-form estimator for metaanalysis and surrogate markers evaluation. Journal of Biopharmaceutical Statistics, 29(2) 318-332. ## Author(s) Dries De Witte

FederatedApproachStage2 function