# Cluster-Period GEE for Estimating the Mean and Correlation Parameters in Cross-Sectional SW-CRTs cpgeeSWD implements the cluster-period GEE developed for cross-sectional stepped wedge cluster randomized trials (SW-CRTs). It provides valid estimation and inference for the treatment effect and intraclass correlation parameters within the GEE framework, and is computationally efficient for SW-CRTs with large cluster sizes. The program currently only allows for a marginal mean model with discrete period effects and the intervention indicator without additional covariates. The program offers bias-corrected ICC estimates as well as bias-corrected sandwich variances for both the treatment effect parameter and the ICC parameters. The technical details of the cluster-period GEE approach are provided in Li et al. (2020+). ```r cpgeeSWD( y, X, id, m, corstr, family = "binomial", maxiter = 500, epsilon = 0.001, printrange = TRUE, alpadj = FALSE, rho.init = NULL ) ``` ## Arguments - `y`: a vector specifying the cluster-period means (proportions) - `X`: design matrix for the marginal mean model, including period indicator and intervention indicator - `id`: a vector specifying cluster identifier - `m`: a vector of the cluster-period sizes - `corstr`: correlation structure specified for the individual-level outcomes, could be `'exchangeable'`, `'nest_exch'` or `'exp_decay'` - `family`: See corresponding documentation to `glm`. The current version only supports `family = 'binomial'` - `maxiter`: maximum number of iterations for Fisher scoring updates - `epsilon`: tolerance for convergence - `printrange`: print details of range violations when `family = 'binomial'`. The default is `TRUE` - `alpadj`: if `TRUE`, performs bias adjustment for the alpha estimating equations. The default is `FALSE` - `rho.init`: user-specified initial value for the decay parameter when `corstr = 'exp_decay'` ## Returns `outbeta` estimates of marginal mean model parameters and standard errors with different finite-sample bias corrections. The current version supports model-based standard error (MB), the sandwich standard error (BC0) extending Zhao and Prentice (2001), the sandwich standard errors (BC1) extending Kauermann and Carroll (2001), the sandwich standard errors (BC2) extending Mancl and DeRouen (2001), and the sandwich standard errors (BC3) extending the Fay and Graubard (2001). A summary of these bias-corrections can also be found in Lu et al. (2007), and Li et al. (2018). `outalpha` estimates of correlation parameters and standard errors with different finite-sample bias corrections. The current version supports the sandwich standard error (BC0) extending Zhao and Prentice (2001), the sandwich standard errors (BC1) extending Kauermann and Carroll (2001), the sandwich standard errors (BC2) extending Mancl and DeRouen (2001), and the sandwich standard errors (BC3) extending the Fay and Graubard (2001). A summary of these bias-corrections can also be found in Preisser et al. (2008). `beta` a vector of estimates for marginal mean model parameters `alpha` a vector of estimates of correlation parameters `MB` model-based covariance estimate for the marginal mean model parameters `BC0` robust sandwich covariance estimate of the marginal mean model and correlation parameters `BC1` robust sandwich covariance estimate of the marginal mean model and correlation parameters with the Kauermann and Carroll (2001) correction `BC2` robust sandwich covariance estimate of the marginal mean model and correlation parameters with the Mancl and DeRouen (2001) correction `BC3` robust sandwich covariance estimate of the marginal mean model and correlation parameters with the Fay and Graubard (2001) correction `niter` number of iterations used in the Fisher scoring updates for model fitting ## Examples ```r # Simulated SW-CRT example with binary outcome ######################################################################## ### Example 1): simulated SW-CRT with smaller cluster-period sizes (5~10) ######################################################################## sampleSWCRT <- sampleSWCRTSmall ############################################################# ### cluster-period id, period, outcome, and design matrix ### ############################################################# ### id, period, outcome id <- sampleSWCRT$id period <- sampleSWCRT$period y <- sampleSWCRT$y_bin X <- as.matrix(sampleSWCRT[, c("period1", "period2", "period3", "period4", "treatment")]) m <- as.matrix(table(id, period)) n <- dim(m)[1] t <- dim(m)[2] clp_mu <- tapply(y, list(id, period), FUN = mean) y_cp <- c(t(clp_mu)) ### design matrix for correlation parameters trt <- tapply(X[, t + 1], list(id, period), FUN = mean) trt <- c(t(trt)) time <- tapply(period, list(id, period), FUN = mean) time <- c(t(time)) X_cp <- matrix(0, n * t, t) s <- 1 for (i in 1:n) { for (j in 1:t) { X_cp[s, time[s]] <- 1 s <- s + 1 } } X_cp <- cbind(X_cp, trt) id_cp <- rep(1:n, each = t) m_cp <- c(t(m)) ##################################################### ### cluster-period matrix-adjusted estimating equations (MAEE) ### with exchangeable, nested exchangeable and exponential decay correlation structures ### ##################################################### # exchangeable est_maee_exc <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "exchangeable", alpadj = TRUE ) print(est_maee_exc) # nested exchangeable est_maee_nex <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "nest_exch", alpadj = TRUE ) print(est_maee_nex) # exponential decay est_maee_ed <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "exp_decay", alpadj = TRUE ) print(est_maee_ed) ##################################################### ### cluster-period GEE ### with exchangeable, nested exchangeable and exponential decay correlation structures ### ##################################################### # exchangeable est_uee_exc <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "exchangeable", alpadj = FALSE ) print(est_uee_exc) # nested exchangeable est_uee_nex <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "nest_exch", alpadj = FALSE ) print(est_uee_nex) # exponential decay est_uee_ed <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "exp_decay", alpadj = FALSE ) print(est_uee_ed) ######################################################################## ### Example 2): simulated SW-CRT with larger cluster-period sizes (20~30) ######################################################################## sampleSWCRT <- sampleSWCRTLarge ############################################################# ### cluster-period id, period, outcome, and design matrix ### ############################################################# ### id, period, outcome id <- sampleSWCRT$id period <- sampleSWCRT$period y <- sampleSWCRT$y_bin X <- as.matrix(sampleSWCRT[, c("period1", "period2", "period3", "period4", "period5", "treatment")]) m <- as.matrix(table(id, period)) n <- dim(m)[1] t <- dim(m)[2] clp_mu <- tapply(y, list(id, period), FUN = mean) y_cp <- c(t(clp_mu)) ### design matrix for correlation parameters trt <- tapply(X[, t + 1], list(id, period), FUN = mean) trt <- c(t(trt)) time <- tapply(period, list(id, period), FUN = mean) time <- c(t(time)) X_cp <- matrix(0, n * t, t) s <- 1 for (i in 1:n) { for (j in 1:t) { X_cp[s, time[s]] <- 1 s <- s + 1 } } X_cp <- cbind(X_cp, trt) id_cp <- rep(1:n, each = t) m_cp <- c(t(m)) ##################################################### ### cluster-period matrix-adjusted estimating equations (MAEE) ### with exchangeable, nested exchangeable and exponential decay correlation structures ### ##################################################### # exchangeable est_maee_exc <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "exchangeable", alpadj = TRUE ) print(est_maee_exc) # nested exchangeable est_maee_nex <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "nest_exch", alpadj = TRUE ) print(est_maee_nex) # exponential decay est_maee_ed <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "exp_decay", alpadj = TRUE ) print(est_maee_ed) ##################################################### ### cluster-period GEE ### with exchangeable, nested exchangeable and exponential decay correlation structures ### ##################################################### # exchangeable est_uee_exc <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "exchangeable", alpadj = FALSE ) print(est_uee_exc) # nested exchangeable est_uee_nex <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "nest_exch", alpadj = FALSE ) print(est_uee_nex) # exponential decay est_uee_ed <- cpgeeSWD( y = y_cp, X = X_cp, id = id_cp, m = m_cp, corstr = "exp_decay", alpadj = FALSE ) print(est_uee_ed) ``` ## References Zhao, L. P., Prentice, R. L. (1990). Correlated binary regression using a quadratic exponential model. Biometrika, 77(3), 642-648. Mancl, L. A., DeRouen, T. A. (2001). A covariance estimator for GEE with improved small sample properties. Biometrics, 57(1), 126-134. Kauermann, G., Carroll, R. J. (2001). A note on the efficiency of sandwich covariance matrix estimation. Journal of the American Statistical Association, 96(456), 1387-1396. Fay, M. P., Graubard, B. I. (2001). Small sample adjustments for Wald type tests using sandwich estimators. Biometrics, 57(4), 1198-1206. Lu, B., Preisser, J. S., Qaqish, B. F., Suchindran, C., Bangdiwala, S. I., Wolfson, M. (2007). A comparison of two bias corrected covariance estimators for generalized estimating equations. Biometrics, 63(3), 935-941. Preisser, J. S., Lu, B., Qaqish, B. F. (2008). Finite sample adjustments in estimating equations and covariance estimators for intracluster correlations. Statistics in Medicine, 27(27), 5764-5785. Li, F., Turner, E. L., Preisser, J. S. (2018). Sample size determination for GEE analyses of stepped wedge cluster randomized trials. Biometrics, 74(4), 1450-1458. Li, F. (2020). Design and analysis considerations for cohort stepped wedge cluster randomized trials with a decay correlation structure. Statistics in Medicine, 39(4), 438-455. Li, F., Yu, H., Rathouz, P., Turner, E. L., Preisser, J. S. (2021). Marginal modeling of cluster-period means and intraclass correlations in stepped wedge designs with binary outcomes. Biostatistics, kxaa056. ## Author(s) Hengshi Yu , Fan Li , Paul Rathouz , Elizabeth L. Turner , John Preisser

cpgeeSWD function