# Bayesian regression model fitting for areal and areal spatio-temporal data. Calculates parameter estimates, validation statistics, and estimated values of several Bayesian model choice criteria. ```r Bcartime( formula, data, family, link = NULL, trials = NULL, offsetcol = NULL, formula.omega = NULL, scol = NULL, tcol = NULL, package = "CARBayes", model = "glm", AR = 1, W = NULL, adj.graph = NULL, residtype = "response", interaction = TRUE, Z = NULL, W.binary = NULL, changepoint = NULL, knots = NULL, validrows = NULL, prior.mean.delta = NULL, prior.mean.beta = NULL, prior.var.beta = NULL, prior.mean.gamma = NULL, prior.var.gamma = NULL, prior.sigma2 = NULL, prior.tau2 = c(2, 1), prior.delta = NULL, prior.var.delta = NULL, prior.lambda = NULL, prior.nu2 = c(2, 1), epsilon = 0, G = NULL, ind.area = NULL, trends = NULL, rho.T = NULL, rho.S = NULL, rho = NULL, rho.slo = NULL, rho.int = NULL, MALA = FALSE, N = 2000, burn.in = 1000, thin = 10, rseed = 44, Nchains = 4, verbose = TRUE, plotit = TRUE ) ``` ## Arguments - `formula`: An object of class "formula" (or one that can be coerced to that class): a symbolic description of the regression model to be fitted. - `data`: The data frame for which the model formula is to be fitted. The data frame should be in long format having one row for each location and time combination. The data frame must be ordered by time within each site, and should optionally have a column, named s.index, providing the site indices. Thus the data, with n sites and T times within each site, should be organized in the order: (s1, t1), (s1, t2), ... (s1, T), ... (sn, t1), ... (sn, T). The data frame should also contain two columns giving the coordinates of the locations for spatio temporal model fitting. - `family`: One of either "gaussian", "binomial","poisson" or "zip", which respectively specify a Gaussian, binomial likelihood model with the logistic link function, a Poisson likelihood model with a log link function, or a zero-inflated Poisson model with a log link function. - `link`: The link function to use for INLA based model fitting. This is ignored for the CARBayes and CARBayesST models. - `trials`: Only used if family="binomial". Either the name (character) or number of the column in the supplied data frame containing the total number of trials The program will try to access data[, trials] to get the binomial denominators. - `offsetcol`: Only used in INLA based modeling. The column name or number in the data frame that should be used as the offset. - `formula.omega`: A one-sided formula object with no response variable (left side of the "~") needed, specifying the covariates in the logistic regression model for modelling the probability of an observation being a structural zero. Each covariate (or an offset) needs to be a vector of length K*1. Only required for zero-inflated Poisson models. - `scol`: Either the name (character) or number of the column in the supplied data frame identifying the spatial units. The program will try to access data[, scol] to identify the spatial units. If this is omitted, no spatial modeling will be performed. - `tcol`: Like the `scol` argument for the time identifier. Either the name (character) or number of the column in the supplied data frame identifying the time indices. The program will try to access data[, tcol] to identify the time points. If this is omitted, no temporal modeling will be performed. - `package`: Which package is to be used in model fitting? Currently available packages are: * "inla": INLA model fitting for areal data. See if(!exists(".Rdpack.currefs")) .Rdpack.currefs \\<-new.env();Rdpack::insert_citeOnly(keys="gomezrubio;textual",package="bmstdr",cached_env=.Rdpack.currefs) . * "CARBayes": All possible models in this package can be fitted. See if(!exists(".Rdpack.currefs")) .Rdpack.currefs \\<-new.env();Rdpack::insert_citeOnly(keys="LeeCARBayes2021;textual",package="bmstdr",cached_env=.Rdpack.currefs) . * "CARBayesST": All possible models in this package can be fitted. See if(!exists(".Rdpack.currefs")) .Rdpack.currefs \\<-new.env();Rdpack::insert_citeOnly(keys="CarBayesST;textual",package="bmstdr",cached_env=.Rdpack.currefs) . Further details and more examples are provided in Chapters 10 and 11 of the book if(!exists(".Rdpack.currefs")) .Rdpack.currefs \\<-new.env();Rdpack::insert_citeOnly(keys="Sahubook;textual",package="bmstdr",cached_env=.Rdpack.currefs) . - `model`: The specific spatio temporal model to be fitted. If the package is "INLA" then the model argument should be a vector with two elements giving the spatial model as the first component. The alternatives for the spatial model are: "bym", "bym2", "besag", "besag2", "besagproper", "besagproper2", "iid" and "none". The temporal model as the second second component can be one of "iid", "rw1", "rw2", ar1" or "none". In case the model component is "none" then no spatial/temporal random effects will be fitted. No temporal random effects will be fitted in case `model` is supplied as a scalar, i.e. not a vector of two values. - `AR`: The order of the autoregressive time series process that must be either 1 or 2. - `W`: A non-negative K by K neighborhood matrix (where K is the number of spatial units). Typically a binary specification is used, where the jkth element equals one if areas (j, k) are spatially close (e.g. share a common border) and is zero otherwise. The matrix can be non-binary, but each row must contain at least one non-zero entry. This argument may not need to be specified if `adj.graph` is specified instead. - `adj.graph`: Adjacency graph which may be specified instead of the adjacency matrix matrix. This argument is used if `W` has not been supplied. The argument `W` is used in case both `W` and `adj.graph` are supplied. - `residtype`: Residual type, either "response" or "pearson", in GLM fitting with the packages CARBayes and CARBayesST. Default is "response" type observed minus fitted. The other option "pearson" is for Pearson residuals in GLM. For INLA based model fitting only the default response residuals are calculated. - `interaction`: TRUE or FALSE indicating whether the spatio-temporal interaction random effects should be included. Defaults to TRUE unless family="gaussian" in which case interactions are not allowed. - `Z`: A list, where each element is a K by K matrix of non-negative dissimilarity metrics. - `W.binary`: Logical, should the estimated neighbourhood matrix have only binary (0,1) values. - `changepoint`: A scalar indicating the position of the change point should one of the change point trend functions be included in the trends vector, i.e. if "CP" or "CT" is specified. - `knots`: A scalar indicating the number of knots to use should one of the monotonic cubic splines trend functions be included in the trends vector, i.e. if "MD" or "MI" is specified. - `validrows`: A vector providing the rows of the data frame which should be used for validation. The default NULL value instructs that validation will not be performed. - `prior.mean.delta`: A vector of prior means for the regression parameters delta (Gaussian priors are assumed) for the zero probability logistic regression component of the model. Defaults to a vector of zeros. - `prior.mean.beta`: A vector of prior means for the regression parameters beta (Gaussian priors are assumed). Defaults to a vector of zeros. - `prior.var.beta`: A vector of prior variances for the regression parameters beta (Gaussian priors are assumed). Defaults to a vector with values 100,000. - `prior.mean.gamma`: A vector of prior means for the temporal trend parameters (Gaussian priors are assumed). Defaults to a vector of zeros. - `prior.var.gamma`: A vector of prior variances for the temporal trend parameters (Gaussian priors are assumed). Defaults to a vector with values 100,000. - `prior.sigma2`: The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for sigma2. Defaults to c(1, 0.01). - `prior.tau2`: The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for tau2. Defaults to c(1, 0.01). - `prior.delta`: The prior maximum for the cluster smoothing parameter delta. Defaults to 10. - `prior.var.delta`: A vector of prior variances for the regression parameters delta (Gaussian priors are assumed) for the zero probability logistic regression component of the model. Defaults to a vector with values 100000. - `prior.lambda`: A vector of prior samples sizes for the Dirichlet prior controlling the probabilities that each trend function is chosen. The vector should be the same length as the trends vector and defaults to a vector of ones. - `prior.nu2`: The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for nu2. Defaults to c(1, 0.01) and only used if family="Gaussian". - `epsilon`: Diagonal ridge parameter to add to the random effects prior precision matrix, only required when rho = 1, and the prior precision is improper. Defaults to 0. Only used for adaptive model fitting in CARBayesST. - `G`: The maximum number of distinct intercept terms (groups) to allow in the localised model. - `ind.area`: A vector of integers the same length as the number of data points (individuals) giving which spatial unit (nunmbered from 1 to K to align with the rows of the W matrix) each individual belongs to. - `trends`: A vector containing the temporal trend functions to include in the model, which include: constant ("Constant""); linear decreasing ("LD"); linear increasing ("LI"); Known change point, where the trend can increase towards the change point before subsequently decreasing ("CP"); or decrease towards the change point before subsequently increasing ("CT"); and monotonic cubic splines which are decreasing ("MD") or increasing ("MI"). At least two trends have to be selected, with the constant trend always included. To avoid identifiability problems only one of "LI" or "MI" can be included at a given time (similarily for "LD" and "MD"). - `rho.T`: The value in the interval [0, 1] that the temporal dependence parameter rho.T is fixed at if it should not be estimated. If this argument is NULL then rho.T is estimated in the model. - `rho.S`: The value in the interval [0, 1] that the spatial dependence parameter rho.S is fixed at if it should not be estimated. If this argument is NULL then rho.S is estimated in the model. - `rho`: The value in the interval [0, 1] that the spatial dependence parameter rho is fixed at if it should not be estimated. If this argument is NULL then rho is estimated in the model. Setting rho=1, reduces the random effects prior to the intrinsic CAR model but does require epsilon>0. - `rho.slo`: The value in the interval [0, 1] that the spatial dependence parameter for the slope of the linear time trend, rho.slo, is fixed at if it should not be estimated. If this argument is NULL then rho.slo is estimated in the model. - `rho.int`: The value in the interval [0, 1] that the spatial dependence parameter for the intercept of the linear time trend, rho.int, is fixed at if it should not be estimated. If this argument is NULL then rho.int is estimated in the model. - `MALA`: Logical, should the function use Metropolis adjusted Langevin algorithm (MALA) updates (TRUE) or simple random walk (FALSE, default) updates for the regression parameters and random effects. - `N`: MCMC sample size. - `burn.in`: How many initial iterations to discard. Only relevant for MCMC based model fitting, i.e., when package is spBayes or Stan. - `thin`: The level of thinning to apply to the MCMC samples to reduce their temporal autocorrelation. Defaults to 1 (no thinning). - `rseed`: Random number seed that controls the starting point for the random number stream. A set value is required to help reproduce the results. - `Nchains`: The number of parallel Markov chains to be used in the Metropolis coupled Markov chain Monte Carlo (MCMCMC) simulations. Defaults to 4. - `verbose`: Logical scalar value: whether to print various estimates and statistics. - `plotit`: Logical scalar value: whether to plot the predictions against the observed values. ## Returns A list containing: * params - A table of parameter estimates * fit - The fitted model object. * fitteds - A vector of fitted values. * mchoice - Calculated model choice statistics if those have been requested by the input argument `mchoice=T`. Not all model fits will contain all the model choice statistics. * residuals - A vector of residual values. * stats - The four validation statistics: rmse, mae, crps and coverage. This is present only if model validation has been performed. * yobs_preds - A data frame containing the validation rows of the model fitting data frame. The last five columns of this data frame contains the validation prediction summaries: mean, sd, median, and 95% prediction interval. This is present only if model validation has been performed. * valpreds - A matrix containing the MCMC samples of the validation predictions. The dimension of this matrix is the number of validations times the number of retained MCMC samples. This is present only if model validation has been performed. * validationplots - Present only if validation has been performed. Contains three validation plots with or without segment and an ordinary plot. See `obs_v_pred_plot` for more. * sn - The number of areal units used in fitting. * tn - The number of time points used in fitting. * formula - The input formula for the regression part of the model. * scale.transform - It is there for compatibility with `Bsptime` output. * package - The name of the package used for model fitting. * model - The name of the fitted model. * call - The command used to call the model fitting function. * computation.time - Computation time required to run the model fitting. ## Examples ```r # Set the validation row numbers vs <- sample(nrow(engtotals), 5) # Total number of iterations N <- 60 # Number of burn in iterations burn.in <- 10 # Number of thinning iterations thin <- 1 # Set the model formula for binomial distribution based modeling f1 <- noofhighweeks ~ jsa + log10(houseprice) + log(popdensity) + sqrt(no2) ## Independent error logistic regression M1 <- Bcartime(formula = f1, data = engtotals, family = "binomial", trials = "nweek", N = N, burn.in = burn.in, thin = thin, verbose = TRUE) summary(M1) # Leroux model M1.leroux <- Bcartime(formula = f1, data = engtotals, scol = "spaceid", model = "leroux", W = Weng, family = "binomial", trials = "nweek", N = N, burn.in = burn.in, thin = thin) summary(M1.leroux) # BYM model M1.bym <- Bcartime(formula = f1, data = engtotals, scol = "spaceid", model = "bym", W = Weng, family = "binomial", trials = "nweek", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M1.bym) # Validation for the Leroux model M1.leroux.v <- Bcartime(formula = f1, data = engtotals, scol = "spaceid", model = "leroux", W = Weng, family = "binomial", trials = "nweek", validrows = vs, N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M1.leroux.v) ## Poisson Distribution based models #################################### # Model formula f2 <- covid ~ offset(logEdeaths) + jsa + log10(houseprice) + log(popdensity) + sqrt(no2) # Independent error Poisson regression M2 <- Bcartime(formula = f2, data = engtotals, family = "poisson", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M2) ## Poisson regression with Leroux Model M2.leroux <- Bcartime(formula = f2, data = engtotals, scol = "spaceid", model = "leroux", family = "poisson", W = Weng, N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M2.leroux) # Poisson regression with BYM Model M2.bym <- Bcartime(formula = f2, data = engtotals, scol = "spaceid", model = "bym", family = "poisson", W = Weng, N = N, burn.in = burn.in, thin = thin) summary(M2.bym) ## Gaussian distribution based models ############### f3 <- sqrt(no2) ~ jsa + log10(houseprice) + log(popdensity) # Independent error model M3 <- Bcartime(formula = f3, data = engtotals, family = "gaussian", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M3) # Leroux model M3.leroux <- Bcartime(formula = f3, data = engtotals, scol = "spaceid", model = "leroux", family = "gaussian", W = Weng, N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M3.leroux) ## Validation M3.leroux.v <- Bcartime(formula = f3, data = engtotals, scol = "spaceid", model = "leroux", family = "gaussian", W = Weng, N = N, burn.in = burn.in, thin = thin, validrows = vs, verbose = FALSE) summary(M3.leroux.v) ## Spatio-temporal modeling ################################################## head(engdeaths) dim(engdeaths) colnames(engdeaths) vs <- sample(nrow(engdeaths), 5) ## Binomial distribution engdeaths$nweek <- rep(1, nrow(engdeaths)) f1 <- highdeathsmr ~ jsa + log10(houseprice) + log(popdensity) M1st_linear <- Bcartime(formula = f1, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", trials = "nweek", W = Weng, model = "linear", family = "binomial", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = TRUE) summary(M1st_linear) M1st_sepspat <- Bcartime(formula = f1, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", trials = "nweek", W = Weng, model = "sepspatial", family = "binomial", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M1st_sepspat) M1st_ar <- Bcartime(formula = f1, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", trials = "nweek", W = Weng, model = "ar", AR = 1, family = "binomial", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M1st_ar) # Model validation M1st_ar.v <- Bcartime(formula = f1, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", trials = "nweek", W = Weng, model = "ar", AR = 1, family = "binomial", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, validrows = vs, verbose = FALSE) summary(M1st_ar.v) ## Spatio temporal Poisson models################################### colnames(engdeaths) f2 <- covid ~ offset(logEdeaths) + jsa + log10(houseprice) + log(popdensity) + n0 M2st_linear <- Bcartime(formula = f2, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "linear", family = "poisson", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M2st_linear) M2st_anova <- Bcartime(formula = f2, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "anova", family = "poisson", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M2st_anova) M2st_anova_nointer <- Bcartime(formula = f2, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "anova", interaction = FALSE, family = "poisson", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M2st_anova_nointer) M2st_sepspat <- Bcartime(formula = f2, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "sepspatial", family = "poisson", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M2st_sepspat) M2st_ar <- Bcartime(formula = f2, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "ar", AR = 1, family = "poisson", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M2st_ar) M2st_ar.v <- Bcartime(formula = f2, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "ar", family = "poisson", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, validrows = vs, verbose = FALSE) M2st_anova.v <- Bcartime(formula = f2, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "anova", family = "poisson", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, validrows = vs, verbose = FALSE) summary(M2st_ar.v) summary(M2st_anova.v) ## Spatio-temporal Normal models ############################### colnames(engdeaths) f3 <- sqrt(no2) ~ jsa + log10(houseprice) + log(popdensity) M3st_linear <- Bcartime(formula = f3, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "linear", family = "gaussian", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M3st_linear) M3st_anova <- Bcartime(formula = f3, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "anova", family = "gaussian", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M3st_anova) M3st_anova_nointer <- Bcartime(formula = f3, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "anova", interaction = FALSE, family = "gaussian", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M3st_anova_nointer) M3st_ar <- Bcartime(formula = f3, data = engdeaths, scol = "spaceid", tcol = "Weeknumber", W = Weng, model = "ar", AR = 2, family = "gaussian", package = "CARBayesST", N = N, burn.in = burn.in, thin = thin, verbose = FALSE) summary(M3st_ar) # Execute the following examples if INLA is available if (require(INLA)) { N <- 55 burn.in <- 5 thin <- 1 vs <- sample(nrow(engtotals), 5) # Spatial Binomial GLM f1 <- noofhighweeks ~ jsa + log10(houseprice) + log(popdensity) + sqrt(no2) M1.inla.bym <- Bcartime(data = engtotals, formula = f1, W = Weng, scol = "spaceid", model = c("bym"), package = "inla", family = "binomial", link="logit", trials = "nweek", N = N, burn.in = burn.in, thin = thin) summary(M1.inla.bym) ## Spatial only Poisson f2inla <- covid ~ jsa + log10(houseprice) + log(popdensity) + sqrt(no2) M2.inla.bym <- Bcartime(data = engtotals, formula = f2inla, W = Weng, scol = "spaceid", offsetcol = "logEdeaths", model = c("bym"), package = "inla", link = "log", family = "poisson", N = N, burn.in = burn.in, thin = thin) summary(M2.inla.bym) ## Normal models f3 <- sqrt(no2) ~ jsa + log10(houseprice) + log(popdensity) ## Fit BYM with iid random effect M3.inla.bym <- Bcartime(data = engtotals, formula = f3, W = Weng, scol = "spaceid", model = c("bym"), package = "inla", family = "gaussian", N = N, burn.in = burn.in, thin = thin) summary(M3.inla.bym) # validation M3.inla.bym.v <- Bcartime(data = engtotals, formula = f3, W = Weng, scol = "spaceid", model = c("bym"), package = "inla", family = "gaussian", validrows = vs, N = N, burn.in = burn.in, thin = thin) summary(M3.inla.bym.v) ###### Spatio-temporal INLA models f1 <- highdeathsmr ~ jsa + log10(houseprice) + log(popdensity) nweek <- rep(1, nrow(engdeaths)) engdeaths$nweek <- rep(1, nrow(engdeaths)) ## INLA Binomial model <- c("bym", "ar1") M1st_inla.bym <- Bcartime(data = engdeaths, formula = f1, W = Weng, scol = "spaceid", tcol = "Weeknumber", model = model, trials = "nweek", family = "binomial", link="logit", package = "inla", N = N, burn.in = burn.in, thin = thin) summary(M1st_inla.bym) M1st_inla_v <- Bcartime(data = engdeaths, formula = f1, W = Weng, scol = "spaceid", tcol = "Weeknumber", offsetcol = "logEdeaths", model = model, trials = "nweek", family = "binomial", link="logit", package = "inla", validrow = vs, N = N, burn.in = burn.in, thin = thin) summary(M1st_inla_v) model <- c("bym", "none") M1st_inla.bym.none <- Bcartime(data = engdeaths, formula = f1, W = Weng, scol = "spaceid", tcol = "Weeknumber", model = model, trials = "nweek", family = "binomial", link="logit", package = "inla", N = N, burn.in = burn.in, thin = thin) summary(M1st_inla.bym.none) model <- c("bym") M1st_inla.bym.none <- Bcartime(data = engdeaths, formula = f1, W = Weng, scol = "spaceid", tcol = "Weeknumber", model = model, trials = "nweek", family = "binomial", link="logit", package = "inla", N = N, burn.in = burn.in, thin = thin) summary(M1st_inla.bym.none) ## Poisson models f2inla <- covid ~ jsa + log10(houseprice) + log(popdensity) + n0 + n1 + n2 + n3 model <- c("bym", "ar1") M2stinla <- Bcartime(data = engdeaths, formula = f2inla, W = Weng, scol = "spaceid", tcol = "Weeknumber", offsetcol = "logEdeaths", model = model, link = "log", family = "poisson", package = "inla", N = N, burn.in = burn.in, thin = thin) summary(M2stinla) M2stinla.v <- Bcartime(data = engdeaths, formula = f2inla, W = Weng, scol = "spaceid", tcol = "Weeknumber", offsetcol = "logEdeaths", model = model, link = "log", family = "poisson", package = "inla", validrows = vs, N = N, burn.in = burn.in, thin = thin) summary(M2stinla.v) ## Normal models f3 <- sqrt(no2) ~ jsa + log10(houseprice) + log(popdensity) model <- c("bym", "iid") M3inla.bym.iid <- Bcartime(data = engdeaths, formula = f3, W = Weng, scol = "spaceid", tcol = "Weeknumber", model = model, family = "gaussian", package = "inla", validrows = vs, N = N, burn.in = burn.in, thin = thin) summary(M3inla.bym.iid) model <- c("bym", "ar1") M3inla.bym.ar1 <- Bcartime(data = engdeaths, formula = f3, W = Weng, scol = "spaceid", tcol = "Weeknumber", model = model, family = "gaussian", package = "inla", validrows = vs, N = N, burn.in = burn.in, thin = thin) summary(M3inla.bym.ar1) } ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs)

Bcartime function