# Predicted Probabilities using Bayesian MCMC estimates for the Average of Observed Cases Implements R function to calculate the predicted probabilities for "observed" cases after a Bayesian logit or probit model, following Hanmer & Kalkan (2013) (2013, American Journal of Political Science 57(1): 263-277). ```r mcmcObsProb( modelmatrix, mcmcout, xcol, xrange, xinterest, link = "logit", ci = c(0.025, 0.975), fullsims = FALSE ) ``` ## Arguments - `modelmatrix`: model matrix, including intercept (if the intercept is among the parameters estimated in the model). Create with model.matrix(formula, data). Note: the order of columns in the model matrix must correspond to the order of columns in the matrix of posterior draws in the `mcmcout` argument. See the `mcmcout` argument for more. - `mcmcout`: posterior distributions of all logit coefficients, in matrix form. This can be created from rstan, MCMCpack, R2jags, etc. and transformed into a matrix using the function as.mcmc() from the coda package for `jags` class objects, as.matrix() from base R for `mcmc`, `mcmc.list`, `stanreg`, and `stanfit` class objects, and `object$sims.matrix` for `bugs` class objects. Note: the order of columns in this matrix must correspond to the order of columns in the model matrix. One can do this by examining the posterior distribution matrix and sorting the variables in the order of this matrix when creating the model matrix. A useful function for sorting column names containing both characters and numbers as you create the matrix of posterior distributions is `mixedsort()` from the gtools package. - `xcol`: column number of the posterior draws (`mcmcout`) and model matrices that corresponds to the explanatory variable for which to calculate associated Pr(y = 1). Note that the columns in these matrices must match. - `xrange`: name of the vector with the range of relevant values of the explanatory variable for which to calculate associated Pr(y = 1). - `xinterest`: semi-optional argument. Name of the explanatory variable for which to calculate associated Pr(y = 1). If `xcol` is supplied, this is not needed. If both are supplied, the function defaults to `xcol` and this argument is ignored. - `link`: type of generalized linear model; a character vector set to `"logit"` (default) or `"probit"`. - `ci`: the bounds of the credible interval. Default is `c(0.025, 0.975)` for the 95% credible interval. - `fullsims`: logical indicator of whether full object (based on all MCMC draws rather than their average) will be returned. Default is `FALSE`. Note: The longer `xrange` is, the larger the full output will be if `TRUE` is selected. ## Returns if `fullsims = FALSE` (default), a tibble with 4 columns: * x: value of variable of interest, drawn from `xrange` * median_pp: median predicted Pr(y = 1) when variable of interest is set to x * lower_pp: lower bound of credible interval of predicted probability at given x * upper_pp: upper bound of credible interval of predicted probability at given x if `fullsims = TRUE`, a tibble with 3 columns: * Iteration: number of the posterior draw * x: value of variable of interest, drawn from `xrange` * pp: average predicted Pr(y = 1) of all observed cases when variable of interest is set to x ## Details This function calculates predicted probabilities for "observed" cases after a Bayesian logit or probit model following Hanmer and Kalkan (2013, American Journal of Political Science 57(1): 263-277) ## Examples ```r if (interactive()) { ## simulating data set.seed(12345) b0 <- 0.2 # true value for the intercept b1 <- 0.5 # true value for first beta b2 <- 0.7 # true value for second beta n <- 500 # sample size X1 <- runif(n, -1, 1) X2 <- runif(n, -1, 1) Z <- b0 + b1 * X1 + b2 * X2 pr <- 1 / (1 + exp(-Z)) # inv logit function Y <- rbinom(n, 1, pr) df <- data.frame(cbind(X1, X2, Y)) ## formatting the data for jags datjags <- as.list(df) datjags$N <- length(datjags$Y) ## creating jags model model <- function() { for(i in 1:N){ Y[i] ~ dbern(p[i]) ## Bernoulli distribution of y_i logit(p[i]) <- mu[i] ## Logit link function mu[i] <- b[1] + b[2] * X1[i] + b[3] * X2[i] } for(j in 1:3){ b[j] ~ dnorm(0, 0.001) ## Use a coefficient vector for simplicity } } params <- c("b") inits1 <- list("b" = rep(0, 3)) inits2 <- list("b" = rep(0, 3)) inits <- list(inits1, inits2) ## fitting the model with R2jags library(R2jags) set.seed(123) fit <- jags(data = datjags, inits = inits, parameters.to.save = params, n.chains = 2, n.iter = 2000, n.burnin = 1000, model.file = model) ### observed value approach library(coda) xmat <- model.matrix(Y ~ X1 + X2, data = df) mcmc <- as.mcmc(fit) mcmc_mat <- as.matrix(mcmc)[, 1:ncol(xmat)] X1_sim <- seq(from = min(datjags$X1), to = max(datjags$X1), length.out = 10) obs_prob <- mcmcObsProb(modelmatrix = xmat, mcmcout = mcmc_mat, xrange = X1_sim, xcol = 2) } ``` ## References Hanmer, Michael J., & Ozan Kalkan, K. (2013). Behind the curve: Clarifying the best approach to calculating predicted probabilities and marginal effects from limited dependent variable models. American Journal of Political Science, 57(1), 263-277. https://doi.org/10.1111/j.1540-5907.2012.00602.x

mcmcObsProb function