Predicted Probabilities using Bayesian MCMC estimates for the "Average" Case

This function calculates predicted probabilities for "average" cases after a Bayesian logit or probit model. As "average" cases, this function calculates the median value of each predictor. For an explanation of predicted probabilities for "average" cases, see e.g. King, Tomz & Wittenberg (2000, American Journal of Political Science 44(2): 347-361).

mcmcAveProb(
  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.

Value

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, holding all other predictors to average (median) values
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) when variable of interest is set to x, holding all other predictors to average (median) values

Details

This function calculates predicted probabilities for "average" cases after a Bayesian logit or probit model. For an explanation of predicted probabilities for "average" cases, see e.g. King, Tomz & Wittenberg (2000, American Journal of Political Science 44(2): 347-361)

References

King, Gary, Michael Tomz, and Jason Wittenberg. 2000. “Making the Most of Statistical Analyses: Improving Interpretation and Presentation.” American Journal of Political Science 44 (2): 347–61. http://www.jstor.org/stable/2669316

Examples

# \donttest{
if (interactive()) {
  ## simulating data
  set.seed(123)
  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)

### average 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)
ave_prob <- mcmcAveProb(modelmatrix = xmat,
                        mcmcout = mcmc_mat,
                        xrange = X1_sim, 
                        xcol = 2)
}
# }