Zero Truncated Binomial. param_estimate and stats_tbl

[spsanderson]

Parameter Estimate Function

Function:

#' Estimate Zero Truncated Binomial Parameters
#'
#' @family Parameter Estimation
#' @family Binomial
#' @family Zero Truncated Distribution
#'
#' @author Steven P. Sanderson II, MPH
#'
#' @details This function will attempt to estimate the zero truncated  
#' binomial size and prob parameters given some vector of values.
#'
#' @description The function will return a list output by default, and  if the parameter
#' `.auto_gen_empirical` is set to `TRUE` then the empirical data given to the
#' parameter `.x` will be run through the `tidy_empirical()` function and combined
#' with the estimated binomial data.
#'
#' One method of estimating the parameters is done via:
#' -  MLE via \code{\link[stats]{optim}} function.
#'
#' @param .x The vector of data to be passed to the function.
#' @param .auto_gen_empirical This is a boolean value of TRUE/FALSE with default
#' set to TRUE. This will automatically create the `tidy_empirical()` output
#' for the `.x` parameter and use the `tidy_combine_distributions()`. The user
#' can then plot out the data using `$combined_data_tbl` from the function output.
#'
#' @examples
#' library(dplyr)
#' library(ggplot2)
#' library(actuar)
#' 
#' x <- as.integer(mtcars$mpg)
#' output <- util_zero_truncated_binomial_param_estimate(x)
#'
#' output$parameter_tbl
#'
#' output$combined_data_tbl |>
#'   tidy_combined_autoplot()
#'
#' set.seed(123)
#' t <- rztbinom(100, 10, .1)
#' util_zero_truncated_binomial_param_estimate(t)$parameter_tbl
#'
#' @return
#' A tibble/list
#' 
#' @name util_zero_truncated_binomial_param_estimate
NULL

#' @export
#' @rdname util_zero_truncated_binomial_param_estimate

util_zero_truncated_binomial_param_estimate <- function(.x, .auto_gen_empirical = TRUE) {
  
  # Check if actuar library is installed
  if (!requireNamespace("actuar", quietly = TRUE)) {
    stop("The 'actuar' package is needed for this function. Please install it with: install.packages('actuar')")
  }
  
  # Tidyeval ----
  x_term <- as.numeric(.x)
  sum_x <- sum(x_term, na.rm = TRUE)
  minx <- min(x_term)
  maxx <- max(x_term)
  m <- mean(x_term, na.rm = TRUE)
  n <- length(x_term)
  
  # Negative log-likelihood function for zero-truncated binomial distribution
  nll_func <- function(par) {
    n <- par[1]
    p <- par[2]
    if (n <= 0 || p <= 0 || p >= 1) {
      return(-Inf)
    }
    -sum(actuar::dztbinom(x_term, size = n, prob = p, log = TRUE))
  }
  
  # Initial parameter guesses 
  initial_params <- c(size = max(x_term), prob = 0.5)  # Adjust based on your data
  
  # Optimization using optim()
  optim_result <- optim(
    par = initial_params, 
    fn = nll_func
  ) |>
    suppressWarnings()
  
  # Extract estimated parameters
  mle_size <- optim_result$par[1]
  mle_prob <- optim_result$par[2]
  mle_msg  <- optim_result$message
  
  # Create output tibble
  ret <- tibble::tibble(
    dist_type = "Zero-Truncated Binomial",
    samp_size = n,
    min = minx,
    max = maxx,
    mean = m,
    method = "MLE_Optim",
    size = mle_size,
    prob = mle_prob,
    message = mle_msg
  )
  
  # Attach attributes
  attr(ret, "tibble_type") <- "parameter_estimation"
  attr(ret, "family") <- "zero_truncated_binomial"
  attr(ret, "x_term") <- .x
  attr(ret, "n") <- n
  
  if (.auto_gen_empirical) {
    # Generate empirical data
    # Assuming tidy_empirical and tidy_combine_distributions functions exist
    te <- tidy_empirical(.x = .x)
    td <- tidy_zero_truncated_binomial(
      .n = n,
      .size = round(mle_size, 3),
      .prob = round(mle_prob, 3)
    )
    combined_tbl <- tidy_combine_distributions(te, td)
    
    output <- list(
      combined_data_tbl = combined_tbl,
      parameter_tbl = ret
    )
  } else {
    output <- list(
      parameter_tbl = ret
    )
  }
  
  return(output)
}

Example:

> library(dplyr)
> library(ggplot2)
> library(actuar)
> 
> x <- as.integer(mtcars$mpg)
> output <- util_zero_truncated_binomial_param_estimate(x)
> 
> output$parameter_tbl
# A tibble: 1 × 8
  dist_type               samp_size   min   max  mean method     size  prob
  <chr>                       <int> <dbl> <dbl> <dbl> <chr>     <dbl> <dbl>
1 Zero-Truncated Binomial        32    10    33  19.7 MLE_Optim    33 0.597
> 
> output$combined_data_tbl |>
+   tidy_combined_autoplot()
> 
> set.seed(123)
> t <- rztbinom(100, 10, .1)
> util_zero_truncated_binomial_param_estimate(t)$parameter_tbl
# A tibble: 1 × 8
  dist_type               samp_size   min   max  mean method     size  prob
  <chr>                       <int> <dbl> <dbl> <dbl> <chr>     <dbl> <dbl>
1 Zero-Truncated Binomial       100     1     4  1.53 MLE_Optim  4.00 0.500

AIC

Function:

#' Calculate Akaike Information Criterion (AIC) for Zero-Truncated Binomial Distribution
#'
#' This function calculates the Akaike Information Criterion (AIC) for a 
#' zero-truncated binomial (ZTB) distribution fitted to the provided data.
#'
#' @family Utility
#' @author Steven P. Sanderson II, MPH
#'
#' @description
#' This function estimates the parameters (`size` and `prob`) of a ZTB
#' distribution from the provided data using maximum likelihood estimation 
#' (via the `optim()` function), and then calculates the AIC value based on the 
#' fitted distribution. 
#'
#' @param .x A numeric vector containing the data (non-zero counts) to be 
#'   fitted to a ZTB distribution.
#'
#' @details
#' **Initial parameter estimates:** The choice of initial values for `size` 
#' and `prob` can impact the convergence of the optimization. Consider using 
#' prior knowledge or method of moments estimates to obtain reasonable starting 
#' values. 
#'
#' **Optimization method:** The default optimization method used is 
#' "L-BFGS-B," which allows for box constraints to keep the parameters within 
#' valid bounds. You might explore other optimization methods available in 
#' `optim()` for potentially better performance or different constraint 
#' requirements.
#'
#' **Data requirements:** The input data `.x` should consist of non-zero counts, 
#' as the ZTB distribution does not include zero values. Additionally, the 
#' values in `.x` should be less than or equal to the estimated `size` parameter.
#'
#' **Goodness-of-fit:** While AIC is a useful metric for model comparison, it's 
#' recommended to also assess the goodness-of-fit of the chosen ZTB model using
#' visualization (e.g., probability plots, histograms) and other statistical 
#' tests (e.g., chi-square goodness-of-fit test) to ensure it adequately 
#' describes the data.
#'
#' @examples
#' library(actuar)
#' 
#' # Example data
#' set.seed(123)
#' x <- rztbinom(30, size = 10, prob = 0.4)
#' 
#' # Calculate AIC
#' util_zero_truncated_binomial_aic(x)
#'
#' @return The AIC value calculated based on the fitted ZTB distribution to 
#'   the provided data.
#'
#' @name util_zero_truncated_binomial_aic
NULL

#' @export
#' @rdname util_zero_truncated_binomial_aic

util_zero_truncated_binomial_aic <- function(.x) {
  # Check if actuar library is installed
  if (!requireNamespace("actuar", quietly = TRUE)) {
    stop("The 'actuar' package is needed for this function. Please install it with: install.packages('actuar')")
  }
  
  # Tidyeval
  x <- as.numeric(.x)
  
  # Negative log-likelihood function for zero-truncated binomial distribution
  neg_log_lik_rztbinom <- function(params, data) {
    size <- params[1]
    prob <- params[2]
    n <- length(data)
    -sum(actuar::dztbinom(data, size = size, prob = prob, log = TRUE))
  }
  
  # Initial parameter guesses (adjust if needed)
  pe <- util_zero_truncated_binomial_param_estimate(x)$parameter_tbl
  
  # Fit zero-truncated binomial distribution to data
  fit_rztbinom <- optim(
    par = c(size = round(pe$size, 3), prob = round(pe$prob, 3)), 
    fn = neg_log_lik_rztbinom,
    data = x
  ) |>
    suppressWarnings()
  
  # Extract log-likelihood and number of parameters
  logLik_rztbinom <- round(-fit_rztbinom$value, 4)
  k_rztnbinom <- 2  # Number of parameters (size and prob)
  
  # Calculate AIC
  AIC_rztbinom <- 2 * k_rztnbinom - 2 * logLik_rztbinom
  
  # Return AIC value
  return(AIC_rztbinom)
}

Example:

> library(actuar)
> 
> # Example data
> set.seed(123)
> x <- rztbinom(30, size = 10, prob = 0.4)
> 
> # Calculate AIC
> util_zero_truncated_binomial_aic(x)
[1] 116.5688

Stats Tibble

Function:

Example: