bin2norm: A user-friendly interface to estimate normal distribution parameters from dichotomized data

Description

This function handles two data-collection settings for estimating normal parameters from threshold-based (dichotomized) data:

• Single-threshold per study: Each of I studies reports one threshold c_i, a sample size n_i, and the observed proportion p_i^{obs} of samples above that threshold. We assume one normal distribution \mathcal{N}(\mu,\sigma^2) across all studies. Methods include "MLE" and "probit".

• Multiple-thresholds per study: Each study i reports K_i thresholds \{c_{ij}\}, each with an observed proportion p_{ij}^{obs}. We assume the study-specific mean \mu_i \sim \mathcal{N}(\mu_0,\tau^2) and within-study variance \sigma^2. Because each study has multiple cutpoints, one can estimate \mu_0, \sigma, \tau. Methods include "MLE_integration", "GLMM", or "Bayesian" (MCMC).

Usage

bin2norm(
  scenario = c("single_threshold", "multiple_thresholds"),
  method = NULL,
  n_i = NULL,
  c_i = NULL,
  p_i_obs = NULL,
  data_list = NULL,
  ...
)

Arguments

Value

A list of estimated parameters, depending on the data-collection setting (scenario) and the chosen method. Typically includes:

• mu or mu0

• sigma

• tau (only for multiple-threshold methods)

Examples

# Single-threshold example
n_i <- c(100, 120, 80)
c_i <- c(1.2, 1.0, 1.5)
p_i_obs <- c(0.30, 0.25, 0.40)
bin2norm(scenario="single_threshold", method="MLE", n_i=n_i, c_i=c_i, p_i_obs=p_i_obs)

# Multiple-thresholds example
data_list <- list(
  n_i = c(100, 120),
  c_ij = list(c(1.0,1.2), c(0.8,1.5,2.0)),
  p_ij_obs = list(c(0.20,0.30), c(0.15,0.40,0.55))
)

# MLE with numeric integration
bin2norm(scenario="multiple_thresholds", method="MLE_integration",
         data_list=data_list, gh_points=5)

# GLMM approximation
# library(lme4)
bin2norm(scenario="multiple_thresholds", method="GLMM",
         data_list=data_list, use_lme4=TRUE)

# Bayesian MCMC approach
# library(rstan)
bin2norm(scenario="multiple_thresholds", method="Bayesian",
         data_list=data_list, iter=1000, chains=2)

Get initial values from data

Description

Get initial values from data

Usage

estimate_initial_values_from_data(data_list)

Arguments

Value

a named list of initial values

GLMM (Multiple Thresholds per Study, Probit Link, Random Intercepts)

Description

Creates a single data frame stacking all thresholds from all studies, then calls lme4::glmer(..., family=binomial(link='probit')) to fit a random-intercept model:

k_{ij} \sim \mathrm{Binomial}\bigl(n_i, \Phi(\alpha_i + \beta\, c_{ij})\bigr),

with \alpha_i \sim \mathcal{N}(0, \sigma_\alpha^2).

Interpreting results: \sigma = 1/|\,\beta\,|, \tau^2 = \sigma^2 \times \sigma_\alpha^2, \mu_0 = (\mathrm{Intercept}) \times \sigma (if not forced to 0).

Usage

estimate_multiThresh_GLMM(data_list, use_lme4 = TRUE)

Arguments

Value

A list with mu0, sigma, tau, method="GLMM_probit".

Bayesian MCMC (Multiple Thresholds per Study) using rstan

Description

Builds an inline Stan model for multiple thresholds per study. The user must have the rstan package installed. We place random effects \mu_i = \mu_0 + \tau * mu\_raw[i] and use a binomial likelihood for each threshold. By default, uses simple weakly informative priors.

Usage

estimate_multiThresh_MCMC(data_list, iter = 2000, chains = 2)

Arguments

Value

a list containing stan_fit (the full Stan fit object), plus mu0_est, sigma_est, tau_est as posterior means, and method="Bayesian_MCMC".

MLE with Numeric Integration (Multiple Thresholds per Study)

Description

Each study i has thresholds \{c_{ij}\}, each with an observed proportion p_{ij}^{obs}. We assume \mu_i \sim \mathcal{N}(\mu_0,\tau^2) and X_{ij} \sim \mathcal{N}(\mu_i,\sigma^2). The log-likelihood integrates out \mu_i via Gauss-Hermite quadrature.

Usage

estimate_multiThresh_MLE(data_list, gh_points = 20)

Arguments

Value

A list with mu0, sigma, tau, method="MLE_integration".

MLE (Single Threshold per Study)

Description

Treats the count of "above threshold" in study i as binomial with probability 1 - \Phi((c_i - \mu)/\sigma). This uses numerical optimization (optim) to maximize the binomial likelihood. Optionally uses Weighted OLS estimates as starting values to improve convergence.

Usage

estimate_singleThresh_MLE(n_i, c_i, p_i_obs, use_wols_init = TRUE)

Arguments

Value

A list with mu, sigma, method="MLE".

Weighted OLS (Initial value in Single Threshold per Study MLE)

Description

Implements the formula c_i = \mu + \sigma * \Phi^{-1}(1 - p_i^{obs}) in a weighted least-squares sense, with weights = n_i.

Usage

estimate_singleThresh_WOLS(n_i, c_i, p_i_obs)

Arguments

Value

A list with mu, sigma.

GLM probit (Single Threshold per Study)

Description

For each group i, we assume the data follows:

\Pr(Y_i = 1) = \Phi\left( \frac{\mu - c_i}{\sigma} \right)

where c_i is a known threshold, and \Phi is the standard normal CDF (the probit link). The function reconstructs individual binary outcomes based on observed probabilities, and estimates the parameters using generalized linear modeling with a probit link.

Usage

estimate_singleThresh_probit(n_i, c_i, p_i_obs)

Arguments

Value

A list with mu, sigma, method="probit".

Minimal Gauss-Hermite Quadrature

Description

Returns (nodes, weights) for approximating \int f(x) e^{-x^2} dx, ignoring any normalizing constant. This is a simple demonstration; for serious applications, more robust libraries or expansions might be used.

Usage

gaussHermite(n)

Arguments

Value

list with nodes and weights