BMDS Modeling Methods

BMDS employs two general modeling approaches: Bayesian for dichotomous endpoints only, and non-Bayesian methods (i.e., maximum likelihood estimation) for dichotomous, continuous, and nested dichotomous endpoints.

Both Bayesian and non-Bayesian methods estimate parameters of a statistical model based on observed data. Both approaches use the likelihood function, which measures how well the model explains the observed data and provides key statistics such as point estimates (BMD) and bounds (BMDL). However, the precise way the likelihood is used differs somewhat between the two approaches.

Maximum Likelihood and Related Non-Bayesian Methods

The BMDS non-Bayesian methods are maximum likelihood estimation (MLE)[1] and statistics that rely on optimized likelihood values, particularly profile likelihood bounds and the Akaike Information Criterion (AIC) for model-selection. In this documentation, the term MLE method is used for the group of methods associated with likelihood maximization, recognizing that the term may be considered to apply most specifically to point estimation.

The MLE approach is based on finding the parameter values that maximize the likelihood function. In simpler terms, it chooses the parameters that make the observed data most probable. Models fit by these methods report associated bounds determined by profile likelihood approaches. Crump and Howe (1985)

Models not labeled as Bayesian are, by default, MLE.

Specifics on the MLE model equations are presented in the following sections:

• Mathematical Details for Models for Continuous Endpoints in Simple Designs

• Mathematical Details for Models for Dichotomous Endpoints in Simple Designs

• Mathematical Details for Models for Nested Dichotomous Endpoints

Bayesian

Bayesian methods combine prior information about parameters with the observed data to update the probability of an event of interest, as more evidence becomes available. Bayesian analysis can be more computationally intensive than MLE methodology but offers a richer view of parameter uncertainty.

Following the Bayesian approach, distributions describing the a priori uncertainty in the parameter values (the so-called prior distributions) are updated using the data under consideration to yield a posteriori distributions (the so-called posterior distributions). A quantile of the posterior BMD distribution, for example the 5th percentile, may be used for a Bayesian BMDL.[2]

A Bayesian point estimation procedure analogous to maximum likelihood is maximum a posteriori probability (MAP) estimation.[3]

The likelihood function plays a critical, formal role in Bayesian inference, different from its role in maximum likelihood.

Note

At this time, EPA does not offer technical guidance on Bayesian modeling or Bayesian model averaging.

To see a full list of Bayesian model priors and contraints, refer to Bayesian Dichotomous Models and their Respective Parameter Priors.

To see how model parameter estimates are reported in the BMDS results, refer to Model Parameters.

Bayesian analysis is described in more detail in Bayesian Dichotomous Analysis, including Model Averaging.

Optimization Algorithms Used in BMDS

BMDS uses the NLopt optimization library for MLE analyses and some Bayesian computations. Somewhat differently constrained optimization methods are involved for BMD and BMDL computations (Bayesian or MLE).

Several optimization algorithms available in the library are used to ensure reliability of the estimation:

• For global optimization involving the maximum likelihood or maximum a posteriori estimation subject to bounds on parameters (which are inequality constraints), the L-BFGS[4] method is attempted first. If it fails to converge, gradient-free algorithms (subplex and BOBYQA[5]) are then attempted. Note that the parameter bounds will be carried forward into the calculations that follow.

• The Bayesian BMDL in BMDS, as well as profile likelihood, involves profiling, wherein parameters are optimized with the BMD fixed at specific values, in addition to the inequality constraints on the individual parameters. Fixing the BMD will add a linear or nonlinear equality constraint.

• For equality-constrained optimizations, the augmented Lagrangian algorithm is used and either the L-BFGS, BOBYQA, or subplex algorithm is used in the local optimization step. When two approaches produce different results, the values producing the larger optimum are used.

• For optimizations involving only inequality constraints, the COBYLA[6] and MMA[7] approaches are used and compared. In the case the methods return different optimum, the values producing the larger of the two are used.

NLopt is available for download from the NLopt GitHub site. For more information regarding the algorithms, refer to the NLopt documentation site.

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[1]

BMDS has in the past referred to the non-Bayesian approach variously as “frequentist,” “Maximum Likelihood,” or “Maximum Likelihood Approach.”

[2]

Bayesian upper or lower bounds computed in this way define “credible” intervals, which are Bayesian analogues of classical confidence intervals.

[3]

The MAP estimate of a parameter is the value that maximizes the posterior probability density for the parameter.

[4]

Limited-memory BFGS (L-BFGS or LM-BFGS) is an optimization algorithm in the family of quasi-Newton methods that approximates the Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS).

[5]

Bound Optimization by Quadratic Approximation (BOBYQA) is a numerical optimization method.

[6]

Constrained Optimization by Linear Approximation (COBYLA) is a numerical optimization method.

[7]

Method of Moving Asymptotes (MMA) is a method for structural optimization.