A Symmetric Extended Logistic Model with Applications to Experimental Toxicity Data

The fit of the logit and probit models for quantal response data can be improved by embedding these classical models within a richer parametric family indexed by one or two shape parameters. In this paper, a symmetric extended logistic model indexed by a shape parameter λ is discussed with application to dose response curves. The usual maximum likelihood method is employed to estimate the parameters of the model. The need to include the shape parameter λ is illustrated by analyzing a set of real experimental data and comparing the fit of the extended logistic model to those obtained by the standard logit and probit models.     

Testing the fit of a quantal model of neurotransmission

Many studies of synaptic transmission have assumed a parametric model to estimate the mean quantal content and size or the effect upon them of manipulations such as the induction of long-term potentiation. Classical tests of fit usually assume that model parameters have been selected independently of the data. Therefore, their use is problematic after parameters have been estimated. We hypothesized that Monte Carlo (MC) simulations of a quantal model could provide a table of parameter-independent critical values with which to test the fit after parameter estimation, emulating Lilliefors's tests. However, when we tested this hypothesis within a conventional quantal model, the empirical distributions of two conventional goodness-of-fit statistics were affected by the values of the quantal parameters, falsifying the hypothesis. Notably, the tests' critical values increased when the combined variances of the noise and quantal-size distributions were reduced, increasing the distinctness of quantal peaks. Our results support two conclusions. First, tests that use a predetermined critical value to assess the fit of a quantal model after parameter estimation may operate at a differing unknown level of significance for each experiment. Second, a MC test enables a valid assessment of the fit of a quantal model after parameter estimation.     

Maximum likelihood estimation of non-uniform transmitter release probabilities at the crayfish neuromuscular junction

The classical model of quantal release of neurotransmitter assumes that a fixed number of quantal units are available for release in the presynaptic terminal, and that each unit has the same probability of being released. This model also assumes that different units are released independently of one another. We consider two variations of the classical model. In the first case we assume that release is independent, but with potentially different release probabilities at different sites. In the second case we allow for dependence among the release units. A maximum likelihood procedure for the estimation of model parameters is developed, and an estimator of the number of quantal units is proposed. The performance of the method is assessed through a simulation study, and the procedures are applied to the analysis of a sequence of post-synaptic potentials recorded intracellularly at the crayfish neuromuscular junction. Goodness of fit and hypothesis test procedures reject the classical model in favor of an independent release mechanism with differing release probabilities. A more general release mechanism, allowing for dependence in the release process, also provides a good fit to the data analyzed.     

A Generalized Logistic Model for Quantal Response Bioassay

In bioassay, where different levels of the stimulus may represent different doses of a drug, the binary response is the death or survival of an individual receiving a specified dose. In such applications, it is common to model the probability of a positive response P at the stimulus level x by P = F(x′β), where F is a cumulative distribution function and β is a vector of unknown parameters which characterize the response function. The two most popular models used for modelling binary response bioassay involve the probit model [BLISS (1935), FINNEY (1978)], and the logistic model [BERKSON (1944), BROWN (1982)]. However, these models have some limitations. The use of the probit model involves the inverse of the standard normal distribution function, making it rather intractable. The logistic model has a simple form and a closed expression for the inverse distribution function, however, neither the logistic nor the probit can provide a good fit to response functions which are not symmetric or are symmetric but have a steeper or gentler incline in the central probability region. In this paper we introduce a more realistic model for the analysis of quantal response bioassay. The proposed model, which we refer to it as the generalized logistic model, is a family of response curves indexed by shape parameters m₁ and m₂. This family is rich enough to include the probit and logistic models as well as many others as special cases or limiting distributions. In particular, we consider the generalized logistic three parameter model where we assume that m₁ = m, m is a positive real number, and m₂ = 1. We apply this model to various sets of data, comparing the fit results to those obtained previously by other dose-response curves such as the logistic and probit, and showing that the fit can be improved by using the generalized logistic.     

Quantile Function Models for Quantal Response Analysis

The usual analysis of quantal response data occurring in diverse fields such as economics, medicine, psychology and toxicology use probit and logit models or their extensions with generalized least squares or the principle of likelihood as the method of statistical inference. The symmetric alternative models lead to practically comparable results and the choice of model or method is determined by considerations of familiarity and computational convenience. Recent attempts at improvement involve larger parametric families of tolerance distributions and employ the method of maximum likelihood in analysis. In this paper we consider models with the tolerance distributions based upon the Tukey-lambda distributions which are described in terms of their quantile functions. The likelihood methods for fitting the models and testing their adequacies are developed and illustrated using classical data due to BLISS (1935) and ASHFORD and SMITH (1964).     

Repeated probit regression when covariates are measured with error

This paper develops a model for repeated binary regression when a covariate is measured with error. The model allows for estimating the effect of the true value of the covariate on a repeated binary response. The choice of a probit link for the effect of the error-free covariate, coupled with normal measurement error for the error-free covariate, results in a probit model after integrating over the measurement error distribution. We propose a two-stage estimation procedure where, in the first stage, a linear mixed model is used to fit the repeated covariate. In the second stage, a model for the correlated binary responses conditional on the linear mixed model estimates is fit to the repeated binary data using generalized estimating equations. The approach is demonstrated using nutrient safety data from the Diet Intervention of School Age Children (DISC) study.     

The adjustment for a natural response rate in probit analysis

A revised scheme of computation is suggested for the fitting of a probit regression line to quantal response data which have to be adjusted because of the occurrence of natural responses not caused by the stimulus under test. The calculations lead to the same results as those proposed when the method was first introduced, but have the advantages of close similarity with multiple regression calculations and of simplifying the test of heterogeneity. The new scheme is illustrated on the example used in the earlier paper.
A table of weighting coefficients for use with high natural response rates is presented.     

Estimating the parameters in the two-component model for cell survival from experimental quantal response data

A statistical technique is given which can be used to estimate the parameters of the two-component model for cell survival from quantal response multifraction data. The method is a nonlinear logistic regression and relies on a mild assumption relating the probability of death to cell survival level. The method is demonstrated on mouse colon data, where more efficient estimates of the parameters are known, and the agreement is good. Also for some mouse lung LD50 data we obtain estimates of the parameters, and the fit to the data is shown to be better than that of linear-quadratic model.     

A Discrete Delta Method for Estimating Standard Errors for Estimators in Quantal Bioassay

A nonparametric discrete delta method for estimating standard errors of percentile estimators in quantal bioassay is described. A simulation study of confidence intervals for EDx in probit analysis shows the discrete delta method compared favorably with intervals based on maximum likelihood and also some parametric bootstrap methods.     

Statistical Estimation of Distribution of Quantal Release of Transmitter in Neuromuscular Synapse of the Lobster Homarus americanus

A new approach to estimation of quantal release distribution of transmitter under conditions of high synaptic activity is presented. Postsynaptic responses of neuromuscular excitatory synapse in muscle-opener of nipper of the lobster, which are obtained by focal extracellular recording, are used as original data set. Based on two data groups (value of evoked and spontaneous postsynaptic responses), the linear regression model is constructed. Parameters of this model describe completely the quantal release distribution. To evaluate the parameters, biased modifications of the least squares method—the penalized least squares method and the principal components method—were applied. As a result, it was possible to achieve estimations of the quantal release distribution with sufficiently low standard errors. Modeling studies have shown that the gain of accuracy of the estimation due to a decrease of the standard error exceeds considerably losses caused by its bias.     

Trimmed logit method for estimating the ED50 in quantal bioassay

Trimmed nonparametric procedures such as the trimmed Spearman-Karber method have been proposed in the literature for overcoming the deficiencies of the probit and logit models in the analysis of quantal bioassay data. However, there are situations where the median effective dose (ED50) is not calculable with the trimmed Spearman-Karber method, but is estimable with a parametric model. Also, it is helpful to have a parametric model for estimating percentiles of the dose-response curve such as the ED10 and ED25. A trimmed logit method that combines the advantages of a parametric model with that of trimming in dealing with heavy-tailed distributions is presented here. These advantages are substantiated with examples of actual bioassay data. Simulation results are presented to support the validity of the trimmed logit method, which has been found to work well in our experience with over 200 data sets. A computer program for computing the ED50 and associated 95% asymptotic confidence interval, based on the trimmed logit method, can be obtained from the authors.     

Tractable Bayes of skew-elliptical link models for correlated binary data

Correlated binary response data with covariates are ubiquitous in longitudinal or spatial studies. Among the existing statistical models, the most well-known one for this type of data is the multivariate probit model, which uses a Gaussian link to model dependence at the latent level. However, a symmetric link may not be appropriate if the data are highly imbalanced. Here, we propose a multivariate skew-elliptical link model for correlated binary responses, which includes the multivariate probit model as a special case. Furthermore, we perform Bayesian inference for this new model and prove that the regression coefficients have a closed-form unified skew-elliptical posterior with an elliptical prior. The new methodology is illustrated by an application to COVID-19 data from three different counties of the state of California, USA. By jointly modeling extreme spikes in weekly new cases, our results show that the spatial dependence cannot be neglected. Furthermore, the results also show that the skewed latent structure of our proposed model improves the flexibility of the multivariate probit model and provides a better fit to our highly imbalanced dataset.     

Maximum likelihood vs. minimum chi-square

In minimum chi-square logit or probit analysis of quantal bioassay data, a requirement for proper asymptotic behavior of the estimates made is that test-group sizes get indefinitely large. Inconsistent estimates result if group sizes are small, however numerous the groups. Maximum likelihood estimates do not show this inconsistent behavior, even if all the many group sizes are only unity. The inconsistent behavior for minimum chi-square results from a bias toward 0.5 for response probabilities. At 0.5 the binomial variance is at a maximum of 0.25, so tending to minimize the calculated value of chi square. The principle of minimum chi-square should not be confused with the principle of least squares.     

Normal frailty probit model for clustered interval‐censored failure time data

Clustered interval‐censored data commonly arise in many studies of biomedical research where the failure time of interest is subject to interval‐censoring and subjects are correlated for being in the same cluster. A new semiparametric frailty probit regression model is proposed to study covariate effects on the failure time by accounting for the intracluster dependence. Under the proposed normal frailty probit model, the marginal distribution of the failure time is a semiparametric probit model, the regression parameters can be interpreted as both the conditional covariate effects given frailty and the marginal covariate effects up to a multiplicative constant, and the intracluster association can be summarized by two nonparametric measures in simple and explicit form. A fully Bayesian estimation approach is developed based on the use of monotone splines for the unknown nondecreasing function and a data augmentation using normal latent variables. The proposed Gibbs sampler is straightforward to implement since all unknowns have standard form in their full conditional distributions. The proposed method performs very well in estimating the regression parameters as well as the intracluster association, and the method is robust to frailty distribution misspecifications as shown in our simulation studies. Two real‐life data sets are analyzed for illustration.     

Threshold dose-response models in toxicology

After a brief discussion of the threshold concept in toxicology, we consider models for the estimation of thresholds in the case where the observed response is binary or quantal. A generalization of the four-parameter Tukey-lambda family of distributions is proposed as a useful class of models for threshold estimation. Properties of these models are discussed and the process of model fitting and evaluation is illustrated using a number of data sets. The discussion includes consideration of background or spontaneous response, and comparison with other models. One of these is the linear-plateau or hockey-stick model, which has been used in a number of toxicological studies.     

Quantitative risk assessment for multivariate continuous outcomes with application to neurotoxicology: the bivariate case

The neurotoxic effects of chemical agents are often investigated in controlled studies on rodents, with multiple binary and continuous endpoints routinely collected. One goal is to conduct quantitative risk assessment to determine safe dose levels. Such studies face two major challenges for continuous outcomes. First, characterizing risk and defining a benchmark dose are difficult. Usually associated with an adverse binary event, risk is clearly definable in quantal settings as presence or absence of an event; finding a similar probability scale for continuous outcomes is less clear. Often, an adverse event is defined for continuous outcomes as any value below a specified cutoff level in a distribution assumed normal or log normal. Second, while continuous outcomes are traditionally analyzed separately for such studies, recent literature advocates also using multiple outcomes to assess risk. We propose a method for modeling and quantitative risk assessment for bivariate continuous outcomes that address both difficulties by extending existing percentile regression methods. The model is likelihood based; it allows separate dose-response models for each outcome while accounting for the bivariate correlation and overall characterization of risk. The approach to estimation of a benchmark dose is analogous to that for quantal data without the need to specify arbitrary cutoff values. We illustrate our methods with data from a neurotoxicity study of triethyl tin exposure in rats.     

Estimation of radiation resistance values of microorganisms in food products

Several statistical methods, including the conventional technique of Schmidt and Nank, were evaluated for estimating radiation resistance values of various strains of Clostridium botulinum by the use of partial spoilage data from an inoculated ham pack study. Procedures based on quantal response were preferred. The tedious but rigorous probit maximum likelihood determination was used as a standard of comparison. Weibull's graphical treatment was the method of choice because it is simple to utilize, it is mathematically sound, and its ld(50) values agreed closely with the reference standard. In addition, it offers a means for analyzing the type of microbial death kinetics that occur in the pack (exponential, normal, log normal, or mixed distributions), and it predicts the probability of microbial death with any radiation dose used, as well as the dose needed to destroy any given number of organisms, without the need to assume the death pattern of the partial spoilage data. The Weibull analysis indicated a normal type kinetics of death for C. botulinum spores in irradiated cured ham rather than an exponential order of death, as assumed by the Schmidt-Nank formula. The Weibull 12D equivalent of a radiation process, or the minimal radiation dose (MRD), for cured ham was consistently higher than both the experimental sterilizing dose (ESD) and the Schmidt-Nank average MRD. The latter calculation was lower than the ESD in three of the five instances examined, which seems unrealistic. The Spearman-K?rber estimate was favored as the arithmetic technique on the bases of ease of computation, close agreement with the reference method, and providing confidence limits for the ld(50) values.     

Generalized Confidence Intervals for Ratios of Regression Coefficients with Applications to Bioassays

The problem of constructing a confidence interval for the ratio of two regression coefficients is addressed in the context of multiple regression. The concept of a Generalized Confidence Interval is used, and the resulting confidence interval is shown to perform well in terms of coverage probability. The proposed methodology always results in an interval, unlike the confidence region generated from Fieller's theorem. The procedure can easily be implemented for parallel‐line assays, slope‐ratio assays, and quantal assays under a probit model. Furthermore, this approach can also be extended to compute confidence intervals based on data from multiple bioassays. The results are illustrated using several examples.     

A Probit Latent Class Model with General Correlation Structures for Evaluating Accuracy of Diagnostic Tests

Summary Traditional latent class modeling has been widely applied to assess the accuracy of dichotomous diagnostic tests. These models, however, assume that the tests are independent conditional on the true disease status, which is rarely valid in practice. Alternative models using probit analysis have been proposed to incorporate dependence among tests, but these models consider restricted correlation structures. In this article, we propose a probit latent class model that allows a general correlation structure. When combined with some helpful diagnostics, this model provides a more flexible framework from which to evaluate the correlation structure and model fit. Our model encompasses several other PLC models but uses a parameter‐expanded Monte Carlo EM algorithm to obtain the maximum‐likelihood estimates. The parameter‐expanded EM algorithm was designed to accelerate the convergence rate of the EM algorithm by expanding the complete‐data model to include a larger set of parameters and it ensures a simple solution in fitting the PLC model. We demonstrate our estimation and model selection methods using a simulation study and two published medical studies.     

Bayesian multivariate logistic regression

Bayesian analyses of multivariate binary or categorical outcomes typically rely on probit or mixed effects logistic regression models that do not have a marginal logistic structure for the individual outcomes. In addition, difficulties arise when simple noninformative priors are chosen for the covariance parameters. Motivated by these problems, we propose a new type of multivariate logistic distribution that can be used to construct a likelihood for multivariate logistic regression analysis of binary and categorical data. The model for individual outcomes has a marginal logistic structure, simplifying interpretation. We follow a Bayesian approach to estimation and inference, developing an efficient data augmentation algorithm for posterior computation. The method is illustrated with application to a neurotoxicology study.