Incorporating historical control information into quantal bioassay with Bayesian approach

doi:10.1016/j.csda.2010.01.023

Computational Statistics & Data Analysis

Volume 54, Issue 6, 1 June 2010, Pages 1646-1656

https://doi.org/10.1016/j.csda.2010.01.023 Get rights and content

Abstract

A Bayesian approach with an iterative reweighted least squares is used to incorporate historical control information into quantal bioassays to estimate the dose–response relationship, where the logit of the historical control responses are assumed to have a normal distribution. The parameters from this normal distribution are estimated from both empirical and full Bayesian approaches with a marginal likelihood function being approximated by Laplace’s Method. A comparison is made using real data between estimates that include the historical control information and those that do not. It was found that the inclusion of the historical control information improves the efficiency of the estimators. In addition, this logit-normal formulation is compared with the traditional beta-binomial for its improvement in parameter estimates. Consequently the estimated dose–response relationship is used to formulate the point estimator and confidence bands for $E D (100 p)$ for various values of risk rate $p$ and the potency for any dose level.

Introduction

In pharmaceutical and toxicological bioassay experiments, a type of quantal bioassay experiment is designed in which a response, such as tumor or mortality, in a group of objects is recorded under different dose levels in the course of the experiment. Specifically, in these experiments, a stimulus (for example, the dose of a drug) is applied to $n$ experimental units and $r$ of them respond and $n - r$ do not respond. This type of quantal response bioassay belongs to the class of qualitative indirect bioassay.

In such bioassays, a set of doses from a substance (such as a drug, chemical, or toxin) is administered to experimental subjects. The primary interest is to estimate the dose–response relationship along with others, such as the estimation of the dose that produces a fixed effect level. This estimated dose–response relationship is used to determine the risks or response rates (such as tumor or mortality) as a function of the dose levels. Traditional methods appear in Finney (1978), Hubert (1992), Morgan (1992), Chen et al. (1999) and Chen, 2007a, Chen, 2007b. In addition, Fung et al. (1996) summarized the statistical test for trends with historical controls in carcinogen bioassay.

If historical control information is available and reliable, then this information should be incorporated into the modelling process. The historical control information is defined as the information on the response variable for a control dose ( $d = 0$ ) from a series of prior experiments. This information can be used and compared with the information from the current experiment. Since most of the current experiments are small, containing only a few dose levels (usually 3 to 4 and less than 6), the historical control information may be useful in establishing the statistical consideration for the dose–response relationship, especially for the marginally significant with historical information incorporated (Tarone, 1982, Hoel, 1983, Breslow and Clayton, 1993, Smythe et al., 1986, Prentice et al., 1992).

There are several issues that can arise with historical controls. Haseman (1992) raised the issue of the validity of the assumption of independent prior information in historical control experiments. He stressed that the past data on the control group may depend on the weight of the animal, the sex of the animal, and when and who does the experiment. Also, the definition of an effect may change over time or may be better detected as time progresses. Thus the use of historical control information is a controversial issue today in toxicology and risk assessment which deserves further attention. However, in experiments where the historical control information is reliable and representative of the population currently under study, improved estimators result from incorporating this information into the model.

This paper offers an alternative methodology for incorporating historical control information based on both empirical and full Bayesian approaches and compares estimators with and without the inclusion of this historical control information. Section 2 addresses an iterated reweighted least squares (IRLS) procedure for estimating the dose–response model without historical control adjustment. Section 3 presents a method for incorporating historical control information of the random effects by assuming that the logit of the historical control responses is normally distributed with both an empirical and a full Bayesian approaches to estimate the parameters from the normal distribution, where the marginal likelihood function is approximated by Laplace’s Method. This approach is different from the conventional approaches to model the historical control information as a Beta prior distribution resulting a beta-binomial likelihood function as well as the multivariate normal approximation from the first-order Taylor expansion of the likelihood function in Dempster et al. (1983). Section 4 presents two real examples and the application of the proposed procedure to estimate the dose–response relationship and the estimation of the proportion (or potency) for particular values of the dose level and the $E D (100 p)$ . Section 5 concludes with a discussion.

Section snippets

Quantal bioassays without historical control data

Consider a dose–response study with $k$ nonzero treatment levels $0 < d_{1} < \dots < d_{k}$ and a control dose $d_{0} = 0$ . Suppose that $n_{i}$ subjects are assigned to dose $i = 0, 1, \dots, k$ and that $x_{i}$ of these subjects respond. Let $P (d | θ)$ denote the probability of a response at dose $d$ with unknown parameter $θ$ .

The logistic model will be used in this paper but the results can be generalized to other models. The form of the logistic model is given by $P (d | α, β) = \frac{e^{α + β d}}{1 + e^{α + β d}} .$

This model is a special case of the generalized linear

Incorporating historical controls by an empirical Bayesian approach

The empirical Bayesian approach usually consists of two stages. In the first stage, only the historical control data are used to model its distribution and to estimate the associated parameters to be incorporated into the dose–response model. In this case, the control information involves the observed response rates $p_{0 j} = \frac{y_{0 j}}{N_{0 j}}, j = 0, 1, \dots, L$ where $j = 1, \dots, L$ are for the historical control information and $j = 0$ is for the current control information.

It is generally agreed that the historical control data

Data description

Table 1 gives two data sets. The first data set (thereafter referred as Data 1) is for the number of alveolar/bronchiolar adenomas induced in mice following 102 weeks exposure to pivalolactone in the diet (National Cancer Institute, 1978). This data is from Smythe et al. (1986) and has been discussed in Prentice et al. (1992). This is a small experiment with only three dose levels. It would be desirable to utilize relevant information from historical data. Fortunately, data from other similar

Discussion

The use of historical control information is a controversial issue in toxicology, carcinogenicity and risk assessment because of the validity in the historical control data. However, the historical control information could be useful in conjunction with the information from the current experiment to improve the dose–response modelling, especially in situations where the current situation is different markedly from historical control information.

A comparative study was conducted in this paper on

Acknowledgements

The author would like to thank Professor Tom Roe, the Associate Editor and two referees for his comments and suggestions, which significantly improved this manuscript.

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Incorporating historical control information into quantal bioassay with Bayesian approach

Abstract

Introduction

Section snippets

Quantal bioassays without historical control data

Incorporating historical controls by an empirical Bayesian approach

Data description

Discussion

Acknowledgements

Computational Statistics and Data Analysis

Statistics and Probability Letters

Approximate inference in generalized linear mixed models

Journal of American Statistical Association

Dose–time–response cumulative multinomial generalized linear model

Journal of Biopharmaceutical Statistics

Empirical Bayes estimation for combination of multivariate bioassays

Biometrics

Modelling Binary Data

Combining historical and randomized control for assessing trends in proportions