Bayesian inference for disease prevalence using negative binomial group testing

Group testing, also known as pooled testing, and inverse sampling are both widely used methods of data collection when the goal is to estimate a small proportion. Taking a Bayesian approach, we consider the new problem of estimating disease prevalence from group testing when inverse (negative binomial) sampling is used. Using different distributions to incorporate prior knowledge of disease incidence and different loss functions, we derive closed form expressions for posterior distributions and resulting point and credible interval estimators. We then evaluate our new estimators, on Bayesian and classical grounds, and apply our methods to a West Nile Virus data set.     

Unbiased and efficient log-likelihood estimation with inverse binomial sampling

The fate of scientific hypotheses often relies on the ability of a computational model to explain the data, quantified in modern statistical approaches by the likelihood function. The log-likelihood is the key element for parameter estimation and model evaluation. However, the log-likelihood of complex models in fields such as computational biology and neuroscience is often intractable to compute analytically or numerically. In those cases, researchers can often only estimate the log-likelihood by comparing observed data with synthetic observations generated by model simulations. Standard techniques to approximate the likelihood via simulation either use summary statistics of the data or are at risk of producing substantial biases in the estimate. Here, we explore another method, inverse binomial sampling (IBS), which can estimate the log-likelihood of an entire data set efficiently and without bias. For each observation, IBS draws samples from the simulator model until one matches the observation. The log-likelihood estimate is then a function of the number of samples drawn. The variance of this estimator is uniformly bounded, achieves the minimum variance for an unbiased estimator, and we can compute calibrated estimates of the variance. We provide theoretical arguments in favor of IBS and an empirical assessment of the method for maximum-likelihood estimation with simulation-based models. As case studies, we take three model-fitting problems of increasing complexity from computational and cognitive neuroscience. In all problems, IBS generally produces lower error in the estimated parameters and maximum log-likelihood values than alternative sampling methods with the same average number of samples. Our results demonstrate the potential of IBS as a practical, robust, and easy to implement method for log-likelihood evaluation when exact techniques are not available.     

Bias-corrected maximum likelihood estimator of the negative binomial dispersion parameter

We derive a first-order bias-corrected maximum likelihood estimator for the negative binomial dispersion parameter. This estimator is compared, in terms of bias and efficiency, with the maximum likelihood estimator investigated by Piegorsch (1990, Biometrics46, 863-867), the moment and the maximum extended quasi-likelihood estimators investigated by Clark and Perry (1989, Biometrics45, 309-316), and a double-extended quasi-likelihood estimator. The bias-corrected maximum likelihood estimator has superior bias and efficiency properties in most instances. For ease of comparison we give results for the two-parameter negative binomial model. However, an example involving negative binomial regression is given.     

Sample size calculation for multiple testing in microarray data analysis

Microarray technology is rapidly emerging for genome-wide screening of differentially expressed genes between clinical subtypes or different conditions of human diseases. Traditional statistical testing approaches, such as the two-sample t-test or Wilcoxon test, are frequently used for evaluating statistical significance of informative expressions but require adjustment for large-scale multiplicity. Due to its simplicity, Bonferroni adjustment has been widely used to circumvent this problem. It is well known, however, that the standard Bonferroni test is often very conservative. In the present paper, we compare three multiple testing procedures in the microarray context: the original Bonferroni method, a Bonferroni-type improved single-step method and a step-down method. The latter two methods are based on nonparametric resampling, by which the null distribution can be derived with the dependency structure among gene expressions preserved and the family-wise error rate accurately controlled at the desired level. We also present a sample size calculation method for designing microarray studies. Through simulations and data analyses, we find that the proposed methods for testing and sample size calculation are computationally fast and control error and power precisely.     

Sample size for testing differences in proportions for the paired-sample design

Miettinen (1968, Biometrics 24, 339-352) presented an approximation for power and sample size for testing the differences between proportions in the matched-pair case. Duffy (1984, Biometrics 40, 1005-1015) gave the exact power for this case and showed that Miettinen's approximation tends to slightly overestimate the power or underestimate the sample size necessary for the design power. A simple alternative approximation that is more conservative is presented here. In many cases, the sample size for the independent-sample case provides a conservative approximation for the matched-pair design.     

Use of binomial group testing in tests of hypotheses for classification or quantitative covariables

In group testing, the test unit consists of a group of individuals. If the group test is positive, then one or more individuals in the group are assumed to be positive. A group observation in binomial group testing can be, say, the test result (positive or negative) for a pool of blood samples that come from several different individuals. It has been shown that, when the proportion (p) of infected individuals is low, group testing is often preferable to individual testing for identifying infected individuals and for estimating proportions of those infected. We extend the potential applications of group testing to hypothesis-testing problems wherein one wants to test for a relationship between p and a classification or quantitative covariable. Asymptotic relative efficiencies (AREs) of tests based on group testing versus the usual individual testing are obtained. The Pitman ARE strongly favors group testing in many cases. Small-sample results from simulation studies are given and are consistent with the large-sample (asymptotic) findings. We illustrate the potential advantages of group testing in hypothesis testing using HIV-1 seroprevalence data.     

Sample size re‐estimation in clinical trials

Adaptive clinical trials are becoming very popular because of their flexibility in allowing mid‐stream changes of sample size, endpoints, populations, etc. At the same time, they have been regarded with mistrust because they can produce bizarre results in very extreme settings. Understanding the advantages and disadvantages of these rapidly developing methods is a must. This paper reviews flexible methods for sample size re‐estimation when the outcome is continuous.     

Small-sample estimation of negative binomial dispersion, with applications to SAGE data

We derive a quantile-adjusted conditional maximum likelihood estimator for the dispersion parameter of the negative binomial distribution and compare its performance, in terms of bias, to various other methods. Our estimation scheme outperforms all other methods in very small samples, typical of those from serial analysis of gene expression studies, the motivating data for this study. The impact of dispersion estimation on hypothesis testing is studied. We derive an "exact" test that outperforms the standard approximate asymptotic tests.     

Sample size determination for testing whether an identified treatment is best

Laska and Meisner (1989, Biometrics 45, 1139-1151) dealt with the problem of testing whether an identified treatment belonging to a set of k + 1 treatments is better than each of the other k treatments. They calculated sample size tables for k = 2 when using multiple t-tests or Wilcoxon-Mann-Whitney tests, both under normality assumptions. In this paper, we provide sample size formulas as well as tables for sample size determination for k > or = 2 when t-tests under normality or Wilcoxon-Mann-Whitney tests under general distribution assumptions are used.     

A score test for testing a zero-inflated Poisson regression model against zero-inflated negative binomial alternatives

Count data often show a higher incidence of zero counts than would be expected if the data were Poisson distributed. Zero-inflated Poisson regression models are a useful class of models for such data, but parameter estimates may be seriously biased if the nonzero counts are overdispersed in relation to the Poisson distribution. We therefore provide a score test for testing zero-inflated Poisson regression models against zero-inflated negative binomial alternatives.     

Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes

We present a method for computing sample size for cluster-randomized studies involving a large number of clusters with relatively small numbers of observations within each cluster. For multivariate survival data, only the marginal bivariate distribution is assumed to be known. The validity of this assumption is also discussed.