Estimating a Proportion Based on Group Testing for Correlated Binary Response

Abstract

When the sampling scheme is in clusters and when the pools (of size k) within a cluster are assumed not to be independent, the Dorfman model for estimating the proportion under the binomial model is incorrect. The purpose of this paper is to propose a method for analyzing correlated binary data under the group testing framework. First, assuming that the probability of an individual varies according to a beta distribution, we derived an analytic expression for the probability of a positive pool and the correlation between two pools in each cluster. Second, we derived the exact probability mass function of the number of positive pools in each cluster that should be used to obtain the maximum likelihood estimate (MLE) of the proportion of individuals with a positive outcome. However, this MLE is not efficient in terms of computational resources. For this reason, we proposed another estimator based on the beta-binomial model for obtaining the approximate MLE of the proportion of interest. Based on a simulation study, the approximate estimator produced results that are very close to the exact MLE of the proportion of interest, with the advantage that this approach is computationally more efficient.Liu et al. [6] provide confidence interval procedures for estimating proportions estimated by group testing with groups of unequal size adjusted for overdispersion (extra-binomial variation). They used a quasi-likelihood approach to correct for the presence of overdispersion. However, in this case, heterogeneity in pool responses is induced by using different pool sizes (k) and may be due to the number of pools per cluster used in the group testing method. In their study, Liu et al.[6] introduced heterogeneity by assuming three clusters (m=3) and using a different pool size (k 1 ,k 2 ,k 3 ) in each cluster, with the following number of pools per cluster: N 1 =5, N 2 =10 and N 3 =15. For example, when k 1 =20, k 2 =10 and k 3 =5, they observed that if Y 1 =5, Y 2 =7 and 3 =4, then 2 1. 28 00 σ = , where Y i denote the number of positive pools observed, i=1,2,3, and 2 σ denotes the estimated dispersion parameter.However, if Y 1 =1, Y 2 =7 and Y 3 =3, then 2 4. 33 07 σ = , which indicates that the proportion of group testing varies widely for specific combinations; this also implies the presence of overdispersion. Here it is important to point out that the outcomes of the units in each cluster are assumed to be independent, identically distributed (i.i.d.) binomial distributions Table 2: Relative bias (RB) and relative mean squared error (RMSE) for the beta-binomial model and Relative bias (RB E ) for the exact distribution (Eq. 3), using various combinations of π, δ and N with k=25 and n l =10, l=1,2,…,N.Relative bias (RB) for the beta-binomial model (in black) and for the true model (in red) with various combinations of π, Citation: Montesinos-López OA, Montesinos-López A, Eskridge K, Crossa J (2014) Estimating a Proportion Based on Group Testing for Correlated Binary Response. J Biomet Biostat 5: 185.

Estimation of Proportion of a Trait by Batch Testing Model in a Quality Control Process

“…Note that he procedure used in testing for GMO in seeds is well detailed by [26]. The paper looks at three scenarios are commonly encountered in practice; number of batches is limited, batch size is restricted and number of units to be tested is limited.…”

Correlation Improves Group Testing

“…Hung and Swallow (1999) uses a two-state Markov Chain to model the serial correlation between neighboring samples and examined the accuracy of prevalence estimates under different pool sizes. Montesinos-López et al (2014) considers a setting where pools are formed within larger clusters of pairwise correlated individuals (e.g., samples of the same plant species may show different prevalence levels of a disease when they are taken in different geographical locations) and proposed a statistical approach for estimating the prevalence and correlation within each cluster.…”