Controlling critical mistag-associated false discoveries in metagenetic data

Metagenetic methods are commonplace within ecological and environmental research. One concern with these methods is the phenomenon of critical mistagging, where sequences from one sample are erroneously inferred to have originated from another sample due to errors in the attachment, PCR replication or sequencing of sample-specific dual-index tags. For studies using PCR-based library preparation on large sample sizes, the most cost-effective approach to limiting mistag-associated false detections involves using an unsaturated Latin square dual-indexing design. This allows researchers to estimate mistagging rates during sequencing but the statistical procedures for filtering out detections using this mistag rate have received little attention. We propose a straightforward method to limit mistag-associated false discoveries during metabarcoding applications. We analyzed two Illumina metabarcoding datasets produced using unsaturated Latin square designs to explore the distribution of mistagged sequences across dual-index combinations on a per taxon basis. We tested these data for conformity to the assumptions that 1) mistagging follows a binomial distribution [i.e., X ~ B(n, p)] where p, the probability of a sequence being mistagged, varies minimally across taxa and 2) mistags are distributed uniformly across dual-index combinations. We provide R functions that estimate the 95th percentile of expected mistags per dual-index combination for each taxon under these assumptions. We show that mistagging rates were consistent across taxa within the datasets analyzed and that modelling mistagging as a binomial process with uniform distribution across dual-index combinations enabled robust control of mistag-associated false discoveries. We propose that this method of taxon-specific filtering of detections based on the maximum mistags expected per dual-index combination should be broadly accepted during metagenetic analysis, provided that experimental and control sequence abundances per taxon are strongly correlated. When this assumption is violated, data may be better fit by assuming that the distribution of mistags across combinations follows Poisson characteristics [i.e., X ~ Pois(𝜆)], with 𝜆 empirically estimated from the abundance distribution of mistags among control samples. We provide a second R function for this case, though we have yet to observe such a dataset. Both functions and demonstrations associated with this work are freely available at https://github.com/RTRichar/ModellingCriticalMistags.

A generalised Dickman distribution and the number of species in a negative binomial process model

We derive the large-sample distribution of the number of species in a version of Kingman’s Poisson–Dirichlet model constructed from an $\alpha$ -stable subordinator but with an underlying negative binomial process instead of a Poisson process. Thus it depends on parameters $\alpha\in (0,1)$ from the subordinator and $r>0$ from the negative binomial process. The large-sample distribution of the number of species is derived as sample size $n\to\infty$ . An important component in the derivation is the introduction of a two-parameter version of the Dickman distribution, generalising the existing one-parameter version. Our analysis adds to the range of Poisson–Dirichlet-related distributions available for modeling purposes.

Measuring capability of a binomial process

Many product characteristics are qualitative in nature, e.g. colour, brightness, surface finish etc. The manufacturing process of such products is usually described in terms of fraction nonconforming or conforming which is assumed to follow binomial distribution. Measuring capability of a binomial process implies assessing to what extent the fraction nonconforming or conforming in the continuous stream of lots conform to the specification limits. The Cp or Cpl of a binomial process can be estimated using several approaches. However, these approaches generally give widely varying assessment about the capability of a given binomial process. Consequently, a user of the index may inadvertently be led to erroneous decision making based on an inaccurate estimate of the index. In this paper, a procedure is proposed for assessing accuracies of estimates of Cpu or Cpl obtained by different methods. Subsequently, the best method for evaluating capability of a binomial process is identified based on analysis of multiple case studies, and also the methods giving inaccurate estimates are highlighted. Keywords: Process capability index, binomial process, fraction nonconforming, nonconforming lot (NL), predicted NL%, prediction error

Quality Improvement in XY Process at ABC Corporation

ABC Corporation is a Taiwanese company that manufactures metal, other related products, and industry in need of thin sheet metal fabrication. The main objective of the study is to improve the M450G1010Z08 that occurred in press brake section, which is the primary problem of the company. The researchers used the Six Sigma Methodology as a technique of reducing quality issues. The research design used in the study is applied research. Based on the result of the study, the researchers found out through stratification process that the main problem was the wrong dimension of M450G1010Z08. In the measure phase, the researchers measured the wrong dimension defects using binomial process capability, which resulted in processing Z value of 1.08, indicating the process is not capable. In the analyze phase, researchers found out that the root cause of the problem is the open size of mould that exceeds 0.20 mm in the front side. In the improve phase, the procedure for modification of mould was done to meet the project target from 14.12% to 4%. After the implementation, the researchers found out that the wrong dimension of M450G1010Z08 was reduced from 14.12% to 2.12%. Also, the process Z increased from 1.08 to 2.03 and investigated that the process performance is capable after the implementation of the improvement. The improvement was done to improve work instruction for modification of the open size of the mould.

Fractional negative binomial and Pólya processes

In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

On the long-range dependence of fractional Poisson and negative binomial processes

We discuss the short-range dependence (SRD) property of the increments of the fractional Poisson process, called the fractional Poissonian noise. We also establish that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property. Our definitions of the SRD/LRD properties are similar to those for a stationary process and different from those recently used in Biard and Saussereau (2014).