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The question of testing for equality in distribution between two linear models, each consisting of sums of distinct discrete independent random variables with unequal numbers of observations, has emerged from the biological research. In this case, the computation of classical χ 2 statistics, which would not include all observations, results in loss of power, especially when sample sizes are small. Here, as an alternative that uses all data, the maximum likelihood estimator for the distribution of sum of discrete and independent random variables, which we call the convolution statistic, is proposed and its limiting normal covariance matrix determined. To challenge null hypotheses about the distribution of this sum, the generalized Wald's method is applied to define a testing statistic whose distribution is asymptotic to a χ 2 with as many degrees of freedom as the rank of such covariance matrix. Rank analysis also reveals a connection with the roots of the probability generating functions associated to the addend variables of the linear models. A simulation study is performed to compare the convolution test with Pearson's χ 2 , and to provide usage guidelines. [ABSTRACT FROM AUTHOR]

Interpreting forensic DNA signal is arduous since the total intensity is a cacophony of signal from noise, artifact, and allele from an unknown number of contributors (NOC). An alternate to traditional bulk‐processing pipelines is a single‐cell one, where the sample is collected, and each cell is sequestered resulting in n single‐source, single‐cell EPGs (scEPG) that must be interpreted using applicable strategies. As with all forensic DNA interpretation strategies, high quality electropherograms are required; thus, to enhance the credibility of single‐cell forensics, it is necessary to produce an efficient direct‐to‐PCR treatment that is compatible with prevailing downstream laboratory processes. We incorporated the semi‐automated micro‐fluidic DEPArray™ technology into the single‐cell laboratory and optimized its implementation by testing the effects of four laboratory treatments on single‐cell profiles. We focused on testing effects of phosphate buffer saline (PBS) since it is an important reagent that mitigates cell rupture but is also a PCR inhibitor. Specifically, we explored the effect of decreasing PBS concentrations on five electropherogram‐quality metrics from 241 leukocytes: profile drop‐out, allele drop‐out, allele peak heights, peak height ratios, and scEPG sloping. In an effort to improve reagent use, we also assessed two concentrations of proteinase K. The results indicate that decreasing PBS concentrations to 0.5X or 0.25X improves scEPG quality, while modest modifications to proteinase K concentrations did not significantly impact it. We, therefore, conclude that a lower than recommended proteinase K concentration coupled with a lower than recommended PBS concentration results in enhanced scEPGs within the semi‐automated single‐cell pipeline. [ABSTRACT FROM AUTHOR]