Getting a Grip on Variability.

Because science is a modeling enterprise, a key question for educators is: What kind of repertoire can initiate students into the practice of generating, revising, and critiquing models of the natural world? Based on our 20 years of work with teachers and students, we nominate variability as a set of connected key ideas that bridge mathematics and science and are fundamental for equipping youngsters for the posing and pursuit of questions about science. Accordingly, we describe a sequence for helping young students begin to reason productively about variability. Students first participate in random processes, such as repeated measure of a person's outstretched arms, that generate variable outcomes. Importantly, these processes have readily discernable sources of variability, so that relations between alterations in processes and changes in the collection of outcomes can be easily established and interpreted by young students. Following these initial steps, students invent and critique ways of visualizing and measuring distributions of the outcomes of these processes. Visualization and measure of variability are then employed as conceptual supports for modeling chance variation in components of the processes. Ultimately, students reimagine samples and inference in ways that support reasoning about variability in natural systems.

Accounting for Parcel-Allocation Variability in Practice: Combining Sources of Uncertainty and Choosing the Number of Allocations.

Item parceling remains widely used under conditions that can lead to parcel-allocation variability in results. Hence, researchers may be interested in quantifying and accounting for parcel-allocation variability within sample. To do so in practice, three key issues need to be addressed. First, how can we combine sources of uncertainty arising from sampling variability and parcel-allocation variability when drawing inferences about parameters in structural equation models? Second, on what basis can we choose the number of repeated item-to-parcel allocations within sample? Third, how can we diagnose and report proportions of total variability per estimate arising due to parcel-allocation variability versus sampling variability? This article addresses these three methodological issues. Developments are illustrated using simulated and empirical examples, and software for implementing them is provided.

Sampling variability in forensic likelihood-ratio computation: A simulation study.

Recently, in the forensic biometric community, there is a growing interest to compute a metric called "likelihood-ratio" when a pair of biometric specimens is compared using a biometric recognition system. Generally, a biometric recognition system outputs a score and therefore a likelihood-ratio computation method is used to convert the score to a likelihood-ratio. The likelihood-ratio is the probability of the score given the hypothesis of the prosecution, Hp (the two biometric specimens arose from a same source), divided by the probability of the score given the hypothesis of the defense, Hd (the two biometric specimens arose from different sources). Given a set of training scores under Hp and a set of training scores under Hd, several methods exist to convert a score to a likelihood-ratio. In this work, we focus on the issue of sampling variability in the training sets and carry out a detailed empirical study to quantify its effect on commonly proposed likelihood-ratio computation methods. We study the effect of the sampling variability varying: 1) the shapes of the probability density functions which model the distributions of scores in the two training sets; 2) the sizes of the training sets and 3) the score for which a likelihood-ratio is computed. For this purpose, we introduce a simulation framework which can be used to study several properties of a likelihood-ratio computation method and to quantify the effect of sampling variability in the likelihood-ratio computation. It is empirically shown that the sampling variability can be considerable, particularly when the training sets are small. Furthermore, a given method of likelihood-ratio computation can behave very differently for different shapes of the probability density functions of the scores in the training sets and different scores for which likelihood-ratios are computed.

Guaranteed performance of individual control chart used in gamma passing rate-based patient-specific quality assurance.

PURPOSE: To assess the effect of sampling variability on the performance of individual charts (I-charts) for PSQA and provide a robust and reliable method for unknown PSQA processes. MATERIALS AND METHODS: A total of 1327 pretreatment PSQAs were analyzed. Different datasets with samples in the range of 20-1000 were used to estimate the lower control limit (LCL). Based on the iterative "Identify-Eliminate-Recalculate" and direct calculation without any outlier filtering procedures, five I-charts methods, namely the Shewhart, quantile, scaled weighted variance (SWV), weighted standard deviation (WSD), and skewness correction (SC) method, were used to compute the LCL. The average run length (ARL0) and false alarm rate (FAR0) were calculated to evaluate the performance of LCL. RESULTS: The ground truth of the values of LCL, FAR0, and ARL0 obtained via in-control PSQAs were 92.31%, 0.135%, and 740.7, respectively. Further, for in-control PSQAs, the width of the 95% confidence interval of LCL values for all methods tended to decrease with the increase in sample size. In all sample ranges of in-control PSQAs, only the median LCL and ARL0 values obtained via WSD and SWV methods were close to the ground truth. For the actual unknown PSQAs, based on the "Identify-Eliminate-Recalculate" procedure, only the median LCL values obtained by the WSD method were closest to the ground truth. CONCLUSIONS: Sampling variability seriously affected the I-chart performance in PSQA processes, particularly for small samples. For unknown PSQAs, the WSD method based on the implementation of the iterative "Identify-Eliminate-Recalculate" procedure exhibited sufficient robustness and reliability.

How many replicates to accurately estimate fish biodiversity using environmental DNA on coral reefs?

Quantifying fish species diversity in rich tropical marine environments remains challenging. Environmental DNA (eDNA) metabarcoding is a promising tool to face this challenge through the filtering, amplification, and sequencing of DNA traces from water samples. However, because eDNA concentration is low in marine environments, the reliability of eDNA to detect species diversity can be limited. Using an eDNA metabarcoding approach to identify fish Molecular Taxonomic Units (MOTUs) with a single 12S marker, we aimed to assess how the number of sampling replicates and filtered water volume affect biodiversity estimates. We used a paired sampling design of 30 L per replicate on 68 reef transects from 8 sites in 3 tropical regions. We quantified local and regional sampling variability by comparing MOTU richness, compositional turnover, and compositional nestedness. We found strong turnover of MOTUs between replicated pairs of samples undertaken in the same location, time, and conditions. Paired samples contained non-overlapping assemblages rather than subsets of one another. As a result, non-saturated localized diversity accumulation curves suggest that even 6 replicates (180 L) in the same location can underestimate local diversity (for an area <1 km). However, sampling regional diversity using ~25 replicates in variable locations (often covering 10 s of km) often saturated biodiversity accumulation curves. Our results demonstrate variability of diversity estimates possibly arising from heterogeneous distribution of eDNA in seawater, highly skewed frequencies of eDNA traces per MOTU, in addition to variability in eDNA processing. This high compositional variability has consequences for using eDNA to monitor temporal and spatial biodiversity changes in local assemblages. Avoiding false-negative detections in future biomonitoring efforts requires increasing replicates or sampled water volume to better inform management of marine biodiversity using eDNA.

Variation in trace element content between liver lobes in cattle. How important is the sampling site?

The aim of the present study was to establish the pattern of lobular distribution of trace elements in the liver of cattle. The objective was to determine which part of the liver would provide accurate estimation of the trace element content of the whole organ. Liver samples were obtained from 10 Holstein-Friesian (HF), 10 Galician Blond (GB) and 10 GBxHF crosses (all aged 10 months) at slaughter. Samples were taken from 6 regions of the liver: the internal and external faces of the right lobe (IR and ER respetively); the left lobe (L), caudate lobe (CAU), quadrate lobe (QUA) and the processus papillaris (PP). The samples were acid digested and trace elements were determined by inductively coupled plasma mass spectrometry. The distribution of all trace elements, except cobalt and zinc, varied significant across the liver. In all cases, the concentrations were highest in L and lowest in CAU. Variations in the distribution between the other areas of the liver (ER, IR, QUA, PP) were not significant. The distribution of trace elements may be related to oxygen perfusion. Moreover, the trace element content of CAU was weakly correlated with those of the other lobes, and the capacity of L to accumulate high levels of trace elements would only be observed at high levels of exposure. Taking into account the main findings of the study, a single sample of liver taken from the same anatomical region (excluding CAU and L) would be adequate for determining the trace element status of cattle.