Effective Sample Size with the Bivariate Gaussian Common Component Model

Hulson, P-J. F., Hanselman, D. H., and Quinn, T. J. II. 2011. Effects of process and observation errors on effective sample size of fishery and survey age and length composition using variance ratio and likelihood methods. – ICES Journal of Marine Science, 68: 1548–1557. Observations of age or length composition from fisheries or research surveys are modelled frequently with the multinomial distribution. Violations of multinomial assumptions in data collection usually cause overdispersion of observations and consequent underestimation of uncertainty. This has led to the adoption of an effective sample size less than the actual sample size to approximate the likelihood function for age or length composition better in, for example, fishery stock assessment models. The behaviour of effective sample size is examined under different scenarios for population age distribution and sampling design. Effective sample size was approximated with three approaches: (i) the ratio of multinomial to empirical variance; (ii) sampling estimation; and (iii) the Dirichlet likelihood. The most significant changes in effective sample size were attributable to process error involving aggregation of ages within schools. In terms of observation error, effective sample size can be increased by increasing the number of tows from which samples are obtained for age or length composition, then, because of the reduced uncertainty in effective sample size, the Dirichlet likelihood can be integrated into the objective function of fishery stock assessment models to estimate the effective sample size in future assessments.

Bayesian methods allow borrowing of historical information through prior distributions. The concept of prior effective sample size (prior ESS) facilitates quantification and communication of such prior information by equating it to a sample size. Prior information can arise from historical observations; thus, the traditional approach identifies the ESS with such a historical sample size. However, this measure is independent of newly observed data, and thus would not capture an actual "loss of information" induced by the prior in case of prior-data conflict. We build on a recent work to relate prior impact to the number of (virtual) samples from the current data model and introduce the effective current sample size (ECSS) of a prior, tailored to the application in Bayesian clinical trial designs. Special emphasis is put on robust mixture, power, and commensurate priors. We apply the approach to an adaptive design in which the number of recruited patients is adjusted depending on the effective sample size at an interim analysis. We argue that the ECSS is the appropriate measure in this case, as the aim is to save current (as opposed to historical) patients from recruitment. Furthermore, the ECSS can help overcome lack of consensus in the ESS assessment of mixture priors and can, more broadly, provide further insights into the impact of priors. An R package accompanies the paper.

We examine various methods to estimate the effective sample size for construction of confidence intervals for survival probabilities. We compare the effective sample sizes of Cutler and Ederer and Peto et al., as well as a modified Cutler-Ederer effective sample size. We investigate the use of these effective sample sizes in the common situation of many censored observations that intervene between the time point of interest and the last death before this time. We note that there is no a priori reason to treat upper and lower confidence intervals in a symmetric fashion since censored survival data are by nature asymmetric. We recommend the use of the Cutler-Ederer effective sample size in construction of upper confidence intervals and the Peto effective sample size in construction of lower confidence intervals. Two examples with real data demonstrate the differences between confidence intervals formed with different effective sample sizes. This study also illustrates the need for caution in the application of simulation studies to real problems.

As características químicas dos solos do Cerrado brasileiro são facilmente alteradas pelas intervenções humanas. O objetivo desse estudo foi avaliar a variabilidade espacial das propriedades químicas em um Neossolo Quartzarênico Órtico típico, em função da substituição do Cerrado por Pastagem, na região de Araguaína (TO), Brasil. Foram feitas amostragens do solo em uma malha irregular em área de 13,11 ha, na profundidade de 0 a 20 cm, para determinação dos teores de matéria orgânica (MO), Al3+, P, K+, Ca2+, Mg2+ , valores de pH, soma de bases (SB), capacidade de troca de cátions em pH 7,0 (CTCpH 7), capacidade de troca de cátions efetiva (CTCe), saturação por bases - V% e saturação por alumínio - m%. As propriedades químicas do solo foram analisadas por estatística descritiva e tiveram suas médias comparadas pelo teste de Tukey, a 5% probabilidade, utilizando geoestatístico e interpolação por krigagem ordinária. Os alcances de dependência espacial dos atributos químicos pH e Ca2+ do solo foram menores que os demais (MO, Al3+, P, K+, Mg2+, SB, CTCpH 7, CTCe e V%), indicando que eles apresentam menor continuidade na distribuição espacial em Neossolo Quartzarênico Órtico típico, em função da conversão do Cerrado em Pastagem.

Log Pearson III distribution — Effect of dependence, distribution parameters and sample size on peak annual flood estimates

Loss of weight on ignition (LOI) at 360°C for 2 h was compared with organic matter (OM) determined by “Walkley‐Black”; titration in 356 Wisconsin soils. The effects of sample size, number of samples heated at one time, and time of heating were studied using these soils. The number of samples heated at one time in a muffle furnace did not affect LOI. For an organic soil (34.2% OM), beaker size (20‐or 50‐mL) was unimportant, but LOI increased as sample size decreased. Sample size was not significant for a mineral soil (3.62% OM). Time of heating was significant for both soils. A coefficient of determination (R2) of 0.97 was obtained for the linear regression between OM and LOI of the entire group, which had OM ranging from 0.1 to 54.4%. When only samples containing less than 10% OM were considered (n=316), the regression equation was: OM = 0.973 LOI‐0.33 (R2 = 0.90).

Information on spatial variability of soil chemical properties is allowing an increasingly efficient management of soil fertility. This study was conducted in the municipality of Santa Rosa do Tocantins, TO, in the 2018/2019 crop year. The objectives of the study were to characterize the spatial variability of chemical properties of a dystrophic Red Latosol in the Cerrado of Tocantins using geostatistics and optimize the management of soil fertility by means of precision agronomy techniques, for more efficient input use in agricultural production areas. For the experiment, 49 soil samples were collected at 0.2 m depth, from equidistant points in a regular grid 100 m apart, over an area of ??150 ha. Each sample was composed of 5 subsamples. The soil properties evaluated included pH, bases sum, potential acidity, organic matter, total cation exchange capacity, base saturation, phosphorus, sulfur, potassium, calcium, magnesium, boron, copper, iron, manganese, and zinc. A descriptive analysis was carried out, highlighting the mean, median, minimum, and maximum values for each soil variable. In addition, the coefficients of variation, asymmetry, kurtosis, and the normality test of Kolmogorov-Smirnov were performed. The area presented significant variations in chemical and macronutrient attributes and little variation in micronutrients, except for zinc. The study reveals variations in different soil attributes and the need for correction depending on the requirements of the crop.

Effective sample size: Quick estimation of the effect of related samples in genetic case–control association analyses

The asymptotic behavior of estimates and information criteria in linear models are studied in the context of hierarchically correlated sampling units. The work is motivated by biological data collected on species where autocorrelation is based on the species' genealogical tree. Hierarchical autocorrelation is also found in many other kinds of data, such as from microarray experiments or human languages. Similar correlation also arises in ANOVA models with nested effects. I show that the best linear unbiased estimators are almost surely convergent but may not be consistent for some parameters such as the intercept and lineage effects, in the context of Brownian motion evolution on the genealogical tree. For the purpose of model selection I show that the usual BIC does not provide an appropriate approximation to the posterior probability of a model. To correct for this, an effective sample size is introduced for parameters that are inconsistently estimated. For biological studies, this work implies that tree-aware sampling design is desirable; adding more sampling units may not help ancestral reconstruction and only strong lineage effects may be detected with high power.

Determining the sample size of an experiment can be challenging, even more so when incorporating external information via a prior distribution. Such information is increasingly used to reduce the size of the control group in randomized clinical trials. Knowing the amount of prior information, expressed as an equivalent prior effective sample size (ESS), clearly facilitates trial designs. Various methods to obtain a prior's ESS have been proposed recently. They have been justified by the fact that they give the standard ESS for one-parameter exponential families. However, despite being based on similar information-based metrics, they may lead to surprisingly different ESS for nonconjugate settings, which complicates many designs with prior information. We show that current methods fail a basic predictive consistency criterion, which requires the expected posterior-predictive ESS for a sample of size N to be the sum of the prior ESS and N. The expected local-information-ratio ESS is introduced and shown to be predictively consistent. It corrects the ESS of current methods, as shown for normally distributed data with a heavy-tailed Student-t prior and exponential data with a generalized Gamma prior. Finally, two applications are discussed: the prior ESS for the control group derived from historical data and the posterior ESS for hierarchical subgroupanalyses.

Uma população é composta por indivíduos, e a amostragem correta dos indivíduos estima adequadamente as características da população. Porém, para avaliar a fertilidade do solo, quem seria o indivíduo solo (unidades de amostra) e qual sua dimensão? Com o objetivo de definir a dimensão do indivíduo solo componente de determinada população, sob plantio direto (PD) ou sob plantio convencional antes (PCAA) ou depois da aração (PCDA), visando avaliar a fertilidade do solo e desenvolver um método de amostragem de solos, determinando o número de amostras simples necessário à formação de uma amostra composta que caracterize o indivíduo solo (unidade de amostra), foram coletadas amostras simples de solo (5,4 cm de diâmetro x 10 cm de profundidade) sobre as semidiagonais de cinco hexágonos delimitados sobre cada uma das áreas selecionadas para amostragem (PD, PCAA e PCDA). Os hexágonos de amostragem apresentavam 2 m de lado e, em cada uma das seis semidiagonais dos mesmos, foram coletadas dez amostras simples de solo (55 por hexágono) nas seguintes distâncias, a partir de uma amostra simples central, medidas até o centro do orifício de coleta: 12,5; 25,0; 37,5; 50,0; 75,0; 100,0; 112,5; 150,0 e 200,0 cm, sendo a amostra simples central comum às seis semidiagonais. Foram determinados o pH (H2O), os teores de P e K disponíveis e de Ca2+ e Mg2+, H + Al, P-rem e a matéria orgânica. As amostras simples foram agrupadas sucessivamente a partir do centro do hexágono, formando nove unidades de amostra: A (até 18,75), B (até 31,25), C (até 43,75), D (até 56,25), E (até 81,25), F (até 106,25), G (até 118,75), H (até 156,25) e I (até 206,25 cm). Foi realizada análise de regressão das médias e dos desvios-padrão das características avaliadas, considerando possíveis dimensões da unidade de amostra de solo. A partir dos resultados obtidos, pôde-se concluir que: para a caracterização do indivíduo solo (unidade de amostra), devem-se coletar 25 amostras simples, necessárias à formação de uma amostra composta representativa, num hexágono de 68,75 cm de lado e área de 1,228 m².

Clinical prediction models are estimated using a sample of limited size from the target population, leading to uncertainty in predictions, even when the model is correctly specified. Generally, not all patient profiles are observed uniformly in model development. As a result, sampling uncertainty varies between individual patients' predictions. We aimed to develop an intuitive measure of individual prediction uncertainty. The variance of a patient's prediction can be equated to the variance of the sample mean outcome in hypothetical patients with the same predictor values. This hypothetical sample size can be interpreted as the number of similar patients that the prediction is effectively based on, given that the model is correct. For generalized linear models, we derived analytical expressions for the effective sample size. In addition, we illustrated the concept in patients with acute myocardial infarction. In model development, can be used to balance accuracy versus uncertainty of predictions. In a validation sample, the distribution of indicates which patients were more and less represented in the development data, and whether predictions might be too uncertain for some to be practically meaningful. In a clinical setting, the effective sample size may facilitate communication of uncertainty about predictions. We propose the effective sample size as a clinically interpretable measure of uncertainty in individual predictions. Its implications should be explored further for the development, validation and clinical implementation of prediction models.

P-values are misunderstood, but do not confound

As the length scale of sample dimensions is reduced to the micron and sub-micron scales, the strength of various materials has been observed to increase with decreasing size, a fact commonly referred to as the ‘sample size effect’. In this work, the influence of temperature on the sample size effect in copper is investigated using in situ microcompression testing at 25, 200 and 400 °C in the SEM on vacuum-annealed copper structures, and the resulting deformed structures were analysed using X-ray μLaue diffraction and scanning electron microscopy. For pillars with sizes between 0.4 and 4 μm, the size effect was measured to be constant with temperature, within the measurement precision, up to half of the melting point of copper. It is expected that the size effect will remain constant with temperature until diffusion-controlled dislocation motion becomes significant at higher temperatures and/or lower strain rates. Furthermore, the annealing treatment of the copper micropillars produced structures which yielded at stresses three times greater than their un-annealed, FIB-machined counterparts.

The paper describes an investigation of parasite richness in relation to host life history and ecology using data from an extensive survey of helminth parasites (cestodes, trematodes and nematodes) in Soviet birds. Correlates of parasite richness (number of parasite species per host species) were sought among 13 life-history variables, 13 ecological variables and one non-biological variable (number of host individuals examined) across a sample of 158 species of host. A statistical method to control for the effects of phylogenetic association was adopted throughout. Parasite richness correlates positively with the number of hosts examined (sample size) in all three parasite groups. Positive correlations (after controlling for the effects of sample size) were also found between host body weight and parasite richness for trematodes and nematodes, but not for cestodes. A number of ecological variables were associated with parasite richness. However, when the effects of sample size and body weight were controlled for, only a single significant correlation (an association between trematode richness and aquatic habitat) remained. Similarly, a number of significant correlates of parasite richness were found among the life-history variables examined. Though several of these were robust to the confounding effects of sample size, all could be explained by the co-variation between life-history traits and body weight among the host species under investigation.