A comparison of fixed-step-size and Bayesian staircases for sensory threshold estimation

Fixed-step-size (FSS) and Bayesian staircases are widely used methods to estimate sensory thresholds in 2AFC tasks, although a direct comparison of both types of procedure under identical conditions has not previously been reported. A simulation study and an empirical test were conducted to compare the performance of optimized Bayesian staircases with that of four optimized variants of FSS staircase differing as to up-down rule. The ultimate goal was to determine whether FSS or Bayesian staircases are the best choice in experimental psychophysics. The comparison considered the properties of the estimates (i.e. bias and standard errors) in relation to their cost (i.e. the number of trials to completion). The simulation study showed that mean estimates of Bayesian and FSS staircases are dependable when sufficient trials are given and that, in both cases, the standard deviation (SD) of the estimates decreases with number of trials, although the SD of Bayesian estimates is always lower than that of FSS estimates (and thus, Bayesian staircases are more efficient). The empirical test did not support these conclusions, as (1) neither procedure rendered estimates converging on some value, (2) standard deviations did not follow the expected pattern of decrease with number of trials, and (3) both procedures appeared to be equally efficient. Potential factors explaining the discrepancies between simulation and empirical results are commented upon and, all things considered, a sensible recommendation is for psychophysicists to run no fewer than 18 and no more than 30 reversals of an FSS staircase implementing the 1-up/3-down rule.     

Ordering genes: controlling the decision-error probabilities.

Determination of the relative gene order on chromosomes is of critical importance in the construction of human gene maps. In this paper we develop a sequential algorithm for gene ordering. We start by comparing three sequential procedures to order three genes on the basis of Bayesian posterior probabilities, maximum-likelihood ratio, and minimal recombinant class. In the second part of the paper we extend sequential procedure based on the posterior probabilities to the general case of g genes. We present a theorem that states that the predicted average probability of committing a decision error, associated with a Bayesian sequential procedure that accepts the hypothesis of a gene-order configuration with posterior probability equal to or greater than pi *, is smaller than 1 - pi *. This theorem holds irrespective of the number of genes, the genetic model, and the source of genetic information. The theorem is an extension of a classical result of Wald, concerning the sum of the actual and the nominal error probabilities in the sequential probability ratio test of two hypotheses. A stepwise strategy for ordering a large number of genes, with control over the decision-error probabilities, is discussed. An asymptotic approximation is provided, which facilitates the calculations with existing computer software for gene mapping, of the posterior probabilities of an order and the error probabilities. We illustrate with some simulations that the stepwise ordering is an efficient procedure.     

Designing occupancy surveys and interpreting non‐detection when observations are imperfect

Aim Conservation practitioners use biological surveys to ascertain whether or not a site is occupied by a particular species. Widely used statistical methods estimate the probability that a species will be detected in a survey of an occupied site. However, these estimates of detection probability are alone not sufficient to calculate the probability that a species is present given that it was not detected. The aim of this paper is to demonstrate methods for correctly calculating (1) the probability a species occupies a site given one or more non‐detections, and (2) the number of sequential non‐detections necessary to assert, with a pre‐specified confidence, that a species is absent from a site. Location Occupancy data for a tree frog in eastern Australia serve to illustrate methods that may be applied anywhere species’ occupancy data are used and detection probabilities are < 1. Methods Building on Bayesian expressions for the probability that a site is occupied by a species when it is not detected, and the number of non‐detections necessary to assert absence with a pre‐specified confidence, we estimate occupancy probabilities across tree frog survey locations, drawing on information about where and when the species was detected during surveys. Results We show that the number of sequential non‐detections necessary to assert that a species is absent increases nonlinearly with the prior probability of occupancy, the probability of detection if present, and the desired level of confidence about absence. Main conclusions If used more widely, the Bayesian analytical approaches illustrated here would improve collection and interpretation of biological survey data, providing a coherent way to incorporate detection probability estimates in the design of minimum survey requirements for monitoring, impact assessment and distribution modelling.     

Detection of significant disease risks using a spatial conditional autoregressive model

SUMMARY: The conditional autoregressive (CAR) model is widely used to describe the geographical distribution of a specific disease risk in lattice mapping. Successful developments based on frequentist and Bayesian procedures have been extensively applied to obtain two-stage disease risk predictions at the subregional level. Bayesian procedures are preferred for making inferences, as the posterior standard errors (SE) of the two-stage prediction account for the variability in the variance component estimates; however, some recent work based on frequentist procedures and the use of bootstrap adjustments for the SE has been undertaken. In this article we investigate the suitability of an analytical adjustment for disease risk inference that provides accurate interval predictions by using the penalized quasilikelihood (PQL) technique to obtain model parameter estimates. The method is a first-order approximation of the naive SE based on a Taylor expansion and is interpreted as a conditional measure of variability providing conditional calibrated prediction intervals, given the data. We conduct a simulation study to demonstrate how the method can be used to estimate the specific subregion risk by interval. We evaluate the proposed methodology by analyzing the commonly used example data set of lip cancer incidence in the 56 counties of Scotland for the period 1975-1980. This evaluation reveals a close similarity between the solutions provided by the method proposed here and those of its fully Bayesian counterpart.     

Self-designing two-stage trials to minimize expected costs

In the design of clinical trials, the sample size for the trial is traditionally calculated from estimates of parameters of interest, such as the mean treatment effect, which can often be inaccurate. However, recalculation of the sample size based on an estimate of the parameter of interest that uses accumulating data from the trial can lead to inflation of the overall Type I error rate of the trial. The self-designing method of Fisher, also known as the variance-spending method, allows the use of all accumulating data in a sequential trial (including the estimated treatment effect) in determining the sample size for the next stage of the trial without inflating the Type I error rate. We propose a self-designing group sequential procedure to minimize the expected total cost of a trial. Cost is an important parameter to consider in the statistical design of clinical trials due to limited financial resources. Using Bayesian decision theory on the accumulating data, the design specifies sequentially the optimal sample size and proportion of the test statistic's variance needed for each stage of a trial to minimize the expected cost of the trial. The optimality is with respect to a prior distribution on the parameter of interest. Results are presented for a simple two-stage trial. This method can extend to nonmonetary costs, such as ethical costs or quality-adjusted life years.     

Accuracy and selection success in yield trial analyses

Summary Yield trials serve research purposes of estimation and selection. Order statistics are used here to quantify the successes or problems to be expected in selection tasks commonly encountered in breeding and agronomy. Greater accuracy of yield estimates implies greater selection success. A New York soybean yield trial serves as a specific example. The Additive Main effects and Multiplicative Interaction (AMMI) statistical model is used to increase the accuracy of these soybean yield estimates, thereby increasing the probability of successfully selecting, on the basis of the empirical yield data, that genotype which has the maximum true mean. The statistical strategy for increasing accuracy is extremely cost effective relative to the alternative strategy of increasing the number of replications. Better selections increase the speed and effectiveness of breeding programs, and increase the reliability of variety recommendations. Selection tasks are frequently more difficult than may be suspected.This research was supported by the Rhizobotany Project of the USDA-ARS     

Normalizing upper trapezius EMG amplitude: Comparison of ramp and constant force procedures

The present study compared three procedures for normalization of upper trapezius surface electromyographic (EMG) amplitudes: (a) a ramp procedure (providing data in per cent of maximal voluntary contraction, MVC); (b) a constant force procedure based on two reference contractions (two-force procedure) (%MVC) and (c) a procedure expressing muscle activation in per cent of a reference voluntary electrical activity (%RVE). The study also evaluated the repeatability of the ramp and the RVE procedures and estimated the force exertion (%MVC) corresponding to the RVE. To illustrate the ergonomic effect of different normalization procedures, trapezius EMG during two work tasks was compared after normalization by the two-force and the RVE procedures. Fifteen subjects participated in the whole study. We found that force estimates obtained by the ramp procedure equation could be translated to force estimates obtained by the two-force procedure by the equation: %MVC_2force = − 0.6 + 0.9*%MVC_ramp, although with a considerable imprecision due to large inter-individual differences. In the ramp procedure, the intra-individual test-retest coefficient of variation (CV) depended on the force level; it was 45% at 5% MVC and 10% at 30% MVC. The CV of the RVE was 15%. The reference contraction used in the RVE procedure corresponded from 13–79% MVC (median 33%MVC). The load reducing effect of an ergonomic intervention was less obvious with the RVE procedure than with the two-force procedure due to a larger inter-individual variation. The advantages and disadvantages of the different procedures are discussed.     

Improved two‐stage group sequential procedures for testing a secondary endpoint after the primary endpoint achieves significance

In two‐stage group sequential trials with a primary and a secondary endpoint, the overall type I error rate for the primary endpoint is often controlled by an α‐level boundary, such as an O'Brien‐Fleming or Pocock boundary. Following a hierarchical testing sequence, the secondary endpoint is tested only if the primary endpoint achieves statistical significance either at an interim analysis or at the final analysis. To control the type I error rate for the secondary endpoint, this is tested using a Bonferroni procedure or any α‐level group sequential method. In comparison with marginal testing, there is an overall power loss for the test of the secondary endpoint since a claim of a positive result depends on the significance of the primary endpoint in the hierarchical testing sequence. We propose two group sequential testing procedures with improved secondary power: the improved Bonferroni procedure and the improved Pocock procedure. The proposed procedures use the correlation between the interim and final statistics for the secondary endpoint while applying graphical approaches to transfer the significance level from the primary endpoint to the secondary endpoint. The procedures control the familywise error rate (FWER) strongly by construction and this is confirmed via simulation. We also compare the proposed procedures with other commonly used group sequential procedures in terms of control of the FWER and the power of rejecting the secondary hypothesis. An example is provided to illustrate the procedures.     

Fast inference of interactions in assemblies of stochastic integrate-and-fire neurons from spike recordings

We present two Bayesian procedures to infer the interactions and external currents in an assembly of stochastic integrate-and-fire neurons from the recording of their spiking activity. The first procedure is based on the exact calculation of the most likely time courses of the neuron membrane potentials conditioned by the recorded spikes, and is exact for a vanishing noise variance and for an instantaneous synaptic integration. The second procedure takes into account the presence of fluctuations around the most likely time courses of the potentials, and can deal with moderate noise levels. The running time of both procedures is proportional to the number S of spikes multiplied by the squared number N of neurons. The algorithms are validated on synthetic data generated by networks with known couplings and currents. We also reanalyze previously published recordings of the activity of the salamander retina (including from 32 to 40 neurons, and from 65,000 to 170,000 spikes). We study the dependence of the inferred interactions on the membrane leaking time; the differences and similarities with the classical cross-correlation analysis are discussed.     

Antral follicle count reliably predicts number of morphologically healthy oocytes and follicles in ovaries of young adult cattle

Methods to predict numbers of healthy oocytes in the ovaries of young adults could have important diagnostic relevance in family planning and animal agriculture. We have observed that peak antral follicle count (AFC) determined by serial ovarian ultrasonography during follicular waves is very highly reproducible within individual young adult cattle, despite 7-fold variation among animals. Herein, we tested the hypothesis that AFC is positively associated with the number of morphologically healthy oocytes and follicles in ovaries and with serum concentrations of anti-Müllerian hormone (AMH), an indirect marker for number of healthy follicles and oocytes in ovaries. In the present study, age-matched young adult cattle (12-18 mo old) were subjected to serial ultrasonography to identify animals with a consistently high (> or =25 follicles that were > or =3 mm in diameter) or low (< or =15 follicles) AFC during follicular waves. Differences in serum AMH concentrations, ovary weight, and number of morphologically healthy and atretic follicles and oocytes were determined. The phenotypic classifications of cattle based on AFC during follicular waves or AMH concentrations both predict reliably the relative number of morphologically healthy follicles and oocytes in ovaries of age-matched young adult cattle.     

Estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials

For confirmatory trials of regulatory decision making, it is important that adaptive designs under consideration provide inference with the correct nominal level, as well as unbiased estimates, and confidence intervals for the treatment comparisons in the actual trials. However, naive point estimate and its confidence interval are often biased in adaptive sequential designs. We develop a new procedure for estimation following a test from a sample size reestimation design. The method for obtaining an exact confidence interval and point estimate is based on a general distribution property of a pivot function of the Self-designing group sequential clinical trial by Shen and Fisher (1999, Biometrics55, 190-197). A modified estimate is proposed to explicitly account for futility stopping boundary with reduced bias when block sizes are small. The proposed estimates are shown to be consistent. The computation of the estimates is straightforward. We also provide a modified weight function to improve the power of the test. Extensive simulation studies show that the exact confidence intervals have accurate nominal probability of coverage, and the proposed point estimates are nearly unbiased with practical sample sizes.     

Bayesian adaptive biased-coin designs for clinical trials with normal responses

Adaptive designs are used in phase III clinical trials for skewing the allocation pattern toward the better treatments. We use optimum design theory to derive a skewed Bayesian biased-coin procedure for sequential designs with continuous responses. The skewed designs are used to provide adaptive designs, the performance of which is studied numerically and theoretically. Important properties are loss and the proportion of allocation to the better treatment.     

Meta-analysis of diagnostic test accuracy assessment studies with varying number of thresholds

Current meta-analytic methods for diagnostic test accuracy are generally applicable to a selection of studies reporting only estimates of sensitivity and specificity, or at most, to studies whose results are reported using an equal number of ordered categories. In this article, we propose a new meta-analytic method to evaluate test accuracy and arrive at a summary receiver operating characteristic (ROC) curve for a collection of studies evaluating diagnostic tests, even when test results are reported in an unequal number of nonnested ordered categories. We discuss both non-Bayesian and Bayesian formulations of the approach. In the Bayesian setting, we propose several ways to construct summary ROC curves and their credible bands. We illustrate our approach with data from a recently published meta-analysis evaluating a single serum progesterone test for diagnosing pregnancy failure.     

Bayesian detection and modeling of spatial disease clustering

Many current statistical methods for disease clustering studies are based on a hypothesis testing paradigm. These methods typically do not produce useful estimates of disease rates or cluster risks. In this paper, we develop a Bayesian procedure for drawing inferences about specific models for spatial clustering. The proposed methodology incorporates ideas from image analysis, from Bayesian model averaging, and from model selection. With our approach, we obtain estimates for disease rates and allow for greater flexibility in both the type of clusters and the number of clusters that may be considered. We illustrate the proposed procedure through simulation studies and an analysis of the well-known New York leukemia data.     

DNA extraction procedure: a critical issue for bacterial diversity assessment in marine sediments

In order to evaluate whether different DNA extraction procedures can affect estimates of benthic bacterial diversity, based on 16S rRNA gene terminal restriction fragment length polymorphism (T-RFLP) fingerprinting technique, we compared two in situ lysis procedures (a SDS-based protocol and a commercial kit for DNA recovery) and one cell-extraction protocol on a variety of marine sediments. Despite the two in situ lysis procedures resulted in significantly different DNA yields (highest with the SDS in situ lysis), estimates of bacterial diversity provided a not significantly different ribotype richness, as well as similar values of the Shannon-Wiener (H') and Margalef (d) indices of biodiversity and of evenness (Pielou index, J). Conversely, the cell-extraction procedure for DNA extraction resulted always in a significantly lower ribotype richness and diversity. The analysis of similarities (anosim) among the T-RFLP electropherograms allowed concluding that ribotypes composition did not change significantly using different protocols. However, the analysis of beta-diversity (turnover diversity) revealed that a large number of ribotypes was observed exclusively with one of the three protocols utilized. When unshared ribotypes from in situ lysis and cell extraction were pooled together, total ribotype richness resulted much higher (up to 80%). Our results indicate that estimates of ribotype diversity based on a single protocol of DNA extraction can significantly underestimate the total number of bacterial ribotypes present in the benthic domain. We recommend that future studies will not only integrate different DNA extraction procedures, but also will explore the possibility of integrating two or more different genetic markers in order to increase our ability to detect the actual bacterial diversity in environmental samples.     

Cyclic kinetics and mathematical expression of the primary immune response

The kinetics of antibody-forming cells (AFC) in the spleen of rats immunized with Salmonella typhi O-antigen was investigated. The number of nucleated cells of spleen and blood serum antibody titres in passive haemagglutination were determined in parallel. Cyclic changes in the number of antibody-forming cells were detected as three peaks on the 4th, 9th, and 13th days following immunization. The fluctuations of their number were not related to the total number of nucleated cells of spleen. The antibody titres reached their peak on the 10th day following immunization, decreased by the 14th day and rose again on the 16th day after immunization. Repeated increases of the number of AFC were probably due to the regular, not accidental, recruitment of committed precursors cells (memory cells).     

Bayesian sample size calculations for comparing two strategies in SMART studies

In the management of most chronic conditions characterized by the lack of universally effective treatments, adaptive treatment strategies (ATSs) have grown in popularity as they offer a more individualized approach. As a result, sequential multiple assignment randomized trials (SMARTs) have gained attention as the most suitable clinical trial design to formalize the study of these strategies. While the number of SMARTs has increased in recent years, sample size and design considerations have generally been carried out in frequentist settings. However, standard frequentist formulae require assumptions on interim response rates and variance components. Misspecifying these can lead to incorrect sample size calculations and correspondingly inadequate levels of power. The Bayesian framework offers a straightforward path to alleviate some of these concerns. In this paper, we provide calculations in a Bayesian setting to allow more realistic and robust estimates that account for uncertainty in inputs through the ‘two priors’ approach. Additionally, compared to the standard frequentist formulae, this methodology allows us to rely on fewer assumptions, integrate pre-trial knowledge, and switch the focus from the standardized effect size to the MDD. The proposed methodology is evaluated in a thorough simulation study and is implemented to estimate the sample size for a full-scale SMART of an internet-based adaptive stress management intervention on cardiovascular disease patients using data from its pilot study conducted in two Canadian provinces.     

Accurate and rapid estimation of phosphene thresholds (REPT)

To calibrate the intensity of transcranial magnetic stimulation (TMS) at the occipital pole, the phosphene threshold is used as a measure of cortical excitability. The phosphene threshold (PT) refers to the intensity of magnetic stimulation that induces illusory flashes of light (phosphenes) on a proportion of trials. The existing PT estimation procedures lack the accuracy and mathematical rigour of modern threshold estimation methods. We present an improved and automatic procedure for estimating the PT which is based on the well-established Ψ Bayesian adaptive staircase approach. To validate the new procedure, we compared it with another commonly used procedure for estimating the PT. We found that our procedure is more accurate, reliable, and rapid when compared with an existing PT measurement procedure. The new procedure is implemented in Matlab and works automatically with the Magstim Rapid(2) stimulator using a convenient graphical user interface. The Matlab program is freely available for download.     

Estimating the probability for major gene Alzheimer disease.

Alzheimer disease (AD) is neuropsychiatric illness caused by multiple etiologies. Prediction of whether AD is genetically based in a given family is problematic because of censoring bias among unaffected relatives as a consequence of the late onset of the disorder, diagnostic uncertainties, heterogeneity, and limited information in a single family. We have developed a method based on Bayesian probability to compute values for a continuous variable that ranks AD families as having a major gene form of AD (MGAD). In addition, we have compared the Bayesian method with a maximum-likelihood approach. These methods incorporate sex- and age-adjusted risk estimates and allow for phenocopies and familial clustering of age at onset. Agreement is high between the two approaches for ranking families as MGAD (Spearman rank [r] = .92). When either method is used, the numerical outcomes are sensitive to assumptions of the gene frequency and cumulative incidence of the disease in the population. Consequently, risk estimates should be used cautiously for counseling purposes; however, there are numerous valid applications of these procedures in genetic and epidemiological studies.     

Sequential hypothesis testing with spatially correlated presence-absence data

A pest management decision to initiate a control treatment depends upon an accurate estimate of mean pest density. Presence-absence sampling plans significantly reduce sampling efforts to make treatment decisions by using the proportion of infested leaves to estimate mean pest density in lieu of counting individual pests. The use of sequential hypothesis testing procedures can significantly reduce the number of samples required to make a treatment decision. Here we construct a mean-proportion relationship for Oligonychus perseae Tuttle, Baker, and Abatiello, a mite pest of avocados, from empirical data, and develop a sequential presence-absence sampling plan using Bartlett's sequential test procedure. Bartlett's test can accommodate pest population models that contain nuisance parameters that are not of primary interest. However, it requires that population measurements be independent, which may not be realistic because of spatial correlation of pest densities across trees within an orchard. We propose to mitigate the effect of spatial correlation in a sequential sampling procedure by using a tree-selection rule (i.e., maximin) that sequentially selects each newly sampled tree to be maximally spaced from all other previously sampled trees. Our proposed presence-absence sampling methodology applies Bartlett's test to a hypothesis test developed using an empirical mean-proportion relationship coupled with a spatial, statistical model of pest populations, with spatial correlation mitigated via the aforementioned tree-selection rule. We demonstrate the effectiveness of our proposed methodology over a range of parameter estimates appropriate for densities of O. perseae that would be observed in avocado orchards in California.