Wald's Sequential Probability Ratio Research Articles

Improving the safety of radiotherapy treatment processes via incident-driven FMEA feedback loops.

Failure mode and effects analysis (FMEA) is a valuable tool for radiotherapy risk assessment, yet its outputs might be unreliable due to failures not being identified or due to a lack of accurate error rates. A novel incident reporting system (IRS) linked to an FMEA database was tested and evaluated. The study investigated whether the system was suitable for validating a previously performed analysis and whether it could provide accurate error rates to support the expert occurrence ratings of previously identified failure modes. Twenty-three pre-identified failure modes of our external beam radiotherapy process, covering the process steps from patient admission to treatment delivery, were proffered on dedicated FMEA feedback and incident reporting terminals generated by the IRS. The clinical setting involved a computed tomography scanner, dosimetry, and five linacs. Incoming reports were used as basis to identify additional failure modes or confirm initial ones. The Kruskal-Wallis H test was applied to compare the risk priorities of the retrospective and prospective failure modes. Wald's sequential probability ratio test was used to investigate the correctness of the experts' occurrence ratings by means of the number of incoming reports. Over a 15-month period, 304 reports were submitted. There were 0.005 (confidence interval [CI], 0.0014-0.0082) reported incidents per imaging study and 0.0006 (CI, 0.0003-0.0009) reported incidents per treatment fraction. Sixteen additional failure modes could be identified, and their risk priorities did not differ from those of the initial failure modes (p=0.954). One failure mode occurrence rating could be increased, whereas the other 22 occurrence ratings could not be disproved. Our approach is suitable for validating FMEAs and deducing additional failure modes on a continual basis. Accurate error rates can only be provided if a sufficient number of reports is available.

Development of a sequential sampling plan for classifying Thrips tabaci (Thysanoptera: Thripidae) populations in onion fields.

Onion thrips, Thrips tabaci Lindeman, is a global pest of onion crops, causing substantial economic damage by diminishing bulb yields and transmitting plant pathogens. Insecticides are used to manage T. tabaci infestations with control decisions traditionally based on action thresholds that require visually counting thrips on a fixed, predetermined number of onion plants per field. However, this approach for treatment decisions is inefficient when thrips populations are well above or below the action threshold. The aim of this research was to develop a sequential sampling plan that would provide a rapid and reliable classification of thrips populations in commercial onion fields above or below prespecified management thresholds. The study was conducted in a total of 24 commercial onion fields in New York in 2021 and 2022. Taylor's power law and Wald's Sequential Probability Ratio Test were used in concert to develop each sampling plan. Simulated and historical field data of thrips populations were used to further validate the efficacy of each sampling plan. Results demonstrated the sequential sampling plan required an average of 78% fewer samples to make a control decision compared with the traditional fixed-sampling approach. Treatment decisions were reached in 72% of cases after inspecting only 10 plants, while only 6% of the cases required examining more than 25 plants. Comparisons with fixed-sample sizes ranging from 23 to 68 plants revealed a 96% agreement in decision-making and a 78% reduction in sampling effort when using the sequential sampling plans.

Bibliography of Sequential Sampling Plans in Insect Pest Management Based on Wald's Sequential Probability Ratio Test

Bibliography of Sequential Sampling Plans in Insect Pest Management Based on Wald's Sequential Probability Ratio Test

A note on efficient performance evaluation of the Cumulative Sum chart and the Sequential Probability Ratio Test

We establish a simple connection between certain in‐control characteristics of the Cumulative Sum (CUSUM) Run Length and their out‐of‐control counterparts. The connection is in the form of paired integral (renewal) equations. The derivation exploits Wald's likelihood ratio identity and the well‐known fact that the CUSUM chart is equivalent to repetitive application of Wald's Sequential Probability Ratio Test (SPRT). The characteristics considered include the entire Run Length distribution and all of the corresponding moments, starting from the zero‐state average run length. A particular practical benefit of our result is that it enables the in‐control and out‐of‐control characteristics of the CUSUM Run Length to be computed concurrently. Moreover, owing to the equivalence of the CUSUM chart to a sequence of SPRTs, the Average Sample Number and Operating Characteristic functions of an SPRT under the null and under the alternative can all be computed simultaneously as well. This would double up the efficiency of any numerical method that one may choose to devise to carry out the actual computations. Copyright © 2016 John Wiley & Sons, Ltd.

Spatial Distribution and Sequential Sampling of the Banana Root Borer

The banana root borer, Cosmopolites sordidus Germar (Coleoptera: Curculionidae), is a major pest in many banana (Musa spp.)‐producing regions of the world, causing yield losses of up to 100%. The aims of this study were to define the spatial distribution pattern of C. sordidus in banana plantations, to determine the probability distribution model that best describes the sampling data, and to develop a sequential sampling plan for this pest. Aggregation was investigated using aggregation indices and theoretical frequency distributions. To quantify the number of insects, 80 traps were distributed in a commercial banana plantation with an area of 4 ha and inspections were performed every 2 wk from 2005 to 2008. The sequential sampling plan was based on Wald's sequential probability ratio test. The calculated indices indicated moderate aggregation, with a common k parameter of the negative binomial distribution equal to 3.29. The best model for predicting the spatial distribution of the insects was the negative binomial distribution, which fit 93% of the inspections, followed by the Neyman Type A distribution, which fit 90% of the surveys, and Poisson's distribution, which fit only one survey. A sequential sampling plan was generated with Type I and II error rates of 5%, for which 19 was the maximum expected number of samples required to reach a decision for an infestation mean of approximately 3.7 insects per trap. The validation showed that the constructed plan is reliable and provides correct recommendations for decision‐making in pest control. The banana root borer has an aggregated spatial distribution in the field. Negative binomial distribution is the best model to explain species spatial pattern. A sequential sampling plan was developed and validated. The sequential sampling plan showed reliable decision rates. The sequential sampling plan showed a reasonable number of required sample units.

Decision Making in the Arrow of Time

We show that the steady-state entropy production rate of a stochastic process is inversely proportional to the minimal time needed to decide on the direction of the arrow of time. Here we apply Wald's sequential probability ratio test to optimally decide on the direction of time's arrow in stationary Markov processes. Furthermore, the steady-state entropy production rate can be estimated using mean first-passage times of suitable physical variables. We derive a first-passage time fluctuation theorem which implies that the decision time distributions for correct and wrong decisions are equal. Our results are illustrated by numerical simulations of two simple examples of nonequilibrium processes.

Exact boundaries in sequential testing for phase-type distributions

Consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. Exact decision boundaries for given type-I and type-II errors are derived by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated with a certain one-sided Markov additive process. By suitable transformations, the results also apply to other types of distributions, including some distributions with regularly varying tails.

Exact boundaries in sequential testing for phase-type distributions

Consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. Exact decision boundaries for given type-I and type-II errors are derived by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated with a certain one-sided Markov additive process. By suitable transformations, the results also apply to other types of distributions, including some distributions with regularly varying tails.

Sequential sampling for panicle caterpillars (Lepidoptera: Noctuidae) in sorghum.

Panicle caterpillars comprise an economically important insect pest complex of sorghum throughout the Great Plains of the United States, particularly in Kansas, Oklahoma, and Texas. The sorghum panicle caterpillar complex consists of larvae of two polyphagous lepidopteran species: the corn earworm, Helicoverpa zea (Boddie), and fall armyworm, Spodoptera frugiperda (J.E. Smith) (Lepidoptera: Noctuidae). Sampling for panicle caterpillars in sorghum fields is usually accomplished by the beat bucket sampling technique with a fixed sample size of 30 beat bucket samples of one sorghum panicle each per 16.2 ha of field. We used Wald's sequential probability ratio test for a negative binomial distribution to develop a sequential sampling plan for panicle caterpillars. In total, 115 sorghum fields were sampled in Kansas, Oklahoma, and Texas from June to August 2010. Panicle caterpillars had an aggregated distribution of counts confirmed by Pearson's chi-square statistic for lack of fit to the negative binomial distribution for each sampled field. A sequential sampling plan was developed using a high threshold (an economic threshold) of 0.5 caterpillars per sorghum panicle, a low threshold (a safe level) of 0.20 caterpillars per panicle, and fixed error rates (alpha = 0.10 and beta = 0.05). At caterpillar densities > 0.45 and < 0.12 per panicle, the average number of panicles inspected to make a decision was less than the current recommendation of 30. In a 2013 validation test of 25 fields, the expected number of samples taken from average sample number curve was in close agreement with the number of samples required using the sequential plan (r2 = 0.93), and all fields were correctly classified when compared with a fixed sample size result. The plan improved upon current sampling recommendations for panicle caterpillars in sorghum because at known acceptable fixed error rates fewer samples were required when caterpillars are scarce or abundant, whereas more samples were required to make decisions with the same acceptable error rates when densities were near the economic thresholds.

On the Wald's Sequential Probability Ratio Test for Lévy Processes

The Wald's sequential probability ratio test (SPRT) of two simple hypotheses regarding the Lévy-Khintchine triplet of a wide family of Lévy processes is analyzed: we concentrate on continuous paths and pure increasing jump Lévy processes. Appealing to the theory of Markov processes, we employ a general method for determining the stopping boundaries and the expected length of the SPRT for a given admissible pair (α, β) of error probabilities. The well-known results of the Wiener and Poisson sequential testing can be derived accordingly. The explicit solution for the SPRT of two simple hypotheses about the parameter p ∈ (0, 1) of a Lévy negative binomial process is shown.

Random sampler M-estimator algorithm with sequential probability ratio test for robust function approximation via feed-forward neural networks.

This paper addresses the problem of fitting a functional model to data corrupted with outliers using a multilayered feed-forward neural network. Although it is of high importance in practical applications, this problem has not received careful attention from the neural network research community. One recent approach to solving this problem is to use a neural network training algorithm based on the random sample consensus (RANSAC) framework. This paper proposes a new algorithm that offers two enhancements over the original RANSAC algorithm. The first one improves the algorithm accuracy and robustness by employing an M-estimator cost function to decide on the best estimated model from the randomly selected samples. The other one improves the time performance of the algorithm by utilizing a statistical pretest based on Wald's sequential probability ratio test. The proposed algorithm is successfully evaluated on synthetic and real data, contaminated with varying degrees of outliers, and compared with existing neural network training algorithms.

The Use of Wald’s Sequential Probability Ratio Test (SPRT) in Cocoa Pod Borer Management

Sequential sampling plan (SSP) has been widely used in many engineering and quality control study. The advantage of using the SSP is the sample size is not fixed in advance, but is determined in part by the results of the sampling process. The application of the SSP is not limited to the industry, it also being used in pest management. This is because the SSP are generally more cost effective than plans based on a fixed sample size. Wald's Sequential Probability Ratio Test (SPRT) is one of the most common sequential sampling plans in insect pest management. It has been used to determine pest status at one time which could be used, through time, to monitor the status of the population and to take action when the pest density exceeded the Economic Threshold Level (ETL). Cocoa pod borer or CPB (Conopomorpha cramerella (Snell)) is a pest of concern to the cocoa growers in South-East Asia that has become one of the major factors that dampened the interest expressed in cocoa cultivation because of the heavy losses of cocoa pods due to the pest if effective control measure is not employed. The chemical control appeared to be one of the effective control measure used by the growers to control the CPB. Current practice used by the growers in applying the chemical is biweekly spraying which could see the increasing cost of chemical and labor used. The study was conducted to develop the Wald’s SPRT to monitor the CPB infestation level relative to the ETL based on counting the CPB eggs found on the pods aged 1 month before ripen or pod length 131 mm to 150 mm. The performance of the Wald’s SPRT was validated using independent data sets collected from Cocoa Research and Development Center (CRDC) Madai.

Closed Sequential and Multistage Inference on Binary Responses With or Without Replacement

We consider closed sequential or multistage sampling, with or without replacement, from a lot of N items, where each item can be identified as defective (in error, tainted, etc.) or not. The goal is to make inference on the proportion, π, of defectives in the lot, or equivalently on the number of defectives in the lot D = Nπ. It is shown that exact inference on π using closed (bounded) sequential or multistage procedures with general prespecified elimination boundaries is completely tractable and not at all inconvenient using modern statistical software. We give relevant theory and demonstrate functions for this purpose written in R (R Development Core Team 2005, available as online supplementary material). Applicability of the methodology is illustrated in three examples: (1) sharpening of Wald's (1947) sequential probability ratio test used in industrial acceptance sampling, (2) two-stage sampling for auditing Medicare or Medicaid health care providers, and (3) risk-limited sequential procedures for election audits.

On SPRT and RSPRT for the Unknown Mean in a Normal Distribution with Equal Mean and Variance

We consider a sequence of independent observations from a N(θ, θ) distribution with 0 < θ, c < ∞. Here, θ remains unknown. Given independent observations X 1,…, X n from N(θ, θ), the statistic is complete and sufficient for θ and T has a noncentral with λ =nθ when θ is the true parameter value. The problem is one of testing H 0: θ = θ0 vs. H 1: θ = θ1 where θ0, θ1 (θ0 ≠ θ1) are specified positive numbers with target type-I and type-II error probabilities 0 < α <1 and 0 < β <1, respectively, α + β <1. We first adapt Wald's sequential probability ratio test (SPRT) and describe some crucial characteristics. However, observations may instead arrive with random group sizes sequentially rather than one at a time. Thus, as a follow-up, next we implement Mukhopadhyay and de Silva's (2008) notion of a random SPRT (RSPRT). In the case of both SPRT and RSPRT, we address the issue of truncation. The proposed methodologies are illustrated with appropriate statistical data analysis. In the end, we briefly explore the problem of confidence interval estimation for the parameter θ after termination of the SPRT.

Cost-Effective Binomial Sequential Sampling of Western Bean Cutworm, Striacosta albicosta (Lepidoptera: Noctuidae), Egg Masses in Corn

Striacosta albicosta (Smith) (Lepidoptera: Noctuidae), is a native pest of dry beans (Phaseolus vulgaris L.) and corn (Zea mays L.). As a result of larval feeding damage on corn ears, S. albicosta has a narrow treatment window; thus, early detection of the pest in the field is essential, and egg mass sampling has become a popular monitoring tool. Three action thresholds for field and sweet corn currently are used by crop consultants, including 4% of plants infested with egg masses on sweet corn in the silking-tasseling stage, 8% of plants infested with egg masses on field corn with approximately 95% tasseled, and 20% of plants infested with egg masses on field corn during mid-milk-stage corn. The current monitoring recommendation is to sample 20 plants at each of five locations per field (100 plants total). In an effort to develop a more cost-effective sampling plan for S. albicosta egg masses, several alternative binomial sampling plans were developed using Wald's sequential probability ratio test, and validated using Resampling for Validation of Sampling Plans (RVSP) software. The benefit-cost ratio also was calculated and used to determine the final selection of sampling plans. Based on final sampling plans selected for each action threshold, the average sample number required to reach a treat or no-treat decision ranged from 38 to 41 plants per field. This represents a significant savings in sampling cost over the current recommendation of 100 plants.

Efficient Sequential Correspondence Selection by Cosegmentation

In many retrieval, object recognition, and wide-baseline stereo methods, correspondences of interest points (distinguished regions) are commonly established by matching compact descriptors such as SIFTs. We show that a subsequent cosegmentation process coupled with a quasi-optimal sequential decision process leads to a correspondence verification procedure that 1) has high precision (is highly discriminative), 2) has good recall, and 3) is fast. The sequential decision on the correctness of a correspondence is based on simple statistics of a modified dense stereo matching algorithm. The statistics are projected on a prominent discriminative direction by SVM. Wald's sequential probability ratio test is performed on the SVM projection computed on progressively larger cosegmented regions. We show experimentally that the proposed sequential correspondence verification (SCV) algorithm significantly outperforms the standard correspondence selection method based on SIFT distance ratios on challenging matching problems.

Sampling Schemes for Sequential Detection With Dependent Observations

Several sampling schemes and their corresponding sequential detection procedures in autoregressive noise are presented in this paper. Two of them use uniform sampling procedures with high and low sampling rates, respectively. The other two employ groups of samples, which are separated by long intergroup delays such that the intergroup correlations are negligible. One of the group-sampling schemes also employs optimal signaling waveforms to further improve its energy-efficiency. In all the schemes, data sampling and transformation are designed in such a way that Wald's sequential probability ratio test (SPRT) can still be implemented. The performances of different schemes, in terms of average termination time (ATT), are derived analytically. When all the schemes employ the same sampling interval and under a constant signal amplitude constraint, their performances are compared through analytical and numerical methods. In addition, under a constant power constraint, their ATTs and energy-efficiency are compared. It is theoretically proved that the scheme using groups of samples with the optimal signaling waveform is the most energy-efficient.

Mitigating primary user emulation attacks in dynamic spectrum access networks using hypothesis testing

We present a Neyman-Pearson composite hypothesis test (NPCHT) and a Wald's sequential probability ratio test (WSPRT) to detect primary user emulation attacks (PUEA) in cognitive radio networks. Most approaches in the literature on PUEA assume the presence of underlying sensor networks for localization of the malicious nodes. There are no analytical studies available in the literature to study PUEA in the presence of multiple malicious users in fading wireless environments. We present an NPCHT and WSPRT based analysis to detect PUEA in fading wireless channels in the presence of multiple randomly located malicious users. We show that there is a range of network radii in which PUEA are most successful. Results also show that for the same desired threshold on the probability of missing the primary, WSPRT can achieve a probability of successful PUEA 50% less than that obtained by NPCHT.

Action Thresholds for Summer Fungicide Sprays and Sequential Classification of Apple Scab Incidence.

The objectives of this study were to establish scab incidence thresholds at which summer fungicide sprays should be initiated and to develop sequential sampling for classification plans to facilitate decision making. The relationship between proportion of scabbed leaves per shoot and percent scabbed fruits at harvest was established based on data collected in both experimental and commercial orchards. Action threshold at which fungicide spray program should be initiated in order to maintain fruit scab at harvest below 2% was established at 0.006 scabbed leaves per shoot; hence incidence thresholds of 0.005 and 0.01 scabbed leaves per shoot were evaluated. Sequential sampling for classification procedures (SSCP), using the Wald's sequential probability ratio test (SPRT), were developed based on a beta-binomial distribution and using parameter estimates of the binary power law. Monte Carlo simulations were used to establish the probability of classifying mean scab incidence as less than the action thresholds (pth = 0.005 and pth = 0.01). Operating characteristic (OC) and average sample number (ASN) curves were established for eight combinations of stop lines and error levels (α and β). Based on the results of the Monte Carlo simulations, four sets of stop lines (two for each pth threshold) were selected for further evaluation. Bootstrap simulations of 50 data sets indicated that both the OC and ASN curves for each of the four pairs of stop lines were similar to OC and ASN values determined by Monte Carlo simulation. When validated with data not used to construct the models, the SSCPs provided the appropriate classification in 98 and 95% of the simulations for the pth = 0.005 and pth = 0.01 (α = 0.05 and β = 0.10), respectively. The sequential sampling plans developed in this study should allow for rapid and accurate classification of the incidence of apple scab on leaves, and aid in sampling for summer scab management decision making.

Robust detection of selfish misbehavior in wireless networks

The CSMA/CA protocols are designed under the assumption that all participant nodes would abide to the protocol rules. This is of particular importance in distributed protocols such as the IEEE 802.11 distributed coordinating function (DCF), in which nodes control their own backoff parameters. In this work, we propose a method to detect selfish misbehaving terminals that may deliberately modify its backoff window to gain unfair access to the network resources. We develop nonparametric batch and sequential detectors based on the Kolmogorov-Smirnov (K-S) statistics that do not require any modification on the existing CSMA/CA protocols, and we apply it to detect misbehaviors in an IEEE 802.11 DCF network using the ns-2 simulator. We compare the performance of the proposed detectors with the optimum detectors with perfect information about the misbehavior strategy, for both the batch case (based on the Neyman-Pearson test), and the sequential case (based on Wald's sequential probability ratio test). We show that the proposed nonparametric detectors have a performance comparable to the optimum detectors for the majority of misbehaviors (the more severe) without any knowledge of the misbehavior strategies.