The use of Wald's sequential probability ratio test to develop composite three-decision sampling plans

1985 ◽  
Vol 15 (2) ◽  
pp. 326-330
Author(s):  
Gary W. Fowler
Keyword(s):  
Monte Carlo ◽  
Negative Binomial ◽  
Sequential Sampling ◽  
Ratio Test ◽  
Sample Number ◽  
Sampling Plans ◽  
Probability Ratio ◽  
Sequential Probability ◽  
Error Probabilities ◽  
Decision Boundaries

Many sequential sampling plans used in forest sampling are composite three-decision plans based on the simultaneous use of two of Wald's sequential probability ratio tests (SPRTs). Wald's operating characteristic (OC) and average sample number (ASN) equations for each SPRT are used to describe the properties of the composite sampling plan. Wald's equations are only approximate because of "overshooting" of the decision boundaries of the SPRTs and the two SPRTs operate simultaneously in the composite plan. Wald's and Monte Carlo OC and ASN functions were developed for (i) two SPRTs used to develop a three-decision composite plan and (ii) the three-decision composite plan based on the negative binomial distribution. Wald's equations, in general, overestimate the true error probabilities and underestimate the true ASN for a given SPRT. Wald's equations are less accurate in describing the properties of the three-decision plan. Monte Carlo functions are more accurate than Wald's functions. Recommendations are made regarding the choice between Wald's and Monte Carlo functions. A Monte Carlo procedure to modify the decision boundaries of the plan to yield actual error probabilities approximately equal to the desired error probabilities is suggested.

Accuracy of sequential sampling plans based on Wald's sequential probability ratio test

1983 ◽  
Vol 13 (6) ◽  
pp. 1197-1203 ◽  
Author(s):  
Gary W. Fowler
Keyword(s):  
Monte Carlo ◽  
Sequential Sampling ◽  
Sequential Probability Ratio Test ◽  
Ratio Test ◽  
Type I ◽  
Sample Number ◽  
Sampling Plans ◽  
Probability Ratio ◽  
Sequential Probability ◽  
Relative Errors

Monte Carlo operating characteristic (OC) and average sample number (ASN) functions were compared with Wald's OC and ASN equations for sequential sampling plans based on Wald's sequential probability ratio test (SPRT) using the binomial, negative binomial, normal, and Poisson distributions. This comparison showed that the errors inherent in Wald's equations as a result of "overshooting" the decision boundaries of the SPRT can be large. Relative errors increased for the OC and ASN equations as the difference between the null (θ0)) and alternative (θ1) test parameter values increased. Relative errors also increased for the ASN equation as the probabilities of type I (α) and type II (β) errors increased. For discrete distributions, the relative errors also increased as θ0 increased with θ1/θ0 fixed. Wald's equations, in general, overestimate the true error probabilities and underestimate the true ASN. For the values of θ0, θ1, α, and β used in many sequential sampling plans in forestry, Wald's equations may not be adequate. For those cases where the errors in Wald's equations are important compared with the other errors associated with the sampling plan, two alternative Monte Carlo OC and ASN functions are proposed.

scholarly journals Spatial Distribution and Development of Sequential Sampling Plans for Diaphorina citri Kuwayama (Hemiptera: Liviidae)

Agronomy ◽  
2021 ◽  
Vol 11 (7) ◽  
pp. 1434
Author(s):  
Gabriel Díaz-Padilla ◽  
J. Isabel López-Arroyo ◽  
Rafael A. Guajardo-Panes ◽  
Ignacio Sánchez-Cohen
Keyword(s):  
Spatial Distribution ◽  
Sequential Sampling ◽  
Field Evaluation ◽  
Production Costs ◽  
Ratio Test ◽  
Operational Characteristic ◽  
Diaphorina Citri ◽  
Sample Number ◽  
Sampling Plans ◽  
Sequential Probability

Vector control in huanglongbing management has been conducted on a calendar basis resulting in high production costs. We addressed this issue and proposed a sequential sampling plan to support decision making for intervention against Diaphorina citri Kuwayama, which is involved in the transmission of the bacteria Candidatus Liberibacter asiaticus, associated with such lethal disease. We analyzed 3,264,660 records from samples gathered from the Mexican trapping program for the monitoring of D. citri; it included weekly inspection of 86,004 yellow sticky traps distributed in the country. Spatial distribution of the insect, estimation of a common k (kc), and sequential sampling plans based on Sequential Probability Ratio Test (SPRT) were determined. Taylor’s power law coefficients were ≥1 indicating aggregation in the spatial distribution of the insect. Common k ranged from 0.0183 to 0.2253 and varied independently of geographic zone or citrus species. We obtained 18 sequential sampling plans, one for each state. In the Average Sample Number (ASN) function, the minimal number of samples to make a decision ranged from 17 to 65. In the Operational Characteristic (OC) function, probabilities for a correct intervention at the threshold of 0.2 D. citri adults/trap in most cases were above 80%. In a field evaluation, the application of sampling plans yielded savings obtained by reduction in the number of interventions for insect control.

SEQUENTIAL SAMPLING OF ADULT NORTHERN AND WESTERN CORN ROOTWORMS (COLEOPTERA: CHRYSOMELIDAE) IN SOUTHERN ONTARIO

1987 ◽  
Vol 119 (6) ◽  
pp. 577-585 ◽  
Author(s):  
H.J. McAuslane ◽  
C.R. Ellis ◽  
O.B. Allen
Keyword(s):  
Negative Binomial ◽  
Sequential Sampling ◽  
Ratio Test ◽  
Economic Threshold ◽  
Diabrotica Barberi ◽  
Southern Ontario ◽  
Sampling Plans ◽  
Field Corn ◽  
Corn Rootworms ◽  
Sequential Probability

AbstractThree sequential-sampling plans were developed for adult Diabrotica barberi Smith and Lawrence and D. virgifera virgifera LeConte in field corn in southern Ontario. The distribution of both species was well approximated by the negative binomial distribution and obeyed Taylor’s power law. Two plans, following the methods of Kuno (1969) and Green (1970), estimated populations with predetermined precision. The third plan, based on Wald’s (1947) sequential probability ratio test, categorized populations relative to an economic threshold. Different sequential-sampling plans were constructed for 1st-year fields and for all other fields because the parameter k, measuring the extent of aggregation of the beetles, differed between these two classes of fields. Decision equations for Wald’s plan in 1st-year fields were: D1 = 0.98n − 17.75 and D2 = 0.98n + 13.82. Decision equations for 2nd-year or older fields were: D1 = 0.99n − 13.42 and D2 = 0.99n + 10.45. The use of Wald’s plan reduced the required sample size by 55% in 1st-year fields, and by 48% in all other fields. The savings associated with the other sequential-sampling plans were not significant.

A sequential decision plan for the management of the eastern hemlock looper, Lambdinafiscellariafiscellaria (Lepidoptera: Geometridae), in Newfoundland

1989 ◽  
Vol 19 (7) ◽  
pp. 911-916 ◽  
Author(s):  
E. J. Dobesberger
Keyword(s):  
Negative Binomial ◽  
Eastern Hemlock ◽  
Ratio Test ◽  
Sequential Decision ◽  
Sample Number ◽  
Average Sample Number ◽  
Probability Ratio ◽  
Sequential Probability ◽  
Hemlock Looper ◽  
Average Sample

A sequential decision plan based on Wald's sequential probability ratio test for the negative binomial distribution was derived for eastern hemlock looper (Lambdinafiscellariafiscellaria (Guen.)) egg populations in Newfoundland. An average sample number of not more than six midcrown branches was feasible, and both α and β error rates were defined. Monte Carlo simulation of operating characteristic and average sample number values for static and dynamic K of the negative binomial showed that Wald's sequential probability ratio test was acceptable. More eggs were found on midcrown balsam fir (Abiesbalsamea (L.) Mill.) branches than on other sampling substrates, such as ground mosses (mainly comprising Hylocomiumsplendens (Hedw.) B.S.G., Pleuroziumschreberi (Brid.) Mitt., and Ptiliumcrista-castrensis (Hedw.) De Not.), loose bark from paper birch (Betulapapyrifera Marsh.), and crown lichens (primarily Usnealongissima Ach.).

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

Plant Disease ◽  
2009 ◽  
Vol 93 (5) ◽  
pp. 490-498 ◽  
Author(s):  
Odile Carisse ◽  
Catherine Meloche ◽  
Guy Boivin ◽  
Tristan Jobin
Keyword(s):  
Decision Making ◽  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Sequential Sampling ◽  
Apple Scab ◽  
Parameter Estimates ◽  
Ratio Test ◽  
Sample Number ◽  
Action Thresholds

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.

The impact of misspecification of the negative binomial shape parameter in sequential sampling plans

1986 ◽  
Vol 16 (3) ◽  
pp. 608-611 ◽  
Author(s):  
W. G. Warren ◽  
Pin Whei Chen
Keyword(s):  
Operating Characteristic ◽  
Shape Parameter ◽  
Negative Binomial ◽  
Sequential Sampling ◽  
Forest Pests ◽  
Sample Number ◽  
Sampling Plans ◽  
Average Sample Number ◽  
The Impact ◽  
Average Sample

Standard sequential sampling plans for determining whether infestations of forest pests have attained critical levels are commonly based on the assumption that the counts follow a negative binomial distribution for which the shape parameter, k, which must be specified, may be difficult to estimate and may well be unstable. This paper studies the effect of misspecification of this parameter on the operating characteristic and average sample number functions of a sequential sampling plan. It appears that slight underestimation of the shape parameter can improve the operating characteristic at little cost, i.e., with only small increase in the average sample number.

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