Selecting Significant Effects in Factorial Designs Taking Type II Errors into Account

When analysing the effects of a factorial design, it is customary to take into account the probability of making a Type I error (the probability of considering an effect significant when it is non‐significant), but not to consider the probability of making a Type II error (the probability of considering an effect as non‐significant when it is significant). Making a Type II error, however, may lead to incorrect decisions regarding the values that the factors should take or how subsequent experiments should be conducted. In this paper, we introduce the concept of minimum effect size of interest and present a visualization method for selecting the critical value of the effects, the threshold value above which an effect should be considered significant, which takes into account the probability of Type I and Type II errors. Copyright © 2006 John Wiley & Sons, Ltd.     

The effects of inspection error on single sampling inspection plans

Acceptance sampling plans are designed under the assumption of perfect inspection. However, inspection tasks are not, even under ideal inspection conditions, free of error. In this paper we consider the effects of inspection error on probability of acceptance, average outgoing quality, and average total inspection. These measures are examined under both replacement and non -replacement assumptions. Also, a method is presented whereby an acceptance sampling plan may be designed which explicitly considers inspection error.     

A review of: “Melal Cutting Principles.” By MILTON C. SHAW. (Clarendon Press, 1984.) [Pp. 594.] Price £60.

This paper develops formulas that can be used in the design of multiple criteria sampling plans or charts for fraction nonconforming, with sampling on variables or attributes. Products often have multiple requirements and the usual acceptance tests or charts do not take this into account, and hence the overall quality of the product may be in poor control. We design tests or charts on the basis of probability of Type I and Type II errors (α and β) that refer to acceptable and rejectable levels of the overall fraction of the product that is nonconforming. Further, recognizing that the average proportion of the product that is actually nonconforming on each of the characteristics may vary independently of the other characteristics, our formulas give protection on a ‘worst case’ basis.     

The Dodge CSP-1 Continuous Sampling Plan Under Inspection Error1

This paper examines the results of Dodge's CSP-1 continuous sampling plan under inspection error. Both type I and II errors are explicitly included. Their effect on the average number of items inspected in a 100% screening sequence, average number of items passed during sampling inspection, average fraction of items inspected, proportion of items passed under sampling, and average outgoing quality are examined. Further, the relationships necessary to determine compensating sampling plans, considering inspection error, which yield the desired actual AOQL, are developed. Use of the compensating plan method is illustrated through an example.     

Acceptance control and guardbanding for error‐prone individual measurements

The concept of fractional nonconformance was recently proposed to assess the probability of conformance when measurements are error‐prone. Applications of fractional nonconformance assessment include acceptance sampling inspection and short‐run process control. In this study, we introduce a fractional nonconformance‐based one‐sided acceptance control chart to monitor a short‐run process as well as decide the acceptability of products manufactured from the process. The guardband technique is also incorporated in the proposed approach to reduce the impact of measurement errors. Guardband selection is investigated for both independent and autocorrelated processes. Our analysis shows that guardbanding is beneficial for short‐run production environments. The optimum guardbands obtained under risk and cost models are also found to be consistent.     

The Distribution of Observed Defectives in Attribute Acceptance Sampling Plans Under Inspection Error

The Type B Operating Characteristic curve in attributes acceptance sampling is simply the probability of lot acceptance versus process fraction defective. The binomial mass function is appropriate for assessing this acceptance probability. Articles on inspection error in attributes acceptance sampling frequently assume the marginal distribution of the number of observed defectives in the sample is binomial with the true fraction defective replaced by the apparent fraction defective. This assumption, however, has been questioned and it is the object of this paper to show explicitly that the existing literature is quite correct. Following this development, additional results are presented which may be useful to others performing research in this area.     

A model for evaluating the effectiveness of motor vehicle inspection programs

A statistical model for the evaluation of the effectiveness of motor vehicle inspection programs in reducing highway crashes is presented. The model is based on the assumption that the waiting time between highway crashes follows an exponential distribution. Since highway crashes are relatively rare events, it is assumed that the length of the study period is such that censoring occurs. Under these assumptions, maximum likelihood estimates of the mean waiting time θ until a crash for the non-inspected (inspected) vehicles is obtained and the corresponding test statistic is derived. As mechanically-caused accidents are but a small part of the overall accident picture and since inspection should only affect this portion, sample size requirements are investigated for various combinations of θ, Δ (increase in average time until a crash due to the effect of inspection), L (length of study period), and = β (probability of Type I error equalling probability of Type II error). For reasonable Δ, the sample required is indeed sizable.     

Bayesian optimum life testing plans under progressive Type‐I interval censoring scheme

In many industrial applications, it is not always feasible to continuously monitor the life testing experiments to collect lifetime data. Moreover, intermediate removals of the test units from the life testing experiment are sometimes essential. Progressive Type‐I interval censoring schemes are useful in these scenarios. Optimal planning of such progressive Type‐I interval censoring schemes is an important issue to the experimenter, as the optimal plans can achieve the desired objectives using much lesser resources. This article provides Bayesian D‐optimal progressive Type‐I interval censoring schemes, assuming that the lifetime follows a log‐normal distribution. An algorithm is provided to find the optimal censoring schemes and the number of inspections. The algorithm is then used to obtain the optimal Bayesian progressive Type‐I interval censoring schemes in 2 different contexts. The resulting optimal Bayesian censoring schemes are compared with the corresponding locally optimal censoring schemes. A detailed sensitivity analysis is performed to investigate the effect of prior information. The sampling variation associated with the optimal censoring schemes is visualized through a simulation study.     

Optimal control limits for CCC charts in the presence of inspection errors

The control chart based on cumulative count of conforming (CCC) items between the occurrence of two non‐conforming ones, or the CCC chart, has been shown to be very useful for monitoring high‐quality processes. However, as in the implementation of other Shewhart‐type control charts, it is usually assumed that the inspection is free of error. This assumption may not be valid and this may have a significant impact on the interpretation of the control chart and the setting of control limits. This paper first investigates the effect of inspection errors and discusses the setting of control limits in such cases. Even if inspection errors are considered, the average time to alarm increases in the beginning when the process deteriorates. Since this is undesirable, the control limits in the presence of inspection errors should be set so as to maximize the average run length when the process is at the normal level. A procedure is presented for solving this problem. Copyright © 2003 John Wiley & Sons, Ltd.     

Accelerated testing of on‐board diagnostics

Modern products frequently feature monitors designed to detect actual or impending malfunctions. False alarms (Type I errors) or excessive delays in detecting real malfunctions (Type II errors) can seriously reduce monitor utility. Sound engineering practice includes physical evaluation of error rates. Type II error rates are relatively easy to evaluate empirically. However, adequate evaluation of a low Type I error rate is difficult without using accelerated testing concepts, inducing false alarms using artificially low thresholds and then selecting production thresholds by appropriate extrapolation, as outlined here. This acceleration methodology allows for informed determination of detection thresholds and confidence in monitor performance with substantial reductions over current alternatives in time and cost required for monitor development. Copyright © 2006 John Wiley & Sons, Ltd.     

Comparison of Weibull small samples using Monte Carlo simulations

The evaluation of the functional reliability of different designs is a common task and times to failure can be compared using the likelihood ratio test. In the microelectronics industry, as in many others, the high cost of testing places severe restrictions on the sample size. Moreover, the products in these tests are often new and do not have previous reliability histories. These factors make the selection of the Type I and Type II errors in comparison tests very difficult. This paper presents the Monte Carlo simulation results of Type II errors for the likelihood ratio test of comparison as a function of the Type I error and the (small) sample size. Our conclusions are summarized as follows: (1) the common microelectronics industry standard sample size of 32 is often insufficient to reach satisfactory conclusions; (2) small sample tests should only be used for prescreening for significant differences; and (3) when only small samples are available, the Type I and the Type II errors must be selected carefully to prevent misleading conclusions. Copyright © 2006 John Wiley & Sons, Ltd.     

THE EFFECTS OF INSPECTION ERRORS TO THE IMPERFECT EMQ MODEL

The presence of Type I and Type II inspection errors may seriously affect the product quality. In this paper we incorporate them into the EMQ (Economic Manufacturing Quantity) model under the imperfect production system. We derive the expected total cost when the shift of the production process follows a general distribution and the inspection interval is arbitrary. The objective is to determine the optimal production cycle length and optimal inspection number while minimizing the total cost. Results for special shift distributions such as exponential and Weibull are established. Sensitivity analysis of the optimal production cycle length and analysis of the cost surplus for incorporating inspection errors under exponential shift distribution are also provided.     

Attribute Control Charts with Optimal Limits

A new attribute control chart is presented to monitor processes that generate count data. The economic objective of the chart is to minimize the total cost of its errors, a linear function of errors Type I and II. The proposed chart can be applied to Poisson, geometric, and negative binomial assumptions. Control limits are calculated optimally, because they are based on exact probability distributions and used to detect defined directional shifts in a process. Some numerical results are provided, and expected costs of the new chart are compared with those of a one‐sided c‐chart. Other effects such as changing the cost structure are shown graphically. Copyright © 2015 John Wiley & Sons, Ltd.     

Applying Bootstrap method to the types I–II errors in the measurement system

This paper attempts to evaluate different methods of calculating the type‐I and type‐II errors in the measurement system. Furthermore, we apply the Bootstrap method to construct the confidence intervals for the type‐I and type‐II errors. Also, the proposed method is compared with the generalized inference method. Several factors such as the sample size, the measurement error, the process mean, and the process variation are simulated to validate the performance. The simulation results show that both methods almost have the same performance. In addition, we develop a computer program that can evaluate the error of measurement system without changing information or data. Two case studies of the nano measurement data are used to demonstrate the application. The simulation results indicate that the sample size has an influence for all cases. The type‐I and type‐II errors are decreased when the measurement error is increased. The type‐I and type‐II errors are affected by the measurement error, the process mean, and the process deviation. The case studies show that the development of nano technology requires the immediate attention of the measurement capability. Copyright © 2007 John Wiley & Sons, Ltd.     

Selecting relevant effects in factorial designs

Industrial contexts tend to be as much or more concerned about the probability of ignoring an effect when its influence on the response is relevant (Type II error) than about the probability of considering an effect to be active when in fact, it is not (Type I error). Here, we present a methodology for taking into account both types of error by fixing an effect value that is considered large enough to control the probability of it going unnoticed. In addition, we propose a plot to visualize the results obtained.     

A Comparison of Inspector Performance Measures

Using experimental data from their signal-detection study, the authors examine traditional measures of inspector performance along with measures based on Bayes'rule. Results indicate that the inspector's decision process is affected by the a priori probability of nonconforming product in the inspection lot and that inspector decision error may result in a tighter OC curve for a sampling plan. The conclusions are that performance measures should be selected relative to performance evaluation objectives and that graphs describing acceptance sampling plans for ideal inspectors should be modified to reflect performance of real inspectors under actual conditions.     

Effect of estimation under nonnormality on the phase II performance of linear profile monitoring approaches

The number of studies about control charts proposed to monitor profiles, where the quality of a process/product is expressed as function of response and explanatory variable(s), has been increasing in recent years. However, most authors assume that the in‐control parameter values are known in phase II analysis and the error terms are normally distributed. These assumptions are rarely satisfied in practice. In this study, the performance of EWMA‐R, EWMA‐3, and EWMA‐3(d²) methods for monitoring simple linear profiles is examined via simulation where the in‐control parameters are estimated and innovations have a Student's t distribution or gamma distribution. Instead of the average run length (ARL) and the standard deviation of run length, we used average and standard deviation of the ARL as performance measures in order to capture the sampling variation among different practitioners. It is seen that the estimation effect becomes more severe when the number of phase I profiles used in estimation decreases, as expected, and as the distribution deviates from normality to a greater extent. Besides, although the average ARL values get closer to the desired values as the amount of phase I data increases, their standard deviations remain far away from the acceptable level indicating a high practitioner‐to‐practitioner variability.     

Supplier Selection Critical Decision Values for Processes with Multiple Independent Lines

The process yield is the most common criterion considered for decision making in supplier selection problem. For normally distributed processes with multiple independent lines, the index provides an exact measurement for the overall yield. Therefore, the index can be implemented to deal with the supplier selection problem with processes having multiple independent lines. In this article, a test statistic obtained by a division method is employed to establish a hypothesis testing procedure, with two phases, which is developed to determine whether two suppliers are equally capable or not. The sampling distribution and the probability density function of the test statistic are derived. For various minimum requirements of process capability, number of lines, sample sizes, magnitudes of the difference between the two suppliers and the type I error, the critical values for decision making are presented. The required sample sizes for various designated powers at given type I error are tabulated. A thin‐film transistor type liquid‐crystal display application example is provided to demonstrate the testing procedure. Copyright © 2012 John Wiley & Sons, Ltd.     

Effects of measurement errors on sample monitoring reliability

Results are presented on the effects of measurement errors on items in routine production as regards the reliability of results on sampling inspection obtained by Monte Carlo simulation. The customers and the makers risks are examined in relation to the distributions of the measurement errors and the values of them.Translated from Izmeritelnaya Tekhnika, No. 12, pp. 8–11, December, 2004.This revised version was published online in April 2005 with a corrected cover date.     

An Integrated Optimization Model for Inventory and Quality Control Problems

In this paper, we develop an integrated economic model for inventory and quality control problems, extending the work of Rahim ( IIE Transactions 1994; 26(6): 2–11) and Rahim and Ben-Daya (IJPR 1998; 36(1): 277–289). The production process is subject to an assignable cause which shifts the process from an in-control state to an out-of-control state. We consider the shifts in both the process mean and the process variance. When a signal for an assignable cause is triggered, a search is initiated and is terminated upon finding the cause within a pre-specified target time. The process is then brought back to an in-control state by repair. However, if the assignable cause is not discovered within the pre-specified time, production is allowed to continue until the next sampling or warning, whichever occurs first. In this case, either the alarm is considered to be false with a probability of Type I error, or the assignable cause has not been eliminated with a probability of Type II error. In the latter case, the process produces products in an out-of-control state until the next sampling or warning, whichever occurs first. However, this state does not indicate any severe damage to the system. Joint X and R charts are used for monitoring both process mean and variance. Under these conditions, a generalized economic model for the joint determination of production quantity, an inspection schedule, and the design of the X and R control charts are developed. A direct search optimization method is used to determine the optimal decision variables of the economic model.