X̄ chart with variable sample size and sampling intervals

Recent theoretical studies have shown that the X̄ chart with variable sampling intervals (VSI) and the X̄ chart with variable sample size (VSS) are quicker than the traditional X̄ chart in detecting shifts in the process. This article considers the X̄ chart with variable sample size and sampling intervals (VSSI). It is assumed that the amount of time the process remains in control has exponential distribution. The properties of the VSSI X̄ chart are obtained using Markov chains. The VSSI X̄ chart is even quicker than the VSI or VSS X̄ charts in detecting moderate shifts in the process.

(X)OVER-BAR CHARTS WITH VARIABLE SAMPLE-SIZE

The usual practice in using a control chart to monitor a process is to take samples of size n from the process every h hours This article considers the properties of the XBAR chart when the size of each sample depends on what is observed in the preceding sample. The idea is that the sample should be large if the sample point of the preceding sample is close to but not actually outside the control limits and small if the sample point is close to the target. The properties of the variable sample size (VSS) XBAR chart are obtained using Markov chains. The VSS XBAR chart is substantially quicker than the traditional XBAR chart in detecting moderate shifts in the process.

X̄ charts with variable sample size

The usual practice in using a control chart to monitor a process is to take samples of size n from the process every h hours. This article considers the properties of the X̄ chart when the size of each sample depends on what is observed in the preceding sample. The idea is that the sample should be large if the sample point of the preceding sample is close to but not actually outside the control limits and small if the sample point is close to the target. The properties of the variable sample size (VSS) X̄ chart are obtained using Markov chains. The VSS X̄ chart is substantially quicker than the traditional X̄ chart in detecting moderate shifts in the process.

The Variable Sample Size (X)over-bar Chart with Estimated Parameters

The VSS X chart, dedicated to the detection of small to moderate mean shifts in the process, has been investigated by several researchers under the assumption of known process parameters. In practice, the process parameters are rarely known and are usually estimated from an in-control Phase I data set. In this paper, we evaluate the (run length) performances of the VSS chart when the process parameters are estimated, we compare them in the case where the process parameters are assumed known and we propose specific optimal control chart parameters taking the number of Phase I samples into account.

Joint X̄ and R Charts with Variable Sample Sizes and Sampling Intervals

Recent studies have shown that the X̄ chart with variable sampling intervals (VSI) and/or with variable sample sizes (VSS) detects process shifts faster than the traditional X̄ chart. This article extends these studies for processes that are monitored by both the X̄ and R charts. A Markov chain model is used to determine the properties of the joint X and R charts with variable sample sizes and sampling intervals (VSSI). The VSSI scheme improves the joint X̄ and R control chart performance in terms of the speed with which shifts in the process mean and/or variance are detected.

Bivariate control charts with double sampling

In this article, we consider the T(2) chart with double sampling to control bivariate processes (BDS chart). During the first stage of the sampling, n(1) items of the sample are inspected and two quality characteristics (x; y) are measured. If the Hotelling statistic T(1)(2) for the mean vector of (x; y) is less than w, the sampling is interrupted. If the Hotelling statistic T(1)(2) is greater than CL(1), where CL(1) > w, the control chart signals an out-of-control condition. If w < T(1)(2) <= CL(1), the sampling goes on to the second stage, where the remaining n(2) items of the sample are inspected and T(2)(2) for the mean vector of the whole sample is computed. During the second stage of the sampling, the control chart signals an out-of-control condition when the statistic T(2)(2) is larger than CL(2). A comparative study shows that the BDS chart detects process disturbances faster than the standard bivariate T(2) chart and the adaptive bivariate T(2) charts with variable sample size and/or variable sampling interval.

Synthetic control charts with two-stage sampling for monitoring bivariate processes

In this article, we consider the synthetic control chart with two-stage sampling (SyTS chart) to control bivariate processes. During the first stage, one item of the sample is inspected and two correlated quality characteristics (x;y) are measured. If the Hotelling statistic T1 2 for these individual observations of (x;y) is lower than a specified value UCL 1 the sampling is interrupted. Otherwise, the sampling goes on to the second stage, where the remaining items are inspected and the Hotelling statistic T2 2 for the sample means of (x;y) is computed. When the statistic T2 2 is larger than a specified value UCL2, the sample is classified as nonconforming. According to the synthetic control chart procedure, the signal is based on the number of conforming samples between two neighbor nonconforming samples. The proposed chart detects process disturbances faster than the bivariate charts with variable sample size and it is from the practical viewpoint more convenient to administer.

Joint (X)over-bar and R charts with variable sample sizes and sampling intervals

Recent studies have shown that the (X) over bar chart with variable sampling intervals (VSI) and/or with variable sample sizes (VSS) detects process shifts faster than the traditional (X) over bar chart. This article extends these studies for processes that are monitored by both the (X) over bar and R charts. A Markov chain model is used to determine the properties of the joint (X) over bar and R charts with variable sample sizes and sampling intervals (VSSI). The VSSI scheme improves the joint (X) over bar and R control chart performance in terms of the speed with which shifts in the process mean and/or variance are detected.

A hierarchy of adaptive (X)over-bar control charts

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Weed control in corn and weed sample size for growth evaluations

The objectives of this study were to evaluate baby corn yield, green corn yield, and grain yield in corn cultivar BM 3061, with weed control achieved via a combination of hoeing and intercropping with gliricidia, and determine how sample size influences weed growth evaluation accuracy. A randomized block design with ten replicates was used. The cultivar was submitted to the following treatments: A = hoeings at 20 and 40 days after corn sowing (DACS), B = hoeing at 20 DACS + gliricidia sowing after hoeing, C = gliricidia sowing together with corn sowing + hoeing at 40 DACS, D = gliricidia sowing together with corn sowing, and E = no hoeing. Gliricidia was sown at a density of 30 viable seeds m-2. After harvesting the mature ears, the area of each plot was divided into eight sampling units measuring 1.2 m² each to evaluate weed growth (above-ground dry biomass). Treatment A provided the highest baby corn, green corn, and grain yields. Treatment B did not differ from treatment A with respect to the yield values for the three products, and was equivalent to treatment C for green corn yield, but was superior to C with regard to baby corn weight and grain yield. Treatments D and E provided similar yields and were inferior to the other treatments. Therefore, treatment B is a promising one. The relation between coefficient of experimental variation (CV) and sample size (S) to evaluate growth of the above-ground part of the weeds was given by the equation CV = 37.57 S-0.15, i.e., CV decreased as S increased. The optimal sample size indicated by this equation was 4.3 m².

Sample size considerations in active-control non-inferiority trials with binary data based on the odds ratio

This paper presents an approximate closed form sample size formula for determining non-inferiority in active-control trials with binary data. We use the odds-ratio as the measure of the relative treatment effect, derive the sample size formula based on the score test and compare it with a second, well-known formula based on the Wald test. Both closed form formulae are compared with simulations based on the likelihood ratio test. Within the range of parameter values investigated, the score test closed form formula is reasonably accurate when non-inferiority margins are based on odds-ratios of about 0.5 or above and when the magnitude of the odds ratio under the alternative hypothesis lies between about 1 and 2.5. The accuracy generally decreases as the odds ratio under the alternative hypothesis moves upwards from 1. As the non-inferiority margin odds ratio decreases from 0.5, the score test closed form formula increasingly overestimates the sample size irrespective of the magnitude of the odds ratio under the alternative hypothesis. The Wald test closed form formula is also reasonably accurate in the cases where the score test closed form formula works well. Outside these scenarios, the Wald test closed form formula can either underestimate or overestimate the sample size, depending on the magnitude of the non-inferiority margin odds ratio and the odds ratio under the alternative hypothesis. Although neither approximation is accurate for all cases, both approaches lead to satisfactory sample size calculation for non-inferiority trials with binary data where the odds ratio is the parameter of interest.

Double sampling (x)over-bar control chart for a first-order autoregressive moving average process model

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Control charts for monitoring the mean vector and the covariance matrix of bivariate processes

In this article, we propose new control charts for monitoring the mean vector and the covariance matrix of bivariate processes. The traditional tools used for this purpose are the T (2) and the |S| charts. However, these charts have two drawbacks: (1) the T (2) and the |S| statistics are not easy to compute, and (2) after a signal, they do not distinguish the variable affected by the assignable cause. As an alternative to (1), we propose the MVMAX chart, which only requires the computation of sample means and sample variances. As an alternative to (2), we propose the joint use of two charts based on the non-central chi-square statistic (NCS statistic), named as the NCS charts. Once the NCS charts signal, the user can immediately identify the out-of-control variable. In general, the synthetic MVMAX chart is faster than the NCS charts and the joint T (2) and |S| charts in signaling processes disturbances.

Joint (X)over-bar and R charts with two-stage samplings

When joint (X) over bar and R charts are in use, samples of fixed size are regularly taken from the process, and their means and ranges are plotted on the (X) over bar and R charts, respectively. In this article, joint (X) over bar and R charts have been used for monitoring continuous production processes. The sampling is performed, in two stages. During the first stage, one item of the sample is inspected and, depending on the result, the sampling is interrupted if the process is found to be in control; otherwise, it goes on to the second stage, where the remaining sample items are inspected. The two-stage sampling procedure speeds up the detection of process disturbances. The proposed joint (X) over bar and R charts are easier to administer and are more efficient than the joint (X) over bar and R charts with variable sample size where the quality characteristic of interest can be evaluated either by attribute or variable. Copyright (C) 2004 John Wiley Sons, Ltd.

VSSI (X)over-bar charts with sampling at fixed times

A standard (X) over bar chart for controlling the process mean takes samples of size no at specified, equally-spaced, fixed-time points. This article proposes a modification of the standard (X) over bar chart that allows one to take additional samples, bigger than no, between these fixed times. The additional samples are taken from the process when there is evidence that the process mean moved from target. Following the notation proposed by Reynolds (1996a) and Costs (1997) we shortly call the proposed (X) over bar chart as VSSIFT (X) over bar chart: where VSSIFT means variable sample size and sampling intervals with fixed times. The (X) over bar chart with the VSSIFT feature is easier to be administered than a standard VSSI (X) over bar chart that is not constrained to sample at the specified fixed times. The performances of the charts in detecting process mean shifts are comparable.