(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.

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.

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.

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.

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.

Sample size determination for kernel regression estimation using sequential fixed-width confidence bands

We consider a random design model based on independent and identically distributed pairs of observations (Xi, Yi), where the regression function m(x) is given by m(x) = E(Yi|Xi = x) with one independent variable. In a nonparametric setting the aim is to produce a reasonable approximation to the unknown function m(x) when we have no precise information about the form of the true density, f(x) of X. We describe an estimation procedure of non-parametric regression model at a given point by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient of at least 1−. Here, d(> 0) and 2 (0, 1) are two preassigned values. Fixed-width confidence intervals are developed using both Nadaraya-Watson and local linear kernel estimators of nonparametric regression with data-driven bandwidths. The sample size was optimized using the purely and two-stage sequential procedures together with asymptotic properties of the Nadaraya-Watson and local linear estimators. A large scale simulation study was performed to compare their coverage accuracy. The numerical results indicate that the confi dence bands based on the local linear estimator have the better performance than those constructed by using Nadaraya-Watson estimator. However both estimators are shown to have asymptotically correct coverage properties.

Optimal image sample size for corneal nerve morphometry

Purpose Arbitrary numbers of corneal confocal microscopy images have been used for analysis of corneal subbasal nerve parameters under the implicit assumption that these are a representative sample of the central corneal nerve plexus. The purpose of this study is to present a technique for quantifying the number of random central corneal images required to achieve an acceptable level of accuracy in the measurement of corneal nerve fiber length and branch density. Methods Every possible combination of 2 to 16 images (where 16 was deemed the true mean) of the central corneal subbasal nerve plexus, not overlapping by more than 20%, were assessed for nerve fiber length and branch density in 20 subjects with type 2 diabetes and varying degrees of functional nerve deficit. Mean ratios were calculated to allow comparisons between and within subjects. Results In assessing nerve branch density, eight randomly chosen images not overlapping by more than 20% produced an average that was within 30% of the true mean 95% of the time. A similar sampling strategy of five images was 13% within the true mean 80% of the time for corneal nerve fiber length. Conclusions The “sample combination analysis” presented here can be used to determine the sample size required for a desired level of accuracy of quantification of corneal subbasal nerve parameters. This technique may have applications in other biological sampling studies.

A variable step-size FxLMS algorithm for feedforward active noise control systems based on a new online secondary path modelling technique

Several approaches have been introduced in literature for active noise control (ANC) systems. Since FxLMS algorithm appears to be the best choice as a controller filter, researchers tend to improve performance of ANC systems by enhancing and modifying this algorithm. This paper proposes a new version of FxLMS algorithm. In many ANC applications an online secondary path modelling method using a white noise as a training signal is required to ensure convergence of the system. This paper also proposes a new approach for online secondary path modelling in feedfoward ANC systems. The proposed algorithm stops injection of the white noise at the optimum point and reactivate the injection during the operation, if needed, to maintain performance of the system. Benefiting new version of FxLMS algorithm and not continually injection of white noise makes the system more desirable and improves the noise attenuation performance. Comparative simulation results indicate effectiveness of the proposed approach.

Sample size considerations and augmentation of computer experiments

Computer Experiments, consisting of a number of runs of a computer model with different inputs, are now common-place in scientific research. Using a simple fire model for illustration some guidelines are given for the size of a computer experiment. A graph is provided relating the error of prediction to the sample size which should be of use when designing computer experiments. Methods for augmenting computer experiments with extra runs are also described and illustrated. The simplest method involves adding one point at a time choosing that point with the maximum prediction variance. Another method that appears to work well is to choose points from a candidate set with maximum determinant of the variance covariance matrix of predictions.

Genetics of anisotropy asymmetry: Registration and sample size effects

Brain asymmetry has been a topic of interest for neuroscientists for many years. The advent of diffusion tensor imaging (DTI) allows researchers to extend the study of asymmetry to a microscopic scale by examining fiber integrity differences across hemispheres rather than the macroscopic differences in shape or structure volumes. Even so, the power to detect these microarchitectural differences depends on the sample size and how the brain images are registered and how many subjects are studied. We fluidly registered 4 Tesla DTI scans from 180 healthy adult twins (45 identical and fraternal pairs) to a geometrically-centered population mean template. We computed voxelwise maps of significant asymmetries (left/right hemisphere differences) for common fiber anisotropy indices (FA, GA). Quantitative genetic models revealed that 47-62% of the variance in asymmetry was due to genetic differences in the population. We studied how these heritability estimates varied with the type of registration target (T1- or T2-weighted) and with sample size. All methods consistently found that genetic factors strongly determined the lateralization of fiber anisotropy, facilitating the quest for specific genes that might influence brain asymmetry and fiber integrity.

Nintendo Wii Fit as an adjunct to physiotherapy following lower limb fractures: Preliminary feasibility, safety and sample size considerations

Objective The Nintendo Wii Fit integrates virtual gaming with body movement, and may be suitable as an adjunct to conventional physiotherapy following lower limb fractures. This study examined the feasibility and safety of using the Wii Fit as an adjunct to outpatient physiotherapy following lower limb fractures, and reports sample size considerations for an appropriately powered randomised trial. Methodology Ambulatory patients receiving physiotherapy following a lower limb fracture participated in this study (n = 18). All participants received usual care (individual physiotherapy). The first nine participants also used the Wii Fit under the supervision of their treating clinician as an adjunct to usual care. Adverse events, fracture malunion or exacerbation of symptoms were recorded. Pain, balance and patient-reported function were assessed at baseline and discharge from physiotherapy. Results No adverse events were attributed to either the usual care physiotherapy or Wii Fit intervention for any patient. Overall, 15 (83%) participants completed both assessments and interventions as scheduled. For 80% power in a clinical trial, the number of complete datasets required in each group to detect a small, medium or large effect of the Wii Fit at a post-intervention assessment was calculated at 175, 63 and 25, respectively. Conclusions The Nintendo Wii Fit was safe and feasible as an adjunct to ambulatory physiotherapy in this sample. When considering a likely small effect size and the 17% dropout rate observed in this study, 211 participants would be required in each clinical trial group. A larger effect size or multiple repeated measures design would require fewer participants.

Statistical power calculation and sample size determination for environmental studies with data below detection limits

Power calculation and sample size determination are critical in designing environmental monitoring programs. The traditional approach based on comparing the mean values may become statistically inappropriate and even invalid when substantial proportions of the response values are below the detection limits or censored because strong distributional assumptions have to be made on the censored observations when implementing the traditional procedures. In this paper, we propose a quantile methodology that is robust to outliers and can also handle data with a substantial proportion of below-detection-limit observations without the need of imputing the censored values. As a demonstration, we applied the methods to a nutrient monitoring project, which is a part of the Perth Long-Term Ocean Outlet Monitoring Program. In this example, the sample size required by our quantile methodology is, in fact, smaller than that by the traditional t-test, illustrating the merit of our method.

A Bayesian decision approach for sample size determination in phase II trials

Stallard (1998, Biometrics 54, 279-294) recently used Bayesian decision theory for sample-size determination in phase II trials. His design maximizes the expected financial gains in the development of a new treatment. However, it results in a very high probability (0.65) of recommending an ineffective treatment for phase III testing. On the other hand, the expected gain using his design is more than 10 times that of a design that tightly controls the false positive error (Thall and Simon, 1994, Biometrics 50, 337-349). Stallard's design maximizes the expected gain per phase II trial, but it does not maximize the rate of gain or total gain for a fixed length of time because the rate of gain depends on the proportion: of treatments forwarding to the phase III study. We suggest maximizing the rate of gain, and the resulting optimal one-stage design becomes twice as efficient as Stallard's one-stage design. Furthermore, the new design has a probability of only 0.12 of passing an ineffective treatment to phase III study.