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.

The Endothelial Sample Size Analysis in Corneal Specular Microscopy Clinical Examinations

Purpose: To evaluate endothelial cell sample size and statistical error in corneal specular microscopy (CSM) examinations. Methods: One hundred twenty examinations were conducted with 4 types of corneal specular microscopes: 30 with each BioOptics, CSO, Konan, and Topcon corneal specular microscopes. All endothelial image data were analyzed by respective instrument software and also by the Cells Analyzer software with a method developed in our lab(US Patent). A reliability degree (RD) of 95% and a relative error (RE) of 0.05 were used as cut-off values to analyze images of the counted endothelial cells called samples. The sample size mean was the number of cells evaluated on the images obtained with each device. Only examinations with RE<0.05 were considered statistically correct and suitable for comparisons with future examinations. The Cells Analyzer software was used to calculate the RE and customized sample size for all examinations. Results: Bio-Optics: sample size, 97 +/- 22 cells; RE, 6.52 +/- 0.86; only 10% of the examinations had sufficient endothelial cell quantity (RE<0.05); customized sample size, 162 +/- 34 cells. CSO: sample size, 110 +/- 20 cells; RE, 5.98 +/- 0.98; only 16.6% of the examinations had sufficient endothelial cell quantity (RE<0.05); customized sample size, 157 +/- 45 cells. Konan: sample size, 80 +/- 27 cells; RE, 10.6 +/- 3.67; none of the examinations had sufficient endothelial cell quantity (RE>0.05); customized sample size, 336 +/- 131 cells. Topcon: sample size, 87 +/- 17 cells; RE, 10.1 +/- 2.52; none of the examinations had sufficient endothelial cell quantity (RE>0.05); customized sample size, 382 +/- 159 cells. Conclusions: A very high number of CSM examinations had sample errors based on Cells Analyzer software. The endothelial sample size (examinations) needs to include more cells to be reliable and reproducible. The Cells Analyzer tutorial routine will be useful for CSM examination reliability and reproducibility.

Parameter recovery for a skew-normal IRT model under a Bayesian approach: hierarchical framework, prior and kernel sensitivity and sample size

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is an interest in studying latent variables (or latent traits). Usually such latent traits are assumed to be random variables and a convenient distribution is assigned to them. A very common choice for such a distribution has been the standard normal. Recently, Azevedo et al. [Bayesian inference for a skew-normal IRT model under the centred parameterization, Comput. Stat. Data Anal. 55 (2011), pp. 353-365] proposed a skew-normal distribution under the centred parameterization (SNCP) as had been studied in [R. B. Arellano-Valle and A. Azzalini, The centred parametrization for the multivariate skew-normal distribution, J. Multivariate Anal. 99(7) (2008), pp. 1362-1382], to model the latent trait distribution. This approach allows one to represent any asymmetric behaviour concerning the latent trait distribution. Also, they developed a Metropolis-Hastings within the Gibbs sampling (MHWGS) algorithm based on the density of the SNCP. They showed that the algorithm recovers all parameters properly. Their results indicated that, in the presence of asymmetry, the proposed model and the estimation algorithm perform better than the usual model and estimation methods. Our main goal in this paper is to propose another type of MHWGS algorithm based on a stochastic representation (hierarchical structure) of the SNCP studied in [N. Henze, A probabilistic representation of the skew-normal distribution, Scand. J. Statist. 13 (1986), pp. 271-275]. Our algorithm has only one Metropolis-Hastings step, in opposition to the algorithm developed by Azevedo et al., which has two such steps. This not only makes the implementation easier but also reduces the number of proposal densities to be used, which can be a problem in the implementation of MHWGS algorithms, as can be seen in [R.J. Patz and B.W. Junker, A straightforward approach to Markov Chain Monte Carlo methods for item response models, J. Educ. Behav. Stat. 24(2) (1999), pp. 146-178; R. J. Patz and B. W. Junker, The applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, J. Educ. Behav. Stat. 24(4) (1999), pp. 342-366; A. Gelman, G.O. Roberts, and W.R. Gilks, Efficient Metropolis jumping rules, Bayesian Stat. 5 (1996), pp. 599-607]. Moreover, we consider a modified beta prior (which generalizes the one considered in [3]) and a Jeffreys prior for the asymmetry parameter. Furthermore, we study the sensitivity of such priors as well as the use of different kernel densities for this parameter. Finally, we assess the impact of the number of examinees, number of items and the asymmetry level on the parameter recovery. Results of the simulation study indicated that our approach performed equally as well as that in [3], in terms of parameter recovery, mainly using the Jeffreys prior. Also, they indicated that the asymmetry level has the highest impact on parameter recovery, even though it is relatively small. A real data analysis is considered jointly with the development of model fitting assessment tools. The results are compared with the ones obtained by Azevedo et al. The results indicate that using the hierarchical approach allows us to implement MCMC algorithms more easily, it facilitates diagnosis of the convergence and also it can be very useful to fit more complex skew IRT models.

Power Calculations for Preclinical Studies Using a K-Sample Rank Test and the Lehmann Alternative Hypothesis

Power calculations in a small sample comparative study, with a continuous outcome measure, are typically undertaken using the asymptotic distribution of the test statistic. When the sample size is small, this asymptotic result can be a poor approximation. An alternative approach, using a rank based test statistic, is an exact power calculation. When the number of groups is greater than two, the number of calculations required to perform an exact power calculation is prohibitive. To reduce the computational burden, a Monte Carlo resampling procedure is used to approximate the exact power function of a k-sample rank test statistic under the family of Lehmann alternative hypotheses. The motivating example for this approach is the design of animal studies, where the number of animals per group is typically small.

Sample size in orthodontic randomized controlled trials: are numbers justified?

Sample size calculations are advocated by the Consolidated Standards of Reporting Trials (CONSORT) group to justify sample sizes in randomized controlled trials (RCTs). This study aimed to analyse the reporting of sample size calculations in trials published as RCTs in orthodontic speciality journals. The performance of sample size calculations was assessed and calculations verified where possible. Related aspects, including number of authors; parallel, split-mouth, or other design; single- or multi-centre study; region of publication; type of data analysis (intention-to-treat or per-protocol basis); and number of participants recruited and lost to follow-up, were considered. Of 139 RCTs identified, complete sample size calculations were reported in 41 studies (29.5 per cent). Parallel designs were typically adopted (n = 113; 81 per cent), with 80 per cent (n = 111) involving two arms and 16 per cent having three arms. Data analysis was conducted on an intention-to-treat (ITT) basis in a small minority of studies (n = 18; 13 per cent). According to the calculations presented, overall, a median of 46 participants were required to demonstrate sufficient power to highlight meaningful differences (typically at a power of 80 per cent). The median number of participants recruited was 60, with a median of 4 participants being lost to follow-up. Our finding indicates good agreement between projected numbers required and those verified (median discrepancy: 5.3 per cent), although only a minority of trials (29.5 per cent) could be examined. Although sample size calculations are often reported in trials published as RCTs in orthodontic speciality journals, presentation is suboptimal and in need of significant improvement.

Sample Size vs. Bias in Defect Prediction

Most empirical disciplines promote the reuse and sharing of datasets, as it leads to greater possibility of replication. While this is increasingly the case in Empirical Software Engineering, some of the most popular bug-fix datasets are now known to be biased. This raises two significants concerns: first, that sample bias may lead to underperforming prediction models, and second, that the external validity of the studies based on biased datasets may be suspect. This issue has raised considerable consternation in the ESE literature in recent years. However, there is a confounding factor of these datasets that has not been examined carefully: size. Biased datasets are sampling only some of the data that could be sampled, and doing so in a biased fashion; but biased samples could be smaller, or larger. Smaller data sets in general provide less reliable bases for estimating models, and thus could lead to inferior model performance. In this setting, we ask the question, what affects performance more? bias, or size? We conduct a detailed, large-scale meta-analysis, using simulated datasets sampled with bias from a high-quality dataset which is relatively free of bias. Our results suggest that size always matters just as much bias direction, and in fact much more than bias direction when considering information-retrieval measures such as AUC and F-score. This indicates that at least for prediction models, even when dealing with sampling bias, simply finding larger samples can sometimes be sufficient. Our analysis also exposes the complexity of the bias issue, and raises further issues to be explored in the future.

Sample size determination in combinatorial chemistry.

Combinatorial chemistry is gaining wide appeal as a technique for generating molecular diversity. Among the many combinatorial protocols, the split/recombine method is quite popular and particularly efficient at generating large libraries of compounds. In this process, polymer beads are equally divided into a series of pools and each pool is treated with a unique fragment; then the beads are recombined, mixed to uniformity, and redivided equally into a new series of pools for the subsequent couplings. The deviation from the ideal equimolar distribution of the final products is assessed by a special overall relative error, which is shown to be related to the Pearson statistic. Although the split/recombine sampling scheme is quite different from those used in analysis of categorical data, the Pearson statistic is shown to still follow a chi2 distribution. This result allows us to derive the required number of beads such that, with 99% confidence, the overall relative error is controlled to be less than a pregiven tolerable limit L1. In this paper, we also discuss another criterion, which determines the required number of beads so that, with 99% confidence, all individual relative errors are controlled to be less than a pregiven tolerable limit L2 (0 < L2 < 1).

A study of sample size in making decisions about instructional materials /

Project No. 711151.01.

On sample size determination; informal report no. 2 to Systems Analysis Division, Chemical Research and Development Laboratories, Edgewood Arsenal, Md., under Contract DA-18-108-CML-7174.

Mode of access: Internet.

Sample size estimation and statistical power analyses

The concept of sample size and statistical power estimation is now something that Optometrists that want to perform research, whether it be in practice or in an academic institution, cannot simply hide away from. Ethics committees, journal editors and grant awarding bodies are now increasingly requesting that all research be backed up with sample size and statistical power estimation in order to justify any study and its findings. This article presents a step-by-step guide of the process for determining sample sizeand statistical power. It builds on statistical concepts presented in earlier articles in Optometry Today by Richard Armstrong and Frank Eperjesi.