Estimation of Sample Size and Power For Quantile Regression

Quantile regression (QR) was first introduced by Roger Koenker and Gilbert Bassett in 1978. It is robust to outliers which affect least squares estimator on a large scale in linear regression. Instead of modeling mean of the response, QR provides an alternative way to model the relationship between quantiles of the response and covariates. Therefore, QR can be widely used to solve problems in econometrics, environmental sciences and health sciences. Sample size is an important factor in the planning stage of experimental design and observational studies. In ordinary linear regression, sample size may be determined based on either precision analysis or power analysis with closed form formulas. There are also methods that calculate sample size based on precision analysis for QR like C.Jennen-Steinmetz and S.Wellek (2005). A method to estimate sample size for QR based on power analysis was proposed by Shao and Wang (2009). In this paper, a new method is proposed to calculate sample size based on power analysis under hypothesis test of covariate effects. Even though error distribution assumption is not necessary for QR analysis itself, researchers have to make assumptions of error distribution and covariate structure in the planning stage of a study to obtain a reasonable estimate of sample size. In this project, both parametric and nonparametric methods are provided to estimate error distribution. Since the method proposed can be implemented in R, user is able to choose either parametric distribution or nonparametric kernel density estimation for error distribution. User also needs to specify the covariate structure and effect size to carry out sample size and power calculation. The performance of the method proposed is further evaluated using numerical simulation. The results suggest that the sample sizes obtained from our method provide empirical powers that are closed to the nominal power level, for example, 80%.

ARL1 of the Attribute c Control Chart with Estimated Parameter

International audience

Sample size for monitoring sirex populations and their natural enemies.

A vespa-da-madeira, Sirex noctilio Fabricius (Hymenoptera: Siricidae) foi introduzida no Brasil em 1988 e tornou-se a principal praga dos plantios de pínus. Encontra-se distribuída em aproximadamente 1.000.000 de ha em diferentes níveis populacionais nos Estados do Rio Grande do Sul, Santa Catarina, Paraná, São Paulo e Minas Gerais. O controle da população da vespa-da-madeira é feito principalmente pela utilização do nematoide Deladenus siricidicola Bedding (Nematoda: Neothylenchidae). A avaliação da eficiência dos inimigos naturais é dificultada por não haver um sistema de amostragem apropriado. Este estudo testou o sistema de amostragem hierárquica para definir o tamanho da amostra para monitorar a população de S. noctilio e também a eficiência dos inimigos naturais, a qual mostrou-se adequada.

Non-Pharmacological Approaches for Pain Management in Sickle Cell Disease: Development of a Mindfulness-Based Intervention

Background: Sickle Cell Disease (SCD) is a genetic hematological disorder that affects more than 7 million people globally (NHLBI, 2009). It is estimated that 50% of adults with SCD experience pain on most days, with 1/3 experiencing chronic pain daily (Smith et al., 2008). Persons with SCD also experience higher levels of pain catastrophizing (feelings of helplessness, pain rumination and magnification) than other chronic pain conditions, which is associated with increases in pain intensity, pain behavior, analgesic consumption, frequency and duration of hospital visits, and with reduced daily activities (Sullivan, Bishop, & Pivik, 1995; Keefe et al., 2000; Gil et al., 1992 & 1993). Therefore effective interventions are needed that can successfully be used manage pain and pain-related outcomes (e.g., pain catastrophizing) in persons with SCD. A review of the literature demonstrated limited information regarding the feasibility and efficacy of non-pharmacological approaches for pain in persons with SCD, finding an average effect size of .33 on pain reduction across measurable non-pharmacological studies. Second, a prospective study on persons with SCD that received care for a vaso-occlusive crisis (VOC; N = 95) found: (1) high levels of patient reported depression (29%) and anxiety (34%), and (2) that unemployment was significantly associated with increased frequency of acute care encounters and hospital admissions per person. Research suggests that one promising category of non-pharmacological interventions for managing both physical and affective components of pain are Mindfulness-based Interventions (MBIs; Thompson et al., 2010; Cox et al., 2013). The primary goal of this dissertation was thus to develop and test the feasibility, acceptability, and efficacy of a telephonic MBI for pain catastrophizing in persons with SCD and chronic pain.

Methods: First, a telephonic MBI was developed through an informal process that involved iterative feedback from patients, clinical experts in SCD and pain management, social workers, psychologists, and mindfulness clinicians. Through this process, relevant topics and skills were selected to adapt in each MBI session. Second, a pilot randomized controlled trial was conducted to test the feasibility, acceptability, and efficacy of the telephonic MBI for pain catastrophizing in persons with SCD and chronic pain. Acceptability and feasibility were determined by assessment of recruitment, attrition, dropout, and refusal rates (including refusal reasons), along with semi-structured interviews with nine randomly selected patients at the end of study. Participants completed assessments at baseline, Week 1, 3, and 6 to assess efficacy of the intervention on decreasing pain catastrophizing and other pain-related outcomes.

Results: A telephonic MBI is feasible and acceptable for persons with SCD and chronic pain. Seventy-eight patients with SCD and chronic pain were approached, and 76% (N = 60) were enrolled and randomized. The MBI attendance rate, approximately 57% of participants completing at least four mindfulness sessions, was deemed acceptable, and participants that received the telephonic MBI described it as acceptable, easy to access, and consume in post-intervention interviews. The amount of missing data was undesirable (MBI condition, 40%; control condition, 25%), but fell within the range of expected missing outcome data for a RCT with multiple follow-up assessments. Efficacy of the MBI on pain catastrophizing could not be determined due to small sample size and degree of missing data, but trajectory analyses conducted for the MBI condition only trended in the right direction and pain catastrophizing approached statistically significance.

Conclusion: Overall results showed that at telephonic group-based MBI is acceptable and feasible for persons with SCD and chronic pain. Though the study was not able to determine treatment efficacy nor powered to detect a statistically significant difference between conditions, participants (1) described the intervention as acceptable, and (2) the observed effect sizes for the MBI condition demonstrated large effects of the MBI on pain catastrophizing, mental health, and physical health. Replication of this MBI study with a larger sample size, active control group, and additional assessments at the end of each week (e.g., Week 1 through Week 6) is needed to determine treatment efficacy. Many lessons were learned that will guide the development of future studies including which MBI strategies were most helpful, methods to encourage continued participation, and how to improve data capture.

AATS for the (X)over-bar chart with variable parameters

A Fortran computer program is given for the computation of the adjusted average time to signal, or AATS, for adaptive (X) over bar charts with one, two, or all three design parameters variable: the sample size, n, the sampling interval, h, and the factor k used in determining the width of the action limits. The program calculates the threshold limit to switch the adaptive design parameters and also provides the in-control average time to signal, or ATS.

AATS for the X̄ Chart with Variable Parameters

A Fortran computer program is given for the computation of the adjusted average time to signal, or AATS, for adaptive X̄ charts with one, two, or all three design parameters variable: the sample size, n, the sampling interval, h, and the factor k used in determining the width of the action limits. The program calculates the threshold limit to switch the adaptive design parameters and also provides the in-control average time to signal, or ATS.

Constrained optimization model for the design of an adaptive (X)over-bar chart

An economic-statistical model is developed for variable parameters (VP) (X) over bar charts in which all design parameters vary adaptively, that is, each of the design parameters (sample size, sampling interval and control-limit width) vary as a function of the most recent process information. The cost function due to controlling the process quality through a VP (X) over bar chart is derived. During the optimization of the cost function, constraints are imposed on the expected times to signal when the process is in and out of control. In this way, required statistical properties can be assured. Through a numerical example, the proposed economic-statistical design approach for VP (X) over bar charts is compared to the economic design for VP (X) over bar charts and to the economic-statistical and economic designs for fixed parameters (FP) (X) over bar charts in terms of the operating cost and the expected times to signal. From this example, it is possible to assess the benefits provided by the proposed model. Varying some input parameters, their effect on the optimal cost and on the optimal values of the design parameters was analysed.

Economic design of (X)over-bar charts with variable parameters: the Markov chain approach

This paper presents an economic design of (X) over bar control charts with variable sample sizes, variable sampling intervals, and variable control limits. The sample size n, the sampling interval h, and the control limit coefficient k vary between minimum and maximum values, tightening or relaxing the control. The control is relaxed when an (X) over bar value falls close to the target and is tightened when an (X) over bar value falls far from the target. A cost model is constructed that involves the cost of false alarms, the cost of finding and eliminating the assignable cause, the cost associated with production in an out-of-control state, and the cost of sampling and testing. The assumption of an exponential distribution to describe the length of time the process remains in control allows the application of the Markov chain approach for developing the cost function. A comprehensive study is performed to examine the economic advantages of varying the (X) over bar chart parameters.

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

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

X̄ Charts with Variable Parameters

Varying the parameters of the X̄ chart has been explored extensively in recent years. In this paper, we extend the study of the X̄ chart with variable parameters to include variable action limits. The action limits establish whether the control should be relaxed or not. When the X̄ falls near the target, the control is relaxed so that there will be more time before the next sample and/or the next sample will be smaller than usual. When the X̄ falls far from the target but not in the action region, the control is tightened so that there is less time before the next sample and/or the next sample will be larger than usual. The goal is to draw the action limits wider than usual when the control is relaxed and narrower than usual when the control is tightened. This new feature then makes the X̄ chart more powerful than the CUSUM scheme in detecting shifts in the process mean.

Economic design of a Vp X̄ chart

We develop an economic model for X̄ control charts having all design parameters varying in an adaptive way, that is, in real time considering current sample information. In the proposed model, each of the design parameters can assume two values as a function of the most recent process information. The cost function is derived and it provides a device for optimal selection of the design parameters. Through a numerical example one can foresee the savings that the developed model possibly provides. © 2001 Elsevier Science B.V. All rights reserved.

The no-central chi-square chart with two-stage samplings

Throughout this article, it is assumed that the no-central chi-square chart with two stage samplings (TSS Chisquare chart) is employed to monitor a process where the observations from the quality characteristic of interest X are independent and identically normally distributed with mean μ and variance σ2. The process is considered to start with the mean and the variance on target (μ = μ0; σ2 = σ0 2), but at some random time in the future an assignable cause shifts the mean from μ0 to μ1 = μ0 ± δσ0, δ >0 and/or increases the variance from σ0 2 to σ1 2 = γ2σ0 2, γ > 1. Before the assignable cause occurrence, the process is considered to be in a state of statistical control (defined by the in-control state). Similar to the Shewhart charts, samples of size n 0+ 1 are taken from the process at regular time intervals. The samplings are performed in two stages. At the first stage, the first item of the i-th sample is inspected. If its X value, say Xil, is close to the target value (|Xil-μ0|< w0σ 0, w0>0), then the sampling is interrupted. Otherwise, at the second stage, the remaining n0 items are inspected and the following statistic is computed. Wt = Σj=2n 0+1(Xij - μ0 + ξiσ 0)2 i = 1,2 Let d be a positive constant then ξ, =d if Xil > 0 ; otherwise ξi =-d. A signal is given at sample i if |Xil-μ0| > w0σ 0 and W1 > knia:tl, where kChi is the factor used in determining the upper control limit for the non-central chi-square chart. If devices such as go and no-go gauges can be considered, then measurements are not required except when the sampling goes to the second stage. Let P be the probability of deciding that the process is in control and P 1, i=1,2, be the probability of deciding that the process is in control at stage / of the sampling procedure. Thus P = P1 + P 2 - P1P2, P1 = Pr[μ0 - w0σ0 ≤ X ≤ μ0+ w 0σ0] P2=Pr[W ≤ kChi σ0 2], (3) During the in-control period, W / σ0 2 is distributed as a non-central chi-square distribution with n0 degrees of freedom and a non-centrality parameter λ0 = n0d2, i.e. W / σ0 2 - xn0 22 (λ0) During the out-of-control period, W / σ1 2 is distributed as a non-central chi-square distribution with n0 degrees of freedom and a non-centrality parameter λ1 = n0(δ + ξ)2 / γ2 The effectiveness of a control chart in detecting a process change can be measured by the average run length (ARL), which is the speed with which a control chart detects process shifts. The ARL for the proposed chart is easily determined because in this case, the number of samples before a signal is a geometrically distributed random variable with parameter 1-P, that is, ARL = I /(1-P). It is shown that the performance of the proposed chart is better than the joint X̄ and R charts, Furthermore, if the TSS Chi-square chart is used for monitoring diameters, volumes, weights, etc., then appropriate devices, such as go-no-go gauges can be used to decide if the sampling should go to the second stage or not. When the process is stable, and the joint X̄ and R charts are in use, the monitoring becomes monotonous because rarely an X̄ or R value fall outside the control limits. The natural consequence is the user to pay less and less attention to the steps required to obtain the X̄ and R value. In some cases, this lack of attention can result in serious mistakes. The TSS Chi-square chart has the advantage that most of the samplings are interrupted, consequently, most of the time the user will be working with attributes. Our experience shows that the inspection of one item by attribute is much less monotonous than measuring four or five items at each sampling.

Joint X̄ and R charts with variable parameters

Recent studies have shown that the X̄chart with variable parameters (Vp X̄ chart) detects process shifts faster than the traditional X̄ chart. This article extends these studies for processes that are monitored by both, X̄ and R charts. Basically, the X̄ and R values establish if the control should or should not be relaxed. When the X̄ and R values fall in the central region the control is relaxed because one will wait more to take the next sample and/or the next sample will be smaller than usual. When the X̄ or R values fall in the warning region the control is tightened because one will wait less to take the next sample and the next sample will be larger than usual. The action limits are also made variable. This paper proposes to draw the action limits (for both charts) wider than usual, when the control is relaxed and narrower than usual when the control is tightened. The Vp feature improves the joint X̄ and R control chart performance in terms of the speed with which the process mean and/or variance shifts are detected. © 1998 IIE.

Effect of measurement error and autocorrelation on the (X)over-bar chart

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A new chart based on sample variances for monitoring the covariance matrix of multivariate processes

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)