Optimal One-Way Random Effects Designs for the Intraclass Correlation Based on Confidence Intervals

ABSTRACT

Confidence intervals for the intraclass correlation coefficient (ρ) are used to determine the optimal allocation of experimental material in one-way random effects models. Designs that produce narrow intervals are preferred since they provide greater precision to estimate ρ. Assuming the total cost and the relative cost of the two stages of sampling are fixed, the authors investigate the number of classes and the number of individuals per class required to minimize the expected length of confidence intervals. We obtain results using asymptotic theory and compare these results to those obtained using exact calculations. The best design depends on the unknown value of ρ. Minimizing the maximum expected length of confidence intervals guards against worst-case scenarios. A good overall recommendation based on asymptotic results is to choose a design having classes of size 2 + √4 + 3r, where r is the relative cost of sampling at the class-level compared to the individual-level. If r = 0, then the overall cost is the sample size and the recommendation reduces to a design having classes of size 4.     

Hypothesis testing for Markov chain Monte Carlo

Testing between hypotheses, when independent sampling is possible, is a well developed subject. In this paper, we propose hypothesis tests that are applicable when the samples are obtained using Markov chain Monte Carlo. These tests are useful when one is interested in deciding whether the expected value of a certain quantity is above or below a given threshold. We show non-asymptotic error bounds and bounds on the expected number of samples for three types of tests, a fixed sample size test, a sequential test with indifference region, and a sequential test without indifference region. Our tests can lead to significant savings in sample size. We illustrate our results on an example of Bayesian parameter inference involving an ODE model of a biochemical pathway.     

Testing for random effect in the Fuller–Battese model

We consider the Fuller–Battese model where random effects are allowed to be from non-normal universes. The asymptotic distribution of the F-statistic in this model is derived as the number of groups tends to infinity (is large) and sample size from any group is either fixed or large. The result is used to establish an approximate test for the significance of the random effect variance component. Robustness of the established approximate test is given.     

Duration of Accrual and Follow-up for Two-Stage Clinical Trials

Group sequential trialswith time to event end points can be complicated to design. Notonly are there unlimited choices for the number of events requiredat each stage, but for each of these choices, there are unlimitedcombinations of accrual and follow-up at each stage that providethe required events. Methods are presented for determining optimalcombinations of accrual and follow-up for two-stage clinicaltrials with time to event end points. Optimization is based onminimizing the expected total study length as a function of theexpected accrual duration or sample size while providing an appropriateoverall size and power. Optimal values of expected accrual durationand minimum expected total study length are given assuming anexponential proportional hazards model comparing two treatmentgroups. The expected total study length can be substantiallydecreased by including a follow-up period during which accrualis suspended. Conditions that warrant an interim follow-up periodare considered, and the gain in efficiency achieved by includingan interim follow-up period is quantified. The gain in efficiencyshould be weighed against the practical difficulties in implementingsuch designs. An example is given to illustrate the use of thesetechniques in designing a clinical trial to compare two chemotherapyregimens for lung cancer. Practical considerations of includingan interim follow-up period are discussed.     

Optimal Statistical Design of a Multivariate EWMA Chart Based on ARL and MRL

Statistical design is applied to a multivariate exponentially weighted moving average (MEWMA) control chart. The chart parameters are control limit H and smoothing constant r. The choices of the parameters depend on the number of variables p and the size of the process mean shift δ. The MEWMA statistic is modeled as a Markov chain and the Markov chain approach is used to determine the properties of the chart. Although average run length has become a traditional measure of the performance of control schemes, some authors have suggested other measures, such as median and other percentiles of the run length distribution to explain run length properties of a control scheme. This will allow a thorough study of the performance of the control scheme. Consequently, conclusions based on these measures would provide a better and comprehensive understanding of a scheme. In this article, we present the performance of the MEWMA control chart as measured by the average run length and median run length. Graphs are given so that the chart parameters of an optimal MEWMA chart can be determined easily.     

Bayesian Estimation of a Multidimensional Additive Graded Response Model for Correlated Traits

The aim of the article is to propose a Bayesian estimation through Markov chain Monte Carlo of a multidimensional item response theory model for graded responses with an additive structure with correlated latent traits. A simulation study is conducted to evaluate the model parameter recovery under different conditions (sample size, test and subtest length, number of response categories, and correlation structure). The results show that the parameters are well reproduced when the sample size is sufficiently large (n = 1, 000), while the worst recovery is observed for small sample size (n = 500), and four response categories with a short number of test items.     

The run length distributions of the R,s and s2 control charts when is estimated

R, s and s2 charts with estimated control limits are widely used in practice. Common practice in control-chart theory is to estimate the control limits using data from the process and, once the process is determined to be in control, to treat the resulting control limits as though fixed. While there are empirical rules for setting up the control charts using past or trial data, little is known about the run length distributions of these charts when the fact that control limits are estimated is taken into account. In this paper, we derive and evaluate the run length distributions associated with the R, s and s2 charts when the process standard deviation a is estimated. The results are then used to discuss the appropriateness of the widely followed empirical rules for choosing the number m of samples and the sample size n.     

Aspects on the control of false alarms in statistical surveillance and the impact on the return of financial decision systems

In systems for online detection of regime shifts, a process is continually observed. Based on the data available an alarm is given when there is enough evidence of a change. There is a risk of a false alarm and here two different ways of controlling the false alarms are compared: a fixed average run length until the first false alarm and a fixed probability of any false alarm (fixed size). The two approaches are evaluated in terms of the timeliness of alarms. A system with a fixed size is found to have a drawback: the ability to detect a change deteriorates with the time of the change. Consequently, the probability of successful detection will tend to zero and the expected delay of a motivated alarm tends to infinity. This drawback is present even when the size is set to be very large (close to one). Utility measures expressing the costs for a false or a too late alarm are used in the comparison. How the choice of the best approach can be guided by the parameters of the process and the different costs of alarms is demonstrated. The technique is illustrated by financial transactions of the Hang Seng Index.     

D-Optimal Two-Level Fractional Factorial Designs of Resolution III with Two Dispersion Factors

The problem of finding D-optimal designs, with two dispersion factors, for the estimation of all location main effects is investigated in the class of regular unreplicated two-level fractional factorial designs of resolution III. Designs having length three words involving both of the dispersion factors in the defining relation are shown to be inferior in terms of D-optimality. Tables of factors that are named as the two dispersion factors so that the resulting design is either D-optimal or has the largest determinant of the information matrix are provided. Rank-order of designs is studied when the number of length three words involving either one of the dispersion factors and the number of length four words involving both of the dispersion factors are fixed. Rank-order of designs when the numbers of aforementioned words are less than or equal to ten is given.     

Using -splines to test the linearity of partially linear models

The nonparametric component in a partially linear model is estimated by a linear combination of fixed-knot cubic B-splines with a second-order difference penalty on the adjacent B-spline coefficients. The resulting penalized least-squares estimator is used to construct two Wald-type spline-based test statistics for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, the first test statistic asymptotically has the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom, under the null hypothesis. The smoothing parameter is determined by specifying a value for the asymptotically expected value of the test statistic under the null hypothesis. When the number of knots is fixed and under the null hypothesis, the second test statistic asymptotically has a chi-squared distribution with K=q+2 degrees of freedom, where q is the number of knots used for estimation. The power performances of the two proposed tests are investigated via simulation experiments, and the practicality of the proposed methodology is illustrated using a real-life data set.     

Invariance,Optimality, and a 1-Observation Confidence Interval for a Normal Mean

In a 1965 Decision Theory course at Stanford University, Charles Stein began a digression with “an amusing problem”: is there a proper confidence interval for the mean based on a single observation from a normal distribution with both mean and variance unknown? Stein introduced the interval with endpoints ± c|X| and showed indeed that for c large enough, the minimum coverage probability (over all values for the mean and variance) could be made arbitrarily near one. While the problem and coverage calculation were in the author’s hand-written notes from the course, there was no development of any optimality result for the interval. Here, the Hunt–Stein construction plus analysis based on special features of the problem provides a “minimax” rule in the sense that it minimizes the maximum expected length among all procedures with fixed coverage (or, equivalently, maximizes the minimal coverage among all procedures with a fixed expected length). The minimax rule is a mixture of two confidence procedures that are equivariant under scale and sign changes, and are uniformly better than the classroom example or the natural interval X ± c|X|?.     

Lack-of-Fit Tests for Generalized Linear Models via Splines

Cubic B-splines are used to estimate the nonparametric component of a semiparametric generalized linear model. A penalized log-likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one df. The smoothing parameter is determined by giving a specified value for its asymptotically expected value under the null hypothesis. A simulation study is conducted to evaluate its power performance; a real-life dataset is used to illustrate its practical use.     

Estimation of multiple correlation coefficient when observations are missing on one of the variables

This paper proposes a control chart with variable sampling intervals (VSI) to detect increases in the expected value of the number of defects in a random sample of constant size n the upper one-sided c-VSI chart

The performance of this chart is evaluated by means of the average time to signal (ATS).The comparisons made between the standard FSI (fixed sampling intervals) and the VSI upper one-sided c - charts indicate that using variable sampling intervals can substantially reduce the average time to signal. Using stochastic ordering we prove that this reduction always occurs.

Special attention is given to the choice of the proposed control chart parameters and to the chart graphical display.     

A queueing model to study the occurrence and duration of ozone exceedances in Mexico City

It is well known that long-term exposure to high levels of pollution is hazardous to human health. Therefore, it is important to study and understand the behavior of pollutants in general. In this work, we study the occurrence of a pollutant concentration's surpassing a given threshold (an exceedance) as well as the length of time that the concentration stays above it. A general N(t)/D/1 queueing model is considered to jointly analyze those problems. A non-homogeneous Poisson process is used to model the arrivals of clusters of exceedances. Geometric and generalized negative binomial distributions are used to model the amount of time (cluster size) that the pollutant concentration stays above the threshold. A mixture model is also used for the cluster size distribution. The rate function of the non-homogeneous Poisson process is assumed to be of either the Weibull or the Musa–Okumoto type. The selection of the model that best fits the data is performed using the Bayes discrimination method and the sum of absolute differences as well as using a graphical criterion. Results are applied to the daily maximum ozone measurements provided by the monitoring network of the Metropolitan Area of Mexico City.     

An economic statistical design of the Poisson EWMA control charts for monitoring nonconformities

An economic statistical model of the exponentially weighted moving average (EWMA) control chart for the average number of nonconformities in the sample is proposed. The statistical and economic performance of proposed design are evaluated using the average run length (ARL) and the hourly expected cost, respectively. A Markov chain approach is applied to derive expressions for ARL. The cost model is established based on the general cost function given in Lorenzen and Vance [The economic design of control charts: a unified approach. Technometrics. 1986;28:3–11]. An example is provided to illustrate the application of the proposed model. A sensitivity analysis is also carried out to investigate the effects of model parameters on the solution of the economic statistical design by using the design of experiments (DOE) technique.     

Robust Confidence Intervals for the Bernoulli Parameter

Despite the simplicity of the Bernoulli process, developing good confidence interval procedures for its parameter—the probability of success p—is deceptively difficult. The binary data yield a discrete number of successes from a discrete number of trials, n. This discreteness results in actual coverage probabilities that oscillate with the n for fixed values of p (and with p for fixed n). Moreover, this oscillation necessitates a large sample size to guarantee a good coverage probability when p is close to 0 or 1.

It is well known that the Wilson procedure is superior to many existing procedures because it is less sensitive to p than any other procedures, therefore it is less costly. The procedures proposed in this article work as well as the Wilson procedure when 0.1 ≤p ≤ 0.9, and are even less sensitive (i.e., more robust) than the Wilson procedure when p is close to 0 or 1. Specifically, when the nominal coverage probability is 0.95, the Wilson procedure requires a sample size 1, 021 to guarantee that the coverage probabilities stay above 0.92 for any 0.001 ≤ min {p, 1 ?p} <0.01. By contrast, our procedures guarantee the same coverage probabilities but only need a sample size 177 without increasing either the expected interval width or the standard deviation of the interval width.     

Inequalities between expected marginal log‐likelihoods,with implications for likelihood‐based model complexity and comparison measures

A multi‐level model allows the possibility of marginalization across levels in different ways, yielding more than one possible marginal likelihood. Since log‐likelihoods are often used in classical model comparison, the question to ask is which likelihood should be chosen for a given model. The authors employ a Bayesian framework to shed some light on qualitative comparison of the likelihoods associated with a given model. They connect these results to related issues of the effective number of parameters, penalty function, and consistent definition of a likelihood‐based model choice criterion. In particular, with a two‐stage model they show that, very generally, regardless of hyperprior specification or how much data is collected or what the realized values are, a priori, the first‐stage likelihood is expected to be smaller than the marginal likelihood. A posteriori, these expectations are reversed and the disparities worsen with increasing sample size and with increasing number of model levels.     

Replacement policies for age and working numbers with random life cycle

It has been modeled for several replacement policies in literatures that the whole life cycle or operating interval of an operating unit should be finite rather than infinite as is done with the traditional method. However, it is more natural to consider the case in which the finite life cycle is a fluctuated parameter that could be used to estimate replacement times, which will be taken up in this article. For this, we first formulate a general model in which the unit is replaced at random age U, random time Y for the first working number, random life cycle S, or at failure X, whichever occurs first. The following models included in the general model, such that replacement done at age T when variable U is a degenerate distribution, and replacement done at working numbers N summed by number N of variable Y, are optimized. We obtain the total expected cost until replacement and the expected replacement cost rate for each model. Optimal age T, working number N, and a pair of (T, N) are discussed analytically and computed numerically.     

The performance of several statistical tests for seasonality in monthly data

The type I and II error rates of several statistical tests for seasonality in monthly data were investigated through a computer simulation study at two nominal significance levels, α=1% and α=5%. Three models were used for the variation: annual sinusoidal; semi—annual sinusoidal; and a curve which is constant in all but three consecutive months of the year, when it exhibits a constant increase (a “one—pulse” model). The statistical tests are compared in terms of the simulation results. These results may be applied to calculate either the sample size required to detect seasonal variation of fixed amplitude or the probability of detecting seasonal variation of variable amplitude with a fixed sample size. A numerical case study is given     

On Statistical Properties of Palindromes in DNA

The main aim of this article is to determine the distribution of the number of palindromes of a fixed length in a DNA sequence. Palindrome in DNA is a part of the DNA sequence which is equal to its complementary sequence read backwards (C ～ G, A ～ T). We examine the general case where the distribution of the nucleobases is arbitrary. Further, we determine conditions in which Normal approximation can be used. Special attention is given to coding and non-coding regions in DNA and to the distribution of bases in those parts of DNA.