Sample Size Considerations for GEE Analyses of Three‐Level Cluster Randomized Trials

Summary Cluster randomized trials in health care may involve three instead of two levels, for instance, in trials where different interventions to improve quality of care are compared. In such trials, the intervention is implemented in health care units (“clusters”) and aims at changing the behavior of health care professionals working in this unit (“subjects”), while the effects are measured at the patient level (“evaluations”). Within the generalized estimating equations approach, we derive a sample size formula that accounts for two levels of clustering: that of subjects within clusters and that of evaluations within subjects. The formula reveals that sample size is inflated, relative to a design with completely independent evaluations, by a multiplicative term that can be expressed as a product of two variance inflation factors, one that quantifies the impact of within‐subject correlation of evaluations on the variance of subject‐level means and the other that quantifies the impact of the correlation between subject‐level means on the variance of the cluster means. Power levels as predicted by the sample size formula agreed well with the simulated power for more than 10 clusters in total, when data were analyzed using bias‐corrected estimating equations for the correlation parameters in combination with the model‐based covariance estimator or the sandwich estimator with a finite sample correction.     

The Effect of Cluster Size Variability on Statistical Power in Cluster-Randomized Trials

The frequency of cluster-randomized trials (CRTs) in peer-reviewed literature has increased exponentially over the past two decades. CRTs are a valuable tool for studying interventions that cannot be effectively implemented or randomized at the individual level. However, some aspects of the design and analysis of data from CRTs are more complex than those for individually randomized controlled trials. One of the key components to designing a successful CRT is calculating the proper sample size (i.e. number of clusters) needed to attain an acceptable level of statistical power. In order to do this, a researcher must make assumptions about the value of several variables, including a fixed mean cluster size. In practice, cluster size can often vary dramatically. Few studies account for the effect of cluster size variation when assessing the statistical power for a given trial. We conducted a simulation study to investigate how the statistical power of CRTs changes with variable cluster sizes. In general, we observed that increases in cluster size variability lead to a decrease in power.     

杂交试验中的最佳样本含量

杂交试验是一项费时费钱的工作,因此在进行试验之前如能进行严密的设计,给出试验所需的样本大小是十分必要的,统计学中常见的估计样本容理的公式不宜应用于杂交试验,本文分两种情况给出了杂交试验中样本容量的估计公式,据此估计出的样本容量安排杂交试验,可在满足试验者要求的条件下,使试验的总成本最低或使试畜的总头数最少。     

Incorporating Correlation for Multivariate Failure Time Data When Cluster Size Is Large

Summary We propose a new estimation method for multivariate failure time data using the quadratic inference function (QIF) approach. The proposed method efficiently incorporates within‐cluster correlations. Therefore, it is more efficient than those that ignore within‐cluster correlation. Furthermore, the proposed method is easy to implement. Unlike the weighted estimating equations in Cai and Prentice (1995, Biometrika 82 , 151–164), it is not necessary to explicitly estimate the correlation parameters. This simplification is particularly useful in analyzing data with large cluster size where it is difficult to estimate intracluster correlation. Under certain regularity conditions, we show the consistency and asymptotic normality of the proposed QIF estimators. A chi‐squared test is also developed for hypothesis testing. We conduct extensive Monte Carlo simulation studies to assess the finite sample performance of the proposed methods. We also illustrate the proposed methods by analyzing primary biliary cirrhosis (PBC) data.     

Power calculation for cross-sectional stepped wedge cluster randomized trials with variable cluster sizes

Standard sample size calculation formulas for stepped wedge cluster randomized trials (SW-CRTs) assume that cluster sizes are equal. When cluster sizes vary substantially, ignoring this variation may lead to an under-powered study. We investigate the relative efficiency of a SW-CRT with varying cluster sizes to equal cluster sizes, and derive variance estimators for the intervention effect that account for this variation under a mixed effects model—a commonly used approach for analyzing data from cluster randomized trials. When cluster sizes vary, the power of a SW-CRT depends on the order in which clusters receive the intervention, which is determined through randomization. We first derive a variance formula that corresponds to any particular realization of the randomized sequence and propose efficient algorithms to identify upper and lower bounds of the power. We then obtain an “expected” power based on a first-order approximation to the variance formula, where the expectation is taken with respect to all possible randomization sequences. Finally, we provide a variance formula for more general settings where only the cluster size arithmetic mean and coefficient of variation, instead of exact cluster sizes, are known in the design stage. We evaluate our methods through simulations and illustrate that the average power of a SW-CRT decreases as the variation in cluster sizes increases, and the impact is largest when the number of clusters is small.     

K‐Sample Test and Sample Size Calculation for Comparing Slopes in Data with Repeated Measurements

Sample size calculations based on two‐sample comparisons of slopes in repeated measurements have been reported by many investigators. In contrast, the literature has paid relatively little attention to the design and analysis of K‐sample trials in repeated measurements studies where K is 3 or greater. Jung and Ahn (2003) derived a closed sample size formula for two‐sample comparisons of slopes by taking into account the impact of missing data. We extend their method to compare K‐sample slopes in repeated measurement studies using the generalized estimating equation (GEE) approach based on independent working correlation structure. We investigate the performance of the sample size formula since the sample size formula is based on asymptotic theory. The proposed sample size formula is illustrated using a clinical trial example. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relative efficiency of unequal versus equal cluster sizes in cluster randomized trials using generalized estimating equation models

There is growing interest in conducting cluster randomized trials (CRTs). For simplicity in sample size calculation, the cluster sizes are assumed to be identical across all clusters. However, equal cluster sizes are not guaranteed in practice. Therefore, the relative efficiency (RE) of unequal versus equal cluster sizes has been investigated when testing the treatment effect. One of the most important approaches to analyze a set of correlated data is the generalized estimating equation (GEE) proposed by Liang and Zeger, in which the “working correlation structure” is introduced and the association pattern depends on a vector of association parameters denoted by ρ. In this paper, we utilize GEE models to test the treatment effect in a two‐group comparison for continuous, binary, or count data in CRTs. The variances of the estimator of the treatment effect are derived for the different types of outcome. RE is defined as the ratio of variance of the estimator of the treatment effect for equal to unequal cluster sizes. We discuss a commonly used structure in CRTs—exchangeable, and derive the simpler formula of RE with continuous, binary, and count outcomes. Finally, REs are investigated for several scenarios of cluster size distributions through simulation studies. We propose an adjusted sample size due to efficiency loss. Additionally, we also propose an optimal sample size estimation based on the GEE models under a fixed budget for known and unknown association parameter (ρ) in the working correlation structure within the cluster.     

Adaptive Cluster Sampling in the Context of Restoration

Abundance is an important population state variable for monitoring restoration progress. Efficient sampling often proves difficult, however, when populations are sparse and patchily distributed, such as early after restoration planting. Adaptive cluster sampling (ACS) can help by concentrating search effort in high density areas, improving the encounter rate and the ability to detect a population change over time. To illustrate the problem, I determined conventional design sample sizes for estimating abundance of 12 natural populations and 24 recently planted populations (divided among two preserves) of Lupinus perennis L. (wild blue lupine). I then determined the variance efficiency of ACS relative to simple random sampling at fixed effort and cost for 10 additional planted populations in two habitats (field vs. shrubland). Conventional design sample sizes to estimate lupine stem density with 10% or 20% margins of error were many times greater than initial sample size and would require sampling at least 90% of the study area. Differences in effort requirements were negligible for the two preserves and natural versus planted populations. At fixed sample size, ACS equaled or outperformed simple random sampling in 40% of populations; this shifted to 50% after correcting for travel time among sample units. ACS appeared to be a better strategy for inter‐seeded shrubland habitat than for planted field habitat. Restoration monitoring programs should consider adaptive sampling designs, especially when reliable abundance estimation under conventional designs proves elusive.     

Sample Size Determination for Designing a Strata-Matched Case-Control Study to Detect Multiple Risk Factors

Investigations of sample size for planning case-control studies have usually been limited to detecting a single factor. In this paper, we investigate sample size for multiple risk factors in strata-matched case-control studies. We construct an omnibus statistic for testing M different risk factors based on the jointly sufficient statistics of parameters associated with the risk factors. The statistic is non-iterative, and it reduces to the Cochran statistic when M = 1. The asymptotic power function of the test is a non-central chi-square with M degrees of freedom and the sample size required for a specific power can be obtained by the inverse relationship. We find that the equal sample allocation is optimum. A Monte Carlo experiment demonstrates that an approximate formula for calculating sample size is satisfactory in typical epidemiologic studies. An approximate sample size obtained using Bonferroni's method for multiple comparisons is much larger than that obtained using the omnibus test. Approximate sample size formulas investigated in this paper using the omnibus test, as well as the individual tests, can be useful in designing case-control studies for detecting multiple risk factors.     

Sample Size Calculations for the Mean in a Two Component Nonstandard Mixture Distribution

We are concerned with calculating the sample size required for estimating the mean of the continuous distribution in the context of a two component nonstandard mixture distribution (i.e., a mixture of an identifiable point degenerate function F at a constant with probability P and a continuous distribution G with probability 1 – P). A common ad hoc procedure of escalating the naïve sample size n (calculated under the assumption of no point degenerate function F) by a factor of 1/(1 – P), has about 0.5 probability of achieving the pre‐specified statistical power. Such an ad hoc approach may seriously underestimate the necessary sample size and jeopardize inferences in scientific investigations. We argue that sample size calculations in this context should have a pre‐specified probability of power ≥1 – β set by the researcher at a level greater than 0.5. To that end, we propose an exact method and an approximate method to calculate sample size in this context so that the pre‐specified probability of achieving a desired statistical power is determined by the researcher. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

暴露评估中样本采集量的模拟研究

选择暴露评估常用的4种右偏分布,就评估关注的高百分位数估计与采样量的关系进行模拟研究;又以对数正态分布为代表,从分布形态和变异的角度做了细致探讨.结果表明：（1）对右偏分布来说,百分位数越高,准确估计所需的采样容量就越大.而其估计值都随采样量的增大而趋近理论值,精度也随之增大,采样量500时,本文考察的4种右偏分布除P99.9外的其它百分位数都得到了较为准确的估计.（2）估计相同的百分位数,对数正态分布所需的样本容量要比正态分布大得多;而其分布变异越大,所需的采样量也就越大.本研究可为暴露评估中数据的采样调查提供借鉴.     

Estimation of Treatment Difference and Standard Deviation with Blinded Data in Clinical Trials

In an ongoing clinical trial and while the randomized treatment codes remain blinded, it is often desirable to estimate the treatment difference and standard deviation for a normally‐distributed response variable. This is particularly useful for estimating the sample size for future trials or for adjusting the sample size for the ongoing trial. We describe the limitations of an available EM algorithm‐based procedure to reestimate the standard deviation without unblinding the codes for the two treatments. We introduce a new procedure and propose a clinical trial design for estimating both the treatment difference and standard deviation without unblinding. The performance of the proposed procedure is evaluated in a simulation study.     

Testing the Intraclass Version of Kappa Coeffcient of Agreement with Binary Scale and Sample Size Determination

The intraclass version of kappa coefficient has been commonly applied as a measure of agreement for two ratings per subject with binary outcome in reliability studies. We present an efficient statistic for testing the strength of kappa agreement using likelihood scores, and derive asymptotic power and sample size formula. Exact evaluation shows that the score test is generally conservative and more powerful than a method based on a chi‐square goodness‐of‐fit statistic (Donner and Eliasziw , 1992, Statistics in Medicine 11 , 1511–1519). In particular, when the research question is one directional, the one‐sided score test is substantially more powerful and the reduction in sample size is appreciable.     

Estimating Population Size for Continuous-Time Capture-Recapture Models Via Sample Coverage

An estimation procedure using the idea of sample coverage is proposed to estimate population size for capture-recapture experiments in continuous time. The capture rates (intensity) are allowed to vary by time and individuals (heterogeneity). Only capture frequency history are sufficient for estimating population size while capture times and sequential orders of animals caught are irrelevant for the analysis. An example is given for illustration. The performance of the proposed estimation procedure is also investigated by simulation.     

Size Matters: Ryanodine Receptor Cluster Size Heterogeneity Potentiates Calcium Waves

Ryanodine receptors (RyRs) mediate calcium (Ca)-induced Ca release and intracellular Ca homeostasis. In a cardiac myocyte, RyRs group into clusters of variable size from a few to several hundred RyRs, creating a spatially nonuniform intracellular distribution. It is unclear how heterogeneity of RyR cluster size alters spontaneous sarcoplasmic reticulum (SR) Ca releases (Ca sparks) and arrhythmogenic Ca waves. Here, we tested the impact of heterogeneous RyR cluster size on the initiation of Ca waves. Experimentally, we measured RyR cluster sizes at Ca spark sites in rat ventricular myocytes and further tested functional impacts using a physiologically detailed computational model with spatial and stochastic intracellular Ca dynamics. We found that the spark frequency and amplitude increase nonlinearly with the size of RyR clusters. Larger RyR clusters have lower SR Ca release threshold for local Ca spark initiation and exhibit steeper SR Ca release versus SR Ca load relationship. However, larger RyR clusters tend to lower SR Ca load because of the higher Ca leak rate. Conversely, smaller clusters have a higher threshold and a lower leak, which tends to increase SR Ca load. At the myocyte level, homogeneously large or small RyR clusters limit Ca waves (because of low load for large clusters but low excitability for small clusters). Mixtures of large and small RyR clusters potentiates Ca waves because the enhanced SR Ca load driven by smaller clusters enables Ca wave initiation and propagation from larger RyR clusters. Our study suggests that a spatially heterogeneous distribution of RyR cluster size under pathological conditions may potentiate Ca waves and thus afterdepolarizations and triggered arrhythmias.     

Minimum Sample Size Ensuring Validity of Classical Confidence Intervals for Means of Skewed and Platykurtic Distributions

COCHRAN (1953) and BARTCH (1957) gave formulae for the magnitude of the sample size (n) ensuring the validity of the limiting normal distribution of the sample mean x(ⁿ) obtained from a non-normal distribution with marked asymmetry and kurtosis. These formulae have been checked empirically in this paper using (a) simulated data with given asymmetry and kurtosis and (b) real data gathered from a coronary heart disease study. We find that our results are in general agreement with Bartch's formula. However, in a number of cases, the asymptotic normal distribution is attained for smaller sample size than that required by Bartch's formula.     

Evaluation of Noether's Method of Sample Size Determination for the Wilcoxon-Mann-Whitney Test

NOETHER (1987) proposed a method of sample size determination for the Wilcoxon-Mann-Whitney test. To obtain a sample size formula, he restricted himself to alternatives that differ only slightly from the null hypothesis, so that the unknown variance o² of the Mann-Whitney statistic can be approximated by the known variance under the null hypothesis which depends only on n. This fact is frequently forgotten in statistical practice. In this paper, we compare Noether's large sample solution against an alternative approach based on upper bounds of σ² which is valid for any alternatives. This comparison shows that Noether's approximation is sufficiently reliable with small and large deviations from the null hypothesis.     

Satellite Telemetry in Avian Research and Management: Sample Size Considerations

Abstract: Satellite tracking is currently used to make inferences to avian populations. Cost of transmitters and logistical challenges of working with some species can limit sample size and strength of inferences. Therefore, careful study design including consideration of sample size is important. We used simulations to examine how sample size, population size, and population variance affected probability of making reliable inferences from a sample and the precision of estimates of population parameters. For populations of >100 individuals, a sample >20 birds was needed to make reliable inferences about questions with simple outcomes (i.e., 2 possible outcomes). Sample size demands increased rapidly for more complex problems. For example, in a problem with 3 outcomes, a sample of >75 individuals will be needed for proper inference to the population. Combining data from satellite telemetry studies with data from surveys or other types of sampling may improve inference strength.     

A semiparametric joint model for cluster size and subunit-specific interval-censored outcomes

Clustered data frequently arise in biomedical studies, where observations, or subunits, measured within a cluster are associated. The cluster size is said to be informative, if the outcome variable is associated with the number of subunits in a cluster. In most existing work, the informative cluster size issue is handled by marginal approaches based on within-cluster resampling, or cluster-weighted generalized estimating equations. Although these approaches yield consistent estimation of the marginal models, they do not allow estimation of within-cluster associations and are generally inefficient. In this paper, we propose a semiparametric joint model for clustered interval-censored event time data with informative cluster size. We use a random effect to account for the association among event times of the same cluster as well as the association between event times and the cluster size. For estimation, we propose a sieve maximum likelihood approach and devise a computationally-efficient expectation-maximization algorithm for implementation. The estimators are shown to be strongly consistent, with the Euclidean components being asymptotically normal and achieving semiparametric efficiency. Extensive simulation studies are conducted to evaluate the finite-sample performance, efficiency and robustness of the proposed method. We also illustrate our method via application to a motivating periodontal disease dataset.     

Power analysis for cluster randomized trials with continuous coprimary endpoints

Pragmatic trials evaluating health care interventions often adopt cluster randomization due to scientific or logistical considerations. Systematic reviews have shown that coprimary endpoints are not uncommon in pragmatic trials but are seldom recognized in sample size or power calculations. While methods for power analysis based on K ($K\ge 2$) binary coprimary endpoints are available for cluster randomized trials (CRTs), to our knowledge, methods for continuous coprimary endpoints are not yet available. Assuming a multivariate linear mixed model (MLMM) that accounts for multiple types of intraclass correlation coefficients among the observations in each cluster, we derive the closed-form joint distribution of K treatment effect estimators to facilitate sample size and power determination with different types of null hypotheses under equal cluster sizes. We characterize the relationship between the power of each test and different types of correlation parameters. We further relax the equal cluster size assumption and approximate the joint distribution of the K treatment effect estimators through the mean and coefficient of variation of cluster sizes. Our simulation studies with a finite number of clusters indicate that the predicted power by our method agrees well with the empirical power, when the parameters in the MLMM are estimated via the expectation-maximization algorithm. An application to a real CRT is presented to illustrate the proposed method.