Imputation strategies for missing continuous outcomes in cluster randomized trials

In cluster randomized trials, intact social units such as schools, worksites or medical practices - rather than individuals themselves - are randomly allocated to intervention and control conditions, while the outcomes of interest are then observed on individuals within each cluster. Such trials are becoming increasingly common in the fields of health promotion and health services research. Attrition is a common occurrence in randomized trials, and a standard approach for dealing with the resulting missing values is imputation. We consider imputation strategies for missing continuous outcomes, focusing on trials with a completely randomized design in which fixed cohorts from each cluster are enrolled prior to random assignment. We compare five different imputation strategies with respect to Type I and Type II error rates of the adjusted two-sample t -test for the intervention effect. Cluster mean imputation is compared with multiple imputation, using either within-cluster data or data pooled across clusters in each intervention group. In the case of pooling across clusters, we distinguish between standard multiple imputation procedures which do not account for intracluster correlation and a specialized procedure which does account for intracluster correlation but is not yet available in standard statistical software packages. A simulation study is used to evaluate the influence of cluster size, number of clusters, degree of intracluster correlation, and variability among cluster follow-up rates. We show that cluster mean imputation yields valid inferences and given its simplicity, may be an attractive option in some large community intervention trials which are subject to individual-level attrition only; however, it may yield less powerful inferences than alternative procedures which pool across clusters especially when the cluster sizes are small and cluster follow-up rates are highly variable. When pooling across clusters, the imputation procedure should generally take intracluster correlation into account to obtain valid inferences; however, as long as the intracluster correlation coefficient is small, we show that standard multiple imputation procedures may yield acceptable type I error rates; moreover, these procedures may yield more powerful inferences than a specialized procedure, especially when the number of available clusters is small. Within-cluster multiple imputation is shown to be the least powerful among the procedures considered.     

A general class of correlation coefficients for the 2 x 2 crossover design

The Pearson correlation coefficient and the Kendall correlation coefficient are two popular statistics for assessing the correlation between two variables in a bivariate sample. We indicate how both of these statistics are special cases of a general class of correlation statistics that is parameterized by gamma element of [0, 1]. The Pearson correlation coefficient is characterized by gamma = 1 and the Kendall correlation coefficient by gamma = 0, so they yield the upper and lower extremes of the class, respectively. The correlation coefficient characterized by gamma = 0.5 is of special interest because it only requires that first-order moments exist for the underlying bivariate distribution, whereas the Pearson correlation coefficient requires that second-order moments exist. We derive the asymptotic theory for the general class of sample correlation coefficients and then describe the use of this class of correlation statistics within the 2 x 2 crossover design. We illustrate the methodology using data from the CLIC trial of the Childhood Asthma Research and Education (CARE) Network.     

General Derivation of Intraclass Correlation Coefficients

A general intraclass correlation coefficient is derived similarly to the general correlation coefficient given by KENDALL (1962). This coefficient II embraces as specific cases three intraclass correlations, related to Pearson's r, Kendall's tau and Spearman's rs, that were dealt with by FISHER (1921), WHITFIELD (1949), SHIRAHATA (1981) and SCHEMPER (1984). Formal relationships are presented and the qualification of the three coefficients for specific applications is discussed.     

Nonparametric estimation of the concordance correlation coefficient under univariate censoring

Assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects. Lin's (1989, Biometrics 45, 255-268) concordance correlation coefficient (CCC) has become a popular measure of agreement for correlated continuous outcomes. However, commonly used estimation methods for the CCC do not accommodate censored observations and are, therefore, not applicable for survival outcomes. In this article, we estimate the CCC nonparametrically through the bivariate survival function. The proposed estimator of the CCC is proven to be strongly consistent and asymptotically normal, with a consistent bootstrap variance estimator. Furthermore, we propose a time-dependent agreement coefficient as an extension of Lin's (1989) CCC for measuring the agreement between survival times among subjects who survive beyond a specified time point. A nonparametric estimator is developed for the time-dependent agreement coefficient as well. It has the same asymptotic properties as the estimator of the CCC. Simulation studies are conducted to evaluate the performance of the proposed estimators. A real data example from a prostate cancer study is used to illustrate the method.     

Effects from shear stress on morphology and growth of early stages of Norway spruce somatic embryos

The shear stress effect on directional expansion of pro embryogenic masses (PEMs) and suspensor cell development of somatic embryos of Norway spruce (Picea abies) at the proliferation stage was studied by a direct and quantitative image analysis system. The experimental system allowed for detailed observations of the effect of hydrodynamic shear stress in rotating and deforming liquid cultures of proliferating Norway spruce somatic embryos. Briefly, somatic embryos at an early development stage comprised only of clusters of meristematic cells without suspensor cells were fixed on an alginate film. The alginate film was affixed on the bottom of a flow cell and the somatic embryos were subjected to laminar flow through the chamber of the flow cell. Magnified images of the cell clusters were collected every 24 h. The image data was processed based on a normalized cross‐correlation method, capable of measuring morphological and size features of individual cell clusters in both temporal and spatial domains. No suspensor cells developed in the cell clusters under shear stress of 140 s^?1 for the duration of the experiments. Cell clusters in the control cultured in stationary liquid conditions developed suspensor cells after 5–9 days in culture. Furthermore, the radial growth of meristematic cell clusters was inhibited by shear rates of 86 and 140 s^?1, corresponding to shear stress of 0.086 and 0.14 N/m², compared to growth under stationary conditions. The shear rate showed a significant negative correlation to growth rate. Control group showed no preference for direction during growth under static conditions. Biotechnol. Bioeng. 2010; 105: 588–599. © 2009 Wiley Periodicals, Inc.     

景观指数之间的相关分析

应用辽宁省1997~1998年的TM 5影像数据,编制了景观类型图,以78个县市区为单位,分割成78个景观,共计算39个景观格局指数,对它们进行了相关分析。总面积是最基本的景观指数,它决定景观总边界长度、斑块数、类型密度等基本指数,同时与多个指数有显著的相关关系(相关系数绝对值大于0.75)。形状指数的独立性强,极少数指数与其它指数有显著的相关关系;多样性指数和蔓延度指数之间信息重复量最多,都表示景观的异质性,但多样性指数以面积百分比表示景观异质性,而蔓延度指数以类型之间相邻边界的百分比表示景观异质性。研究发现,如果两个指数之间存在显著的相关关系,而由它们两个构成的指数与它们之间没有显著的相关关系。如果指数平均值之间存在显著的相关关系,则它们的变异系数之间不存在显著的相关关系。景观指数间的相关系数不仅与景观格局本身有关,还与空间尺度,分类系统、计算公式及其参数、计算单元和生态学意义关系密切。指数之间影响因子的相同之处越多,它们之间存在显著相关关系的概率越大。     

Use of quasi-least squares to adjust for two levels of correlation

This article considers data with two levels of association. Our motivating example is an international intervention trial with repeated observations on subjects who reside within geographically defined clusters. To account for the potential correlation within clusters and within the repeated measurements that pertain to each subject, we apply a method based on generalized estimating equations for a correlation structure proposed by Lefkopoulou, Moore, and Ryan (1989, Journal of the American Statistical Association 84, 810-815).     

Measurement of local mass transfer coefficient in biofilms

Local mass transfer rates for an electrochemically formed microsink in an aerobic biofilm was measured by a mobile microelectrode using limiting current technique. Mass transfer coefficients varied both horizontally and vertically in the biofilm. The results implied the existence of an irregular biofilm structure consisting of microbial cell clusters surrounded by tortuous water channels. An unexpected increase of the local mass transfer coefficient just above the biofilm surface suggested the existence, of local flow instability in this region. As expected, the influence of bulk flow velocity on the local mass transfer rate decreased with increasing depth into the biofilm. Mass transfer coefficients fluctuated significantly inside microbial cell clusters, suggesting the existence of internal channels through which liquid could flow. A new conceptual model of biofilm microbial cluster structure is proposed to account for such biofilm microstructure irregularities. (c) 1995 John Wiley & Sons, Inc.     

Membrane-bound basic peptides sequester multivalent (PIP2), but not monovalent (PS), acidic lipids

Several biologically important peripheral (e.g., myristoylated alanine-rich C kinase substrate) and integral (e.g., the epidermal growth factor receptor) membrane proteins contain clusters of basic residues that interact with acidic lipids in the plasma membrane. Previous measurements demonstrate that the polyvalent acidic lipid phosphatidylinositol 4,5-bisphosphate is bound electrostatically (i.e., sequestered) by membrane-adsorbed basic peptides corresponding to these clusters. We report here three experimental observations that suggest monovalent acidic lipids are not sequestered by membrane-bound basic peptides. 1), Binding of basic peptides to vesicles does not decrease when the temperature is lowered below the fluid-to-gel phase transition. 2), The binding energy of Lys-13 to lipid vesicles increases linearly with the fraction of monovalent acidic lipids. 3), Binding of basic peptides to vesicles produces no self-quenching of fluorescent monovalent acidic lipids. One potential explanation for these results is that membrane-bound basic peptides diffuse too rapidly for the monovalent lipids to be sequestered. Indeed, our fluorescence correlation spectroscopy measurements show basic peptides bound to phosphatidylcholine/phosphatidylserine membranes have a diffusion coefficient approximately twofold higher than that of lipids, and those bound to phosphatidylcholine/phosphatidylinositol 4,5-bisphosphate membranes have a diffusion coefficient comparable to that of lipids.     

A study of the effects of truncation in bivariate normal distribution

The moments of bivariate normal distribution, which is truncated with respect to both the random variables, are obtained by using the orthogonal expansion of the distribution and the properties of HERMITE polynomials. In particular the correlation coefficient of the truncated distribution is derived in terms of the actual correlation coefficient. In order to study the effect of truncation tables have been prepared of this correlation coefficient for certain given values of the actual correlation coefficient and for a few selected values of the points of truncation. A listing of the computer program for this purpose is also given.     

Using matrix of thresholding partial correlation coefficients to infer regulatory network

DNA arrays measure the expression levels for thousands of genes simultaneously under different conditions. These measurements reflect many aspects of the underlying biological processes. A method based on the matrix of thresholding partial correlation coefficients (MTPCC) is proposed for network inference from expression profiles. It includes three main parts: (1) hierarchical cluster analysis, (2) cluster boundaries establishment, and (3) regulatory network inference. The method was applied to the expression data of 2467 genes in Saccharomyces cerevisiae measured under 79 different conditions [Eisen, M.B., Spellman, P.T., Brown, P.O., Botstein, D., 1998. Cluster analysis and display of genome-wide expression patterns. Proc. Natl. Acad. Sci. 95, 14863-14868]. Using hierarchical clustering and cluster boundaries establishment, the 2467 genes were grouped into 12 clusters. The expression profiles of each cluster were expressed as a set of expression levels average over the cluster that constituted genes of each condition. Then the expression data of these clusters were subjected to the analysis of partial correlation, and the significance of each element in the obtained partial correlation coefficient matrix (PCCM) was examined by a permutation test. The corresponding undirected dependency graph (UDG) was obtained as a model of the regulatory network of S. cerevisiae. The veracity of the network was evidenced by the consistency of our results with the collected results from experimental studies.     

Confidence interval estimation of the intraclass correlation coefficient for binary outcome data

We obtain closed-form asymptotic variance formulae for three point estimators of the intraclass correlation coefficient that may be applied to binary outcome data arising in clusters of variable size. Our results include as special cases those that have previously appeared in the literature (Fleiss and Cuzick, 1979, Applied Psychological Measurement 3, 537-542; Bloch and Kraemer, 1989, Biometrics 45, 269-287; Altaye, Donner, and Klar, 2001, Biometrics 57, 584-588). Simulation results indicate that confidence intervals based on the estimator proposed by Fleiss and Cuzick provide coverage levels close to nominal over a wide range of parameter combinations. Two examples are presented.     

Estimating the generalized concordance correlation coefficient through variance components

The intraclass correlation coefficient (ICC) and the concordance correlation coefficient (CCC) are two of the most popular measures of agreement for variables measured on a continuous scale. Here, we demonstrate that ICC and CCC are the same measure of agreement estimated in two ways: by the variance components procedure and by the moment method. We propose estimating the CCC using variance components of a mixed effects model, instead of the common method of moments. With the variance components approach, the CCC can easily be extended to more than two observers, and adjusted using confounding covariates, by incorporating them in the mixed model. A simulation study is carried out to compare the variance components approach with the moment method. The importance of adjusting by confounding covariates is illustrated with a case example.     

Assessment of local influence for the analysis of agreement

The concordance correlation coefficient (CCC) and the probability of agreement (PA) are two frequently used measures for evaluating the degree of agreement between measurements generated by two different methods. In this paper, we consider the CCC and the PA using the bivariate normal distribution for modeling the observations obtained by two measurement methods. The main aim of this paper is to develop diagnostic tools for the detection of those observations that are influential on the maximum likelihood estimators of the CCC and the PA using the local influence methodology but not based on the likelihood displacement. Thus, we derive first‐ and second‐order measures considering the case‐weight perturbation scheme. The proposed methodology is illustrated through a Monte Carlo simulation study and using a dataset from a clinical study on transient sleep disorder. Empirical results suggest that under certain circumstances first‐order local influence measures may be more powerful than second‐order measures for the detection of influential observations.     

Sample size considerations for stepped wedge designs with subclusters

The stepped wedge cluster randomized trial (SW-CRT) is an increasingly popular design for evaluating health service delivery or policy interventions. An essential consideration of this design is the need to account for both within-period and between-period correlations in sample size calculations. Especially when embedded in health care delivery systems, many SW-CRTs may have subclusters nested in clusters, within which outcomes are collected longitudinally. However, existing sample size methods that account for between-period correlations have not allowed for multiple levels of clustering. We present computationally efficient sample size procedures that properly differentiate within-period and between-period intracluster correlation coefficients in SW-CRTs in the presence of subclusters. We introduce an extended block exchangeable correlation matrix to characterize the complex dependencies of outcomes within clusters. For Gaussian outcomes, we derive a closed-form sample size expression that depends on the correlation structure only through two eigenvalues of the extended block exchangeable correlation structure. For non-Gaussian outcomes, we present a generic sample size algorithm based on linearization and elucidate simplifications under canonical link functions. For example, we show that the approximate sample size formula under a logistic linear mixed model depends on three eigenvalues of the extended block exchangeable correlation matrix. We provide an extension to accommodate unequal cluster sizes and validate the proposed methods via simulations. Finally, we illustrate our methods in two real SW-CRTs with subclusters.     

Spearman Correlation Identifies Statistically Significant Gene Expression Clusters in Spinal Cord Development and Injury

An important problem in the analysis of large-scale gene expression data is the validation of gene expression clusters. By examining the temporal expression patterns of 74 genes expressed in rat spinal cord under three different experimental conditions, we have found evidence that some genes cluster together under multiple conditions. Using RT-PCR data from spinal cord development and two sets of microarray data from spinal injury, we applied Spearman correlation to identify clusters and to assign P values to pairs of genes with highly similar temporal expression patterns. We found that 15% of genes occurred in statistically significant pairs in all three experimental conditions, providing both statistical and experimental support for the idea that genes that cluster together are co-regulated. In addition, we demonstrated that DNA microarray and RT-PCR data are comparable, and can be combined to confirm gene expression relationships.     

珙桐群落种间关系的研究

用方差比率法(VR)和在2×2联列表基础上应用Jaccard指数、Dice指数和Pearson相关系数、Spearman秩相关系数对卧龙自然保护区三江珙桐群落种间关系进行了分析,结果表明珙桐群落中主要种群在总体上表现出正关联,群落中种间联结的显著度较低,仅有7对联结显著,珙桐与总状山矾呈显著负联结,与稠李呈极显著负联结。大多数种对间关联程度较低,存在极大排斥性的种对较少,暗示目前珙桐群落处于较成熟阶段。研究种间关系时,种间关联性测定指标结合相关系数分析得出的结果较好,最后对珙桐的保护提出了一些建议。     

蜘蛛毒腺形态解剖的初步研究

本研究对蜘蛛目中几个科的代表种的毒腺形态特征及其位置进行了解剖观察。并在测量其毒腺的形态度量参数与蜘蛛体长和体重的基础上,采用一元线性回归分析和相关性分析的统计学原理,分析了其毒腺的形态度量参数与蜘蛛体长和体重之间所存在的相关性,同时探讨了不同种类蜘蛛毒腺的解剖方法。结果表明：蜘蛛毒腺的位置和形态与蜘蛛的进化、生活方式和蛛体形态有一定关系,即低等及地下穴居种类(如虎纹捕鸟蛛Selenocosmia huwena等)的蜘蛛毒腺多前位(型)于螯基内;其余的多后位(型)于头胸部前段内。蜘蛛毒腺的重量与其体重之间为正相关,如大腹园蛛Araneus ventricosus的相关系数rAl＝0．9846(P＜0．01),一元线性回归方程为:y^＝-0．0038 0．1829x;蜘蛛毒腺的长度与其头胸长之间为正相关,如穴居狼蛛Lycosa singoriensis的相关系数rc3＝0．9858(P＜0．01),方程式为:y^＝-0．1641 0．5325x;蜘蛛毒腺的宽度与其头胸宽之间为负相关,如虎纹捕鸟蛛Selenocosmia huwena的相关系数rD3：＝-0．9704(P＜0．01),方程式为：y^＝0．2498—0．0072x。     

基于相关系数的水文相依性变异分级方法——以自回归模型为例

水文过程演变存在时间相依性,含有相依成分的水文序列无法满足水文计算中的一致性假设,给水问题研究带来诸多困难.针对水文序列相依性变异这一现象,以自相关模型为例,提出基于相关系数的水文相依性变异分级方法.该方法通过计算相依成分与原序列之间的相关系数,并选取合理的相关系数阈值,将相依性变异程度分为无变异、弱变异、中变异、强变异、巨变异.通过推导相关系数与序列各阶自相关系数之间的公式,说明相关系数主要取决于1阶到p阶自相关系数的大小,从而阐明分级方法的理论基础.以一阶和二阶自回归模型为例,利用统计试验验证了公式的合理性,并说明了相关系数和自相关系数的联系.将所提分级方法应用于3个实测径流序列进行分析,结果显示水文过程常常存在相依性与随机性并存的现象.     

Temporal pooling of point transect data increases precision in density estimates of southern chamois

Estimating animal abundances in small areas is a difficult task and because a limited number of observations often results in low-precision estimates whose inaccuracies may even be exacerbated if surveys are focussed on clustered populations and/or are only carried out once a year. In an attempt to overcome this problem, we used point transects to monthly survey two small areas of a game reserve to assess the density of Pyrenean chamois (Rupicapra p. pyrenaica). The coefficient of variation associated with the density estimates after pooling observations by season was still high but decreased to reasonable values (<20%) when observations were over 29 chamois groups (clusters). Our results suggest that Distance Sampling may be a useful way of estimating the population density of mountain ungulates such as Pyrenean chamois in small rugged areas where only a small or moderate number of observations are to be expected.