Equal Numbers Of Clusters Research Articles

Optimal allocation of clusters in stepped wedge designs with a decaying correlation structure.

The cluster randomized stepped wedge design is a multi-period uni-directional switch design in which all clusters start in the control condition and at the beginning of each new period a random sample of clusters crosses over to the intervention condition. Such designs often use uniform allocation, with an equal number of clusters at each treatment switch. However, the uniform allocation is not necessarily the most efficient. This study derives the optimal allocation of clusters to treatment sequences in the cluster randomized stepped wedge design, for both cohort and cross-sectional designs. The correlation structure is exponential decay, meaning the correlation decreases with the time lag between two measurements. The optimal allocation is shown to depend on the intraclass correlation coefficient, the number of subjects per cluster-period and the cluster and (in the case of a cohort design) individual autocorrelation coefficients. For small to medium values of these autocorrelations those sequences that have their treatment switch earlier or later in the study are allocated a larger proportion of clusters than those clusters that have their treatment switch halfway the study. When the autocorrelation coefficients increase, the clusters become more equally distributed across the treatment sequences. For the cohort design, the optimal allocation is almost equal to the uniform allocation when both autocorrelations approach the value 1. For almost all scenarios that were studied, the efficiency of the uniform allocation is 0.8 or higher. R code to derive the optimal allocation is available online.

A SAS Macro for Covariate-Constrained Randomization of General Cluster-Randomized and Unstratified Designs

Ivers et al. (2012) have recently stressed the importance to both statistical power and face validity of balancing allocations to study arms on relevant covariates. While several techniques exist (e.g., minimization, pair-matching, stratification), the covariate-constrained randomization (CCR) approach proposed by Moulton (2004) is favored when clusters can be recruited prior to randomization. CCRA V1.0, a macro published by Chaudhary and Moulton (2006), provides a SAS implementation of CCR for a particular subset of possible designs (those with two arms, small numbers of strata and clusters, an equal number of clusters within each stratum, and constraints that can be expressed as absolute mean differences between arms). This paper presents a more comprehensive macro, CCR, that is applicable across a wider variety of designs and provides statistics describing the range of possible allocations meeting the constraints in addition to performing the actual random assignment.

Stepped-wedge cluster randomised controlled trials: some variations on the common design

Stepped wedge cluster randomised trials (SW-CRTs) are being used with increasing frequency in health service evaluation. Conventionally, these studies are designed with equally spaced steps, with an equal number of clusters randomised at each step, and data collected at each and every step. Here we introduce several variations on this design and consider implications for power. One modifiction we consider is where the number of clusters varies at each step, or where at some steps data are not collected. We show that the parallel CRT with staggered, but balanced randomisation, can be considered a special case of the incomplete SW-CRT. And, we extend these designs to allow for multiple layers of clustering, for example wards within a hospital. Building on results for complete designs, power and detectable difference are derived using Wald-test and obtaining the variance-covariance matrix of the treatment effect assuming a generalised linear mixed model. These variations are illustrated by several real examples.

ON THE EQUIVALENCE OF SOME INDICES OF SIMILARITY: IMPLICATION FOR BINARY PRESENCE/ABSENCE DATA

Summary Cohen’s kappa, a special case of the weighted kappa, is a chance-corrected index used extensively to quantify inter-rater agreement in validation and reliability studies. In this paper, it is shown that in inter-rater agreement for 2 × 2 tables, for two raters having the same number of opposite ratings, the weighted kappa, Cohen’s kappa, Peirce, Yule, Maxwell and Pilliner and Fleiss indices are identical. This implies that the weights in the weighted kappa are less important under such assumptions. Equivalently, it is shown that for two partitions of the same data set, resulting from two clustering algorithms having the same number of clusters with equal cluster sizes, these similarity indices are identical. Hence, an important characterisation is formulated relating equal numbers of clusters with the same cluster sizes to the presence/absence of a trait in a reliability study. Two numerical examples that exemplify the implication of this relationship are presented.

The Globular Cluster Systems around NGC 3311 and NGC 3309

We present extensive new photometry in -->(g', i') of the large globular cluster (GC) system around NGC 3311, the central cD galaxy in the Hydra cluster. Our GMOS data cover a 5.5 -->' field of view and reach a limiting magnitude -->i' 26.0, about 0.5 mag fainter than the turnover point of the GC luminosity function. We find that NGC 3311 has a huge population of 16,000 GCs, closely similar to the prototypical high specific frequency Virgo giant M87. The color-magnitude distribution shows that the metal-poor blue GC sequence and the more metal-rich red sequence are both present, with nearly equal numbers of clusters. Bimodal fits to the color distributions confirm that the blue sequence shows the same trend of progressively increasing metallicity with GC mass that has previously been found in many other large galaxies; the correlation we find corresponds to a scaling of GC metallicity with mass of -->Z ~ M0.6. By contrast, the red sequence shows no change of mean metallicity with mass but does show an upward extension to much higher than normal luminosity into the UCD-like range, strengthening the potential connections between massive GCs and UCDs. The GC luminosity function, which we measure down to the turnover point at -->MI ? 8.4, also has a normal form like those in other giant ellipticals. Within the Hydra field, another giant elliptical NGC 3309 is sitting just 100 -->'' from the cD NGC 3311. We use our data to solve simultaneously for the spatial structure and total GC populations of both galaxies. Their specific frequencies are -->SN(N 3311) = 12.5 ? 1.5 and -->SN(N 3309) = 0.6 ? 0.4. NGC 3311 is completely dominant and entirely comparable with other cD-type systems such as M87 in Virgo.

Horizontal cell morphology in nocturnal and diurnal primates: A comparison between owl-monkey (Aotus) and capuchin monkey (Cebus)

Horizontal cell morphology was studied in the retina of the nocturnal owl-monkey, Aotus, and compared with that of its diurnal, close relative, the capuchin monkey, Cebus . Cells were initially labeled with DiI and the staining was later photoconverted in a stable precipitated using DAB as chromogen. The sizes of cell bodies, dendritic fields, and axon terminals, number of dendritic clusters, intercluster spacing, and intercone spacing were measured at increasing eccentricities. Two distinct morphological classes of horizontal cells were identified, which resembled those of H1 and H3 cells described in diurnal monkeys. A few examples of a third class, possibly corresponding to the H2 cells of diurnal monkeys, were labeled. Both H1 and H3 cells increased in size and had increasing numbers of dendritic clusters with eccentricity. H3 cells were larger and had a larger number of dendritic clusters than H1 cells. Owl-monkey H1 cells had larger dendritic fields than capuchin monkey H1 cells at all quadrants in the central and midperipheral retinal regions, but the difference disappeared in the far periphery. Owl-monkey and capuchin monkey H1 cells had about the same number of dendritic clusters across eccentricity. As owl-monkey H1 cells were larger than capuchin monkey H1 cells, the equal number of clusters in these two primates was due to the fact that they were more spaced in the owl-monkey cells. H1 intercluster distance closely matched intercone spacing for both the owl-monkey and capuchin monkey retinas. On the other hand, H3 intercluster distance was larger than intercone spacing in the retina of both primates. Owl-monkey H1 axon terminals had 2-3 times more knobs than capuchin monkey H1 axon terminals in spite of having about the same size and, consequently, knob density was 2-3 times higher for owl-monkey than capuchin monkey H1 axon terminals across all eccentricities. The differences observed between owl-monkey and capuchin monkey horizontal cells, regarding the morphology of their dendritic trees and axon terminals, may be related to the differences found in the cone-to-rod ratio in the retina of these two primates. They seem to represent retinal specializations to the nocturnal and diurnal life styles of the owl-monkey and capuchin monkey, respectively.

Wide-Field Washington Photometry of the NGC 5128 Globular Cluster System. II. Large-Scale Properties of the System

Building on the CMT1 photometric database presented in Paper I, in this paper we derive the large-scale properties of the globular cluster system (GCS) in NGC 5128, the nearest giant elliptical and the dominant galaxy in the Centaurus group. In global terms, it has a smaller total population than previously thought: we estimate 980 ± 120 clusters over all magnitudes, yielding a specific frequency SN = 1.4 ± 0.2, with a steep projected radial distribution σ ~ r-2. The luminosity distribution of the clusters resembles that of an old, normal GC luminosity function (Gaussian-like with peak at MV -7.4 and dispersion of 1.3 mag), but these parameters are unfortunately quite uncertain because of the system's low population and the heavy field contamination. Using the metallicity-sensitive C - T1 color index, we discuss the metallicity distribution function (MDF) for a subsample of 211 previously identified clusters, all on a homogeneous photometric system. We find the MDF to be strongly bimodal, with metallicity peaks at [Fe/H] = -1.55 and -0.55 and with nearly equal numbers of clusters in each of the metal-poor and metal-rich modes. The combined evidence from the system's low specific frequency, the MDF, and the isophotal shell features in the halo light make a major merger a plausible model for the formation history of this giant E galaxy. However, the progenitor galaxies must have been more gas-rich than in any present-day mergers or starbursts. Finally, we present a list of 327 new cluster candidates not identified in any previous surveys; most of these are in the less well studied bulge region of the galaxy and along the minor axis.

Application of an adjusted chi2 statistic to site-specific data in observational dental studies.

When a binary response is observed on teeth from each subject belonging to 2 or more exposure groups, application of the usual Pearson chi2 tests is invalid, since such responses within the same subject are not independent. Consequently, special statistical methods are needed to control for the correlation among teeth (sites) within the same subject. A simple adjustment to the Pearson chi2 statistic has been proposed for comparing proportions in site-specific data. However, the required assumptions for this statistic have not yet been thoroughly addressed. These assumptions are guaranteed to hold in experimental comparisons, but may be violated in some observational studies. We investigate the conditions under which the adjusted chi2 statistic is valid and examine the performance of the adjusted chi2 statistic when these conditions are violated. Our simulation study shows that the adjusted chi2 statistic generally produces good empirical type I errors under the assumption of a common intracluster correlation coefficient. Even if the intracluster correlations are different, the adjusted statistic performs well when the groups have equal numbers of clusters (subjects). The discussion is illustrated using an observational study of caries on the roots of teeth.

Dynoc?A dynamic optimal cluster-seeking technique

A new technique for automatic clustering of multivariate data is proposed. In this approach a performance index for determining optimal clusters is introduced. This performance index is expressed in terms of the ratio of the minimum interset distance to maximum intraset distance. The optimal clusters are found when the performance index reaches a global maximum. If there are alternative groupings with equal number of clusters, the one with the largest performance index is chosen.

Studies on Grape Variability and Field Sampling

A study has been made of the amount of variation of sugar and acid content in maturing grapes and factors affecting it. Variation in composition between vines was found to be greater than that between clusters within vines. The effect of the position of clusters on the vine and the weight of individual clusters was quite small. The mean sugar content per vine was inversely correlated with the yield of individual vines while the mean acid content was directly proportional to yield. Both the between-vine variance component and the within-vine (i.e. between cluster) variance component were highly correlated with the mean level of acid. The mean sugar content was correlated with the between-cluster variance component but not with the between-vine variance component. Irrigated grapes were more variable than non-irrigated grapes when the sugar content was less than 20 per cent. The most efficient method of sampling a plot of vines was to take a minimum amount of material per vine, i.e. one cluster if cluster sampling is used. The standard error of the mean sugar content in plots of 200 vines was ± 0.25 per cent sugar when 1 cluster was picked from each of 25 random vines or 2 clusters were picked from each of 21 random vines. Single berry sampling gave comparable estimates of the mean sugar content, and is recommended for field assessment of maturity because of the lesser amount of fruit involved. Sample estimates of the mean sugar content, using cluster Sampling, compared well with the total crop. Thirty-four such comparisons were made. Biassed estimates of the mean sugar and acid contents are obtained when equal number of clusters (or berries) are sampled from each vine, due to the correlation between yield and sugar and acid content. Methods of eliminating this bias are given.