Determining the number of clusters by sampling with replacement

A split-sample replication criterion originally proposed by J. E. Overall and K. N. Magee (1992) as a stopping rule for hierarchical cluster analysis is applied to multiple data sets generated by sampling with replacement from an original simulated primary data set. An investigation of the validity of this bootstrap procedure was undertaken using different combinations of the true number of latent populations, degrees of overlap, and sample sizes. The bootstrap procedure enhanced the accuracy of identifying the true number of latent populations under virtually all conditions. Increasing the size of the resampled data sets relative to the size of the primary data set further increased accuracy. A computer program to implement the bootstrap stopping rule is made available via a referenced Web site.     

Sequential validation of cluster analytic subtypes of homeless veterans

To identify subgroups within the homeless population, a number of researchers have employed cluster analytic statistical procedures. Although this is an appropriate application of cluster analysis, many studies have not employed important statistical safeguards against arbitrary results. This study demonstrates a cluster analytic procedure—sequential validation—that enhances the replicability, external validity, and cross-validity of cluster solutions. The procedure is applied to a nationwide sample of 745 homeless veterans. After 12 different clustering procedures were subjected to derivation, replication, external validation, and cross-validation phases, a 4-cluster Ward solution emerged as the most sound. Substantively, the clusters were an alcoholic subtype, a psychiatrically impaired subtype, a best functioning subtype, and a multiproblem subtype. The generalizability of these subgroups to other contexts was assessed by comparing them to subgroups identified in other homelessness research. Suggestions were made for improving the quality of cluster analytic research in community psychology.     

Profiling local optima in K-means clustering: developing a diagnostic technique

Using the cluster generation procedure proposed by D. Steinley and R. Henson (2005), the author investigated the performance of K-means clustering under the following scenarios: (a) different probabilities of cluster overlap; (b) different types of cluster overlap; (c) varying samples sizes, clusters, and dimensions; (d) different multivariate distributions of clusters; and (e) various multidimensional data structures. The results are evaluated in terms of the Hubert-Arabie adjusted Rand index, and several observations concerning the performance of K-means clustering are made. Finally, the article concludes with the proposal of a diagnostic technique indicating when the partitioning given by a K-means cluster analysis can be trusted. By combining the information from several observable characteristics of the data (number of clusters, number of variables, sample size, etc.) with the prevalence of unique local optima in several thousand implementations of the K-means algorithm, the author provides a method capable of guiding key data-analysis decisions.     

Correcting correlation coefficients for group heterogeneity when one variable is a dichotomy

A method is illustrated for estimating correlation coefficients and mean criterion scores in a full-range population from bivariate distributions available in a selected sample when the criterion is a dichotomy. The proposed method requires only the assumptions needed for use of formulas for correcting correlation coefficients for restriction of range when both variables are continuous and is suitable for use when restriction is due to either direct or indirect selection.The research reported in this paper was sponsored by the 6570th Personnel Research Laboratory, AMD, under AFSC Project 7717.     

Validating Clusters with the Lower Bound for Sum-of-Squares Error

Given that a minor condition holds (e.g., the number of variables is greater than the number of clusters), a nontrivial lower bound for the sum-of-squares error criterion in K-means clustering is derived. By calculating the lower bound for several different situations, a method is developed to determine the adequacy of cluster solution based on the observed sum-of-squares error as compared to the minimum sum-of-squares error. The author was partially supported by the Office of Naval Research Grant #N00014-06-0106.     

A Graph Theoretic Criterion for Determining the Number of Clusters in a Data Set

This article is concerned with procedures for determining the number of clusters in a data set. Most of the procedures or stopping rules currently in use involve finding internally coherent and externally isolated clusters, but do not derive from the formal structure of the respective clustering model. Based on the graph theoretic concepts of minimal spanning tree, maximal spanning tree, and homomorphic function, a new criterion is advanced that yields a well-defined clustering solution. Its performance in determining the number of clusters in several empirical data sets is evaluated by comparing it to four prominent stopping rules. It is shown that the proposed criterion not only possesses mathematically attractive properties but also may contribute to solving the number-of-clusters problem.     

DETERMINING JOB GROUPS: APPLICATION OF HIERARCHICAL AGGLOMERATIVE CLUSTER ANALYSIS IN DIFFERENT JOB ANALYSIS SITUATIONS

The present study showed that researchers must consider underlying data structure when using hierarchical agglomerative cluster analysis to group jobs. Five cluster procedures were applied to four simulated data sets constructed to reflect common job analysis situations. The structures contained jobs varying in degree of task overlap, number of tasks performed, and relative number of people doing the jobs. Average linkage/distance was the most accurate procedure when jobs had highly positively correlated task profiles, a situation characteristic of jobs within a career family over a restricted range of levels. Average linkage/correlation was the most accurate for three other structures containing jobs whose profiles were not highly positively correlated. Such are characteristically found when analyzing (a) jobs in different functional units, (b) jobs over a wide range of hierarchical levels such as entry to advanced, and (c) jobs differing markedly in the number of incumbents.     

Spurious Latent Class Problem in the Mixed Rasch Model: A Comparison of Three Maximum Likelihood Estimation Methods under Different Ability Distributions

Recent research has shown that over-extraction of latent classes can be observed in the Bayesian estimation of the mixed Rasch model when the distribution of ability is non-normal. This study examined the effect of non-normal ability distributions on the number of latent classes in the mixed Rasch model when estimated with maximum likelihood estimation methods (conditional, marginal, and joint). Three information criteria fit indices (Akaike information criterion, Bayesian information criterion, and sample size adjusted BIC) were used in a simulation study and an empirical study. Findings of this study showed that the spurious latent class problem was observed with marginal maximum likelihood and joint maximum likelihood estimations. However, conditional maximum likelihood estimation showed no overextraction problem with non-normal ability distributions.     

Audiovisual bounce-inducing effect: When sound congruence affects grouping in vision

Two disks moving from opposite points in space, overlapping, and stopping at one another’s starting point can be seen as either bouncing off one another or streaming through one another. With silent displays, observers report streaming, whereas, if a sound is played when the disks are in the overlap region, observers report bouncing. The change in perception is thought to be modulated by a lack of attention that inhibits the integration of the motion signal when disks overlap and by the sound that increases the congruence of the display, in comparison with a real elastic bounce. Here, we accompanied the disks’ motion with either a bounce-congruent sound (a billiard ball) or with bounce-incongruent sounds (a water drop, a firework). When the sound was switched on 200 msec before the disks’ overlap, (1) all the audiovisual displays induced more bounce responses than did the silent display, but (2) the bounce-congruent sound induced more bounce responses than did the bounceincongruent sounds. However, when the sound was switched on at the disks’ overlap, only the first result was observed. These results highlight both the role of attention and that of sound congruence.     

Design and power analysis: n and confidence intervals of means

In this study, we analyzed the validity of the conventional 80% power. The minimal sample size and power needed to guarantee non-overlapping (1-alpha)% confidence intervals for population means were calculated. Several simulations indicate that the minimal power for two means (m = 2) to have non-overlapping CIs is .80, for (1-alpha) set to 95%. The minimal power becomes .86 for 99% CIs and .75 for 90% CIs. When multiple means are considered, the required minimal power increases considerably. This increase is even higher when the population means do not increase monotonically. Therefore, the often adopted criterion of a minimal power equal to .80 is not always adequate. Hence, to guarantee that the limits of the CIs do not overlap, most situations require a direct calculation of the minimum number of observations that should enter in a study.     

Cluster analysis in community research: Epistemology and practice

Cluster analysis refers to a family of methods for identifying cases with distinctive characteristics in heterogeneous samples and combining them into homogeneous groups. This approach provides a great deal of information about the types of cases and the distributions of variables in a sample. This paper considers cluster analysis as a quantitative complement to the traditional linear statistics that often characterize community psychology research. Cluster analysis emphasizes diversity rather than central tendency. This makes it a valuable tool for a wide range of familiar problems in community research. A number of these applications are considered here, including the assessment of change over time, network composition, network density, person-setting relationships, and community diversity. A User's Guide section is included, which outlines the major decisions involved in a basic cluster analyses. Despite difficulties associated with the identification of optimal cluster solutions, carefully planned, theoretically informed application of cluster analysis has much to offer community researchers. Editor's note: Dr. Edward Seidman served as action editor for this article while serving as Associate Editor for Methodology.     

Impairments in tactile search following superior parietal damage

The superior parietal cortex is critical for the control of visually guided actions. Research suggests that visual stimuli relevant to actions are preferentially processed when they are in peripersonal space. One recent study demonstrated that visually guided movements towards the body were more impaired in a patient with damage to superior parietal cortex. Whereas past studies have explored disordered movement in optic ataxic patients, there has been less exploration of space perception in terms of search capacity in this population. In addition, there is some debate concerning the relationship between deficits of visuomotor control and impaired attention/perception in optic ataxia. Given that the dorsal stream has been implicated in the spatial processing of stimuli in peripersonal space, and damage to this region is known to cause optic ataxia, we felt that further investigation was warranted. We examined tactile search behavior in the fronto-parallel and radial planes in a patient with right superior parietal damage and optic ataxia. We used a pegboard with removable cylindrical pegs that allowed for the reorganization of targets between trials. To better characterize three-dimensional search behavior, we included both horizontal and vertical search conditions. Results showed that the patient spent more time searching, was more accurate and revisited more targets in right versus left space. Interestingly, the patient spent the majority of her time specifically searching the lower right quadrant of the stimulus array. Further analysis revealed lower target detection rates along the outer borders of the pegboard on all sides. The search pattern observed here is unusual considering that all targets were within arm's reach. The present experiment demonstrates that damage to superior parietal cortex impairs tactile search and biases exploration towards lower right peripersonal space.     

The effect of aging on termination of voluntary movement while standing: A study on community-dwelling older adults

For older adults, falls often occur when transitioning from motion to a complete stop, as the motor control required during this phase is very complex and challenging. The purpose of this study was to clarify the effect of aging on the motor control required to terminate motion. Twenty-five healthy older adults (aged >65 years) and 25 healthy young adults (20–23 years) performed a rapid stopping task while standing on a force plate. The rapid stopping task was conducted by analyzing center of pressure (COP) on the force plate during a visually guided tracking experiment. To assess the ability to terminate motion, we measured the velocity waveform for the COP, along with the reaction, propulsion, braking, and total movement times. Both the reaction and movement times of the older-adult group were significantly longer than those of the younger-adult group (all, p < 0.05). There was no significant difference between the groups in regard to the initial backward propulsion time; however, in the subsequent sequence of backward braking, forward propulsion, and backward braking, all times were longer in the older-adult group than in the younger-adult group (p < 0.05). Our results show that the series of time delays shown by older adults when initiating and terminating motion is due to not only delayed reactions but also delayed stopping. Furthermore, our findings suggest that older adults have not only a diminished propulsion ability but also a diminished braking ability.     

An examination of procedures for determining the number of clusters in a data set

A Monte Carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2, 3, 4, or 5 distinct nonoverlapping clusters. To provide a variety of clustering solutions, the data sets were analyzed by four hierarchical clustering methods. External criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level. Thus, the clustering present in the data was quite strong. The simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data. Several procedures worked fairly well, whereas others performed rather poorly. Thus, the latter group of rules would appear to have little validity, particularly for data sets containing distinct clusters. Applied researchers are urged to select one or more of the better criteria. However, users are cautioned that the performance of some of the criteria may be data dependent.The authors would like to express their appreciation to a number of individuals who provided assistance during the conduct of this research. Those who deserve recognition include Roger Blashfield, John Crawford, John Gower, James Lingoes, Wansoo Rhee, F. James Rohlf, Warren Sarle, and Tom Soon.     

Spoken word recognition by eye

Spoken word recognition is thought to be achieved via competition in the mental lexicon between perceptually similar word forms. A review of the development and initial behavioral validations of computational models of visual spoken word recognition is presented, followed by a report of new empirical evidence. Specifically, a replication and extension of Mattys, Bernstein & Auer's (2002) study was conducted with 20 deaf participants who varied widely in speechreading ability. Participants visually identified isolated spoken words. Accuracy of visual spoken word recognition was influenced by the number of visually similar words in the lexicon and by the frequency of occurrence of the stimulus words. The results are consistent with the common view held within auditory word recognition that this task is accomplished via a process of activation and competition in which frequently occurring units are favored. Finally, future directions for visual spoken word recognition are discussed.     

Stopping rules for multi-category computerized classification testing

Computerized classification testing (CCT) aims to classify persons into one of two or more possible categories to make decisions such as mastery/non-mastery or meet most/meet all/exceed. A defining feature of CCT is its stopping criterion: the test terminates when there is enough confidence to make a decision. There is abundant research on CCT with a single cut-off, and two common stopping criteria are the sequential probability ratio test (SPRT) statistic and the generalized likelihood ratio statistic (GLR). However, there is a relative scarcity of research extending the SPRT to the multi-hypothesis case for when there is more than one cut-off. In this paper, we propose a new multi-category GLR (mGLR) statistic as well as a stochastically curtailed version of the CCT with three or more categories. A simulation study was conducted to show that the mGLR statistic outperformed the existing stopping rules by generating shorter average test length without sacrificing classification accuracy. Results also revealed that the stochastically curtailed mGLR successfully increased test efficiency in certain testing conditions.     

Order-Constrained Solutions in <Emphasis Type="Italic">K</Emphasis>-Means Clustering: Even Better Than Being Globally Optimal

This paper proposes an order-constrained K-means cluster analysis strategy, and implements that strategy through an auxiliary quadratic assignment optimization heuristic that identifies an initial object order. A subsequent dynamic programming recursion is applied to optimally subdivide the object set subject to the order constraint. We show that although the usual K-means sum-of-squared-error criterion is not guaranteed to be minimal, a true underlying cluster structure may be more accurately recovered. Also, substantive interpretability seems generally improved when constrained solutions are considered. We illustrate the procedure with several data sets from the literature.     

The variable-criteria sequential stopping rule: Generality to unequal sample sizes,unequal variances,or to large ANOVAs

The variable-criteria sequential stopping rule (SSR) is a method for conducting planned experiments in stages after the addition of new subjects until the experiment is stopped because the p value is less than or equal to a lower criterion and the null hypothesis has been rejected, the p value is above an upper criterion, or a maximum sample size has been reached. Alpha is controlled at the expected level. The table of stopping criteria has been validated for a t test or ANOVA with four groups. New simulations in this article demonstrate that the SSR can be used with unequal sample sizes or heterogeneous variances in a t test. As with the usual t test, the use of a separate-variance term instead of a pooled-variance term prevents an inflation of alpha with heterogeneous variances. Simulations validate the original table of criteria for up to 20 groups without a drift of alpha. When used with a multigroup ANOVA, a planned contrast can be substituted for the global F as the focus for the stopping rule. The SSR is recommended when significance tests are appropriate and when the null hypothesis can be tested in stages. Because of its efficiency, the SSR should be used instead of the usual approach to the t test or ANOVA when subjects are expensive, rare, or limited by ethical considerations such as pain or distress.