An examination of indexes for determining the number of clusters in binary data sets

The problem of choosing the correct number of clusters is as old as cluster analysis itself. A number of authors have suggested various indexes to facilitate this crucial decision. One of the most extensive comparative studies of indexes was conducted by Milligan and Cooper (1985). The present piece of work pursues the same goal under different conditions. In contrast to Milligan and Cooper's work, the emphasis here is on high-dimensional empirical binary data. Binary artificial data sets are constructed to reflect features typically encountered in real-world data situations in the field of marketing research. The simulation includes 162 binary data sets that are clustered by two different algorithms and lead to recommendations on the number of clusters for each index under consideration. Index results are evaluated and their performance is compared and analyzed.Author names are listed in alphabetical order.This piece of research was supported by the Austrian Science Foundation (FWF) under grant SFB#010 (Adaptive Information Systems and Modeling in Economics and Management Science).The authors would like to thank the anonymous reviewers and especially the associate editor for their helpful comments and suggestions.     

A cautionary note on using internal cross validation to select the number of clusters

A highly popular method for examining the stability of a data clustering is to split the data into two parts, cluster the observations in Part A, assign the objects in Part B to their nearest centroid in Part A, and then independently cluster the Part B objects. One then examines how close the two partitions are (say, by the Rand measure). Another proposal is to split the data into k parts, and see how their centroids cluster. By means of synthetic data analyses, we demonstrate that these approaches fail to identify the appropriate number of clusters, particularly as sample size becomes large and the variables exhibit higher correlations.The authors express their thanks to the Sol C. Snider Entrepreneurial Center, Wharton School, for support of this project.     

An algorithm for generating artificial test clusters

An algorithm for generating artificial data sets which contain distinct nonoverlapping clusters is presented. The algorithm is useful for generating test data sets for Monte Carlo validation research conducted on clustering methods or statistics. The algorithm generates data sets which contain either 1, 2, 3, 4, or 5 clusters. By default, the data are embedded in either a 4, 6, or 8 dimensional space. Three different patterns for assigning the points to the clusters are provided. One pattern assigns the points equally to the clusters while the remaining two schemes produce clusters of unequal sizes. Finally, a number of methods for introducing error in the data have been incorporated in the algorithm.     

A monte carlo study of thirty internal criterion measures for cluster analysis

A Monte Carlo evaluation of thirty internal criterion measures for cluster analysis was conducted. Artificial data sets were constructed with clusters which exhibited the properties of internal cohesion and external isolation. The data sets were analyzed by four hierarchical clustering methods. The resulting values of the internal criteria were compared with two external criterion indices which determined the degree of recovery of correct cluster structure by the algorithms. The results indicated that a subset of internal criterion measures could be identified which appear to be valid indices of correct cluster recovery. Indices from this subset could form the basis of a permutation test for the existence of cluster structure or a clustering algorithm.     

An examination of disciplinary procedures with children

In Experiment I the effect of the contingent application of negative attention (a verbal reprimand plus a brief period in which the authority person glares intently at the child) and repeated commands on the noncompliant behavior of children was examined. Thirty-two mother-child pairs served as subjects. The results indicated that negative attention decreased noncompliance whereas repeated commands did not. In Experiment II the effectiveness of negative attention, isolation, ignoring, and a combination of procedures (the alternate use of isolation, ignoring, and negative attention with the same subject) in reducing noncompliance and maintaining it at a low level during a recovery period was examined. Twenty-eight mother-child pairs served as subjects. The results indicated that each of the four disciplinary procedures reduced noncompliance. However, negative attention was associated with less off-task behavior than the isolation and combination procedures when discipline was being imposed and a lower level of noncompliance than ignoring during recovery, whereas the combination condiiton was the only procedure that maintained noncompliance during recovery at the treatment level.     

An examination of the construct validity of posttraumatic stress disorder with veterans using a revised criterion set

Ongoing concerns exist in the literature regarding the construct of posttraumatic stress disorder (PTSD) and how to best conceptualize and measure this disorder. We compared the traditional DSM-IV PTSD symptom criteria (i.e., symptoms from clusters B, C, and D) to a revised criterion set that omits overlapping mood and other anxiety symptoms on PTSD prevalence, PTSD diagnostic caseness, associated psychiatric comorbidity, functional status, and structural validity using a cross-sectional, multi-site primary care sample of 747 veterans. After removing items theorized to overlap with mood and other anxiety disorders, PTSD prevalence was identical using both criterion sets (i.e., 12%). Overall, there were few statistically significant differences in PTSD caseness, associated psychiatric comorbidity, functional status, and structural validity across the two diagnostic criterion sets. These data provide further support that removing items that overlap with other psychiatric disorders does not significantly impact the prevalence of PTSD, its associated comorbidity and functional impairment, or its structural validity. Although the revised criterion set represents a more parsimonious model, the current study findings generally support the strong construct validity of PTSD. The implications of these study findings for research and clinical practice are discussed.     

An examination of interrater reliability for scoring the Rorschach Comprehensive System in eight data sets

In this article, we describe interrater reliability for the Comprehensive System (CS; Exner. 1993) in 8 relatively large samples, including (a) students, (b) experienced re- searchers, (c) clinicians, (d) clinicians and then researchers, (e) a composite clinical sample (i.e., a to d), and 3 samples in which randomly generated erroneous scores were substituted for (f) 10%, (g) 20%, or (h) 30% of the original responses. Across samples, 133 to 143 statistically stable CS scores had excellent reliability, with median intraclass correlations of.85, .96, .97, .95, .93, .95, .89, and .82, respectively. We also demonstrate reliability findings from this study closely match the results derived from a synthesis of prior research, CS summary scores are more reliable than scores assigned to individual responses, small samples are more likely to generate unstable and lower reliability estimates, and Meyer's (1997a) procedures for estimating response segment reliability were accurate. The CS can be scored reliably, but because scoring is the result of coder skills clinicians must conscientiously monitor their accuracy.     

Fuzzy partition models for fitting a set of partitions

Methodology is described for fitting a fuzzy consensus partition to a set of partitions of the same set of objects. Three models defining median partitions are described: two of them are obtained from a least-squares fit of a set of membership functions, and the third (proposed by Pittau and Vichi) is acquired from a least-squares fit of a set of joint membership functions. The models are illustrated by application to both a set of hard partitions and a set of fuzzy partitions and comparisons are made between them and an alternative approach to obtaining a consensus fuzzy partition proposed by Sato and Sato; a discussion is given of some interesting differences in the results.We are grateful to Dr. M.G. Pittau for carrying out the analyses of the macroeconomic data using the method of Sato and Sato (1994).     

An examination of within-person variance in contextual display rules and deviation from display rules

Contextual display rules refer to perceptions of the particular emotions that should be displayed in a specific set of circumstances. As such, it is important to examine within-person variance in such perceptions within the dynamic context of real-life organizations. To this end, experience sampling methodology was used to highlight within-person variance in display rules and examine event-level predictors of these contextual display rules. Thirty-nine university staff members were surveyed four times per day about their most recent interaction. Employee momentary affect (pleasantness and activation) and relationship with the interaction target (solidarity and relative power) were examined as predictors of both contextual display rule perceptions and deviation from those display rules. Pleasantness of affect related positively to the level of expression allowed by the display rule, whereas activation of affect and target solidarity related positively to deviation from display rules. Complex interactions were also found between the predictors, further highlighting the complexity of the emotional labour process.     

An examination of the effectiveness of multiple summary scoring procedures of the BNBAS in detecting group differences

The Brazelton Neonatal Behavioral Assessment Scale is the most widely used and accepted tool in the study of the behavior of the human newborn; yet there remains some controversy over the most appropriate way to score the results. The present research compares three a priori methods of scoring. The sample consisted of 43 infants: (a) 11 preterms, (b) 10 fullterms in intensive care, (c) 7 fullterms with sick mothers, and (d) 15 healthy fullterms. BNBAS exams administered to these four groups of infants were scored using the Als (1978), Lester, Als, and Brazelton (1982), and Jacobsen, Fein, Jacobsen, and Schwartz (1984) clusters. The data from these three methods were analyzed by means of one-way ANOVAs for group differences. In addition, a factor analysis on the BNBAS scores was also completed. These three approaches all detected differences between medically at-risk and healthy infants on those items assessing motor maturity and orientation. These results suggest that at this time no one scoring system can capture the richness of behavior of the newborn and address the question by researchers, clinicians, and parents.     

MMPI-2 as a predictor of change in PTSD symptom clusters: a further analysis of the Forbes et al. (2002) data set

In this study, we reanalyzed the Forbes et al. (2002) data set to examine the Minnesota Multiphasic Personality Inventory (MMPI-2; Butcher, Dahlstrom, Graham, Tellegen, & Kaemmer, 1989) as a differential predictor of change across posttraumatic stress disorder symptom clusters following treatment in 141 Vietnam veterans. A series of partial correlation and linear multivariate regression analyses, controlling for initial symptom severity, identified several scales predictive of symptom change. None of the MMPI-2 scales, however, emerged as predictors of change in reexperiencing symptoms. Social alienation and marital distress were the most potent predictors for avoidance symptoms. Anger, alcohol use, and hypomania were the most potent predictors for the hyperarousal symptoms. Of the personality disorders, borderline personality was the strongest predictor of change in the avoidance and hyperarousal clusters. Further replication of the findings of this article and those reported by Forbes et al. (2002) is required.     

Managing data for integrity: Policies and procedures for ensuring the accuracy and quality of the data in the laboratory

Management of the research data is an extremely important responsibility of the Principal Investigator (PI) and other members of the research team. Without accurate data, no worthwhile conclusions can be drawn from the research study. Integrity in data management is critical to the success of the research group and to public trust in the research outcomes. One of the primary responsibilities of the PI is to provide proper training to the junior members of the lab. This effort can be buttressed by institutional data policies that are implemented at the group level. Extensive and frequent guidance in good research practices by the PI and other senior research staff is critical to the proper training of new scientists.     

An abac for determining the mean deviation of a class from the general mean

The abac is presented with instructions for use.     

Some simple procedures for handling missing data in multivariate analysis

For analyses with missing data, some popular procedures delete cases with missing values, perform analysis with missing value correlation or covariance matrices, or estimate missing values by sample means. There are objections to each of these procedures. Several procedures are outlined here for replacing missing values by regression values obtained in various ways, and for adjusting coefficients (such as factor score coefficients) when data are missing. None of the procedures are complex or expensive.This research was supported by NIH Special Research Resources Grant RR-3. The author expresses his gratitude to Robert I. Jennrich and the referees for their suggestions.     

Analysis of proposed three-class classification decision rules in terms of the ideal observer decision rule

We analyze recently proposed decision rules for three-class classification from the point of view of ideal observer decision theory. We consider three-class decision rules proposed by Scurfield, by Chan et al., and by Mossman. Scurfield's decision rule is shown to be a special case of the three-class ideal observer decision rule in three different situations. Chan et al. start with an ideal observer model and specify its decision-consequence utility structure in a way that causes two of the decision lines used by the ideal observer to overlap and the third line to become undefined. Finally, we show that, for a particular and obvious choice of ideal-observer-related decision variables, the Mossman decision rule cannot be a special case of the ideal observer decision rule. Despite the considerable difficulties presented by the three-class classification task, the three-class ideal observer provides a useful framework for analyzing a variety of three-class decision strategies.     

Assessing categorization performance at the individual level: a comparison of Monte Carlo simulation and probability estimate model procedures

Two analytical procedures for identifying young children as categorizers, the Monte Carlo Simulation and the Probability Estimate Model, were compared. Using a sequential touching method, children aged 12, 18, 24, and 30 months were given seven object sets representing different levels of categorical classification. From their touching performance, the probability that children were categorizing was then determined independently using Monte Carlo Simulation and the Probability Estimate Model. The two analytical procedures resulted in different percentages of children being classified as categorizers. Results using the Monte Carlo Simulation were more consistent with group-level analyses than results using the Probability Estimate Model. These findings recommend using the Monte Carlo Simulation for determining individual categorizer classification.     

There's more in a number than you think: new validity data for the Global Assessment Scale

Holcomb and Otto (1988) recently suggested that the widely used Global Assessment Scale may have ". . . questionable validity as an index of psychopathology." The present paper points out procedural and conceptual problems in their study and presents extensive evidence supporting the concurrent and predictive validity of the scale.