A Riemannian Inexact-Newton Method for Solving the Orthogonal INDSCAL Problem in Multidimensional Scaling

Residency training programs vary across characteristics such as their didactic and clinical experiences, attributes of the incoming residents, faculty characteristics, research conducted, community service performed by the program, and eventual practice choices of graduates. Which of these characteristics are most salient for evaluating the quality of a program? Elliott1 lists characteristics of graduates, cost—effectiveness, fair and ethical treatment of trainees, and meeting societal needs as important quality indicators. Iverson2 takes a dimensional approach. His dimensions, with metrics, are: intake [U.S. Medical Licensing Examination (USMLE) scores of matched applicants]; customer satisfaction [percentage of available positions filled by match, and percentage filled by U.S. medical school graduates (USMGs)]; residency review committee (RRC) quintile scores; and outcome (specialty board pass rates). The Accreditation Council for Graduate Medical Education (ACGME) recently switched its focus from process variables to outcomes, and is encouraging RRCs to evaluate a program on how well it provides for six core competencies: patient care, clinical science; interpersonal skills and communication, professionalism, practice-based learning and improvement, and systems-based practice.3 In 1997, a task force of the American Association of Directors of Psychiatry Residency Training (AADPRT) developed a survey to define the variables important to determining a program's quality from the psychiatry resident's perspective. The 41-item questionnaire was based on feedback from focus groups of psychiatry residents and program directors and a review of the literature. A total of 180 psychiatry residents from 16 programs completed the survey. Quality of supervision and teaching conferences, respect of faculty for residents, responsiveness of the program to feedback from residents, and morale in the department were the items most important to residents' satisfaction. A detailed description of the construction of the survey and its results was published by Elliott.4 In 1998, the AADPRT's survey was repeated with psychiatry residency directors and heads of major rotations to see whether their values agreed with those of the residents.5 This paper describes the use of multidimensional scaling (MDS) of the survey's results to establish whether there were distinct groupings of program directors with different opinions about the determinants of quality in psychiatry residency programs. These groupings might represent types of psychiatry programs (or market niches) as reflected in the values and priorities of their faculty and directors. Method Multidimensional scaling is an analytic technique frequently used in marketing research to identify the psychological dimensions underlying customers' preferences with respect to multiple variables or features of a product.6 In MDS the difference between clusters or groups of variables is predicted by the distance between the variables in psychological "space," with the dimensionality of the space equal to the number of relevant dimensions underlying the data. These dimensions can be thought of as analogous to the latent constructs derived in factor analysis. The scaling algorithm derives the dimensions and plots the coordinates of the variables in the resulting multidimensional space. MDS is an inherently interpretive procedure—it locates variables on dimensions but requires the investigator to determine whether the dimensions can be intelligibly labeled. Individual Differences Scaling (INDSCAL) is an MDS algorithm that models both the overall dimensions that underlie the perceptions of the group of respondents and individual weights on those dimensions for each respondent, allowing individuals to vary in the importance they attach to each of the dimensions. For example, for one individual, dimension A (the educational resources available, for example) may be highly salient, while dimension B (the administration of the program, for example) is relatively unimportant. For another individual, these priorities may be reversed. By examining the distribution of subject weights one can identify clusters of subjects who share similar values regarding the relative importances of the various dimensions. The questionnaire was sent in late 1998 to all psychiatry residency directors listed in the American Medical Association's 1998–1999 Directory of Accredited Graduate Medical Education (GME) Programs. The faculty members who served as the heads of the inpatient and outpatient psychiatry rotations of each program were also surveyed. These are the two major rotations of psychiatry training programs, and the opinions of the heads of these rotations (henceforth referred to as service chiefs) would most likely represent the dominant values of the program. The survey asked directors and service chiefs to rate how important the 41 items of the questionnaire were in determining the quality of a residency program. The anchors were 1 = least important, 4 = average importance, and 7 = most important. Multidimensional scaling using INDSCAL was done on the survey responses. Solutions in two to six dimensions were generated. Results Of the 186 active programs listed in the GME directory, 117 programs (63%) responded to the survey. There were 234 individual responses from the 117 programs. Of these, 142 (61%) were from program directors and 92 (39%) were from service chiefs who were not identified as directors. For some programs the head of inpatient or outpatient services also served as an associate program director, confirming our supposition that these faculty members represent the administrative backbone of the program. The Pearson correlation between the responses of the residency directors and those of the service chiefs was 0.98 (p < 0.01). We therefore pooled data from both chiefs' and residency directors' responses for the following analysis. The two-dimensional INDSCAL solution was degenerate and was discarded. The solutions in three to six dimensions were inspected for goodness of fit and interpretability. The three-dimensional configuration provided the most interpretable dimensions, and accounted for 46.4% of the variance in the data; higher-dimensional solutions accounted for only slightly more variance. Based on examination of the locations of the items, particularly those that had particularly high or low coordinates in each dimension, the dimensions seemed to correspond to three constructs: "curriculum," "quality of the institution," and "supportiveness of the administration of the program." The three dimensions, and the highest-loading items on each, are given in Table 1.TABLE 1: Three Dimensions of Quality of Psychiatry Residency Programs, Based on Multidimensional Scaling of Responses by Residency Directors and Service Chiefs to a 1998 Questionnaire*Subject weights measure the importance or salience of each dimension to each respondent; they range from 0 (completely ignored) to 1 (overwhelmingly important), and need not sum to 1. In addition, each respondent is assigned a "weirdness' value, which measures the similarity of his or her responses to those of the typical respondent, based on the relative importance of each dimension and the goodness of fit for that respondent. There was a great deal of variation in individual preferences, but no distinct clusters were evident. Notably, although weights for the supportiveness of the administration of the program dimension fell between 0.25 and 0.40 for nearly all respondents, the importance attributed to the dimensions of curriculum and quality of the institution varied extensively across individuals. Figure 1 plots the weights of curriculum and quality of the institution against one another for each respondent. The unshaded polygon encloses data for more typical respondents with less than the median weirdness, represented as circles; the two shaded polygons identify two groups of less typical respondents with more than the median weirdness, represented as crosses. Two respondents in the upper left had extreme (outlier) weirdness values.Figure 1: Subject weights for the "curriculum" and "quality of the institution" dimensions of the three-dimensional INDSCAL solution. The unshaded polygon encloses subjects with less than the median weirdness (based on weights in all three dimensions), represented as circles; the two shaded polygons identify two groups of subjects with greater than median weirdness, represented as crosses. Two subjects in the upper left had extreme (outlier) weirdness values."Weirdness" measures how similar each respondent is to the typical respondent, based on the relative importance of each dimension and the goodness of fit for that respondent.While the most typical respondents gave curriculum weights between 0.3 and 0.6, and quality of the institution weights between 0.2 and 0.4, two groups of respondents displayed different weight patterns. One group (lower right) gave curriculum substantially higher weights than typical (ranging from 0.5 to 0.75); the other group (upper left) gave quality of the institution substantially higher weights than typical (ranging from 0.25 to 0.65). On the average, most respondents' data displayed a continuum of weight patterns in which curriculum was considered to be more important than either quality of the institution or supportiveness of the administration of the program; the respondents with the lowest weirdness scores weighted these dimensions in the proportions 1.3:1:1, respectively. Discussion The three dimensions that emerged from the MDS are consistent with the many suggested quality indicators reviewed above.1,2,3 How might we conceptualize this triad? The dimensions of curriculum and supportiveness reflect two different aspects of the process of residency training. The curriculum dimension describes the content of the educational program; the supportiveness dimension reflects the culture or ambiance within which the training occurs. Residency directors and their faculty seem to differentiate between these two aspects of the program, and value both as indicators of the quality of the program. The dimension of institutional quality includes items reflecting the reputation and resources of the institution as well as items generally considered to be outcomes (i.e., board scores of graduates and graduates' job satisfaction). In this instance, these "outcome" items probably serve as proxies for (and a reflection of) the reputation of the institution and the quality of the residents it attracts, rather than as true outcome measures. This dimension could represent a general context factor, reflecting the quality of the facilities, the faculty, and of the residents themselves. This general factor could itself modify the effects of the process variables either up or down, thereby affecting the expected outcomes. Thus, this dimension may reflect the expectation of program directors and chiefs that equivalent processes (curriculum and supportiveness) could lead to better outcomes if they are provided in the context of a higher-quality institution. Donabedian lists input, process, and product as the dimensions of quality in healthcare.7 Interestingly, product (outcomes) did not emerge as a dimension of the quality of a program. Perhaps residency directors and service chiefs focus on process variables as indicators of quality since it is in the process of residency training that they deal. The neglect of outcome measures may also reflect a philosophy that, while the program is responsible for teaching, it is the residents' responsibility to learn. Outcome measures are highly confounded by the abilities and characteristics of the individual residents and, thus, may not be considered an accurate or reliable measure of the quality of the program per se. The ACGME and others who have begun the move towards outcome evaluation may wish to take this as a word of caution. Outcomes should not be considered in a vacuum. For at least some key stakeholders—the residents and faculty of the program—the context, content, and culture of the program are significant as well. No truly distinct clusters or groups of respondents emerged from the multidimensional scaling of the data. While there may be programs with different missions—programs oriented towards research or community psychiatry, for example—these missions do not seem to result in drastically different definitions of quality. This suggests that there is a core concept of quality that holds across contexts and across missions—consistent with the RRC's model of minimum standards. On the other hand, there seems to be a continuum of individual variation, rather than variation based on group membership. The individual respondents differed widely in the dimensions they considered most important. Since the individuals in this case are the faculty leaders and directors of the programs, these priorities are most likely reflected in the programs as a whole. The individual variation can be usefully segregated into three "market niches," corresponding to the three polygons in Figure 1. Thus we could describe three types of programs: (1) programs in which the quality of the sponsoring institution (context) is paramount, (2) programs in which the quality of the curriculum (content) is paramount, and (3) programs with a more typical weighting of the three dimensions. While there was less variability in the importance attached to the supportiveness (culture) of the program, this should not be construed as lack of salience. Rather, all programs should be alert to the importance of this dimension. Residents, too, vary in the levels of importance they attribute to the various features of a training program.4 The context, content, and culture of a program may provide a good conceptual model of the dimensions along which the market varies. Programs may find it useful to identify and market themselves on the basis of these dimensions. This study focused on only one of the many stakeholder groups of residency programs. The needs and expectations of other stake-holders, such as program funders and employers of a program's graduates, remain to be defined. This study also focused only on psychiatry programs, but the dimensions of context, content, and culture would seem to be potentially applicable to other specialties as well. Repeating the study with other specialties will tell us whether indeed this triad is relevant to the quality of programs across specialties. The method of multidimensional scaling is a novel one for determining quality measures in graduate medical education. Repeated use of this technique, across stakeholders and across specialties, can help elucidate the factors most important to the evaluation of residency programs.

This review covers metric and nonmetric multidimensional scaling(MDS)studies. Topics reviewed include metric vs. nonmetric methods, similarities, distances and the Minkowski power metrics, distance vs. content methods, the additive constant problem in the2-way and the3-way MDS, computational problems(gradient vs. rank image methods, the alternating least squares method, the N-way Eckart-Young decomposition, maximum likelihood, convex function analysis, continuous transformation, lincar programming), computer programs, individual difference scaling of similarity and preference data(the3-way scaling and factor analysis, the unfolding vs. vector models), Monte Carlo studies, incmplete designs, interactive scaling, integrated approaches, books on MDS, and a quantitative look at main contributors to the MDS literature. Because of the limited space, applications of MDS are not reviewed, and an extensive bibliography on MDS had to be deleted from(2).

In this note we show that the kernel PCA algorithm of Schölkopf, Smola, and Müller (Neural Computation, 10, 1299–1319.) can be interpreted as a form of metric multidimensional scaling (MDS) when the kernel function k(x, y) is isotropic, i.e. it depends only on ‖x − y‖. This leads to a metric MDS algorithm where the desired configuration of points is found via the solution of an eigenproblem rather than through the iterative optimization of the stress objective function. The question of kernel choice is also discussed.

A natural extension of classical metric multidimensional scaling is proposed. The result is a new formulation of nonmetric multidimensional scaling in which the strain criterion is minimized subject to order constraints on the disparity variables. Innovative features of the new formulation include: the parametrization of the p-dimensional distance matrices by the positive semidefinite matrices of rank ≤p; optimization of the (squared) disparity variables, rather than the configuration coordinate variables; and a new nondegeneracy constraint, which restricts the set of (squared) disparities rather than the set of distances. Solutions are obtained using an easily implemented gradient projection method for numerical optimization. The method is applied to two published data sets.

An amino acid map of inter-residue contact energies using metric multi-dimensional scaling

Localization is a fundamental operation in wireless networks. Location determination is normally accomplished using the Global Positioning System (GPS) for outdoor applications. For indoor localization, GPS does not work due to the lack of the line of sight to satellites. High-precision indoor localization is critical to many personal and business applications. WiFi-based indoor localization was proposed to be a practical method to locate WiFi-enabled devices due to the popularity of WiFi networks. However, it suffers from large localization errors. Our experimental results indicate that this scheme consistently leads to an average error around 3 meters. The existence of different locations with similar WiFi signal strength is the reason behind the large errors. To improve the localization precision, a hybrid indoor localization scheme, HILL, is proposed in this paper. Inspired by the fact that a large number of WiFi-enabled mobile devices have been deployed, HILL uses 3 phases to improve the precision of WiFi-based localization. First of all, it measures the distances between each pair of peer devices through acoustic ranging. Secondly, the Classical Metric Multidimensional Scaling (MDS) method is applied to the collected distances, which results in a graph consistent with the distances. Finally, the graph generated by MDS is embedded onto the graph corresponding to WiFi-based localization in order to achieve high localization precision. Our experimental results indicate that the average localization error of HILL is about 1 meter.

The ideas described in this paper are motivated by a basic problem in Multidimensional Scaling which is solved not quite satisfactory up to now. Suppose that there are given pairwise dissimilarities δij, 1 ≤ i, j ≤ n, as nonnegative real numbers between n objects. These originate from the special type of application and may be, for example in cartography, measurements of pairwise distances disturbed by additive random errors or, in psychology, individual assessments of the deviation in behaviour of n persons measured on a certain real scale. There are many other applications in a variety of fields, a structured overview may be obtained from de Leeuw & Heiser (1980, 1982).

Multidimensional scaling (MDS) is a popular tool for dimensionality reduction and data visualization. Given distances between data points and a target low-dimension, the MDS problem seeks to find a configuration of these points in the low-dimensional space, such that the inter-point distances are preserved as well as possible. We focus on the most common approach to formulate the MDS problem, known as stress minimization, which results in a challenging non-smooth and non-convex optimization problem. In this paper, we propose an inertial version of the well-known SMACOF Algorithm, which we call AI-SMACOF. This algorithm is proven to be globally convergent, and to the best of our knowledge this is the first result of this kind for algorithms aiming at solving the stress MDS minimization. In addition to the theoretical findings, numerical experiments provide another evidence for the superiority of the proposed algorithm.

The study assessed a decade long changes of the coral community structure in the Palk Bay region using multivariate analysis based on observed coral life-forms and its diversity between the years, 2002 and 2018. A total of 11 study sites were selected from Rameswaram Island to Mandapam coast. Live coral cover was found to be 27% (21% non-Acropora and 6% Acropora corals) during the study period 2002–2005. Principal component analysis (PCA) showed that there was a significant variability (PC1: 39.5% variance) in coral cover between the study site, Bison House and the rest of the study sites. In the study of 2014–2018, live coral cover observed was 17% and this included 11% non-Acropora and 6% Acropora corals. PCA showed that the contribution of live coral cover gradually reduced towards the northern part of Rameswaram Island. Metric multi-dimensional Scaling (MDS) showed that there were four clusters formed between the sites with 80% similarity as observed based on the life-form categories. There were three major clusters with 70% similarity observed based on coral diversity. The highest diversity index (Shannon-wiener) was observed at Olaikudah and Villunditheertham as 1.906 and 1.864 respectively. As of 2014–2018, the live coral status showed 37% less than that found during 2002–2005 study period. Variation of reef flat composition was studied using Depth-Invariant Index for the study period 2002–2005. This study has helped to understand the longitudinal gradient of incidence, temporal variations, and distribution pattern of coral reef community categories in the Palk Bay region.

A psychophysical investigation of timbre was undertaken with the intent of deriving quantitative results that could be useful in developing a better human interface to control effects processing for expressiveness in music. Multidimensional perceptual scaling analysis was performed for a set of musical timbres. Two listening experiments were performed in order to determine the most salient perceptual attributes. These two experiments provide data enabling KANSEI engineering for these effects. A commercially available effects processor generated the six stimuli used in these experiments. In the first experiment, listeners were asked to give ratings of the relative difference between two stimuli without regard to any property contributing to that difference. In the second experiment, the listeners were asked to give ratings of each stimulus with respect to a particular property. Dissimilarity ratings of all pair-wise comparisons of the six effects were submitted to INdividual Differences SCALing (INDSCAL) analysis in order to derive a multidimensional perceptual space for the stimuli. Three semantic components were derived using Principal Components Analysis (PCA) of property ratings for the same set of six effects. Multiple Regression Analysis (MRA) was used to relate the perceptual dimensions and semantic components, the results of which provide a control structure for a commercially available effects processor. The study revealed a direct mapping between three perceptual dimensions and three semantic component scales, those being labeled as “sharp-dull, ”“diffusecompact, ”and “thick-thin”(in Japanese, surudoi-nibui, hirogotta-kojinmarishita, and usui-atsui).

Evolution is best viewed as a multivariate process making a working knowledge of linear algebra a great aide in our understanding of this process (Lande & Arnold, 1983; Manley, 1985; Endler, 1986; Phillips & Arnold, 1989). We therefore are in general agreement with the main thrust of Blows' (2007) argument – a fuller appreciation of multivariate methods, particularly those that employ the diagonalization of the G and γ matrices, can aid our understanding of microevolutionary change. Nevertheless, the approach advocated by Blows (2007) is not a panacea that solves all problems in multivariate evolutionary analyses, and there are circumstances where the diagonalization of the G and γ matrices is either inappropriate or inferior to alternatives. In our opinion, the utility of matrix diagonalization is largely determined by the particular question of interest and the biology of the system examined. In support of our arguments, we discuss how matrix diagonalization can obscure the biological interpretation of a complex data set. We then discuss how the benefits of matrix diagonalization in evolutionary studies vary with the specific question addressed. Organisms are made up of many interacting parts that are often shaped by selection to function as integrated units (or character) (Cheverud, 1982, 1996). In order to understand the evolution of such complex multidimensional characters, we must simplify the system to an understandable level that allows for biological interpretation. One of the major strengths of the multivariate approach is that it summarizes the entire pattern of (co)variation in a series of successive orthogonal axes, thereby removing the need to interpret single traits on an element-by-element basis. Given that a symmetric matrix with k traits contains k(k + 1)/2 elements, it can be a formidable task to interpret individual elements even when k is not particularly large. Unfortunately, biology is rarely as simple as the above statistical procedures would dictate. Although matrix diagonalization is mathematically convenient, there are circumstances where it obscures biological interpretation. First, it is unlikely that causal factors will always have orthogonal effects on the phenotype –Houle et al. (2002, pp. 434–435) describe such a cautionary tale. Complex phenotypes are composed of many different measurable components, but it is unlikely (if not impossible) that each of these components will have exactly the same function. Consequently, we often do not know how to define a 'functionally related' character and, therefore, to decide on the appropriate unit of selection (Wagner & Laubichler, 2000). Thus, a misguided solution is to measure everything possible and then use matrix diagonalization to identify the dimensions of variation and define the character. Although each orthogonal axis represents a new 'trait' that can be defined by its eigenvector (Phillips & Arnold, 1989), such axes do not necessarily correspond to multivariate patterns of functional variability (Cheverud, 1982; Mitchell-Olds & Rutledge, 1986). In such instances, matrix diagonalization may actually disrupt or obscure the biologically meaningful units of variation. Consider the case where understanding pleiotropic relationships between functional units of a character, such as those in the mouse mandible (Schwenk, 2001), is the goal. The mouse mandible has clearly identifiable functional units that are genetically and developmentally integrated with a modular structure (Cheverud, 2001). However, despite the modular structure of variation, there are still very large genetic covariances between the modular functional units of the mandible due to whole-mandible pleiotropic effects (Cheverud, 2001). If one diagonalizes G, the first eigenvalue contains significant variation from all functional units due to these pleiotropic effects. In such instances, it is preferable to extract eigenvalues from each of the individual modules and then directly estimate genetic parameters in these dimensions (Kirkpatrick & Meyer, 2004; Meyer & Kirkpatrick, 2005). This reduces the number of dimensions to be examined but keeps the modular structure intact. Alternatively, where the nature of the covariance structure is known a priori, confirmatory factor analysis and/or structural equation modelling may be a more promising approach (Houle, 2001; Houle et al., 2002). A second issue, as noted by Lande & Arnold (1983), is that selection analyses based on orthogonal transformations of the data [such as principle components (PCs)] can obscure the relationship between the individual traits and fitness. This occurs because, in the eigenvectors of the PCs that define the new composite 'traits' created by the transformation, traits that are subject to selection are associated with traits that are not experiencing selection (Lande & Arnold, 1983). Whenever traits that are the targets of selection have a substantial phenotypic covariance with traits unrelated to fitness, the use of orthogonal transformations may dilute the ability to detect the effect of selection (Mitchell-Olds & Shaw, 1987). Thus, selection analyses performed on orthogonal transformations, particularly when associated with substantial multicollinearity in the data set, should always be interpreted with caution (Lande & Arnold, 1983; Mitchell-Olds & Shaw, 1987). The value of orthogonal transformations may be more as a first step for deriving hypotheses that can be followed up with manipulative experiments (Manley, 1985; Endler, 1986; Mitchell-Olds & Shaw, 1987). Finally, in systems with a simple biology, interpretation may be obfuscated by orthogonal transformation of the data. We can illustrate this point using Moore's (1990) dataset examining episodes of sexual selection acting on sexually dimorphic traits in the dragonfly, Libellula luctuosa. Although the traits in this study were simple linear measurements, it is likely that a large proportion of the biological reality was captured – the wings are virtually flat with a simple shape, there are three (virtually rectangular) brown, white and clear areas on the wings, and the correlation between the width of the coloured patches and their area is high. Although the standard selection gradients significantly under-estimated the strength of nonlinear selection (median |γ| = 0.52, median |λ| = 0.72, Wilcoxon test: Z = −2.201, n = 6, P = 0.03), as is commonly the case in studies of selection (Blows & Brooks, 2003), little interpretive value is gained from a canonical analysis of the results. For example, regression analysis of mating success data revealed significant positive directional selection on brown-patch and white-patch size, quadratic (negative γ) selection on body size and positive correlational selection on brown-patch and white-patch size (Table 1; Moore, 1990). Canonical analysis of these data yields two statistically significant axes, m1 and m4, but no new insight is gained (Table 2). The axis m1 describes the pattern of linear and nonlinear selection operating on the size of the brown and white patches, whereas m4 describes the nonlinear selection operating on body size. The result is an individual selection surface with a multivariate saddle (Fig. 1) that is congruent with intuition based on the sign and magnitude of the β and γ values presented in Table 1. Thin-plate spline visualization of the individual selection surface representing linear and nonlinear selection on m1 and m4. (a) a perspective view, (b) a shaded contour plot with supporting data points. Lighter regions indicate higher fitness. Given the importance of G and γ to evolutionary theory (Lande & Arnold, 1983), it is not surprising that many questions in evolutionary biology require an intimate knowledge of their structure. If one is trying to describe the overall variation in G or γ, or even how the dominant axis of G (or gmax) relates to the β vector to examine the importance of genetic constraints (Schluter, 1996), matrix diagonalization is clearly a useful technique. There are, however, a number of questions in evolutionary biology where matrix diagonalization is not useful. If an examination of the causal relationship between traits and fitness is the goal, it is unlikely to be answered convincingly with an observational approach that characterizes patterns of variation in β or γ (Mitchell-Olds & Shaw, 1987). The biological understanding of selection is ultimately the only way to determine how selection acts on the phenotype (Manley, 1985; Endler, 1986). To this end, manipulative studies that either break the functional relationships between traits (e.g. Brooks et al., 2005; Bentsen et al., 2006) or extend phenotypes beyond their natural range (e.g. Anholt, 1991; Sinervo & Basolo, 1996) have been particularly useful. In systems that are not as amenable to experimental manipulation, path analysis may provide an alternative (Scheiner et al., 2000), particularly because it enables analytic models to be constructed (and tested) that are based specifically on the biology of the organism being examined. Sometimes estimating important genetic parameters of evolutionary change, such as the mean phenotypic response to selection () or the change of G due to selection within a generation [ΔG=G(γ−ββT)G], is the goal. In the former case, matrix diagonalization is unnecessary because the vector of mean phenotypic response to selection (i.e. the multivariate breeders' equation) uses G as a transformation matrix to relate selection within a generation (β) to changes in trait means across generations. This gives a concise way of relating the space of G and β in the vector without the need for matrix diagonalization. In the later case, the diagonalization of γ does not provide the correct characterization of the form of nonlinear selection because it is the curvature of the adaptive landscape (i.e. γ−ββT) that determines how selection shapes G, not the curvature of the individual selection surface (i.e. γ) (Phillips & Arnold, 1989). In most cases (when the trait distribution is approximately multivariate normal), the adaptive landscape and the individual selection surface have the same β, but the curvature of the adaptive landscape often differs considerably from γ because it is weighted (and smoothed) by the phenotypic distribution (Phillips & Arnold, 1989). Thus, to understand the relationship between the structure of G and patterns of selection, analyse the structure of γ−ββT not γ. Although Blows (2007) gives a cursory discussion to issues regarding linear selection, nonlinear components of selection need not be emphasized at the cost of linear selection. We reiterate the point made eloquently by Phillips & Arnold (1989): one cannot interpret the consequences of γ without β. For example, the diagonalization of γ may reveal a negative eigenvalue for the major canonical axis suggesting 'stabilizing' selection towards an optimum. However, if there is a strong linear component along this axis, the pattern of selection should be seen as linear selection that has curvature (Lande & Arnold, 1983). Phillips & Arnold (1989) describe how to characterize both linear and nonlinear selection within canonical analysis (the 'A canonical form'). When linear selection is minimal an alternative involving an additional transformation ('B canonical form') allows a full description of selection using the curvature of the individual fitness surface, but additional restrictions and assumptions are associated with the use of form B (Phillips & Arnold, 1989). Finally, if questions of generality and/or comparability are being addressed, then matrix diagonalization may complicate the direct comparison of two or more matrices; e.g. when comparing multiple individual selection surfaces for a common set of phenotypic traits (e.g. Chenoweth & Blows, 2005). Because the axes derived from an orthogonal transformation occupy a unique multivariate subspace that often differs considerably from the original trait space, the canonical rotation of the individual fitness surfaces places the data in different subspaces. Consequently, any biologically meaningful comparison of the M matrices between the sexes requires projection into a common subspace. It is conceptually and computationally simpler to use sequential model building procedures (Draper & John, 1988) to compare selection in the untransformed space. Although we agree that the multivariate approach advocated by Blows (2007) often provides insights into current selection, it should not be viewed as a panacea that solves all problems in evolutionary biology. Although a multivariate view of evolution clearly prevails, the difficulty in defining characters, the plethora of legitimate research goals, and the potential loss of generality or interpretability can supersede the statistical sophistication of matrix diagonalization. By identifying some of the limitations of this methodology we hope that we clarify when its use is appropriate. Our general recommendation is that matrix diagonalization should be used if possible and appropriate – but as always, biological questions must take precedence over statistical techniques. JH thanks Mark Blows and Rob Brooks for clarifying many of the aspects covered in this manuscript during his time in Sydney. AJM & JBW remain influenced in this area by Jim Cheverud and Butch Brodie. We thank David Houle, Patrick Phillips and Richard Preziosi for discussions that helped clarify several important issues. Our research is supported by NERC, BBSRC and Leverhulme.

Individual Compositional Cluster Analysis

Research shows that college students are deficient in statistical skills. There is a need today for the development of learning tools that can help students overcome their anxiety of statistical applications. This study seeks to respond to that need by describing a student assignment involving multivariate statistics. The assignment integrates aspects of perceptual mapping via multidimensional scaling and multi-attribute models. A feedback questionnaire revealed that the assignment was well-received by students and successful in increasing their knowledge of statistics. Students felt the assignment did increase their knowledge of perceptual maps, multi-attribute models, and multidimensional scaling. Students showed an interest in learning more about statistics and marketing research. Since the assignment increased the student's interest in statistics and willingness to learn and use statistics, findings suggest that the students anxiety regarding statistics was reduced.

An investigation of the cognitive organization of body comparison sites in relation to physical appearance related anxiety and drive for thinness

Projective mapping in sensory analysis of ewes milk cheeses: A study on consumers and trained panel performance