the analysis of multitrait multimethod matrices via constrained components analysis multitrait multimethod mtmm matrices are often analyzed by means of confirmatory factor analysis cfa however fitting mtmm models often leads to improper solutions or non convergence in an attempt to overcome these problems various alternative cfa models have been proposed but with none of these the problem of finding improper solutions was solved completely in the present paper an approach is proposed where improper solutions are ruled out altogether and convergence is guaranteed the approach is based on constrained variants of components analysis ca besides the fact that these methods do not give improper solutions they have the advantage that they provide component scores which can later on be used to relate the components to external variables the new methods are illustrated by means of simulated data as well as empirical data sets

correspondence analysis and association models constrained by a conditional independence graph the manner in which the conditional independence graph of a multiway contingency table effects the fitting and interpretation of the goodman association model rc and of correspondence analysis ca is considered estimation of the row and column scores is presented in this context by developing a unified framework that includes both models incorporation of the conditional independence constraints inherent in the graph may lead to equal or additive scores for the corresponding marginal tables depending on the topology of the graph an example of doubly additive scores in the analysis of a burt subtable is given

identification theory for high dimensional static and dynamic factor models abstract high dimensional factor models can involve thousands of parameters the jacobian matrix for identification is of a large dimension it can be difficult and numerically inaccurate to evaluate the rank of such a jacobian matrix we reduce the identification problem to a small rank problem which is easy to check the identification conditions allow both linear and nonlinear restrictions under reasonable assumptions for high dimensional factor models the small rank conditions are shown to be necessary and sufficient for local identification

local identifiability of the factor analysis and measurement error model parameter abstract measurement error regression models are factor analysis models the latent correct regressors are the factors there is however no common statistical method between the factor analysis and the regression model because the covariance elements that are known identifying constituents of the former model are unknown in the latter instead the idea that the data come from the same regression model as with panel data but can be grouped in two or more groups each group having its own different regrressor generating process is shown to supply credible restrictions we generalize and compare relevant identifiability criteria and corresponding asymptotically efficient estimators that are recursive in the number of overidentifying restrictions

analysis of additive dependencies and concurvities using smallest additive principal components additive principal components are a nonlinear generalization of linear principal components their distinguishing feature is that linear forms i a i x i are replaced with additive functions i f i x i a considerable amount of flexibility for fitting data is gained when linear methods are replaced with additive ones our interest is in the smallest principal components which is somewhat uncommon smallest additive principal components amount to data descriptions in terms of approximate implicit equations i f i x i 0 estimation of such equations is a data analytic method in its own right competing in some cases with the more customary regression approaches it is also a diagnostic tool in additive regression for detection of so called concurvity this term describes degeneracies among predictor variables in additive regression models similar to collinearity in linear regression models concurvity may lead to statistically unstable contributions of variables to additive models as an example we show in a reanalysis of the ozone data from breiman and friedman that concurvity does indeed exist in this particular data a fact which may impact the interpretation of the additive fits in the second half of this paper we give some second order theory including the description of null situations and eigenexpansions derived from associated eigenproblems we show how ace and additive principal components are related and we outline some analytical methods for theoretical calculations of additive principal components lastly we consider methods of estimation and computation additive principal components have had a long tradition in psychometric research and correspondence analysis they have started receiving attention by statisticians only in recent years

analysis of contingency tables by correspondence models subject to order constraints abstract inferential correspondence analysis which has gained much attention in recent years is applied here to contingency tables with ordered categories to reflect such order the parameters of the underlying correspondence models are constrained to follow the order induced by the categories of the analyzed table a reparameterization of the correspondence model in terms of a latent variable model is presented this allows a simple and straightforward use of the em algorithm to obtain efficient order restricted estimates a goodness of fit test is also discussed and an example is analyzed a small monte carlo example is presented

fixed distance latent class models for the analysis of sets of two way contingency tables this paper develops and describes the application of modified latent class models for analyzing sets of two way contingency tables the proposed fixed distance models differ from traditional latent class models in that the assumption of local stochastic independence is superseded by allowing interactions of the manifest variables within each class which can be represented by a single association parameter as an example two data sets on eye color by hair color collected in caithness n1 5 387 and aberdeen n2 22 361 and fixed distance models with up to six classes three classes per data set are considered finally leading to satisfactory fit and rather simple interpretation

identifiability of factor analysis some results and open problems abstract identifiability of the factor analysis model is discussed and some recent results and their relation to previous work are surveyed sufficient conditions for local identifiability are presented a counterexample shows that these conditions are not necessary in general a simple counting rule for local identifiability of the factor analysis model is given generically finally some open problems are formulated

models for the analysis of association in multivariate contingency tables abstract the present work describes a family of models that can be used for the analysis of association in multiway cross classifications the models are complementary to the usual i e the so called hierarchical log linear models an advantage of this family of models is that it is frequently possible to give a more parsimonious description of the association than is possible with the usual log linear models approach the models will be especially useful when some or all of the variables are ordinal the models are derived from a reparameterization of the saturated model based on singular value decompositions of interaction terms in the log linear parameterization multivariate association models given in goodman 1979 1986 clogg 1982a b agresti and kezouh 1983 gilula and haberman 1988 and becker and clogg 1989 are obtained as special cases the utility of the models developed in this article is demonstrated with the analysis of two examples the first example a cross classification of

the analysis of three way contingency tables by three mode association models therc m association model goodman 1979 1985 1986 1991 is useful for analyzing the relationship between the variables of a 2 way cross classification the models presented here are generalizations of therc m association model for 3 way tables the family of models proposed here 3 mode association models use tucker s 3 mode components model tucker 1964 1966 kroonenberg 1983 to represent either the three factor interaction or the combined effects of two and three factor interactions an example from a study in developmental psychology kramer gottman 1992 is provided to illustrate the usefulness of the proposed models

identification in restricted factor models and the evaluation of rank conditions abstract this paper presents a rank condition for the local identification in restricted factor analysis models and it provides a method for the computation of the exact rank in the condition the method can be applied to evaluate the rank of rather general jacobian matrices

a finite tame wild trichotomy theorem for tensor diagrams abstract in this paper we consider the problem of determining when two tensor networks are equivalent under a heterogeneous change of basis in particular to a tensor or string diagram in a certain monoidal category which we call tensor diagrams we formulate an associated abelian category of representations each representation corresponds to a tensor network on that diagram we then classify which tensor diagrams give rise to categories that are finite tame or wild in the traditional sense of representation theory for those tensor diagrams of finite and tame type we classify the indecomposable representations our main result is that a tensor diagram is wild if and only if it contains a vertex of degree at least three otherwise it is of tame or finite type

holographic algorithms without matchgates abstract the theory of holographic algorithms which are polynomial time algorithms for certain combinatorial counting problems surprised the complexity community by showing certain problems very similar to p complete problems were in fact in the class p in particular the theory produces algebraic tests for a problem to be in p in this article we describe the geometric basis of these algorithms by i replacing the construction of graph fragments in the procedure by the direct construction of a skew symmetric matrix and ii replacing the computation of weighted perfect matchings of an auxiliary graph by computing the pfaffian of the directly constructed skew symmetric matrix this procedure indicates a more geometric approach to complexity classes it also leads to more general constructions where one replaces the grassmann plucker identities which test for admissibility by other algebraic tests natural problems treatable by these methods have been previously considered in a different context and we present one such example

bayesian estimation of sparse dynamic factor models with order independent and ex post mode identification abstract common variation in n series is extracted into k n dynamic factors we induce sparsity by using a zero point mass normal mixture prior distribution on the loadings estimation and rotational identification are independent of variable ordering sparsity helps identifying the factor space and the factors rotational identification including factor order and sign is obtained by processing the posterior output and based on factor draws rather than factor loading draws simulating data we document sampler and estimation efficiency to illustrate we estimate the model for a large panel of swiss macroeconomic and detailed price data we identify 16 factors with a clear economic interpretation

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

analysis of sets of two way contingency tables using association models abstract a class of models is introduced for the analysis of group differences in the association between two discrete variables the rc m association model for two way tables is reviewed and alternative weighting systems for identifying interaction parameters are presented this model is generalized for the setting where a two way contingency table is available for two or more groups various restricted models can be used to examine possible sources of intergroup heterogeneity in the association these sources pertain to heterogeneity in the intrinsic association and or in the scores for the row and column variables the importance of weights used to identify the row and column scores is emphasized a classical set of data previously analyzed by many authors is used to illustrate the advantages of the models and methods developed here

young diagrammatic methods for the representation theory of the classical groups of type bn cn dn since the appearance of the celebrated article the classical groups of weyl 7 numerous influences have occurred in the representation theory of the classical groups especially it has become folklore that the facts in this field should be expressed only in terms of young diagrams the aim of this paper is one of the realizations of this principle a classical model of our approach is the results for gl n which are known as littlewood richardson coefficients and the kostka coefficients these two coefficients can be stated in the combinatorial manner using only young diagrams independently of the rank n of gl n we generalize these results to the other classical groups over c namely we give explicit formulas to decompose the tensor product of two irreducible representations into irreducible constituents also we have obtained an interesting relationship linking the irreducible characters of sp and so which we call duality relation between the representations of sp and so as for explicit formulas for the multiplicities of weights in the form of polynomials in the rank n we refer to 3

identification of linear stochastic models with covariance restrictions abstract the purpose of this paper is to provide a systematic treatment of the problem of identification in systems of linear structural equations where some of the disturbances are uncorrelated

on some similarities between canonical correlation models and latent class models for two way contingency tables summary relationships between parameters of canonical correlations models and latent class models for two way contingency tables are explored it is found that under certain conditions canonical models and latent models are equivalent as a result a new interpretation of canonical variables in contingency tables is proposed let p be an i x j probability matrix describing the joint distribution of two categorical random variables x and y having i and j categories respectively in canonical analysis e g fisher 1940 williams 1952 and lancaster 1958 p is decomposed into a sum of r 2 which is discussed in 3

computing the tutte polynomial of lattice path matroids using determinantal circuits we give a quantum inspired o n 4 algorithm computing the tutte polynomial of a lattice path matroid where n is the size of the ground set of the matroid furthermore this can be improved to o n 2 arithmetic operations if we evaluate the tutte polynomial on a given input fixing the values of the variables the best existing algorithm found in 2004 was o n 5 and the problem has only been known to be polynomial time since 2003 conceptually our algorithm embeds the computation in a determinant using a recently demonstrated equivalence of categories useful for counting problems such as those that appear in simulating quantum systems

ordinal association in contingency tables some interpretive aspects abstract two families of models for ordered contingency tables goodman s association models and canonical correlation models are investigated and compared with respect to the interpretation of their parameters we show that the two families of models actually refer to different kinds of ordinal association stochastic order extremity for correlation models and stochastic order entropy for association models this difference is related to the way the two models scale interaction the scale difference is proven to be of substantial consequence especially under strong association

a combinatorial proof of the giambelli identity for schur functions the giambelli identity provides a formula for expressing an arbitrary schur function of shape as a determinant of schur functions of hook shapes l 1k we give the first complete combinatorial proof of the giambelli identity we show how to derive various hook formulas from the giambelli identity and show how to extend our methods to derive extensions of the giambelli identity and the hook formula for the number of standard tableaux to certain skew shapes

modelling ordered categorical data recent advances and future challenges this article summarizes recent advances in the modelling of ordered categorical ordinal response variables we begin by reviewing some models for ordinal data introduced in the literature in the past 25 years we then survey recent extensions of these models and related methodology for special types of applications such as for repeated measurement and other forms of clustering we also survey other aspects of ordinal modelling such as small sample analyses power and sample size considerations and availability of software throughout we suggest problem areas for future research and we highlight challenges for statisticians who deal with ordinal data

double instrumental variable estimation of interaction models with big data abstract the factor analysis of a n m matrix of observations y is based on the joint spectral decomposition of the matrix squares y y and y y for principal component analysis pca for very large matrix dimensions n and m this approach has a high level of numerical complexity the big data feature suggests new estimation methods with a smaller degree of numerical complexity the double instrumental variable iv approach uses row and column instruments to estimate consistently the factors via an averaging method we compare the double iv approach to pca in terms of numerical complexity and statistical efficiency the double iv approach can be used for the analysis of recommender systems and provides a new collaborative filtering approach

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

efficient estimation and identification of simultaneous equation models with covariance restrictions this paper is a revision of an earlier version presented at the european econometric meetings 1982

grouping and association in contingency tables an exploratory canonical correlation approach abstract the criteria of homogeneity and structure were proposed by goodman 1981a for determining whether certain rows or columns of a contingency table should be grouped a data based procedure using the canonical form of bivariate distributions is presented in this article to guide exploratory analysis to determine which rows or columns of a table may be grouped this procedure facilitates the application of the homogeneity criterion relationships between the proposed method of grouping and the structural criterion are discussed as well as simultaneous inference for grouped tables the grouping method is extended to multiway tables the use of canonical forms as a model exploratory tool is addressed examples are discussed in detail

canonical analysis of contingency tables with linear constraints a generalized least squares approach is presented for incorporating linear constraints on the standardized row and column scores obtained from a canonical analysis of a contingency table the method is easy to implement and may simplify considerably the interpretation of a data matrix the approach is compared to a restricted maximum likelihood procedure

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

quantum circuits that can be simulated classically in polynomial time a model of quantum computation based on unitary matrix operations was introduced by feynman and deutsch it has been asked whether the power of this model exceeds that of classical turing machines we show here that a significant class of these quantum computations can be simulated classically in polynomial time in particular we show that two bit operations characterized by 4 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time this contrasts with the known universality of two bit operations and demonstrates that efficient quantum computation of restricted classes is reconcilable with the polynomial time turing hypothesis therefore it is possible that the techniques introduced bring the quantum computational model within the realm of algebraic complexity theory in a manner consistent with one view of quantum physics the wave function is simulated deterministically and randomization arises only in the course of making measurements the results generalize the quantum model in that they do not require the matrices to be unitary in a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read twice boolean formulae all our results are based on the use of gates that are defined in terms of their graph matching properties

the order restricted association model two estimation algorithms and issues in testing this paper presents a row column rc association model in which the estimated row and column scores are forced to be in agreement with an a priori specified ordering two efficient algorithms for finding the order restricted maximum likelihood ml estimates are proposed and their reliability under different degrees of association is investigated by a simulation study we propose testing order restricted rc models using a parametric bootstrap procedure which turns out to yield reliablep values except for situations in which the association between the two variables is very weak the use of order restricted rc models is illustrated by means of an empirical example

generalized counting constraint satisfaction problems with determinantal circuits abstract generalized counting constraint satisfaction problems include holant problems with planarity restrictions polynomial time algorithms for such problems include matchgates and matchcircuits which are based on pfaffians in particular they use gates which are expressible in terms of a vector of sub pfaffians of a skew symmetric matrix we introduce a new type of circuit based instead on determinants with seemingly different expressive power in these determinantal circuits a gate is represented by the vector of all minors of an arbitrary matrix determinantal circuits permit a different class of gates applications of these circuits include proofs of theorems from algebraic graph theory including the chung langlands formula for the number of rooted spanning forests of a graph and computing tutte polynomials of certain matroids they also give a strategy for simulating quantum circuits with closed timelike curves monoidal category theory provides a useful language for discussing such counting problems turning combinatorial restrictions into categorical properties we introduce the counting problem in monoidal categories and count preserving functors as a way to study fp subclasses of problems in settings which are generally p hard using this machinery we show that surprisingly determinantal circuits can be simulated by pfaffian circuits at quadratic cost

canonical analysis of contingency tables by maximum likelihood abstract canonical analysis has often been employed instead of log linear models to analyze the relationship of two polytomous random variables however until the last few years analysis has been informal in this article models are examined that place nontrivial restrictions on the values of the canonical parameters so that a parsimonious description of association is obtained maximum likelihood is used to obtain parameter estimates for these restricted models approximate confidence intervals are derived for parameters and chi squared tests are used to check adequacy of models the resulting models may be used to determine the appropriateness of latent class analysis or to determine whether a set of canonical scores has specified patterns results are illustrated through analysis of two tables previously analyzed in the statistical literature comparisons are made with alternate methods of analysis based on a log linear parameterization of cell probabilities it is shown that canonical analysis wh

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

identification with deficient rank loading matrices in confirmatory factor analysis multitrait multimethod models this paper presents some results on identification in multitrait multimethod mtmm confirmatory factor analysis cfa models some mtmm models are not identified when the factorial patterned loadings matrix is of deficient column rank for at least one other mtmm model identification does exist despite such deficiency it is also shown that for some mtmm cfa models howe s 1955 conditions sufficient for rotational uniqueness can fail yet the model may well be identified and rotationally unique implications of these results for cfa models in general are discussed

bayesian analysis of stochastically ordered distributions of categorical variables abstract this article considers a finite set of discrete distributions all having the same finite support the problem of interest is to assess the strength of evidence produced by sampled data for a hypothesis of a specified stochastic ordering among the underlying distributions and to estimate these distributions subject to the ordering we present a bayesian approach that is an alternative to using the posterior probability of the hypothesis and the bayes factor in favor of the hypothesis we develop computational methods for the implementation of bayesian analyses we analyze examples to illustrate inferential and computational developments the methodology used for testing a hypothesis is seen to apply to a wide class of problems in bayesian inference and has some distinct advantages

a dual of macmahon s theorem on plane partitions a classical theorem of macmahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula in this paper we present a counterpart of this formula corresponding to the exterior of a concave hexagon obtained by turning 120 after drawing each side macmahon s hexagon is obtained by turning 60 after each step

relationships among several methods of linearly constrained correspondence analysis this paper shows essential equivalences among several methods of linearly constrained correspondence analysis they include fisher s method of additive scoring hayashi s second type of quantification method ter braak s canonical correspondence analysis nishisato s type of quantification method ter braak s canonical correspondence analysis nishisato s anova of categorical data correspondence analysis of manipulated contingency tables bockenholt and bockenholt s least squares canonical analysis with linear constraints and van der heijden and meijerink s zero average restrictions these methods fall into one of two classes of methods corresponding to two alternative ways of imposing linear constraints the reparametrization method and the null space method a connection between the two is established through khatri s lemma

principal components estimation and identification of static factors a b s t r a c t it is known that the principal component estimates of the factors and the loadings are rotations of the un derlying latent factors and loadings we study conditions under which the latent factors can be estimated asymptotically without rotation we derive the limiting distributions for the estimated factors and factor loadings when n and t are large and make precise how identification of the factors affects inference based on factor augmented regressions we also consider factor models with additive individual and time effects the asymptotic analysis can be modified to analyze identification schemes not considered in this analysis

a note on the identification of restricted factor loading matrices it is shown that problems of rotational equivalence of restricted factor loading matrices in orthogonal factor analysis are equivalent to problems of identification in simultaneous equations systems with covariance restrictions a necessary under a regularity assumption and sufficient condition for local uniqueness is given and a counterexample is provided to a theorem by j algina concerning necessary and sufficient conditions for global uniqueness

the analysis of multivariate contingency tables by restricted canonical and restricted association models abstract restricted canonical models and restricted association models are proposed and applied to multiway contingency tables these models have been previously applied to two way contingency tables however multivariate generalization has been impeded in the past since canonical and association models both depend on singular value decompositions that apply only to two way arrays in this article this restriction to two way arrays is overcome by division of the cross classified variables into explanatory and response variables the explanatory variables are treated as a single polytomous variable and the response variables are treated as a second single polytomous variable in this fashion the multiway table is reduced to a two way array to which traditional canonical and association models may be applied use of linear restrictions on parameters in canonical and association models is especially important in multiway tables if useful models are to be constructed the class of models considered in th

computation of maximum likelihood estimates in association models abstract in association models for cross classified data computation of maximum likelihood estimates mle s is relatively difficult due to the nonlinear constraints on the parameters currently available procedures based on the scoring algorithm for constrained maximum likelihood are relatively unreliable and other cyclic procedures are relatively slow and do not provide estimated asymptotic standard deviations as by products of calculations to facilitate computations it is noted that in standard association models removal of constraints results in underidentification of parameters but does not affect the model itself so that the mle s of cell probabilities and conditional probabilities are unaffected given this observation maximum likelihood estimation may be accomplished by unconstrained maximization of an objective function with two components a log likelihood ratio and a sum of squares representing deviations of parameters from their constraints the objective function is then maximized by u

some models for the analysis of association in multiway cross classifications having ordered categories abstract goodman recently presented a class of models for the analysis of association between two discrete ordinal variables the association was measured in terms of the odds ratios in 2 2 subtables formed from adjacent rows and adjacent columns of the cross classification and models were devised that allowed the odds ratios to depend on an overall effect on row effects on column effects and on other effects this article presents some generalizations of this approach appropriate for multiway cross classifications including a models for the analysis of conditional association b models for the analysis of partial association and c models for the analysis of symmetric association three cross classifications are analyzed with these models and methods and rather simple interpretations of the association in each are provided

extended rc association models allowing for order restrictions and marginal modeling in the context of two way contingency tables the rc model may be seen as a way of modeling the set of log odds ratios for adjacent 2 2 subtables through row and column scores this article presents an extended class of rc models that allows simultaneous modeling of the marginal distributions and the association between two categorical variables these components are parameterized with logits and log odds ratios that may be of different types e g global or continuation to suit the nature of the data and the type of dependence of interest an algorithm for maximum likelihood estimation under equality and inequality constraints is presented the asymptotic distribution of the likelihood ratio statistic to test some interesting hypotheses is derived and extensions to multiway tables are outlined three examples are presented to highlight the features of the new approach

expressiveness of matchgates matchgates have been used to simulate classically in polynomial time assemblies of quantum gates in this paper it is shown that every matchgate for 2 input 2 output functions has to obey a certain set of five polynomial identities it is also shown that no matchgate can realize a nontrivial control gate for any number of inputs on the other hand it is proved that classical boolean formulae can be expressed as matchcircuits of polynomial size

on the investigation of local identifiability a counterexample an assertion that the parameters of a covariance structure are locally identified at a certain point only if the rank of the jacobian matrix at that point equals the number of parameters is shown to be false by means of a counterexample

shifted tableaux and the projective representations of symmetric groups les groupes de representation de s n classes de conjugaison algebres de clifford et la representation spin de base produits induits pour des representations spin une application caracteristique pour des caracteres de spin tableaux decales et q fonctions de schur les caracteres de spin irreductibles de s n et a n une analogue decalee de la regle lr produits tensoriels interieurs

reduced rank models for contingency tables summary reduced rank models for the analysis of two way contingency tables are introduced two classes of reduced rank models are discerned with well known exponents canonical analysis and latent class analysis the relation between these two classes is discussed results on the subject mentioned earlier in the literature are shown to be either redundant or inaccurate