Factor Loading Matrix Research Articles

Factor analysis for gene regulatory networks and transcription factor activity profiles.

BackgroundMost existing algorithms for the inference of the structure of gene regulatory networks from gene expression data assume that the activity levels of transcription factors (TFs) are proportional to their mRNA levels. This assumption is invalid for most biological systems. However, one might be able to reconstruct unobserved activity profiles of TFs from the expression profiles of target genes. A simple model is a two-layer network with unobserved TF variables in the first layer and observed gene expression variables in the second layer. TFs are connected to regulated genes by weighted edges. The weights, known as factor loadings, indicate the strength and direction of regulation. Of particular interest are methods that produce sparse networks, networks with few edges, since it is known that most genes are regulated by only a small number of TFs, and most TFs regulate only a small number of genes.ResultsIn this paper, we explore the performance of five factor analysis algorithms, Bayesian as well as classical, on problems with biological context using both simulated and real data. Factor analysis (FA) models are used in order to describe a larger number of observed variables by a smaller number of unobserved variables, the factors, whereby all correlation between observed variables is explained by common factors. Bayesian FA methods allow one to infer sparse networks by enforcing sparsity through priors. In contrast, in the classical FA, matrix rotation methods are used to enforce sparsity and thus to increase the interpretability of the inferred factor loadings matrix. However, we also show that Bayesian FA models that do not impose sparsity through the priors can still be used for the reconstruction of a gene regulatory network if applied in conjunction with matrix rotation methods. Finally, we show the added advantage of merging the information derived from all algorithms in order to obtain a combined result.ConclusionMost of the algorithms tested are successful in reconstructing the connectivity structure as well as the TF profiles. Moreover, we demonstrate that if the underlying network is sparse it is still possible to reconstruct hidden activity profiles of TFs to some degree without prior connectivity information.

An implicit function approach to constrained optimization with applications to asymptotic expansions

In this article, an unconstrained Taylor series expansion is constructed for scalar-valued functions of vector-valued arguments that are subject to nonlinear equality constraints. The expansion is made possible by first reparameterizing the constrained argument in terms of identified and implicit parameters and then expanding the function solely in terms of the identified parameters. Matrix expressions are given for the derivatives of the function with respect to the identified parameters. The expansion is employed to construct an unconstrained Newton algorithm for optimizing the function subject to constraints. Parameters in statistical models often are estimated by solving statistical estimating equations. It is shown how the unconstrained Newton algorithm can be employed to solve constrained estimating equations. Also, the unconstrained Taylor series is adapted to construct Edgeworth expansions of scalar functions of the constrained estimators. The Edgeworth expansion is illustrated on maximum likelihood estimators in an exploratory factor analysis model in which an oblique rotation is applied after Kaiser row-normalization of the factor loading matrix. A simulation study illustrates the superiority of the two-term Edgeworth approximation compared to the asymptotic normal approximation when sampling from multivariate normal or nonnormal distributions.

A Direct Estimation Method on Analyzing Ipsative Data With Chan and Bentler's (1993) Method

Ipsative data (individual scores subject to a constant-sum constraint), suggested to minimize response bias, are sometimes observed in behavioral sciences. Chan and Bentler (1993, 1996) proposed a method to analyze ipsative data in a single-group case. Cheung and Chan (2002) extended the method to multiple-group analysis. However, these methods require tedious procedures on formulating within- and between-group constraints and recovering the parameter estimates and their standard errors. A direct estimation method, which is equivalent to Chan and Bentler's method with an alternative model specification, is proposed in this article. The 1st-order factor-analytic ipsative model in Chan and Bentler's method is reparameterized as a restricted 2nd-order factor-analytic model with fixed factor loading matrix reflecting the ipsative properties in the direct estimation method. The direct estimation method can be easily extended to test measurement invariance properties in multiple-group analysis. Issues related to ipsative models are also addressed.

Testing main effects and interactions in latent curve analysis.

A key strength of latent curve analysis (LCA) is the ability to model individual variability in rates of change as a function of 1 or more explanatory variables. The measurement of time plays a critical role because the explanatory variables multiplicatively interact with time in the prediction of the repeated measures. However, this interaction is not typically capitalized on in LCA because the measure of time is rather subtly incorporated via the factor loading matrix. The authors' goal is to demonstrate both analytically and empirically that classic techniques for probing interactions in multiple regression can be generalized to LCA. A worked example is presented, and the use of these techniques is recommended whenever estimating conditional LCAs in practice.

Frequency Domain Principal Components Estimation of Fractionally Cointegrated Processes

In this paper we study the zero frequency spectral properties of fractionally cointegrated long memory processes and introduce a new frequency domain principal components estimator of the cointegration space and the factor loading matrix for the long memory factors. We find that for fractionally differenced (fractionally) cointegrated processes the squared multiple coherence at the zero frequency is equal to one, the spectral density matrix at the zero frequency is singular, and the factor loading and cointegrating matrices can be obtained from the eigenvectors of the spectral matrix at the zero frequency, associated with the positive and zero roots, respectively. A Monte Carlo simulation reveals that the proposed principal components estimator has already good properties with relatively small sample sizes. JEL Classification: C22

Techniques of Successful Application of Factor Analysis in Software Measurement

Metrics in a software sense can be roughly divided into two groups, internal metrics and external metrics. Internal metrics can be measured directly, while external metrics are predicted using internal metrics. Factor analysis is a statistical technique which can be applied to analyze the structure of internal metrics and form a prediction model for external metrics A prediction model generated by using factor analysis needs further validation since factor analysis has a non-unique solution for the factor-loading matrix. The computational methodology to establish a model is presented as well as the validation process. This is illustrated by an empirical study of design metrics and a prediction model for design quality.

On the relations among regular, equal unique variances, and image factor analysis models

We investigate under what conditions the matrix of factor loadings from the factor analysis model with equal unique variances will give a good approximation to the matrix of factor loadings from the regular factor analysis model. We show that the two models will give similar matrices of factor loadings if Schneeweiss' condition, that the difference between the largest and the smallest value of unique variances is small relative to the sizes of the column sums of squared factor loadings, holds. Furthermore, we generalize our results and discus the conditions under which the matrix of factor loadings from the regular factor analysis model will be well approximated by the matrix of factor loadings from Joreskog's image factor analysis model. Especially, we discuss Guttman's condition (i.e., the number of variables increases without limit) for the two models to agree, in relation with the condition we have shown, and conclude that Schneeweiss' condition is a generalization of Guttman's condition. Some implications for practice are discussed.

Confirmatory Factor Analysis of the Physical Self-Efficacy Scale With a College-Aged Sample of Men and Women

The 2-factor model proposed by Ryckman, Robbins, Thornton, and Cantrell (1982) for describing responses to the Physical Self-Efficacy Scale (PSES) was tested in a sample of male and female college students (N = 313). The factor structure was tested using confirmatory factor analysis with maximum likelihood estimation in LISREL 8.20 (Jökog & Sörbom, 1996a). The two-factor model for the PSES may not represent an entirely acceptable fit to the data (x²[208, N = 313] = 410, p < .0001, root mean square error of approximation = .06 [90% confidence interval = .05-.07], standardized root mean square residual = .06, Non-Normed Fit Index = .80, Goodness of Fit Index = .89). Examination of the standardized residuals and modification indexes (MIs) in the factor-loading matrix identified possible problems with the proposed 2-factor model. The number of large standardized residuals suggested that the 2-factor model did not accurately predict the relations among several pairs of items, and the MIs suggested that some items may cross-load or even load on a different factor. Because the 2-factor model provided by Ryckman et al. (1982) may not represent an entirely satisfactory fit to the PSES responses of male and female college students, further research proposing and testing various solutions to PSES responses is necessary.

Forecasting stock returns with reference to global capital asset pricing forces

In the paper we provide new evidence on the predictability of Scandinavian stock returns, when utilizing the determinants of global capital asset pricing. Three factors are extracted by principal components factor analysis. The VARIMAX‐rotated factor loadings matrix clearly suggests the presence of geographically distinguished returns generating factors: Europe, Asia and America. The corresponding factor price series are used as driving forces for the Finnish and Swedish market returns. The results indicate that the predictability of Scandinavian stock returns is significantly improved by the world factors.

Standard errors for the class of orthomax-rotated factor loadings: Some matrix results

The partial derivative matrices of the class of orthomax-rotated factor loadings with respect to the unrotated maximum likelihood factor loadings are derived. The reported results are useful for obtaining standard errors of the orthomax-rotated factor loadings, with or without row normalization (standardization) of the initial factor loading matrix for rotation. Using a numerical example, we verify our analytic formulas by comparing the obtained standard error estimates with that from some existing methods. Some advantages of the current approach are discussed.

Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core to simple structure

SIMPLIMAX is proposed as a procedure for oblique rotation of a factor loadings matrix to simple structure. The distinguishing feature of this method is that it rotates the loading matrix so that after rotation the m smallest elements have a minimal sum of squares (where m is specified in advance). In the present paper, the SIMPLIMAX method is generalized to handle three-way arrays: Three-way SIMPLIMAX finds oblique simple structure rotations of the core matrix that results from a three-mode factor analysis. Specifically, three-way SIMPLIMAX minimizes the sum of the m smallest elements of the rotated core array. An algorithm for three-way SIMPLIMAX is presented, the performance of the algorithm is discussed, some applications are shown, and it is indicated how the method can be used to serve additional purposes. Also, it is shown how the method can be used for rotation of solutions of N-mode factor analysis, and for rotation over a subset of the modes of an N-mode core array.

Robustness of bayesian factor analysis estimates

This paper presents the result of a study of the robustness of posterior estimators of the factor loading matrix, the factor scores, and the disturbance covariance matrix (the main model parameters) in a Bayesian factor analysis with respect to variations in the values of the parameters of their prior distributions (the hyperparameter). We adopt the ε - contamination model of Berger and Berliner(1986) to generate prior distributions whose hyper-paramters reflects small variations in the elements of the uncontaminated hyperparameters, and we use directional derivatives to examine the variation of the uncontaminated estimators with respect to changes in the values of the hyperparameters, in the directions of the main model parameters. Several matrix norms are used to measure the closeness of the resulting values. We illustrate the results with a numerical example.

CONVECTIVE RAINFALL REGIONS OF PUERTO RICO

Geographical regions of covariability in hourly precipitation over the island of Puerto Rico are exposed using factor analysis. It is argued that the data are consistent with a common factor model when an orthgonal rotation is applied to the factor loading matrix. The results suggest that Puerto Rico can be divided into six regions, with each region having a similar covariance structure of summer season convective rainfall. The six regions can be grouped into a western area and an eastern area based on contrasting diurnal rainfall signatures. The study is the first step in developing improved forecast guidance for precipitation over the island. It is believed to be one of the first studies attempting geographical regionalization of precipitation on the convective scale.

Classification of pulmonary-function tests by principal component analysis

The structure of pulmonary-function data obtained from 468 randomly selected subjects was analyzed. The subjects included patients with chronic bronchitis, bronchial asthma, chronic pulmonary emphysema, diffuse panbronchiolitis, and idiopathic interstitial pneumonitis, and normal health adults. From among the many possible indices of pulmonary function, 19 were chosen and were used as variables in principal component analysis. Six significant principal components (PCs) were extracted. The first three PCs accounted for 70% of the total information and were termed "ventilation," "volume," and "diffusion." The second three PCs accounted for 17% of the information and were termed "small airway," "lower airway," and "shape." Indices of ventilatory unevenness were not separated from indices of airway obstruction, and were included in the first PC. No other, unknown PC was detected with these pulmonary-function indices. The relationship among indices is displayed in a factor loading matrix, and pulmonary-function tests are classified statistically.

A statistical approach towards a regionalization of daily rainfall in Sri Lanka

AbstractRegionalization of daily rainfall in Sri Lanka was examined using orthogonal factor analysis (OFA) based on daily rainfall data of 42 stations for a 15‐year period (1971–1985). The number of potential rainy days was computed from the original data matrix and subjected to S‐mode OFA. The first 10 orthogonal factors were shown as highly significant, explaining 65.1 per cent of the total variance of the whole data matrix, where the level of eigenvalues represented was &gt; 1.0. Noticeably, the 10 orthogonal factors clearly revealed the different homogeneous daily rainfall regions in Sri Lanka (labelled as A to J), according to the orthogonal factor high loadings matrix. Delimitation of the daily rainfall regions was carried out in accordance with the ‘variance boundaries’ resulting from the rotated factor loadings matrix. The number of potential rainy days has been identified for each region, showing an increasing trend with elevation. The probability of rainy days, wet spells and their mean length have been computed separately on a monthly and yearly basis in order to analyse the nature of the daily rainfall reliability in each region. The main factors for the daily rainfall fluctuation patterns in Sri Lanka are the typical monsoon rainfall, the dominant influence of the Inter Tropical Convergence Zone and topography.

The Asymptotic Normal Distribution of Estimators in Factor Analysis under General Conditions

Asymptotic properties of estimators for the confirmatory factor analysis model are discussed. The model is identified by restrictions on the elements of the factor loading matrix; the number of restrictions may exceed that required for identification. It is shown that a particular centering of the maximum likelihood estimator derived under assumed normality of observations yields an asymptotic normal distribution that is common to a wide class of distributions of the factor vectors and error vectors. In particular, the asymptotic covariance matrix of the factor loading estimator derived under the normal assumption is shown to be valid for the factor vectors containing a fixed part and a random part with any distribution having finite second moments and for the error vectors consisting of independent components with any distributions having finite second moments. Thus the asymptotic standard errors of the factor loading estimators computed by standard computer packages are valid for virtually any type of nonnormal factor analysis. The results are extended to certain structural equation models.

DETFAC: A program for assessing factor indeterminacy

DIAG[I-2(I+..:lt ] . (5) where 'Y represents Guttman's criterion. DETFAC calculates both of these measures. The Formulas. The algorithm used by DETFAC is quite simple. Given the p x n factor loading matrix A, wherep is the number of variables and n is the number of factors, it is relatively trivial to calculate the n x n matrix ..:l: analyst to use. With theavailability of programs forgeneratingcanonical or maximum likelihood factor solutions, these techniques have become all the more feasible. Thepurpose of thispaperis to reporton a portable program,DETFAC, forestimating the indeterminacy of each factor from a given factor loading matrix. McDonald (1974) suggested that the squaredmultiple correlation of the factor score regression equation, e2 , is the most applicable measure of factor indeterminacy. Thisvalue may be seen as the expected correlation between pairs of arbitrary factor solutions (Muliak, 1976). Some authors (Green, 1976; Maxwell, 1972) havesupported claims for theutility of e2 inassessing indeterminacy, whereas others (Elffers et al., 1978; Schonemann, 1971; Schonemann & Wang, 1972) have argued for the more stringent criterion proposed by Guttman (1955). These two measures, although conceptually quite distinct, are related by the formula DETFAC: A program for assessing factor indeterminacy

The asymptotic distribution of a goodness of fit statistic for factorial invariance

Suppose that random factor models with k factors are assumed to hold for m, p-variate populations. A model for factorial invariance has been proposed wherein the covariance or correlation matrices can be written as Σ i = LC i L′ + σ i 2 I, where C i is the covariance matrix of factor variables and L is a common factor loading matrix, i = 1,…, m. Also a goodness of fit statistic has been proposed for this model. The asymptotic distribution of this statistic is shown to be that of a quadratic form in normal variables. An approximation to this distribution is given and thus a test for goodness of fit is derived. The problem of dimension is considered and a numerical example is given to illustrate the results.

Quasi Three-Mode Principal Component Analysis — A Method for Assessing the Factor Change —

This paper describes an exploratory procedure for assessing the factor change, quasi three-mode factor analysis, which is applicable to longitudinal or multitrait-multimethod data. The model is a simplified case of Tucker’s three-mode factor analysis, and formally equivalent to Kroonenberg and de Leeuw’s Tucker 2 model. That is, it intends to decompose the standardized data matrix for each occasion into the product of three matrices, namely, the factor score matrix, the core matrix, and the factor loading matrix. It is illustrated that the patterns to which core matrices conform can distinguish the change of factor loadings from the change of factor scores. A practical least squares algorithm for the model is formulated, and a way of scalings of parameters which facilitates the interpretation of the results and the comparison with the output of other factor analytic methods is discussed. Analysis of a multitrait-multimethod matrix is attached as a numerical example.

Evaluations of Credibility of the Source of a Message

This srudy investigated subjects' perceptions of the credibility of message sources who either agreed or disagreed with them on general opinion items and particular written messages. Sixty undergraduate students enrolled in interpersonal communication classes first completed a 10-item questionnaire on campus issues using seven-point scales. The students were also asked to write short signed essays expressing their feelings about the use of bell-curve grading to combat grade inflation. The subjects' ideas served as the basis for written messages developed by the experimenter which either agreed or disagreed with the use of bell-curve grading. Subjects were then asked to rate the credibility of a hypothetical message source who either agreed or disagreed with their ratings of the opinion items and whose written position on grade inflation either agreed or disagreed with theirs. Fifteen subjects were assigned to each cell of a 2 x 2 design. Source credibility was evaluated on a 32-item scale containing items drawn from others' work (1, 2, 3 ) . An oblique factor analysis indicated a factor structure compatible with those reported by the above researchers. Factor scores were constructed from the complete factor loading matrix and used in a multivariate analysis of variance. A significant main effect for the source's written message on bell-curve grading (F4.m = 11.62, p < .001) was noted. Examination of the univariate P-ratios (based on Z scores since both Liken and semantic differential items appeared on the credibility measure) indicated that message sources were judged equally on the trustworthiness factor regardless of whether the~r written comments agreed or disagreed with the perceivers'. Significant F values were discovered for the charisma (F1.w = 11.19, p < .001), co-orientation (F1.w = 4.83, p < .03), and expertise (F1.m = 34.62, p < .001) factors. Subjects found the message source who disagreed more charismatic ( M = -1.83) than the conformer ( M = -2.60). Conversely, the agreeing message source was rated higher in co-orientation ( M = -7.02) and expertise ( M = -4.40) than the disagreeing source ( A I = -7.84 for coorientation and M = 5 32 for expertise). These findings suggest a source promoting a message that disagrees with the receiver's view is well adv~sed to expect and compensate for low evaluations in coorientation and expertise. As long as a substantive message is presented, the disagreeing source can seem reasonably trustworthy and quite charismatic. V