What are the constraints that are imposed on the parameters in the original model?

to ensure a unique solution for the model unknowns Λ and Ψ additional constraints such as e.g. Λ>Λ or Λ>Ψ−2Λ being a diagonal matrix are imposed on the parameters in the original model (Jöreskog 1977).

What is the equivalent of the Procrustes problem of maximizing trace(B>ZA?

for fixed A, minimizing (25) over B subject to B>B = Ip+k is equivalent to the Procrustes problem of maximizing trace(B>ZA) over B satisfying B>B = Ik+p.

What is the general class of WLS problems?

It can be formulated as the following general class of WLS problems (Bartholomew and Knott 1999):min Λ,Ψ||(Z>Z−ΛΛ> −Ψ2)Γ||2F , (6)where ||X||F = √ trace(X>X) denotes the Frobenius norm of a matrix X and Γ is a matrix of weights.

What is the EFA model for the five socio-economic variables?

The preprocessed measurements are collected in a 12 × 5 matrix Z. According to the classical factor analyses of the five socio-economic variables data, the best EFA model for these data is with two common factors (k = 2) (Harman 1976; SAS Institute 1990).

What is the main drawback of the decomposition models with fixed matrix parameters?

The main drawback of the decomposition models with fixed matrix parameters is that it is not possible to test them by statistical methods.

What is the way to solve the Procrustes problems?

For interpretational reasons and to avoid this rotational indeterminacy one can look for lower triangular matrix of factor loadings Λ (Anderson and Rubin 1956), with a triangle of k(k − 1)/2 zeros, and thus obtain an alternative solution of the Procrustes problems discussed above.

Open AccessJournal Article10.1111/J.1751-5823.2010.00120.X

Simultaneous Parameter Estimation in Exploratory Factor Analysis: An Expository Review

Steffen Unkel, +1 more

- 01 Dec 2010

- International Statistical Review

- Vol. 78, Iss: 3, pp 363-382

TL;DR: A review of methods for simultaneous parameter estimation in exploratory factor analysis can be found in this article, where Harman's five socioeconomic variables data and a high-dimensional data set from genome research are compared.

Abstract: The classical exploratory factor analysis (EFA) finds estimates for the factor loadings matrix and the matrix of unique factor variances which give the best fit to the sample correlation matrix with respect to some goodness-of-fit criterion. Common factor scores can be obtained as a function of these estimates and the data. Alternatively to the classical EFA, the EFA model can be fitted directly to the data which yields factor loadings and common factor scores simultaneously. Recently, new algorithms were introduced for the simultaneous least squares estimation of all EFA model unknowns. The new methods are based on the numerical procedure for singular value decomposition of matrices and work equally well when the number of variables exceeds the number of observations. This paper provides an account that is intended as an expository review of methods for simultaneous parameter estimation in EFA. The methods are illustrated on Harman's five socio-economic variables data and a high-dimensional data set from genome research.

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Most frequently asked questions

1. What are the contributions mentioned in the paper "Simultaneous parameter estimation in exploratory factor analysis: an expository review" ?

This paper provides an account that is intended as an expository review of methods for simultaneous parameter estimation in EFA.

2. What is the simplest way to minimize the residuals?

Minimizing (24) amounts to minimizing the sum of the squares of the residuals defined as the differences between the observed and predicted standardized values of the scores on the variables.

3. What is the LS solution for estimating F,,U simultaneously?

LS solutions for estimating {F,Λ,U,Ψ} simultaneously are obtained by minimizing FDeL in (24), making use of the iterative algorithm of De Leeuw (2004, 2008) presented in Section 4.

4. What is the standard numerical solution of the EFA problems?

The standard numerical solutions of the optimization problems (5) and (6) are iterative, often based on a Newton-Raphson procedure (Jöreskog 1977; Magnus and Neudecker 1988).

Figure 2: Initial scores of the 62 cells on the 1st common factor obtained from EFA of the lymphoma data of Alizadeh et al. (2000) (k = 5).

Table 1: Standard LS solutions for Harman’s five socio-economic variables data.

Table 2: Simultaneous EFA solutions for Harman’s five socio-economic variables data.

Figure 1: Two EFA solutions for Harman’s five-socio-economic variables data: LT solution, Table 2 (solid lines), EVD solution, Table 1 (dotted lines).

Figure 3: Bar plot of the initial loadings of the 4026 genes on the 1st common factor obtained from EFA of the lymphoma data of Alizadeh et al. (2000) (k = 5).

Table 3: Common factor scores for Harman’s five socio-economic variables data.

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Nickolay T. Trendafilov, +2 more

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References

Journal Article•10.1111/J.2517-6161.1977.TB01600.X

Maximum likelihood from incomplete data via the EM algorithm

Arthur P. Dempster, +2 more

- 01 Sep 1977

- Journal of the royal statistical society...

55.2K

•Book

The Elements of Statistical Learning: Data Mining, Inference, and Prediction

Trevor Hastie, +2 more

- 28 Jul 2013

TL;DR: In this paper, the authors describe the important ideas in these areas in a common conceptual framework, and the emphasis is on concepts rather than mathematics, with a liberal use of color graphics.

...read moreread less

21.3K

The elements of statistical learning. 2001

Trevor Hastie, +2 more

- 01 Jan 2001

17.2K

Reference Entry•10.1002/0470013192.BSA501

Principal Component Analysis

Ian T. Jolliffe

- 15 Oct 2005

TL;DR: Principal component analysis (PCA) as discussed by the authors replaces the p original variables by a smaller number, q, of derived variables, the principal components, which are linear combinations of the original variables.

...read moreread less

15.1K

Journal Article•10.1038/35000501

Distinct types of diffuse large B-cell lymphoma identified by gene expression profiling

Ash A. Alizadeh, +29 more

- 03 Feb 2000

- Nature

TL;DR: It is shown that there is diversity in gene expression among the tumours of DLBCL patients, apparently reflecting the variation in tumour proliferation rate, host response and differentiation state of the tumour.

...read moreread less

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...

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Simultaneous Parameter Estimation in Exploratory Factor Analysis: An Expository Review

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AI Agents for this Paper

Most frequently asked questions

1. What are the contributions mentioned in the paper "Simultaneous parameter estimation in exploratory factor analysis: an expository review" ?

2. What is the simplest way to minimize the residuals?

3. What is the LS solution for estimating F,,U simultaneously?

4. What is the standard numerical solution of the EFA problems?

5. What are the constraints that are imposed on the parameters in the original model?

6. What is the equivalent of the Procrustes problem of maximizing trace(B>ZA?

7. What is the general class of WLS problems?

8. What is the EFA model for the five socio-economic variables?

9. What is the main drawback of the decomposition models with fixed matrix parameters?

10. What is the way to solve the Procrustes problems?

Figures

Citations

Multiple factor analysis.

Oil security revisited: An assessment based on complex network analysis

Exploratory factor and principal component analyses: some new aspects

A new method for simultaneous estimation of the factor model parameters, factor scores, and unique parts

Some Mathematical Properties of the Matrix Decomposition Solution in Factor Analysis.

References

Maximum likelihood from incomplete data via the EM algorithm

The Elements of Statistical Learning: Data Mining, Inference, and Prediction

The elements of statistical learning. 2001

Principal Component Analysis

Distinct types of diffuse large B-cell lymphoma identified by gene expression profiling

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