Exploratory factor analysis: rotation (20)

Chair: Michel van del Velden, Wednesday 22nd July, 9.55 - 11.15, Uppercroft, School of Pythagoras. 

Vartan Choulakian, Dépt de Math/Statistique, Université de Moncton, Canada. Some notes on Maxbet. (056)

Kohei Adachi, Graduate School of Human Sciences, Osaka University, Japan. Permutimin: Factor rotation to specified simple structure with least squares permutation. (029)

Doyo Gragn and Nickolay T. Trendafilov, Department of Mathematics and Statistics, The Open University, Milton Keynes, UK. Penalized varimax. (038) ♥

Jyun-Ji Lin, Department of Psychology, National Chung Cheng University,Taiwan, and Wen-Chung Wang, The Hong Kong Institute of Education, Hong Kong. Applying parallel analysis to full-information factor analysis for binary data. (164)

ABSTRACTS 

Some notes on Maxbet. (056)
Vartan Choulakian
The aim of this talk is to discuss the multivariate eigenvalue problem via algebraic methods that arises in the Maxbet method. The algebraic methods that we shall use are Gröbner bases and resultants.

Permutimin: Factor rotation to specified simple structure with least squares permutation. (029)
Kohei Adachi
On the assumption that ideally simple structure can be specified by a p-variables × m-factors binary matrix B, I consider cases where it is unknown how the elements of B correspond to those of the p × m loading matrix A to be rotated. For giving the simple structure in B to the elements of A, I propose an oblique rotation technique called permutimin, in which the rotation of A and the permutation of its rows are performed jointly. The loss function of permutimin is written as LS(P, C, T) = ||B•C−PAT||2 with • denoting the Hadamard product and P a p × p permutation matrix. This function is minimized over P, C, and T subject to diag (T−1T′−1) = Ip, which can be attained by alternately minimizing LS(P, C, T) over each of P, C, and T with the other two matrices fixed. If P is fixed to the identity matrix and B is unknown except the number of zero elements, permutimin would become equivalent to the simplimax rotation. A feature unique to permutimin is to use a least squares permutation procedure for the rotation toward simple structure. A simulation study and real data analysis are reported.

Penalized varimax. (038)
Doyo Gragn and Nickolay T. Trendafilov
A common weakness of all analytical methods for simple structure rotation is that the rotated factors are usually unequally loaded, which may spoil their interpretation. In this paper, a modified varimax criterion is introduced by attaching a penalty term to the original varimax objective function. The penalty term explicitly controls the size of the column sums of squared loadings. As a result, the penalized varimax solution has equal sum of squared loadings for all factors. Two algorithms, a matrix and a sequential one, are proposed for obtaining penalized varimax solutions. The matrix algorithm directly finds an orthogonal rotation matrix to produce the penalized varimax solution. The sequential algorithm finds the orthogonal rotation matrix column by column, making use of the fact that the varimax objective function is separable. Numerical experiments show that the sequential algorithm is more precise than the matrix one. If the penalty term is switched off, the sequential algorithm simply turns into a new efficient way to perform standard varimax rotation. The method is applied on two benchmark data sets and the results are compared to the classical varimax solutions.

Applying parallel analysis to full-information factor analysis for binary data. (164)
Jyun-Ji Lin and Wen-Chung Wang
Determination of the number of factors plays an important role in exploratory factor analysis. In binary data, Bock, Gibbons and Muraki (1988) have developed the full-information factor analysis, which computed item difficulty and item discrimination and then transforms them into factor loading. This approach does not require the calculation of inter-item correlation coefficients. This study attempts to compare the performance of the traditional tetrachoric correlation factor analysis and the full-information factor analysis in binary data combined with parallel analysis. Two simulation studies were conducted, one with uni-threshold items and the other with multi-threshold items. The examinees were sampling from N(0,1). A one-factor model with 20 items was generated. In the uni-threshold situation, the following independent variables were manipulated: (a) estimation method (full-information and tetrachoric correlation), (b) method of determination of the number of factors (parallel analysis with mean eigenvalues,95th percentile eigenvalues, 99th percentile eigenvalues, and eigenvalue >1), (c) sample size, (d) threshold parameters, and (e) slope parameter. In the multi-threshold situation, one factor model consisted of two thresholds, and the other independent variables were identical to those in uni-threshold situation. The results indicated that the full-information factor analysis with parallel analysis was more accurate in the identification of the uni-factor structure than the tetrachoric correlation factor analysis with parallel analysis, and the parallel analysis was superior to the method of Eigenvalue > 1.