Configuration of Measurement Variables in the Common Factor Space

Since factors in common factor space are in generally correlated, two dimensional orthogonal coordinate system on the graphical display of a personal computer is not suitable to present the observed variables in it. The reference axis is defined as the independent vector of the hyperplane constructed the other dimensions. The graphical rotation methods on the reference axis such as the Promax and the Rotoplot are reviewed. The relationships between factor axis and reference axis are also discussed

Factor Pattern, Communality, Reliability, and Factor-Trueness of the Scale Constructed by Item-Factor Analysis

Exploratory factor analysis for items is used to establish meaningful factors underlying the multi-dimensional constructs. Checking the factor pattern matrix, the scales representing the factors are constructed. Automatically, alpha coefficients are reported as the reliabilities of such scales. The purpose of this study is to demonstrate that the scale reliability can directly calculate as the communality of this scale by the factor pattern matrix and inter-factor correlation matrix of item common factors. In single factor model, the square of the factor structure of a scale calculated by the correlation between the scale and the common factor is equal to the coefficient omega for the factor-based reliability of a scale by McDonald (1999). To expand the theory on the communality of a scale in the one dimensional space of items to in the multi-oblique-factor space, the factor structures and the factor patters of common factors of a scale are defined using item factor patterns and factor correlations. The scalar of the row vector of scale factor structures multiplied by the column vector of scale factor patterns is the communality and the reliability of this scale in the item-factor space. Examples of scale communalities and coefficients of factor-trueness are demonstrated with the script of R for scale statistics in item-factor space

Methodology of Extension Factor Analysis: Using the factor structure defined as the correlations between variables and factors

In this note three kinds of the extension factor analysis were reviewed focusing on the term of the factor structure: The correlations between extended variables and factor score estimates (Dwyer, 1937; Mosier, 1938; Horn, 1973), the decomposition method of correlations among variables and factors (Gorsuch, 1997) adapting the regression component analysis (Schönemann & Steiger, 1976), and the measurement model of structural equation modeling adding the extension variables. These methods were discussed relating to the terminology of factor analysis such as the factor loading, the factor pattern, and the factor structure. The application of the extension factor analysis to evaluate the results of the item parceling constructed for the observed variables in the factor analysis or structural equation modeling was also mentioned. The R-scripts of the Gram-Schmidt triangular decomposition for the Gorsuch’ s extension method was also presented

Item Analysis Using the Correlations between Items and Latent Variable: Revisiting the Relationships among Factor Loading, Factor Structure, and Factor Pattern

In the research filed of the item analysis, Richardson (1936) proposed the correlations between items and a test for the estimates of the factor loadings in a test. Guilford (1953) proposed the correction formula to get the correlation between the item and the sum of the remaining n – 1 items. Although the total score of such items is not sufficient for the criterion of the item selection, this rational of the item analysis has been used for the selection the items for scale construction while applying the factor analysis to the same items. In this note, it is suggested that latent variable like the factor score is more suitable than such sum score for the item analysis. Revisiting the definitions or descriptions of the factor loading, the factor structure and the factor pattern by Thurstone (1935), Thomson (1939), and Holzinger & Harman (1941), the estimation method of the reliability in the context of the factor analysis is also discussed

Exploring the Patterns of Skill Development by Mixture Growth Modeling: Using the Batting Average Data on Professional Baseball Players

　Reviewing latent growth modeling for longitudinal data and some results using this methodology on career development, mixture modeling methodologies were introduced for identifying clusters of individuals following similar developmental trajectories. For the latent growth model analysis by Amos and the groupbased trajectory model analysis using SAS Traj procedure, the batting average records of Japanese professional baseball players over ten years were selected from the published offi cial records. Results of latent growth modeling demonstrated that the quadratic form trajectory model fi t the data well. Six subgroups were also clustered by the same quadratic form using the Traj. Findings of these analyses were discussed with particular reference to the utility of the group-based trajectory modeling of mixture model methodology for analyzing career development processes.　縦断的データへの潜在成長モデルとこの方法論を使ったキャリア発達についての結果を概観しながら、混合モデリングの方法論を、類似した発達軌跡に従う個人のクラスタを特定するために、紹介した。Amosによる潜在成長モデル分析とSAS Trajプロシジャを使った集団ベースの軌跡モデル分析のために、10年間を越える記録を持つ日本のプロ野球選手の打撃成績記録を公開されている公式記録から取り出した。潜在成長モデルの結果は、2次形式軌跡モデルがデータにうまく適合することを示した。6集団が、また、TRAJを使って、同じ2次形式によってクラスタ化された。これらの分析からの見いだしたことを、混合モデル方法論の集団ベースの軌跡モデル化の有用性をキャリア発達過程の解析と関連づけて議論した

因子分析によるテスト構成

日本テスト学会第3回大会 シンポジウム『心理テストの効用をめぐって－21世紀を展望する－』2005年8月15

Misuse and Artifact in Factor Analytic Research

The theory of factor analysis has been developed for incorporating mathematical　statistical theories such as the maximum likelihood method and asymptotic methods. However, there have been several instances of misuse while employing procedures for factor analysis studies. In several studies, factor analysis has been performed by deleting items exhibiting the ceiling effect or floor effect. The number of samples required for factor analysis is not well known. Kaiser-Guttman criterion cannot be applied for determining the number of factors. Furthermore, various studies have employed Scree Graphs and Parallel Analysis for the said purpose, but no definitive method exists for the same. Orthogonal rotation methods such as Varimax cannot be considered as a conclusive solution. However, Geomin has been considered as a better rotation method not only for simple structure but also for more complex factor configuration. Simple structure and bifactor structure are discussed in connection to factor rotation problem. Although there are various artifacts associated with the usage of factor analysis, this issue can be addressed by verifying factorial invariance through multi-group simultaneous analysis incorporated by SEM programs such as Mplus and R Package.因子分析の理論は、最尤法と漸近的方法のような数理統計学的理論を組み込んだ形で発展してきた。しかしながら、因子分析研究の手順にはまだ誤用がみられる。いくつかの研究において、天井効果や床効果を示す項目を削除して因子分析が行われている。因子分析に必要なサンプル数は明確ではない。因子の数を決定するためにKaiser-Guttman 基準は使うことはできない。そして、この目的でScree Graph とParallel Analysis を使用している研究は数多くあるが、そのための決定的な方法はない。Varimax のような直交回転は最終的な解と考えることはできない。しかしながら、Geomin は単純構造だけでなくより複雑な因子の布置に対しても優れた回転方法と考えられている。因子回転問題を考慮した単純構造とbifactor 構造について議論した。因子分析の使い方には多くのartifacts があるが、この問題は、Mplus やR Package などのSEMプログラムによって組み込まれた複数集団の同時分析によって因子的不変性を検証することによって対処することができる

Item factor analysis with target rotation for the one hundred and twenty items of the Yatabe Guilford Personality Inventory

The Yatabe Guilford Personality Inventory was developed for measuring the twelve traits of personality (Tsujioka, 1957). Although ten items for each trait were selected using the item analysis method (item-total correlation), some researchers have questioned the dimensionality of the one hundred and twenty items of this inventory. In this research, we conducted item factor analysis for the one hundred and twenty items using the target rotation method with Weighted Least Square Mean and Variance adjusted of Mplus. The results of twelve factors were evaluated according to the principle of factor-trueness by Cattell and Tsujioka (1964). The communality of each scale of ten items was compared with alpha coefficient as an index for estimating the reliability of the scale. By examining the sixty-six figures of two-dimensional plotting for the factor patterns of one hundred and twenty items and twelve scales, the implications of the methodology for scale development were discussed.YG性格検査は、12のパーソナリティの特性を測定するために開発された (辻岡、1957) 。各特性の10個の項目は、項目分析法 (項目－合計相関) によって選択されたにもかかわらず、一部の研究者は、この検査の120個の項目の次元性について疑問を提起した。この研究では、MplusのWeighted Least Square Mean and Variance adjustedで解を推定し、ターゲット回転法を使用して、120個の項目の項目因子分析を行った。12因子の結果を、Cattell and Tsujioka (1964) による因子的真理の原理の観点から評価した。尺度の信頼性を推定するための指標という観点から、10項目からなる尺度の共通性をアルファ係数と比較した。120項目と12尺度の因子パターンについての66個の2次元プロット図を吟味しながら、尺度開発の方法論について議論した