0412 factor nalysis

Philosophy (1/8) Greek philosophers

1. Greek rationalist and atomists - appearance is to be explained by something not observed.
2. Aristotle - induction (歸納) and seeking common features of things to explain them.

Aristotle: syllogistic deduction (三段論)

• Hence he believed that in trying to explain something, one must demonstrate that out of which it was made (material cause), the essential features of its form (formal cause), the agent that brought about that form (efficient cause), and the end (or final form) toward which the object's form was directed (final cause).

Philosophy (2/8) Bacon and auto methods

3. Bacon - auto algorithm for inductively discovering common cause.

• syllogistic logistic (三段論) as a fruitless enterprise.
• Bacon proposed a "new" method of induction that would begin without hypotheses or speculations and move to more general truths through an automatic procedure or algorithm.

Philosophy (3/8) Descartes and universal methods

4. Descartes (笛卡爾) - (1) orthogonal or linearly independent factors, (2) reproducing the corr matrix from the factors.

笛卡兒在《談談方法》中指出，研究問題的方法分四個步驟：

• 沒有經過實證的結論，都可以懷疑。這就是著名的「懷疑一切」理論。例如亞里士多德曾下結論說，女人比男人少兩顆牙齒。但事實並非如此。
• 將要研究的複雜問題，分解為多個簡單的小問題，分開解決。
• 將這些小問題從簡單到複雜排列，先從容易解決的問題著手。
• 將所有問題解決後，再綜合起來檢驗，看是否完全，是否將問題徹底解決了。

Philosophy (4/8) Kant

5. Kant - not to discard inductive inferences but to treat them as hypotheses that must test against additional data to establish objectivity.

• And so, the conclusions of exploratory factor analyses are never complete without a subsequent confirmatory analysis with additional variables and new data.

Philosophy (5/8) British scientists

Galton - regression,

Pearson - correlation

Philosophy (6/8) Spearman and common factor analysis

6. British empiricist (Galton, Pearson, Yule) - exploratory, descriptive methods.

• the inductivist fallacy (歸納謬誤)- factor analysis can not yield unique and unambiguous knowledge w/o prior assumption. —> factor indeterminacy (因素不確定性)
• two-factor” (general and specific factors) theory
• the first common-factor-analysis model

\[Y_j = a_j G + \Psi_j\]

Philosophy (7/8) Thurstone and simple structure

• simple structure
• Search for factors related to the observed variables so that each factor pertained as much as possible to one non-overlapping subset of the observed variables.
• Method of multiple-factor analysis. By this time, the latent-ability variables had come to be called “factors” owing to the usage of Spearman (1927).

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A graphic representation of the 3 oblique factors from the Thurstone data using fa.diagram. Factors were transformed to an oblique solution using oblimin.

Philosophy (8/8) Factor Indeterminancy

• It is hard to avoid indeterminacy even after factor rotations and loadings are defined. Cuz the interpretation process is inductive, and data are not the only thing for determining inductive inferences. ()

Methodology (1/6) Factor extraction

centroid method

• Until the advent of computers made the principal-axes method of factoring the preferred method, the centroid method was the most frequently used for extracting the factors. –- Burt , Thurstone (1947)
• The centroid method seeks to find a variable that is an unweighted composite of the residual variables after previous factors have been extracted.

principal axes

finding that particular combination of weights which, when applied to the scores of the tests in the test battery, determines the most representative single score. The solution of this kind of problem is the essence of the principal-axes methods of factoring.

Methodology (2/6) Rotation methods

• promax

after computer

• varimax ()
• oblimax ()

Methodology (3/6) Numereical methods

aftrer computer

• eigenvalues and eigenvectors
• derivatives of matrix
• MLE for factor analysis

Methodology (4/6) Confirmatory methods

• A primitive form of confirmatory factor analysis is multiple-group factor analysis.（但沒有受到重視） –> 最早這是一種 factor extration 的方法。
• 早期 Thurstone (1944)、Guttman (1952)等提出的 multiple group method (一種處理相關矩陣的方法)。
• “Group” here refers to “group factor” and not to “group of subjects.”

Thurstone (1945)
The result of this method of factoring is a factor matrixF which satisfies the fundamental relation \(FF'=R\).

• 這是一種 principal-axes solution（cited from Thurstone, 1944）
• 1960s 之後轉為流行 eigenvectors 與 eigenvalue 的作法。

Methodology (5/6) Confirmatory methods

exploratory factor analysis, which may be regarded as an “abductive,” “hypothesis-generating” methodology

• LISREL
• RAM

Steiger. (). General Models for Covariance Structures

Methodology (6/6) Confirmatory methods

note: EFA之後正交轉軸成 simple structure，跟 CFA 直接設定。

1) SEM

2) common factor analysis

In the common factor model:

1. the dependent variables are observed only, hence no \(\eta\) ;
2. no dependent variable is a function of other dependent variables, hence no \(A\begin{bmatrix} \eta\\y \end{bmatrix}\).
3. there are no manifest exogenous (外生) variables, hence no \(x\); and
4. since there are no endogenous (內生) variables that are functions of other endogenous variables, there are no \(\zeta\) and thus no \(\Delta\).