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Cov( ,F) = E( F′) = 0 (pxm)
Orthogonal Factor Model with m Common Factors
X (px1) = µ (px1) + L (pxm) F (mx1) + (px1) .................................................................. (3)
µ i = mean of variable i
i = ith specific factor
F j = jth common factor
l ij = loading of the ith variable on the jth factor
The orthogonal factor model provides the covariance structure for X. By definition and using the
model equation (3) we have
∑ = Cov (X) = E (X - µ)(X - µ)′
= LE (FF′) L′ + E ( F′) L′ + LE (F ′) + ′
= LL′ + Ψ
Also by independence, Cov ( F) =E ( F′) = 0
Communalities
The important assumption of linearity is inherent in the formulation of the factor model. Hence the
portion of the variance of the i th variable contributed by the m common factors is called the i th
communality denoted by Var (X i ) = δ ii. This introduces uniqueness in definition of t-he i th specific
variance. Thus the δ ii is composed of two components as shown below.
The i th communality denoted by h 2 i is given as
δ ii = l 2 i1 + l 2 i2 + … + l 2 im + Ψ i
and δ ii = h 2 i + Ψ i , where i = 1, 2, ….., p, h 2 i = l 2 i1 + l 2 i2 + ….. + l 2 im
Principal Component Solution for the Factor Model
Principal Component Analysis (PCA)
Principal component analysis (PCA) is commonly thought of as a statistical technique for data
reduction. It helps to reduce the number of variables in an analysis by describing a series of
uncorrelated linear combinations of the variables that contain most of the variance. PCA was
introduced by Pearson (1901) and later developed by Hotelling (1933) who described the variation in a
set of multivariate data in terms of a set of uncorrelated variables.
The objective of PCA is to find unit-length linear combinations of the variables with the greatest
variance. The first principal component has a maximal overall variance. The second principal
component has maximal variance among all unit length linear combinations that are uncorrelated to the
first principal component, etc. The last principal component has the smallest variance among all unit
length linear combinations of the variables. All principal components combined contain the same
information as the original variables, but the important information is partitioned over the components
in a particular way such that the components are orthogonal, and earlier components contain more
information than later components. PCA thus conceived is just a linear transformation of the data.
The purpose of factor analysis is to describe the covariance relationship among many variables in
terms of a few underlying on observable random quantity called factors. Now with the principal
component factor analysis of the sample covariance matrix S is specified in terms of its eigenvalueeigenvector
pairs (λ 1 , 1 ), (λ 2 , 2 ),.............. (λ p , p), where λ 1 ≥ λ 2 ≥ ....... ≥ λ p . Let m < p be the
number of common factors. The principal component factor analysis of the sample correlation matrix
is obtained by starting with R in the place of S, where R is the subset of S and of a full rank. For the
principal component solution, the estimated loadings for a given factor do not change as the number of
factors is increased.
Rotating Factor Loadings
The factor loadings are obtained from the initial loadings by an orthogonal transformation while
retaining the same ability to reproduce the covariance (or correlation) matrix. An orthogonal
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