# Factor Analysis — Overview

 X1: Convenient location X2: Near home F1 F2 F3 0.954 -0.234 -0.236 0.942 0.254 0.325 0.251 0.723 -0.221 0.124 0.884 -0.251 -0.132 0.952 0.122 0.114 0.231 0.945 -0.122 0.341 0.789 0.181 -0.332 0.678 0.238 0.102 0.988

#### Exhibit 33.31 Factor analysis summarizes variables into factors. Example — supermarket shopping attributes.

Factor analysis is a generic term referring to a class of statistical methods for investigating whether a number of variables of interest are linearly related to a smaller number of unobservable factors.

The prime objective of this inter-dependence technique in marketing models (e.g., models for brand equity and customer satisfaction), is to simplify the data. Based on patterns in the data, the technique summarises numerous variables into a few factors.

For example, the 9 (n=9) variables (attributes) in Exhibit 33.31, are summarized as 3 (k=3) factors. It is assumed that each variable (X1, X2 … Xn) is linearly related to the factors (F1, F2 … Fk) as shown below:

$$X_1 = \beta_{10}+\beta_{11}F_1+\beta_{12}F_2+ … \beta_{1k}F_k + e_1$$ $$X_2 = \beta_{20}+\beta_{21}F_1+\beta_{22}F_2+ … \beta_{2k}F_k + e_2$$ $$…$$ $$X_n = \beta_{n0}+\beta_{n1}F_1+\beta_{n2}F_2+ … \beta_{nk}F_k + e_n$$

The error terms e1, e2 etc. indicate that these relationships are not exact.

The parameters β ij are referred to as loadings, i.e., β 11 is called the loading of variable X1 on factor F1.

For mathematical convenience, it is assumed that the factors are in standardized form, i.e., E(Fj) = 0 and Var(Fj) = 1. With this assumption, the variance of Xi may be computed as:

$$Var(X_i) = \beta^2_{i1}Var(F_1)+ \beta^2_{i2}Var(F_2) + … + Var(e_i)$$ $$Var(X_i) = \sum_{j=1}^k \beta^2_{ij} + Var(e_i)$$

The portion of the variance that is explained by the common factors ∑(βij2) is called the communality of the variable. The greater the communality, the better the ability of the postulated model in explaining the variable.

Factor analysis methods such as principal component analysis, seek values of the loadings that bring the estimate of the total communality as close as possible to the total of the observed variances.

#### Exhibit 33.32 Variables grouped according to the factors that they define.

Variables with high loading help define the factor. For instance, as seen from Exhibit 33.31, the variables ‘value for money’, ‘attractive promotions’ and ‘low prices’ move in concert and are associated more strongly with F2. These variables that define the same factor are usually grouped under their respective factors in shown in Exhibit 33.32.

Since loading can be interpreted like standardized regression coefficients, the factor loading is the correlation between the variable and the factor. The variable ‘convenient location’, for instance, has a correlation of 0.954 with factor F1.

There often exists some common meaning among the variables that define a factor. Factor naming is a subjective process that combines understanding of market with inspection of variables that define the factor. For instance, in Exhibit 33.32, factor F1 has been labelled ‘location’, since the variables that define it allude to proximity of store.

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