# Factor Analysis Homework

The key statistics associated with factor analysis are as follows:

**Bartlett’s test of sphericity.** Bartlett’s test of sphericity is a test statistic used to examine the hypothesis that the variables are uncorrelated in the population. In other words, the population correlation matrix is an identity matrix; each variable correlates perfectly with itself (r = I) but has no correlation with the other variables (r = 0).

**Correlation matrix.** A correlation matrix is a lower triangle matrix showing the simple correlations, r. between all possible pairs of variables included in the analysis. The diagonal elements, which are all I, are usually omitted.

**Communality.** Communality is the amount of variance a variab\e there with all the other variables being considered. This is also the proportion of variance explained by the common factors

**Eigenvalue**. The eigenvalue represents the total variance explained by each factor. Factor loadings. Factor loadings are simple correlations between the variables and the factors. Factor loading plot. A factor loading plot is a plot of the original variables using the factor loadings as coordinates.

**Factor matrix.** A factor matrix contains the factor loadings of all the variables on all the factors extracted.

**Factor scores.** Factor scores are composite scores estimated for each respondent on the derived factors

Factor scores coefficient matrix. This matrix contains the weights, or factor score coefficients, used to combine the standardized variables to obtain factor scores .

Kaiser-Meyer-Olkin (KMOj measure of sampling adequacy. The Kaiser-Meyer-Olkin (KMO) measure of samplint.(dequacy is an index used to examine the appropriateness of factor analysis. High values (between 0.5 and 1.0) indicate factor analysis is appropriate. Values below 05 imply that factor analysis may not be appropriate.

**Percentage of variance.** This is the percentage of the total variance attributed to each factor

**Residuals.** Residuals are the differences between the observed correlations, as given in the input correlation matrix, and the reproduced correlations, as estimated from the factor matrix.

**Scree plot.** A scree plot is a plot of the eigenvalues against the number of factors in order of extraction.

In the next section, we describe the uses of these statistics in the context of the procedure for conducting factor analysis.

Factor analysis can also be used for preference mapping of respondents for a set of alternatives. Factor analysis-based preference-mapping technique facilitates marketers to understand the competitive structure of their markets. Based on the insight obtained from the map, they can position their offerings to achieve a favorable response from their target segment.

A preference map differs from a perceptual map by a way in which the attribute-based similarity, either derived or stated, ratings of respondents are obtained, and then consumer perception about these attributes are mapped. In preference mapping, the preference responses for each brand or alternative are obtained and these preferences are mapped. Such preference map allows the identification of consumer-preference positioning for various alternatives (brands).

This helps the marketers in identifying whom to compete against in the market and also aids in developing a positioning strategy for the new product. Like other mapping techniques, it also offers only the partial explanation of the consumers’ preference, constrained by a selected set of alternatives or brands, and does not suggest any underlying reason for preference

**Case Problem**

Consumers while buying an apparel product, particularly a business suit, focus on the brand name as that only signals both product attributes as well as symbolic values like status and esteem. With fast westernization of Indian culture and the increasing need for a formal business wear, a business suit has become more popular in India than ever before. Suppose, we want to explore how different business-suit brands are positioned based on consumers’ preference, we can use factor analysis-based preference mapping. To demonstrate this, we have taken eight business-suit brands (Peter England, Van Heusen, Koutons, Raymond, Reid & Taylor, Siyaram’s, Louis Philippe, and Blackberry), and these brands are rated on a preference scale of 1-5 (1 = Not at all preferred; 5 = Very much preferred). This is a direct approa~h of.collecting the preference data for a set of brands. For illustration, we have taken a sample of 40 students studying in an Indian premier business school to rate the business-suit brands on preference. The retained input data are shown in Table 21.4

Sample Data Set

**Procedure for Plotting Preference Map**

Table 21.5 presents the detailed principal-component-analysis result of preference rating of eight selected brands by 40 respondents. Bartlett’s test of sphericity suggests that factor analysis can be done as brand preferences are significantly correlated. The value of Kaiser-Meyer- measure of sampling adequacy is 0.583, which is more than 0.5, indicating a correlation among the slated brand preferences, and can be explained by some factors. Factor-analysis result using varimax rotation (Table 21.5) has identified three factors of brand preference (Eigenvalues of these three factors are greater than I). These three identified factors can explain eight different brands on consumer preference by forming three distinct groups and also explain 65.73% of variance of brand preference. After identifying the factors, the next step is to estimate the factor scores of the individual respondents. To estimate an individual’s factor score on a particular factor, the individual’s preference rating on brands are standardized, weighted by a generated factor score coefficient for “the factor under consideration, and then summed across all brands.

There are many software packages like SPSS which estimates the factor score directly. Mathematically, an individual factor score can be calculated as follows

Output of Principal-Component Analysis (Results of Principal-Component Analysis with Varimax Rotation)

Standardized Factor Scores of Respondents

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In our example, since each respondent has rated eight suit brands, he or she will have eight factor-score coefficients on each of the emerging factors, one for each brand. These factor-score coefficients are multiplied by a standardized individual-preference rating. We illustrate the factor-score calculation of the I st respondent for the 1st factor by putting the value in the factor-score-calculation equation, as follows

formula

**Interpretation of Preference Plot**

Interpretation of all joint maps must be done in two steps: first, individually maps are to be interpreted and then, interpretation of all maps should be combined to draw a meaningful insight on the consumer preference. Positions of individual respondents and brands on the maps indicate the marketing structure of the suit brands and the preference pattern of the individual respondents for different suit brands. It can be noted that each respondent and each suit brand has a unique point on the maps. The spatial proximity between the respondent and the suit brand indicates the respondent’s higher preference for that brand. The closeness among the suit brands suggests the similar consumer preference for the close brands. The relative distance of the brand’s position from the origin, along with a dimension, indicates the strength of association of the brand with the dimension. It can also be seen from the Figure 21.10 that Raymond, Reid & Taylor, and Siyaram’s are strongly associated with Fl, and Peter England and Koutons with F2. It can also be noted that Blackberry is not associated with any of these two factors. Factor-analysis result indicated that Raymond, Reid & Taylor, and Siyaram’s brands can be grouped as one factor based on consumer preference while Peter England and Koutons can be grouped as another factor. The higher density of respondents’ points around a brand indicates the higher preference for that brand. It can also be observed from Figure 21.1 0 that a more number of

Preference Map of Respondents and Brands on Factor 1 and Factor’2

Preference Map of Respondents and Brands on Factor and Factor 3

individual respondents lay around Raymond, Reid & Taylor, and Siyaram’s brands (Factor I). This indicates that a larger percentage of respondents strongly prefer the above three brands. Similarly, the plot with Fl and F3 dimension (Figure 21.11) can be interpreted. While making an inference of maps, an analysis of both the plots must be combined to make any informed decision about a brand.

Factor analysis can also be utilized to plot an attribute-based perceptual map. In this approach, each respondent rates n brands on m attributes. Brands can be positioned on the basis of factor scores and attributes on the basis of factor loading in the perceptual map. The perceptual map can be developed at an individual level as well as aggregate level, by taking the average factor score of each. of respondents. Factor analysis based Perceptual map is based on both perceived dissimilarity between brands and venation among consumers’ perceptions of brands. Therefore, it offers a richer solution. use more of the attributes, and result in more dimensions. Several past studies provide empirical evidence that factor-analysis dimensions provide more interpretative value than discriminant-analysis dimensions

**Sample Question (Part of the Questionnaire)**

Please rate your preference for the following business-suit brand:

Sample Question (Part of the Questionnaire)

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