Windows 95 transformed the PC software market and established Microsoft as the dominant player. Can Microsoft do it again? Can it use the release of Windows 8 to elevate its entire brand image? More importantly to some of us, can Microsoft use statistical modeling to help it achieve its goal?
Companies, like Microsoft, with a product portfolio seek some degree of synergy from being able to provide a comprehensive solution rather than a single product. They want the “good” achieved with one product (e.g., Xbox Kinect) to transfer to the brand first and then filter out to the other products it offers (e.g., MS Office). Can statistical modeling help? Can we collect brand ratings from customers and model these connections between the product and the brand and then between the brand and other products?
To be clear, I am not suggesting that we ever ask customers to rate Microsoft. Brand image is a latent construct that cannot be directly measured. Of course, there is nothing to stop us from asking customers to rate Microsoft, but we have no idea what they are rating. Perhaps with a less wellknown company with fewer products, we might believe that customers have no information except what they have learned from using the company’s products. But this is not the case with Microsoft. What are customers thinking about when they provide Microsoft ratings? We cannot just allow them to fill in the “blanks” and make their own decision about what to include under the Microsoft label. Do some respondents focus on the corporate side of Microsoft appearing in the news and the business reports? Are others thinking about productivity software, or the operating system, or email and internet explorer? Do gamers include Xbox? What products are businesses including under the Microsoft heading? And by businesses, do we mean owners, decision makers, or IT administers?
We are going to be looking at a correlation matrix. Often we will be examining a very large correlation matrix with lots of ratings and lots of products. In order to keep it simple in this introduction, let us suppose that we asked respondents to rate three different Microsoft products on three different brand image items. This would yield a 9×9 correlation matrix. Can we see an underlying structure that would account for the pattern of correlations in this matrix?
A1

A2

A3

B1

B2

B3

C1

C2

C3


A1

1.00

0.64

0.64

0.00

0.00

0.00

0.00

0.00

0.00

A2

0.64

1.00

0.64

0.00

0.00

0.00

0.00

0.00

0.00

A3

0.64

0.64

1.00

0.00

0.00

0.00

0.00

0.00

0.00

B1

0.00

0.00

0.00

1.00

0.64

0.64

0.00

0.00

0.00

B2

0.00

0.00

0.00

0.64

1.00

0.64

0.00

0.00

0.00

B3

0.00

0.00

0.00

0.64

0.64

1.00

0.00

0.00

0.00

C1

0.00

0.00

0.00

0.00

0.00

0.00

1.00

0.64

0.64

C2

0.00

0.00

0.00

0.00

0.00

0.00

0.64

1.00

0.64

C3

0.00

0.00

0.00

0.00

0.00

0.00

0.64

0.64

1.00

A1

A2

A3

B1

B2

B3

C1

C2

C3


A1

1.00

0.59

0.53

0.48

0.41

0.34

0.48

0.41

0.34

A2

0.59

1.00

0.49

0.41

0.35

0.29

0.41

0.35

0.29

A3

0.53

0.49

1.00

0.35

0.30

0.25

0.34

0.29

0.24

B1

0.48

0.41

0.35

1.00

0.67

0.61

0.47

0.40

0.33

B2

0.41

0.35

0.30

0.67

1.00

0.58

0.40

0.34

0.28

B3

0.34

0.29

0.25

0.61

0.58

1.00

0.33

0.28

0.24

C1

0.48

0.41

0.34

0.47

0.40

0.33

1.00

0.77

0.72

C2

0.41

0.35

0.29

0.40

0.34

0.28

0.77

1.00

0.69

C3

0.34

0.29

0.24

0.33

0.28

0.24

0.72

0.69

1.00

We can agree that there are just too many correlations to examine without a factor analytic model to uncover the underlying structure. But, the underlying structure that we are proposing has a very specific set of constraints. We need to run a bifactor model because we believe that these correlations are the outcome of two independent components: a general brand effect impacting all the ratings and separate more specific product effects isolated to each product. If you wish to learn more, I have written two earlier posts describing the bifactor model in more detail:
Halo Effects and Multicollinearity: Separating the General from the Specific
Structural Equation Modeling: Separating the General from the Specific (Part II)
If the bifactor model works, then the path coefficients in this diagram ought to reflect what we observed in the correlation matrix. There are three specific product factors (F1*, F2*, and F3*) and one general brand latent variable (g). The path coefficients for Product C are consistently higher than the path coefficients for Product B, which in turn, are consistently higher than the path coefficients for Product A. This is what we found in the product blocks along the main diagonal of our second correlation matrix. The Product A intercorrelations are in the 0.50’s. The Product B intercorrelations tend to be in the 0.60’s (one exception at 0.58). The Product C intercorrelations range from 0.69 to 0.77. In addition, the general brand effect (labeled “g”) is responsible for some level of correlation among all the ratings. In fact, it is the only reason why ratings from different products are correlated. These path coefficients from the g latent variable seem to be larger for the ratings of the first item, as we noted when we looked at the correlation matrix (these were the higher correlations in red).
However, we do see some differences in the gcoefficients for the three ratings. Across the three products, the first rating has the largest gcoefficient. If this first rating item measured innovation, for example, we would argue that innovation is more central to the brand image and where we might get our best return, all less being equal. That is, improving the innovation perception for Product C would work its way backward to the general brand perception and thus increase all the ratings for all the products proportional to their gcoefficients.
Finally, you can see the advantages of using the bifactor modeling approach. We are able to make specific recommendations about individual ratings for each product in Microsoft’s portfolio. Where should Microsoft focus its efforts? They should focus on those ratings with the highest centrality – a measure from network analysis indicating the ability of that rating to have the most impact on the greatest number of other ratings.
Appendix (R code to run this analysis)
library(psych)
loadings < matrix(c(
.70, .40, .00, .00,
.60, .43, .00, .00,
.50, .45, .00, .00,
.69, .00, .50, .00,
.59, .00, .53, .00,
.49, .00, .55, .00,
.68, .00, .00, .60,
.58, .00, .00, .63,
.48, .00, .00, .65),
nrow=9,ncol=4, byrow=TRUE)
cor_matrix<loadings %*% t(loadings)
diag(cor_matrix)<1
cor_matrix
R<data.frame(cor_matrix)
names(R)<c("A1","A2","A3","B1","B2","B3","C1","C2","C3")
R
m<omega(R)
momega.diagram(m, digits=2, main="Bifactor Structure Underlying Brand Image")
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