Friday, December 4, 2015

The Topology Underlying the Brand Logo Naming Game: Unidimensional or Local Neighborhoods?

You can find the app on iTunes and Google Play. It's a game of trivial pursuits - here's the logo, now tell me the brand. Each item is scored as right or wrong, and the players must take it all very seriously for there is a Facebook page with cheat sheets for improving one's total score.

Psychometrics Sees Everything as a Test

What would a psychometrician make of such a game based on brand logo knowledge? Are we measuring one's level of consumerism ("a preoccupation with and an inclination toward buying consumer goods")? Everyone knows the most popular brands, but only the most involved are familiar with logos of the less publicized products. The question for psychometrics is whether they are able to explain the logos that you can identify correctly by knowing only your level of consumption.

For example, if you were a car enthusiast, then you would be able to name all the car logos in the above table. However, if you did not drive a car or watch commercial television or read car ads in print media, you might be familiar with only the most "popular" logos (i.e., the ones that cannot be avoided because their signage is everywhere you look). We make the assumption that everyone falls somewhere between these two extremes along a consumption continuum and assess whether we can reproduce every individual pattern of answers based solely on their location on this single dimension. Shopping intensity or consumerism is the path, and logo identifications are the sensors along that path.

Specifically, if some number of N respondents played this game, it would not be difficult to rank order the 36 logos in the above table along a line stretching from 0% to 100% correct identification. Next, we examine each respondent, starting by sorting the players from those with the fewest correct identifications to those getting the most right. As shown in an earlier post, a heatmap will reveal the relationship between the ease of identifying each logo and the overall logo knowledge for each individual, as measured by their total score over all the brand logos. [The R code required to simulate the data and produce the heatmap can be found at the end of this post.]

You can begin by noting that blue is correct and red is not. Thus, the least knowledgeable players are in the top rows filled with the most red and the least blue. The logos along the x-axis are sorted by difficulty with the hardest to name on the left and the easiest on the right. In general, better players tend to know the harder logos. This is shown by the formation of a blue triangle as one scans towards the lower, right-hand corner. We call this a Guttman scale, and it suggests that both variation among the logos and the players can be described by a single dimension, which we might call logo familiarity or brand presence. However, one must be wary of suggestive names like "brand presence" for over time we forget that we are only measuring logo familiarity and not something more impactful.

Our psychometrician might have analyzed this same data using the R package ltm for latent trait modeling. A hopefully intuitive introduction to item response modeling was posted earlier on this blog. Those results could be summarized with a series of item characteristic curves displaying the relationship between the probability of answering correctly and the underlying trait, labeled ability by default.

As you see in the above plot, the items are arranged from the easiest (V1) to the hardest (V36) with the likelihood of naming the logo increasing as a logistic function of the unobserved consumerism measured as z-scores and called ability because item response theory (IRT) originated in achievement testing. These curves are simple to read and understand. A player with low consumption (e.g., a z-score near -2) has a better than even chance of identifying the most popular logos, but almost zero probability of naming any of the least familiar logos. All those probabilities move up their respective S-curves together as consumers become more involved.

In this example the function has been specified for I have plotted the item characteristics curves from the one-parameter Rasch model. However, a specific functional form is not required, and we could have used the R package KernSmoothIRT to fit a nonparametric model. The topology remains a unidimensional manifold, something similar to Hastie's principal curve in the R package princurve. Because the term has multiple meanings, I should note that I am using "topology" in a limited sense in order to refer to the shape of the data and not as in topological data analysis.

To be clear, there must be powerful forces at work to constrain logo naming to a one-dimensional continuum. Sequential skills that build on earlier achievements can often be described by a low-dimensional manifold (e.g., learning descriptive statistics before attempting inference since the latter assumes knowledge of the former). We would have needed a different model had our brands been local so that higher shopping intensity would have produced greater familiarity only for those logos available in a given locality (e.g., country-specific brands without an international presence).


The Meaning of Brand Familiarity Depends on Brand Presence in Local Markets

Now, it gets interesting. We started with players differentiated by a single parameter indicating how far they had traveled along a common consumption path. The path markers or sensors are the logos arrayed in decreasing popularity. Everyone shares a common environment with similar exposures to the same brand logos. Most have seen the McDonald's double-arcing M or the Nike swoosh because both brands have spent a considerable amount of money to buy market presence. On the other hand, Hilton's "blue H in the swirl" with less market presence would be recognized less often (fourth row and first column in the above brand logo table).

But what if market presence and thus logo popularity depended on your local neighborhood? Even international companies have differential presence in different countries, as well as varying concentration within the same country. Spending and distribution patterns by national, regional and local brands create clusters of differential market presence. Everyone does not share a common logo exposure so that each cluster requires its own brand list. That is, consumers reside in localities with varying degrees of brand presence so that two individuals with identical levels of consumption intensity or consumerism would not be familiar with the same brand logos. Consequently, we need to add a second parameter to each individual's position along a path specific to their neighborhood. The psychometrician calls this differential item functioning (DIF), and R provides a number of ways of handling the additional mixture parameter.


Overlapping Audiences in the Marketplace of Attention

You may have anticipated the next step as the topology becomes more complex. We began with one pathway marked with brand logos as our sensors. Then, we argued for a mixture model with groups of individuals living in different neighborhoods with different ordering of the brand logos. Finally, we will end by allowing consumers to belong to more than one neighborhood with whatever degree of belonging they desire. We are describing the kind of fragmentation that occurs when consumers seize control and there is more available to them than they can attend to or consider. James Webster outlines this process of audience formation in his book The Marketplace of Attention.

The topology has changed again. There are just too many brand logos, and unless it becomes a competitive game, consumers will derive diminishing returns from continuing search and they typically will stop sooner rather than later. It helps that the market comes preorganized by providers trying to make the sale. Expert reviews and word of mouth guide the search. Yet, it is the consumer who decides what to select from the seemingly endless buffet. In the process, an individual will see and remember only a subset of all possible brand logos. We need a new model - one that simultaneously sorts both rows and columns by grouping together consumers and the brand logos that they are likely to recognize.

A heatmap may help to explain what can be accomplished when we search for joint clusterings of the rows and columns (also known as biclustering). Using an R package for nonnegative matrix factorization (NMF), I will simulate a data set with such a structure and show you the heatmap. Actually, I will display two heatmaps, one without noise so that you can see the pattern and a second with the same pattern but with added noise. Hopefully, the heatmap without noise will enable you to see the same pattern in the second heatmap with additional distortions.



I kept the number of columns at 36 for comparison with the first one-dimensional heatmap that you saw toward the beginning of this post. As before, blue is one, and red is zero. We discover enclaves or silos in the first heatmap without noise (polarization). The boundaries become fuzzier with random variation (fragmentation). I should note that you can see the biclusters in both heatmaps without reordering the rows and columns only because this is how the simulator generates the data. If you wish to see how this can be done with actual data, I have provided a set of links with the code needed to run a NMF in R at the end of my post on Brand and Product Category Representation.

Finally, although we speak of NMF as a form of simultaneous clustering, the cluster membership are graded rather than all-or-none (soft vs. hard clustering). This yields a very flexible and expressive topology, which becomes clear when we review the three alternative representations presented in this post. First, we saw how some highly structured data matrices can be reproduced using a single dimension with rows and columns both located on the same continuum (IRT). Next, we asked if there might be discrete groups of rows with each row cluster having its own unique ordering of the columns (mixed IRT). Lastly, we sought a model of audience formation with rows and columns jointly collected together into blocks with graded membership for both the rows and the columns (NMF).

Knowledge is organized as a single dimension when learning is formalized within a curriculum (e.g., a course at an educational institution) or accumulative (e.g., need to know addition before one can learn multiplication). However, coevolving networks of customers and products cannot be described by any one dimension or even a finite mixture of different dimensions. The Internet creates both microgenres and fragmented audiences that require their own topology.

R Code to Produce Figures in this Post

# use psych package to simulate latent trait data
library(psych)
logos<-sim.irt(nvar=36, n=500, mod="logistic")
 
# Sort data by both item mean
# and person total score
item<-apply(logos$items,2,mean)
person<-apply(logos$items,1,sum)
logos$itemsOrd<-logos$items[order(person),order(item)]
 
# create heatmap
# may need to increase size of plots window in R studio
library(gplots)
heatmap.2(logos$itemsOrd, Rowv=FALSE, Colv=FALSE, 
          dendrogram="none", col=redblue(16), 
          key=T, keysize=1.5, density.info="none", 
          trace="none", labRow=NA)
 
library(ltm)
# two-parameter logistic model
fit<-ltm(logos$items ~ z1)
summary(fit)
 
# item characteristic curves
plot(fit)
 
# constrains slopes to be equal
fit2<-rasch(logos$items)
plot(fit2)
summary(fit2)      
 
library(NMF)
# generate a synthetic dataset with 
# 500 rows and three groupings of
# columns (1-10, 11-20, and 21-36)
n <- 500
counts <- c(10, 10, 16)
 
# no noise
V1 <- syntheticNMF(n, counts, noise=FALSE)
V1[V1>0]<-1
 
# with noise
V2 <- syntheticNMF(n, counts)
V2[V2>0]<-1
 
# produce heatmap with and without noise
heatmap.2(V1, Rowv=FALSE, Colv=FALSE, 
          dendrogram="none", col=redblue(16), 
          key=T, keysize=1.5, density.info="none", 
          trace="none", labRow=NA)
heatmap.2(V2, Rowv=FALSE, Colv=FALSE, 
          dendrogram="none", col=redblue(16), 
          key=T, keysize=1.5, density.info="none", 
          trace="none", labRow=NA)
Created by Pretty R at inside-R.org

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