Statistical Models That Support Design Thinking: Driver Analysis vs. Partial Correlation Networks

We have been talking about design thinking in marketing since Tim Brown's Harvard Business Review article in 2008. It might be easy for the data scientist to dismiss the approach as merely a type of brainstorming for new products or services. Yet, design issues do arise in data visualization where we are concerned with communicating our findings. However, my interest is model selection: Should the analyst select one statistical model over another because the user might find it more helpful in planning interventions or designing new products and services?

For example, the marketing manager who wants to retain current customers seeks guidance from customer satisfaction questionnaires filled with performance ratings and intentions to recommend or purchase again. Motivated by the desire to keep it simple, common practice tends to focus attention on only the most important "causes" of customer retention. As I noted in my first post, Network Visualization of Key Driver Analysis, a more complete picture can be revealed by a correlation graph displaying all the interconnections among all the ratings. The edges or links are colored green or red so that we know if the relationship is positive or negative. The thickest of the path indicates the strength of the correlation. But correlations measure total effects, both those that are direct and those obtained through associations with other ratings.

The designer of intervention strategies aimed at preventing churn could acquire additional insights from the partial correlation graph depicting the effects between all pairs of ratings controlling for all the other ratings in the model. While the correlation map reveals total effects, the partial correlation map removes all but the direct effects. The graph below was created using the R code from my first post to simulate a data set that mimics what is often found when airline passengers complete satisfaction surveys. Once the data were generated, the procedures outlined in my post Undirected Graphs When the Causality is Mutual were followed. The R code is listed at the end of this discussion.

We can pick any node, such as the one labeled "Satisfaction" in the middle of the right-hand side of the figure. A simple way of interpreting this graph is to think of Satisfaction as the dependent variable and the lines radiating from Satisfaction as the weights obtained from the regression of this node on the other 14 ratings. Clearly, overall satisfaction serves as an inclusive summary measure with so many pathways from so many other nodes. Each of the four customer service ratings (below Satisfaction and in light pink) adds its own unique contribution with the greatest impact indicated by the thickest green edge from the Service node. Moreover, Easy Reservation and Ticket Price plus Clean Aircraft with room for people and baggage make incremental improvements in Satisfaction.

The same process can be repeated for any node. Instead of a driver analysis that narrows our thinking to a single dependent variable and its highest regression weights, the partial correlation map opens us to the possibilities. If the goal was customer retention, then the focus would be on the Fly Again node. Recommend seems to have the strongest link to Fly Again. Can the airline induce repeat purchase by encouraging recommendation? What if frequent flyer miles were offered when others entered your name as a recommender? Such a proposal may not be practical in its current form, but the graph supports this type of design thinking.

Because there are no direct paths from the four service nodes to Fly Again, a driver analysis would miss the indirect connection through Satisfaction. And what of this link between Courtesy and Easy Reservation? Do customers infer a "friendly" personality trait that links their perceptions of the way they are treated when they buy a ticket and when they board the plane? Design thinkers would entertain such a possibility and test the hypothesis. Such "cascaded inferences" fill the graph for those willing to look. Perhaps many small and less costly improvements might combine to have a greater impact than concentrating on a single aspect? Encouraging passengers to check their bags would create more overhead storage without reconfiguring the airplane. Let the design thinking begin!

The discussion ends with the identification of "the most important" in a driver analysis. The network, on the other hand, invites creative thought. Isn't this the point of data science? What can we learn from the data? The answer is a good deal more than can be revealed by the largest coefficient in a single regression equation.

# Calculates Sparse Partial Correlation Matrix
sparse_matrix<-EBICglasso(cor(ratings), n=1000)
round(sparse_matrix,2)
 
# Plots results
gr<-list(1:4,5:8,9:12,13:15)
node_color<-c("lightgoldenrod","lightgreen","lightpink","cyan")
qgraph(sparse_matrix, fade = FALSE, layout="spring", groups=gr, 
       color=node_color,labels=names(ratings), label.scale=FALSE, 
       label.cex=1, node.width=.5, edge.width=.25, minimum=.05)

Created by Pretty R at inside-R.org

Mutually Exclusive Clusters Are Boxes within Which Consumers No Longer Fit

Sometimes we force our categories to be mutually exclusive and exhaustive even as the boundaries are blurring rapidly.

Of course, I am speaking of cluster analysis and whether it makes sense to force everyone into one and only one of a set of discrete boxes. Diversity is diverse and requires a more expressive representation than possible in a game of twenty questions. "Is it this or that?" is inadequate when it is a little of this and a lot of that.

How do you classify a love seat? Is it a small sofa or a large chair for two people? Natural categories are not defined by all-or-none criteria. All birds do not possess the same degree of "birdness" (as shown below for the graded structure underlying the classification of birds). Some birds are more "bird" than other birds, and some mammals might be thought of as birds (bats) because they look and behave more like birds than typical mammals.

In an earlier post, I issued a warning that clusters may appear more separated in textbooks than in practice. I urged that we consider other representations for individual variation. Archetypes work because they reside at the periphery of the objects to be described with all the other species in-between (e.g., birds of prey, household pets in cages, winter migrants, evil dark birds, and white birds of peace). In marketing this is the realm of fans, fanatics and -philes in which it is so easy to visualize the extreme users and name everyone else as hybrid combinations of pure types. R makes the analysis doable. Moreover, with dimensions defined as contrasting ideals (liberal vs. conservative), archetypal analysis mimics hidden dimensions such as those from factor analysis and item response theory.

The Forces behind Diversity Do Not Yield Disjoint Clusters

I have found it best to begin with the forces generating diversity. Why doesn't one size fit all? As a marketer, I look toward consumer demand - whether it originates out of individual usage, preference or need or whether it is manufactured by providers introducing new products and services. I discover that demand is seldom contained within a single box. Some cable television viewers want to watch lots of sports, so let us place them in the sport segment. But they also want movies-on-demand, so we need four segments filling in the 2x2 for sports and movies-on-demand. Hopefully, they do not want to hear the business news because now we have 8 segments in our 2x2x2. As the consumer acquires greater control, the old segmentation scheme seems more and more forced as if we are holding onto a simpler world with everyone crammed into one of only a few silos.

Recommendation systems adopt a different metaphor with the marketplace partitioned by user collaboration and the chunking of offerings into micro-genres. Heterogeneity is seen as coevolving networks of consumers and what they buy. A handful of mutually exclusive boxes will not work in a market that is increasingly fragmenting.

When there are so many alternatives within easy reach of the internet, no consumer can attend to it all. The rows in our data matrix, each containing information from a single individual, become both longer with more options and sparser with limited attention. If we want to play in the marketplace of attention, we will need more than K-means or finite mixture models. Researchers will require even more - the type of easy access provided by R packages such as archetype and NMF.

With the appropriate statistical model, one can uncover such generating processes from the data matrix with consumers as rows and what they want or like in the columns. Using figurative language I have called this approach the "ecology of data matrices" and have suggested the need for biclustering. Yet, there is opposition since we are so accustomed to dividing the analysis of rows and columns into two separate procedures. Most cluster analyses input all the columns to calculate distances among the rows. Factor analysis starts with column correlations computed from data including every row. Biclustering, on the other hand, cares about sorting the cells into simultaneous groupings of row-column combinations. The data matrix gets divided into subspaces, possibly overlapping, with this community of consumers similar on only those groupings of variables.

The simplicity of boxes will not work with consumers in control and more available than any one buyer can attend to or know about. An underlying structure remains, but one defined by the joint interaction of rows and columns. Consumers with common needs and experiences are attract to the same purchase channels and learn about offerings from the same sources. This simultaneous clustering of rows and columns are the blocks from which consumers customized their own personal consumption patterns. Nothing forces the consumer to select only one building block. In fact, the opposite is more generally true for most of us play multiple roles (e.g., items purchased for work and play, self and others, necessities and gifts, and the list goes on). To capture such common practices, we need a clustering technique that does not impose a simplistic representation forcing consumers into boxes within which they no longer fit.

Clustering Customer Satisfaction Ratings

We run our cluster analysis with great expectations, hoping to uncover diverse segments with contrasting likes and dislikes of the brands they use. Instead, too often, our K-means analysis returns the above graph of parallel lines indicating that the pattern of high and low ratings are the same for everyone but at different overall levels. The data come from the R package semPLS and look very much like what one sees with many customer satisfaction surveys.

I will not cover any specifics about the data, but instead refer you to earlier discussions of this dataset, first in a post showing its strong one-dimensional structure using biplots and later in an example of an undirected graph or Markov network displaying brand associations.

We will begin with the mean ratings for the four lines in the above graph and include a relatively small fifth segment in the last column with a different narrative. Ordering the 23 items from lowest to highest mean scores over the entire sample makes both the table below and the graph above easier to read.

	not at all	a little	some	a lot	pricey
	9%	27%	34%	21%	10%
FairPrice	4.1	5.1	6.4	8.2	5.6
BuyAgain	3.0	6.9	8.7	9.7	3.8
Responsible	4.3	6.2	6.9	8.1	7.0
GoodValue	4.8	5.7	7.3	8.6	7.0
ComplaintHandling	4.4	6.1	7.2	8.9	7.8
Fulfilled	4.8	6.0	7.5	8.6	7.8
IsIdeal	4.6	6.2	7.7	9.0	7.8
NetworkQuality	5.6	6.2	7.4	8.4	8.0
Recommend	3.8	6.7	8.4	9.6	7.2
ClearInfo	4.6	6.5	7.9	9.2	8.6
Concerned	5.3	6.4	8.1	9.0	8.1
QualityExp	6.0	7.1	7.6	8.6	7.9
CustomerService	4.8	6.7	8.0	9.3	8.5
MeetNeedsExp	6.1	7.1	7.3	8.7	8.4
GoWrongExp	7.0	6.2	7.5	8.5	8.5
Trusted	6.1	6.6	7.8	9.1	8.4
Innovative	5.8	7.4	8.1	9.2	8.2
Reliability	6.1	6.8	7.9	9.2	8.7
RangeProdServ	6.2	7.1	8.0	9.2	8.3
Stable	7.1	6.7	7.8	9.1	8.3
OverallQuality	6.3	7.0	8.2	9.2	8.5
ServiceQuality	6.3	7.1	7.9	9.4	8.6
OverallSat	6.4	7.3	8.2	8.9	8.7

You can pick any row in this table and see that the first four segments with 90% of the customers are ordered the same. The first cluster is simply not at all happy with their mobile phone provider. They give the lowest Buy Again and Recommend ratings. In fact, with only two small exceptions, they uniformly give the lowest scores. For every row the second column is larger (note the two discrepancies already mentioned), followed by an even bigger third column, and then the most favorable fourth column. Successful brands have loyal customers, and at least one out of five customers in this data have "a lot" of love with a mean ratings of 9.7 on a 10-point scale.

You can see why I labeled these four segments with names suggesting differing levels of attraction. Each group has the same profile, as can be seen in the largely parallel lines on our graph. The good news for our providers is that only 9% are definitely at risk. The bad news is that another 10% like the product and the service but will not buy again, perhaps because the price is not perceived as fair (see their graph below with a dip for the second variable, Buy Again, and a much lower score than expected for the first variable, Fair Price, given the elevation of the rest of the curve).

Some might argue that what we are seeing is merely a measurement bias reflecting a propensity among raters to use different portions of the scale. Does this mean that 90% of the customers have identical experiences but give different ratings due to some scale-usage predisposition? If it is a personality trait, does this mean that they use the same range of scale values to rate every brand and every product? Would we have seen individuals using the same narrow range of scores had the items been more specific and more likely to show variation, for example, if they had asked about dropped calls and dead zones rather than network quality?

Given questions without any concrete referent, the uniform patterns of high and low ratings across the items are shaped by a network of interconnected perceptions resulting from a common technology and a shared usage of that technology. In addition, one overhears a good deal of discussion about the product category in the media and from word-of-mouth so that even a nonuser might be aware of the pros and cons. As a result, we tend to find a common ordering of ratings with some customers loving it all "a lot" and others "not at all." Unless customers can provide a narrative (e.g., "I like the product and service, but it costs too much"), they will all reproduce the same profile of strengths and weaknesses at varying levels of overall happiness. That is, satisfied or not, almost everyone seems to rate value and price fairness lower than they score overall quality and satisfaction.

Finally, my two prior posts cited earlier may seem to paint a somewhat contradictory picture of customer satisfaction ratings. On the one hand, we are likely to find a strong first principal component indicating the presence of a single dimension underlying all the ratings. Customer satisfaction tends to be one-dimensional so that we might expect to observe the four clusters with parallel lines of ratings. Satisfaction falls for everyone as features and services become more difficult for any brand to deliver. On the other hand, the graph of the partial correlations suggests a network of interconnected pairs of ratings after controlling for the all the remaining items. One can identify regions with stronger relationships among items measuring quality, product offering, corporate citizenship, and loyalty.

Both appear to be true. Rating with the highest partial intercorrelations form local neighborhoods with thicker edges in our undirected graph. Although some nodes are more closely related, all the variables are still connected either directly with a pairwise edge or indirectly through a separating node. Everything is correlated, but some are more correlated than others.

R code needed to reproduce these tables and plots.

library("semPLS")
data(mobi)
 
# descriptive names for graph nodes
names(mobi)<-c("QualityExp",
               "MeetNeedsExp",
               "GoWrongExp",
               "OverallSat",
               "Fulfilled",
               "IsIdeal",
               "ComplaintHandling",
               "BuyAgain",
               "SwitchForPrice",
               "Recommend",
               "Trusted",
               "Stable",
               "Responsible",
               "Concerned",
               "Innovative",
               "OverallQuality",
               "NetworkQuality",
               "CustomerService",
               "ServiceQuality",
               "RangeProdServ",
               "Reliability",
               "ClearInfo",
               "FairPrice",
               "GoodValue")
 
# kmeans with 5 cluster and 25 random starts
kcl5<-kmeans(mobi[,-9], 5, nstart=25)
 
# cluster profiles and sizes
cluster_profile<-t(kcl5$centers)
cluster_size<-kcl5$size
 
# row and column means
row_mean<-apply(cluster_profile, 1, mean)
col_mean<-apply(cluster_profile, 2, mean)
 
# Cluster profiles ordered by row means
# columns sorted so that 1-4 are increasing means
# and the last column has low only for buyagain & fairprice
# Warning: random start values likely to yield different order
sorted_profile<-cluster_profile[order(row_mean),c(4,3,1,2,5)]
 
# reordered cluster sizes and profiles
cluster_size[c(4,3,1,2,5)]/250
round(sorted_profile,2)
 
# plots for first 4 clusters
matplot(sorted_profile[,-5], type = c("b"), pch="*", lwd=3,
        xlab="23 Brand Ratings Ordered by Average for Total Sample",
        ylab="Average Ratings for Each Cluster")
title("Loves Me Little, Some, A Lot, Not At All")
 
# plot of last cluster
matplot(sorted_profile[,5], type = c("b"), pch="*", lwd=3,
        xlab="23 Brand Ratings Ordered by Average for Total Sample",
        ylab="Average Ratings for Last Cluster")
title("Got to Switch, Costs Too Much")

Created by Pretty R at inside-R.org