Friday, November 14, 2014

In Praise of Substantive Expertise in Data Science

Substantive expertise makes it into the Data Science Venn Diagram from DataCamp's infographic on how to become a data scientist. It's one of the three circles of equal size along with programming and statistics. Regrettably, substantive expertise is never mentioned in the definition of a data scientist as "someone who is better at statistics than any software engineer and better at software engineering than any statistician." And it gets no step. Statistics is the first step, and the remaining steps cover programming in all its varying forms. "Alas, poor Substance! I knew him, DataCamp."

All of this, of course, is to be taken playfully. I have no quarrel with any of DataCamp's 8-step program. I only ask that we recognize that there are three circles of equal value. Some of us come to data science with substantive expertise and seeking new models for old problems. Some even contribute libraries applying those models in their particular areas of substantive expertise. R provides a common language through which we can visit foreign disciplines and see the same statistical models from a different perspective.

John Chambers reminds us in his UseR! 2014 keynote address that R began as a "user-centric scientific software tool" providing "an interface to the very best numerical algorithms." Adding an open platform for user-submitted packages, R also becomes the interface to a diverse range of applications. This is R's unique selling proposition. It is where one goes for new ways of seeing.

Wednesday, November 12, 2014

Building Blocks: A Compelling Image for Clustering with Nonnegative Matrix Factorization (NMF)

Would hierarchical clustering be as popular without the dendrogram? Cannot the same be said of finite mixture modeling with its multidimensional spaces populated by normal distributions? I invite you to move your mouse over the figure on the introductory page of the website for the R package mclust and click through all the graphics that bring mixture modeling to life. So what is the compelling image for nonnegative matrix factorization (NMF)?

Dendrograms are constructed from distance matrices. We have some choice in the distance or divergence metric, but once a variable has been selected, it is included in all the distance calculations. Finite mixtures and k-means avoid such matrices, but still define fit as some variation on the ratio of between-cluster and within-cluster distances, once again computed from all the variables selected for the analysis. These are the clusters pictured in the above links to dendrograms and isodensity curves in low-dimensional spaces derived from the entire set of included variables. 

However, such representations do not exhaust common ways of talking and thinking about similarity. For example, object substitution in a task or an activity is based on a more limited definition of shared functionality. These are goal-derived categories that I discussed at the end of my post showing how NMF can use top-contender rankings to reveal preference patterns for breakfast foods. Will a Danish pastry be good enough when all the donuts have been eaten? The thought of eating the donut evokes the criteria upon which substitutability and hence similarity will be judged (see Norm Theory: Comparing Reality to Its Alternatives). In the context of toast and other options for breakfast, the donut and the Danish may appear more similar in contrast, yet that is not what comes to mind when hungry for donuts. Similarity can only be defined within a context, as noted by Nelson Goodman

Similarity Derived from Building Blocks in Localized Additive Representations

What did you do today? I could give you a list of activities and ask you to indicate how frequently you engaged in each activity. Who else is like you? Should we demand complete agreement across all the activities, or is it sufficient that you share some common task? If my list is a complete inventory, it will include many relatively infrequent activities performed only by specific subgroups. For example, caregivers for very young children come in all ages and gender from diverse backgrounds and with other responsibilities, yet they form a product category with an entire aisle of the supermarket dedicated to their shared needs. Situational demands pull together individuals in rows and activities in columns to mold a building block of relational data.

To be clear, a hierarchical clustering of respondents or the rows of our data matrix averages over all the columns. We start with an nxp data matrix, but perform the analysis with the nxn dissimilarity or distance matrix. Still, the data matrix contains relational data. Individuals are associated with the activities they perform. Instead of ignoring these relationships between the rows and the columns, we could seek simultaneous clustering of individuals and activities to form blocks running along the diagonal of the data matrix (a block diagonal matrix). Consequently, we may wish to alter our initial figure at the beginning of this post to be more precise and push the colored blocks out of a straight line to form a diagonal with each block demarcated by the intersection of individuals and their frequent activities.

A Toy Example

We can see how it is all accomplished in the following NMF analysis. We will begin with a blank data matrix and combine two blocks to form the final data matrix in the lower right of the figure below.

The final data matrix represents 6 respondents in the rows and 4 activities in the columns. The cells indicate the frequency of engaging in the activity ranging from 0=never to 6=daily. Since the rows and columns have been sorted into two 3x2 blocks along the diagonal, we have no problem directly interpreting this small final data matrix. The frequency of the 4 activities is greatest in the first column and least for third column. The first 3 and last 3 respondents are separated by the first 2 and last 2 activities. It appears that the 6x4 data matrix might be produced by only two latent features.

The following R code creates the final data matrix as the matrix product of respondent mixing weights and activity latent feature coefficients. That is, activities get organized into packets of stuff done by the same respondents, and respondents get clustered based on their activities. If you are familiar with the co-evolution of music genre and listening communities, you will not be surprised by the co- or bi-clustering of rows and columns into these diagonal building blocks. In a larger data matrix, however, we would expect to see both purist with nonzero mixing weights for only one latent feature and hybrids that spread their weights across several latent features. As noted in earlier posts, NMF thrives on sparsity in the data matrix especially when there is clear separation into non-overlapping blocks of rows and columns (e.g., violent action films and romantic comedies appealing to different audiences or luxury stores and discount outlets with customers tending to shop at one or the other).

# enter the data for the respondent mixing weights
MX<-matrix(c(3,2,1,0,0,0,0,0,0,1,1,2), ncol=2)
# enter the data for the latent features
LP<-matrix(c(2,1,0,0,0,0,1,2), ncol=4, byrow=TRUE)
# observed data is the matrix product
# load the NMF library
# run with rank=2
fit<-nmf(DATA, 2, "lee", nrun=20)
# output the latent feature coefficients
#output the respondent mixing weights
# reproduce the data matrix using NMF results
# output residuals
# same but only for 1st latent feature
# same but only for 2nd latent feature
# additive representation 

Created by Pretty R at

The extensive comments in this R code reduce the need for additional explanation except to emphasize that you should copy and run the code in R (after installing NMF). I did not set a seed so that the order of the two parts may be switched. The exercise is intended to imprint the building block imagery. In addition, you might wish to think about how NMF deals with differences in respondent and activity intensity. For example, the first three respondents all engage in the first two activities but with decreasing frequency. Moreover, the same latent feature is responsible for the first two activities, yet the first activity is more frequent than the second. 

I would suggest that the answer can be found in the following section of output from the above code. You must, of course, remember your matrix multiplication. The first cell in our data matrix contains a "6" formed by multiplying the first row of mx by the first column of lp or 0.5 x 12 + 0.0 x 0 = 6. Now, it is easy to see that the 0.500, 0.333 and 0.167 in mx reveal the decreasing intensity of the first latent feature. Examining the rest of mx suggest that the last respondent should have higher scores than the previous two and that is what we discover.

> round(lp,3)
      [,1] [,2] [,3] [,4]
[1,]   12    6    0    0
[2,]    0    0    4    8

> round(mx,3)
          [,1] [,2]
[1,] 0.500 0.00
[2,] 0.333 0.00
[3,] 0.167 0.00
[4,] 0.000 0.25
[5,] 0.000 0.25
[6,] 0.000 0.50

Parting Comments

When you see diagrams, such as the following from Wikipedia, you should take them literally. 

The data matrix V is reproduced approximately by a reduced rank matrix of mixing weights W multiplied by a reduced rank matrix of latent features H. These interpretations of W and H depend on V being a respondents-by-variables data matrix. One needs to be careful because many applications of NMF reverse the rows and columns changing the meaning of W and H. 

The number of columns in W and the number of rows in H can be much smaller than the number of observed variables, which is what is meant by data reduction. The same latent features are responsible for the clustering of respondents and variables. This process of co- or bi-clustering has redefined similarity by computing distances within the building blocks instead of across all the rows and columns. Something had to be done if we wish to include a complete inventory of activities. As the number of activities increase, the data become increasingly sparse and distances become more uniform (see Section 3 The Curse of Dimensionality).

The building block imagery seems to work in this example because different people engage in different activities. The data matrix is sparse due to such joint separation of row and columns. Those building blocks, the latent features, provide a localized additive representation from which we can reproduce the data matrix by stacking the blocks, or stated more accurately, by a convex combination of the latent features.

Wednesday, November 5, 2014

Net Promoter Mixture Modeling: Can a Single Likelihood Rating Reveal Customer Segments?

Net Promoter believes that customers come in one of three forms: promoters (happy yellows), passives (neutral grays), or detractors (angry reds). Cluster identification is relatively easy for all you need to do is ask the "ultimate question" concerning likelihood to recommend. As the figure indicates, the top two boxes are promoters, the bottom six boxes are distractors, and the net promoter score (NPS) is the difference in those aggregate percentages. Although there is considerable controversy surrounding this scoring system, I wish to focus on the density estimation question of whether I can look at the distribution of ratings across this 11-point scale and determine whether or not there are clusters (e.g., a bimodal distribution might suggest the mixing of two components).

The mclust R package (Section 6 Density Estimation) illustrates this type of analysis using eruption data from the Old Faithful geyser. The histogram on the left reflects our situation using only the waiting time between eruptions. The distribution appears to be bimodal suggesting that there might be two different sources generating the eruptions. The two-dimensional plot on the right, displaying both waiting times between eruptions and length of the eruptions, was included to show how easily we can generalize to higher dimensional spaces and how an additional dimension can increase separation. It appears that some eruptions occur more quickly and last for a shorter time than those eruptions with greater latencies and longer durations. You can find a more detailed introduction to mclust and all the R code needed to produce such plots in an earlier post.

Evidence for Clusters in the Distribution of Recommendation Scores?

The observed distribution for likelihood to recommend tends to follow a familiar pattern with the entire curve moved up or down the scale depending on the provider's overall performance. High recommendation ratings are not unusual with the best providers having as many as half of their customers giving the highest possible marks. At the other end, we often find a much smaller group who are quite unhappy. Another bump in the distribution appears in the middle with a larger than otherwise expected proportion using the midpoint. In general, the highest peak is toward the upper end with progressively smaller summits as one moves down the scale.

First, we will look at the density plot for a group of more boutique providers with higher overall recommendation scores. Although the data are from but one proprietary study, one consistently finds similar curves for smaller brands appealing to niche segments.  Each of the 11 bars in the histogram represents a different rating from 0 to 10. Over 40% of the customers tell us that they are extremely likely to recommend. We also see a slight "bump" at 5 and another at the very bottom for a score of 0. Net Promoter claims that this distribution of recommendation ratings was generated by three different customer clusters as shown in the first figure and defined by the intervals 9-10, 7-8 and 0-6.

The mclust R package allows us to test that hypothesis, returning the best BIC for 6 components with equal variances spread out at almost two-point intervals. That is, when asked what mixture of normal distributions might have generated this distribution of recommendation likelihood, mclust identified six components spaced at approximately equal intervals. We can specify the number of clusters by inserting G=3 into the densityMclust function. The three normal density curves representing the three clusters from mclust have been added to the histogram: 78% with mean 9.3, 15% in the middle concentrated about 5.8, and 7% averaging 1.4.

Next, we will repeat the previous analysis with a different set of providers and their customers. As shown below, customers from larger mass market providers followed a similar pattern but with more spread at the top and more use of the middle and bottom boxes. Once again mclust returns 6 components, and when forced with G=3, reproduces the three normal density curves: 61% centered at 8.7, 25% close to the midpoint at 5.2, and 14% at the bottom with a mean of 0.9. The pattern seems to be the same but with lower ratings. At no time do we see any empirical support for the Net Promoter cutpoints.

Finally, we can compare the distributions for recommendation and overall satisfaction for our mass market. A correlation of 0.89 between the two suggests that we are measuring the same construct, except that satisfaction gets higher ratings because customers are more reluctant to recommend than they are to express satisfaction. Generally, recommendation is a more difficult achievement for the provider than satisfaction, or said differently, satisfaction is necessary but not sufficient for recommendation (e.g., must be better than competitors to recommend or must be good for everyone and not just you).

Measuring a Single Avoidance-Approach Evaluative Dimension

Customers probably do come from a mixture of clusters. The hostage is looking for an escape route, the advocate enjoys singing praises, and the habitual is oblivious. If we wished to identify such loyalty segments, we would need to ask about actual behaviors in real situations for this is what differentiates the clusters (e.g., a battery of who, where and when did you recommend rather than some abstract propensity to recommend measured out of any context).

We capture none of the underlying segmentation with ratings scales that measure a single evaluative dimension. Recommendation is not the ultimate question but just another indicator of one's orientation toward the brand, specifically a index of avoidance-approach or thumbs-up and thumbs-down. What separates satisfaction, retention and recommendation is their difficulty level for all three measure the same latent variable, which is why they are so highly correlated.

The peaks and valleys in the satisfaction and recommendation distribution are not indicators of customer type but scale usage. Customers make their point when they use the extremes of a rating scale. The dissatisfied let you know by uniformly giving the lowest possible score. The pleased also want to tell you to keep up the good work. Those who are not sure or don't care overuse the midpoint.

R code from mclust to create these plots:
promoters <- densityMclust(recommend, G=3)
summary(promoters, parameters = TRUE)

Created by Pretty R at

Monday, November 3, 2014

Let's Do Some MORE Hierarchical Bayes Choice Modeling in R!

R enables us to "test drive" statistical models by running simulations and analyzing actual data. We can look at the output and get some sense of how the whole thing works. The goal is to provide a realistic application that is not too difficult to follow: here is the data, next reformat that data into a form that the function can handle, now for the code to run the analysis and get the output, and finally a little interpretation.

This was my motivation for the original post "Let's Do Some Hierarchical Bayes Choice Modeling in R!" (without the "more"). In that example, I varied pricing for two brands in a 3x3 complete factorial design and included a "Buy None of These" checkbox in order to measure demand. As long as consumers can leave the product category and buy nothing, we need to keep that option in the choice set.
In addition, I allowed for alternative-specific effects by estimating the impact of price separately for the two brands. I was trying to mimic the differential effects of price changes for national and generic products (e.g., when buying olive oil becomes a choice between a well-known Italian and a store brand). At least one could argue that the types of price variations seen in this experiment are not unlike what one sees in the store with side-by-side comparisons and periodic price discounts for both national and store brands. As with any experimental study, choice modeling can be obtrusive. We need to be careful not to create price sensitivity that is not present in the market by introducing artificial variation and focusing the consumer's attention.

This mimicking of the purchase process where consumers decide among several products with varying descriptions may well be the most common implementation of choice modeling in marketing research, Yet, choice modeling can also be used to tradeoff features within a realistic context without including alternative-specific effects. For example, Huber and Train analyze data from choice sets where households selected the company from which they wanted to buy electricity. Assuming that every household wants electricity, customers were presented offers from four different suppliers and ask to select one.

You can find the dataset in the R package mlogit and a complete description of the analysis, including R code, in Section 3 Mixed Logit Model. Instead of repeating the details of the mlogit analysis, I will focus on how to transform the dataset into the format that bayesm wants and run a hierarchical Bayes (HB), which according to Huber and Train produces similar results.

The R Code to Format the Data for bayesm and Run the HB
data("Electricity", package = "mlogit")
Electr <-, id="id", choice="choice",
  varying=3:26, shape="wide", sep="")
Elec.mxl <- mlogit(choice~pf+cl+loc+wk+tod+seas|0, Electr,
  rpar=c(pf='n', cl='n', loc='n', wk='n', tod='n', seas='n'),
  R=100, halton=NA, print.level=0, panel=TRUE)
for (i in 1:nresp)
  for (j in 1:nobs) {
matplot(trace, type="l")
write.csv(estimate2, file="estimate.csv")
Created by Pretty R at

Once you understand that mlogit wants the data in a long format and that bayesm is looking for a list structure, it will all become a little easier. There is always nesting in a hierarchical design, and in this case, choice sets are nested within respondent id. Things get a little complicated because all respondents do not complete the same number of choice sets. We have a solution because we know the respondent id, which is what each row of Electricity provides. In addition, each row tells us which of the four offers was selected (choice) and the features for each offer: fixed price (pf1, pf2, pf3, pf4), contract length (cl1, cl2, cl3, cl4), local company (loc1, loc2, loc3, loc4), well-known (wk1, wk2, wk3, wk4), time-of-day (tod1, tod2, tod3, tod4), and seasonal rate (seas1, seas2, seas3, seas4). Now, we have all that is necessary to run the analysis using maximum likelihood (mlogit) or hierarchical Bayes (bayesm).

The mlogit package transforms the wide form with all the choices on a single row into a long form with each alternative on its own row using the ) function. You should print out the data for the first respondent in both the wide Electricity (12 rows) and the long Electr (4x12=48 rows). In the long format, choice is not a number but a TRUE or FALSE for each alternative (alt). All of this will seem familiar if you have used the reshape function in R or the reshape R package.

The bayesm package adopts a different formatting approach by placing each respondent into a list structure containing choice as the dependent variable and the features gathered together as a design matrix. In the end, each respondent would look like the following for the first respondent who is the first element of the list. If you look back at the first 12 rows of Electricity, you will see that the choices are listed below in $y and the values of the six features are each of the 12 choice sets are repeated in blocks of four rows in $X. Thus, the first element of $y (option 4 was picked) and the feature profile for option 4 in the first choice set was (0, 5, 0, 1, 1, 0). Had there been a "none of these" option, it would have been given a "5" in $y and a fifth row with all zeros would have been inserted at the end of each choice set in $X, so that $X would now have 60 rows instead of 48.

> lgtdata[[1]]
 [1] 4 3 4 4 1 4 1 3 1 2 3 4

   pf1 cl1 loc1 wk1 tod1 seas1
1    7   5    0   1    0     0
1    9   1    1   0    0     0
1    0   0    0   0    0     1
1    0   5    0   1    1     0
2    7   0    0   1    0     0
2    9   5    0   1    0     0
2    0   1    1   0    1     0
2    0   5    0   0    0     1
3    9   5    0   0    0     0
3    7   1    0   1    0     0
3    0   0    0   1    1     0
3    0   0    1   0    0     1
4    0   1    0   1    0     1
4    9   1    0   0    0     0
4    7   0    1   0    0     0
4    0   5    0   1    1     0
5    0   0    1   0    1     0
5    9   1    0   1    0     0
5    0   0    0   0    0     1
5    7   5    0   1    0     0
6    0   0    0   0    0     1
6    9   0    0   0    0     0
6    0   1    1   0    1     0
6    7   5    0   1    0     0
7    0   5    0   1    0     1
7    9   0    0   0    0     0
7    0   1    0   0    1     0
7    7   5    1   0    0     0
8    9   0    0   1    0     0
8    7   1    0   0    0     0
8    0   1    1   0    0     1
8    0   5    0   0    1     0
9    0   5    0   1    0     1
9    0   5    0   0    1     0
9    9   0    0   1    0     0
9    7   1    0   0    0     0
10   0   1    0   1    0     1
10   0   5    1   0    1     0
10   7   0    0   0    0     0
10   9   0    0   1    0     0
11   9   5    0   1    0     0
11   0   0    0   1    1     0
11   0   5    1   0    0     1
11   7   1    0   0    0     0
12   0   1    0   1    1     0
12   7   5    0   0    0     0
12   9   5    1   0    0     0
12   0   0    0   1    0     1

How did I get the data in this format? I used a loop: for (i in 1:nresp). First I pulled out only the data for that one respondent and placed it in respdata. The choice column went into ty. The length of ty told me how many choice sets the respondent completed, and my loop continued until I reached that number. Although there are more direct ways to do this, I simply created separate design matrices for each alternative using the correct column number for this particular dataset. Then, rbind stacked these choice set design matrices and placed them in tdesign. Finally, ty and tdesign were added as an element of the list structure called lgtdata. This entire process is repeated until each respondent has their own element in the list (e.g., the last respondent is lgtdata[[361]]).

Interpreting the bayesm Output

The heavy lifting is over for we have the data formatted for bayesm. HB estimation requires some values for the MCMC, in this case the number of simulations (R=2000) and how many to keep (keep=10). My earlier post with the almost identical title covers this in more detail. Although I do not recommend buying or using their software, Sawtooth is a good source for introductory papers and videos. The review by Allenby and Rossi should be helpful. You may wish to watch the video analyzing choice data.

The function that does the work in bayesm is a hierarchical multinomial logit with a mixture of normal distributions. I have set ncomp=1 to indicate that all the respondents come from the same multivariate normal density. Specifically, hierarchical Bayes "borrows" data from other respondents. It is not unlike borrowing clothes from your friends. It helps if they are a similar shape and size. Here, I am claiming that all the respondents come from the same distribution of weights. Different respondents can have different weights, but the weights for all the individuals are normally distributed with a common mean and variance. Consequently, I cannot go back after the fact and segment respondents for I have already claimed that there is one segment in order to borrow data from others.

I limited the number of simulations so that bayesm runs in only a few minutes. You should copy and run the code after installing mlogit and bayesm. The plots after the HB will display the changes in the fit (loglike) and the coefficients over the 200 runs that were kept (R=2000 but only each 10th was saved). Increasing the number of runs will take more time and help one become more confident that they have achieved convergence. Personally, as shown in the R code, I like to compare the coefficients after discarding the first 25%, 50% and 75% of the runs. If the results to not fluctuate, I am more comfortable.

It should be noted that the coefficients from mlogit and bayesm will not be identical. There are always scaling factors when the goal is to predict choice (e.g., doubling all the coefficients returns the same result for the highest scoring alternative remains the highest scoring). Huber and Train introduce such a scaling factor to adjust their Bayesian estimates. In addition, you should remember that only some of the features are binary effects. For example, length of the contact (loc) has levels of 0, 1 and 5 years. How important is length of the contract? It depends on the variation in length. How important is price? It depends on the price range, implying that I can make price more important in the choice process by increasing its range. Whenever we speak of the importance of some attribute, there is always an unspecified range of values assumed by the speaker, which is why stated importance is typically of such little value.

Finally, I have given the R code to write out a file that Excel can read. I ask that you output and look at these coefficients. Moreover, now that you have the individual estimates as an R object, why not examine the descriptive statistics and plots to determine if the normality assumptions are reasonable. Section 3 on the mixed logit model from the mlogit vignette covering Kenneth Train's exercises is especially valuable because it provides a window into how econometricians approach choice modeling. The term "mixed" is used as it is in ANOVA to indicate that some of the parameters can be fixed and others can be random. Do we use the same levels (fixed) or sample new levels (random) when the study is replicated? Respondents are random because the study is replicated with different households. The six features are fixed since the same choice sets are used again without alteration.

I have tried to make it easier to get started with choice modeling by explaining the R code needed to obtain individual-level hierarchical Bayes estimates. My experience is that once you are confident that you can analyze data, you will be more willing to spend the time and effort to learn the statistics and the psychology underlying these models. In this particular case, one learns a great deal by comparing different estimation procedures from two R packages, bayesm and mlogit. As a bonus, we are introduced to the different viewpoints from marketing and econometrics. Marketing sees heterogeneity as an opportunity, while econometric tends to view it more as a source of variation that must be acknowledged and dealt with (almost as a nuisance).