Contextual Measurement Is a Game Changer

Adding a context can change one's frame of reference:

Are you courteous? 

Are you courteous at work? 

Decontextualized questions tend to activate a self-presentation strategy and retrieve memories of past positioning of oneself (impression management). Such personality inventories can be completed without ever thinking about how we actually behave in real situations. The phrase "at work" may disrupt that process if we do not have a prepared statement concerning our workplace demeanor. Yet, a simple "at work" may not be sufficient, and we may be forced to become more concrete and operationally define what we mean by courteous workplace behavior (performance appraisal). Our measures are still self-reports, but the added specificity requires that we relive the events described by the question (episodic memory) rather than providing inferences concerning the possible causes of our behavior.

We have such a data set in R (verbal in the difR package). The data come from a study of verbal aggression triggered by some event: (S1) a bus fails to stop for me, (S2) I miss a train because a clerk gave faulty information, (S3) the grocery store closes just as I am about to enter, or (S4) the operator disconnect me when I used up my last 10 cents for a call. Obviously, the data were collected during the last millennium when there were still phone booths, but the final item can be updated as "The automated phone support system disconnects me after working my way through the entire menu of options" (which seems even more upsetting than the original wording).

Alright, we are angry. Now, we can respond by shouting, scolding or cursing, and these verbally aggressive behaviors can be real (do) or fantasy (want to). The factorial combination of 4 situations (S1, S2, S3, and S4) by 2 behavioral modes (Want and Do) by 3 actions (Shout, Scold and Curse) yields the 24 items of the contextualized personality questionnaire. Respondents are given each description and asked "yes" or "no" with "perhaps" as an intermediate point on what might be considered an ordinal scale. Our dataset collapses "yes" and "perhaps" to form a dichotomous scale and thus avoids the issue of whether "perhaps" is a true midpoint or another branch of a decision tree.

David Magis et al. provide a rather detailed analysis of this scale as a problem in differential item functioning (DIF) solved using the R package difR. However, I would like to suggest an alternative approach using nonnegative matrix factorization (NMF). My primary concern is scalability. I would like to see a more complete inventory of events that trigger verbal aggression and a more comprehensive set of possible actions. For example, we might begin with a much longer list of upsetting situations that are commonly encountered. We follow up by asking which situations they have experienced and recalling what they did in each situation. The result would be a much larger and sparser data matrix that might overburden a DIF analysis but that NMF could easily handle.

Hopefully, you can see the contrast between the two approaches. Here we have four contextual triggering events (bus, train, store, and phone) crossed with 6 different behaviors (want and do by curse, scold and shout). An item response model assumes that responses to each item reflect each individual's position on a continuous latent variable, in this case, verbal aggression as a personality trait. The more aggressive you are, the more likely you are to engage in more aggressive behaviors. Situations may be more or less aggression-evoking, but individuals maintain their relative standing on the aggression trait.

Nonnegative matrix factorization, on the other hand, searches for a decomposition of the observed data matrix within the constraint that all the matrices contain only nonnegative values. These nonnegative restrictions tend to reproduce the original data matrix by additive parts as if one were layering one component after the other on top of each other. As an illustration, let us say that our sample could be separated into the shouters, the scolders, and those who curse based on their preferred response regardless of the situation. These three components would be the building blocks and those who shout their curses would have their data rows formed by the overlay of shout and curse components. The analysis below will illustrate this point.

The NMF R code is presented at the end of this post. You are encourage to copy and run the analysis after installing difR and NMF. I will limit my discussion to the following coefficient matrix showing the contribution of each of the 24 items after rescaling to fall on a scale from 0 to 1.

		Want to and Do Scold	Store Closing	Want to and Do Shout	Want to Curse	Do Curse
S2DoScold		1.00	0.19	0.00	0.00	0.00
S4WantScold		0.96	0.00	0.00	0.08	0.00
S4DoScold		0.95	0.00	0.00	0.00	0.11
S1DoScold		0.79	0.37	0.02	0.05	0.15
S3WantScold		0.00	1.00	0.00	0.08	0.00
S3DoScold		0.00	0.79	0.00	0.00	0.00
S3DoShout		0.00	0.15	0.14	0.00	0.00
S2WantShout		0.00	0.00	1.00	0.13	0.02
S1WantShout		0.00	0.05	0.91	0.17	0.04
S4WantShout		0.00	0.00	0.76	0.00	0.00
S1DoShout		0.00	0.12	0.74	0.00	0.00
S2DoShout		0.08	0.00	0.59	0.00	0.00
S4DoShout		0.10	0.00	0.39	0.00	0.00
S3WantShout		0.00	0.34	0.36	0.00	0.00
S1wantCurse		0.13	0.18	0.03	1.00	0.09
S2WantCurse		0.34	0.00	0.08	0.92	0.20
S3WantCurse		0.00	0.41	0.00	0.85	0.02
S2WantScold		0.59	0.00	0.00	0.73	0.00
S1WantScold		0.40	0.22	0.01	0.69	0.00
S4WantCurse		0.31	0.00	0.00	0.62	0.48
S1DoCurse		0.24	0.16	0.01	0.17	1.00
S2DoCurse		0.47	0.00	0.00	0.00	0.99
S4DoCurse		0.46	0.00	0.02	0.00	0.95
S3DoCurse		0.00	0.54	0.00	0.00	0.69

As you can see, I extracted five latent features (the columns of the above coefficient matrix). Although there are some indices in the NMF package to assist in determining the number of latent features, I followed the common practice of fitting a number of different solutions and picking the "best" of the lot. It is often informative to learn how the solutions changes with the rank of the decomposition. In this case similar structures were uncovered regardless of the number of latent features. References to a more complete discussion of this question can be found in an August 29th comment from a previous post on NMF.

Cursing was the preferred option across all the situations, and the last two columns reveal a decomposition of the data matrix with a concentration of respondents who do curse or want to curse regardless of the trigger. It should be noted that Store Closing (S3) tended to generate less cursing, as well as less scolding and shouting. Evidently there was a smaller group that were upset by the store closing, at least enough to scold. This is why the second latent feature is part of the decomposition; we need to layer store closing for those additional individuals who reacted more than the rest. Finally, we have two latent features for those who shout and those who scold across situations. As in principal component analysis, which is also a matrix factorization, one needs to note the size of the coefficients. For example, the middle latent features reveals a higher contribution for wanting to shout over actually shouting.

Contextualized Measurement Alters the Response Generation Process

When we describe ourselves or other, we make use of the shared understandings that enable communication (meeting of minds or brain to brain transfer). These inferences concerning the causes of our own and others behavior are always smoothed or fitted with context ignored, forgotten or never noticed. Statistical models of decontextualized self-reports reflect this organization imposed by the communication process. We believe that our behavior is driven by traits, and as a result, our responses can be fit with an item response model assuming latent traits.

Matrix factorization suggests a different model for contextualized self-reports. The possibilities explode with the introduction of context. Relatively small changes in the details create a flurry of new contexts and an accompanying surge in the alternative actions available. For instance, it makes a differences if the person closing the store as you are about to enter has the option of letting one more person in when you plea that it is for a quick purchase. The determining factor may be an emotional affordance, that is, an immediate perception that one is not valued. Moreover, the response to such a trigger will likely be specific to the situation and appropriately selected from a large repertoire of possible behaviors. Leaving the details out of the description only invites the respondents to fill in the blanks themselves,

You should be able to build on my somewhat limited example and extrapolate to a data matrix with many more situations and behaviors. As we saw here, individuals may have preferred responses that generalize over context (e.g., cursing tends to be overused) or perhaps there will be situation-specific sensitivity (e.g., store closings). NMF builds the data matrix from additive components that simultaneously cluster both the columns (situation-action pairings) and the rows (individuals). These components are latent, but they are not traits in the sense of dimensions over which individuals are ranked ordered. Instead of differentiating dimensions, we have uncovered the building blocks that are layered to reproduce the data matrix.

Although we are not assuming an underlying dimension, we are open to the possibility. The row heatmap from the NMF may follow a characteristic Guttman scale pattern, but this is only one of many possible outcomes. The process might unfold as follows. One could expect a relationship between the context and response with some situations evoking more aggressive behaviors. We could then array the situations by increasing ability to evoke aggressive actions in the same way that items on an achievement test can be ordered by difficulty. Aggressiveness becomes a dimension when situations accumulated like correct answers on an exam with those displaying less aggressive behaviors encountering only the less aggression-evoking situations. Individuals become more aggressive by finding themselves in or by actively seeking increasingly more aggression-evoking situations.


R Code for the NMF Analysis of the Verbal Aggression Data Set

# access the verbal data from difR
library(difR)
data(verbal)
 
# extract the 24 items
test<-verbal[,1:24]
apply(test,2,table)
 
# remove rows with all 0s
none<-apply(test,1,sum)
table(none)
test<-test[none>0,]
 
library(NMF)
# set seed for nmf replication
set.seed(1219)
 
# 5 latent features chosen after
# examining several different solutions
fit<-nmf(test, 5, method="lee", nrun=20)
summary(fit)
basismap(fit)
coefmap(fit)
 
# scales coefficients and sorts
library(psych)
h<-coef(fit)
max_h<-apply(h,1,function(x) max(x))
h_scaled<-h/max_h
fa.sort(t(round(h_scaled,3)))
 
# hard clusters based on max value
W<-basis(fit)
W2<-max.col(W)
 
# profile clusters
table(W2)
t(aggregate(test, by=list(W2), mean))

Created by Pretty R at inside-R.org

Archetypal Analysis: Similarity Defined by Distances from Contrasting Ideals

Carl Jung was at least partially correct. We do tend to think in terms of the extremes as shown in this archetypal wheel with rulers versus outlaws and heroes versus caregivers at different ends of bipolar dimensions. Happily, we are not required to accept Jung's collective unconscious to explain this tendency. Metaphorical thinking works just fine. For example, why not separate all political players into two camps based on our earliest experiences with powerful others: liberals as caregivers (supportive mothers) and conservatives as heroes (demanding and punishing fathers)?

Political ideology was selected as my example because of its universality and because R offers so many ways of analyzing such data. Probably the quickest introduction is through the voteview blog, which relies on a dimensional representation of our liberal and conservative archetypes (such as the following figure showing the polarization in the U.S. Congress).

Two points define a line, and it is seldom difficult to image a continuum between any two bipolar types, in this case between liberals and conservatives. Do we have a dimension or categories? It depends on any separation within the density distribution. Obviously, the distributions in the both the House (light blue Democrats and light red Republicans) and the Senate (dark blue and red) are at least bimodal. Thus, we are free to represent the same data as points along the liberal-conservative dimension or as ratios of mixture coefficients for the two clusters (i.e., odds ratio of membership likelihood in the red or blue clusters).

The mclust R code and a more complete discussion can be found in an earlier post using likelihood to recommend as the dimension and promoters versus distractors as the clusters. In order that there is no misunderstanding, the liberal-conservative continuum is a latent variable derived from a series of votes on a range of issues with the R package basicspace. Recommendation, on the other hand, is an observed likelihood rating along an 11-point scale from 0 to 10. In both cases, we are looking for evidence of separation as if we had a mixture of different generative models.

Given the above figure, liberal and conservative archetypes would be located toward the end points of this scale. That is, instead of describing the two clusters using their centroids positioned near the "humps" in the two curves, archetypal analysis attempts to describe political ideology in terms of idealized liberals and conservatives. These are not necessarily the most extreme points, as the archetypes R package makes clear with displays such as the following showing both the convex hull of the most extreme data points in gray and archetypes as the vertices of the internal red triangle. Three archetypes are necessary to locate any data point in the two-dimensional space.

Before continuing, we ought to review a few examples so that we understand what we mean by an archetype. If you live in a region that receives snow or just watch a lot of Christmas movies and I told you that it was a perfect winter day, that picture you just imagined is an ideal or archetype. All winter days can be described in terms of their distance from that ideal. The same can be said of spring, fall and summer days. If you are familiar with smoggy days, as was Leo Breiman when he introduced archetypal analysis to describe ozone levels in Los Angeles, then you know what a smog alert feels like. We use the shorthand provided by shared archetypes to summarize succinctly a large amount of information.

As you may have noticed, I have interchanged the words "ideal" and "archetype" in my writing. This was deliberate since archetypes tend to be seen when describing ideal instantiation of a category rather than the average category member. Thus, when asked to tell you about a specific athlete, such as a basketball center, you are not likely to describe the average center nor the greatest center that ever played the game. Instead, one thinks about the role that the center plays in the game, lists those defensive and offensive contributions, and distinguishes this position from the other players on the team. Manual Eugster demonstrates how the R package archetypes would uncover such archetypal athletes.

Of course, there is no requirement that forces us to retrieve goal-derived categories and their associated ideals from memory. We could evaluate "on a curve" and think about the average basketball center, as we might if asked to guess the average height of a NBA center. Yet, the center in basketball serves a purpose within a team of other players with other purposes. Not unlike the archetypal wheel that introduced this post, the center is defined in contrast to the other positions on the team. The rules of the game play a role in the clustering of players with similarity measured not by distance from the average but distance from the ideal. Therefore, two centers are similar because they play similar roles in the game, that is, both are close to the ideal center. Moreover, they are seen as even more alike when guards are added into the mix. Similarity is shaped by the context of competing archetypes or ideals.

In one of my first posts, I demonstrated how the R package archetypes would identify features usage types. Repeatedly, we find that usage intensity has the greatest impact differentiating the light from the heavy user. I have reproduced a figure from that previous post showing both the k-means clusters (the K's) and the position of the archetypes (the A's).

The data are 10 feature usage ratings that are projected onto the plane formed by the first two principal components. The points are respondents and the arrows represent the features. All the arrows point to the right indicating that the first principal component reflects usage intensity with heavier usage toward the right in the direction that all the arrows point. As you know, the angles between arrows reflect their correlations, so that the two bundles of arrows suggest a two-factor solution. We can call such a pattern a bifactor solution: a general factor separating light and heavy users and two specific factors distinguishing between those more involved with each of the two bundles of feature sets. It is worth your time to become familiar with this factor structure because it reappears frequently with usage data, as well as preference and satisfaction.

Do you see clusters of data points in the above scatterplot? The three centroids from a K-means clustering follows the path of the first principal component with a low usage (K2), a medium usage (K1) and a high usage (K3) segment. Personally, I find it difficult to separate out clusters in this data cloud. I see a fan-spread distribution with the amount of variation on the second dimension dependent on the value of the first dimension, that is, little or no feature usage among light users and increasing separation of the two feature bundles for heavier users. The archetypes reveal this pattern by forming a triangle with vertices at no usage (A3), bundle A1 usage and bundle A2 usage. K-means yields a restatement of usage intensity along the first dimension, while archetypal analysis summarizes the data as contained with the triangle formed by three usage types.

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.

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)
MX
 
# enter the data for the latent features
LP<-matrix(c(2,1,0,0,0,0,1,2), ncol=4, byrow=TRUE)
LP
 
# observed data is the matrix product
DATA<-MX%*%LP
DATA
 
# load the NMF library
library(NMF)
 
# run with rank=2
fit<-nmf(DATA, 2, "lee", nrun=20)
 
# output the latent feature coefficients
lp<-coef(fit)
round(lp,3)
 
#output the respondent mixing weights
mx<-basis(fit)
round(mx,3)
 
# reproduce the data matrix using NMF results
data<-mx%*%lp
round(data)
# output residuals
round(DATA-data,3)
 
# same but only for 1st latent feature
rank1<-as.matrix(mx[,1])%*%lp[1,]
round(DATA-rank1,3)
 
# same but only for 2nd latent feature
rank2<-as.matrix(mx[,2])%*%lp[2,]
round(DATA-rank2,3)
 
# additive representation 
round(rank1+rank2,3)
round(DATA-rank1-rank2,3)

Created by Pretty R at inside-R.org

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.