# Log Linear Models For Consumer Brand Switching Behavior: What a Manager Can Learn From Studying Standardized Residuals

^{[ to cite ]:}

Dawn Iacobucci and Geraldine Henderson (1997) ,"Log Linear Models For Consumer Brand Switching Behavior: What a Manager Can Learn From Studying Standardized Residuals", in NA - Advances in Consumer Research Volume 24, eds. Merrie Brucks and Deborah J. MacInnis, Provo, UT : Association for Consumer Research, Pages: 375-380.

^{[ direct url ]:}

http://acrwebsite.org/volumes/8073/volumes/v24/NA-24

INTRODUCTION

Loyalty and brand-switching are of interest to marketing managers and researchers alike (Carpenter and Lehmann 1985; Deighton, Henderson, and Neslin 1994; Grover and Srinivasan 1992; Novak 1993; Stephan and Tannenholz 1994). In this paper we hope to demonstrate a clear, illustrative example of the potential of log linear models applied to brand switching data. The methods are somewhat underutilized in analyzing brand switching data, perhaps in part because academic treatments of the topic can be somewhat cryptic and technical, or perhaps do not show the richness of the possible managerial implications. We hope to demonstrate in this paper that the insights from these methods are rich. We consider issues of market share and loyalty, and we illustrate the use of studying standardized resduals to examine comparative competitor structure, customer attraction and retention patterns, and asymmetric trading partners. Finally, we present these methods in a straightforward manner and we offer an appendix with instructions for fitting these models, to further enhance ease of adoption. We have found this treatment of these methods to be taught easily in the MBA classroom, and therefore we might expect the methods to be carried more readily into future industry research studies.

THE DATA

The brand switching data we model in this paper represent the automobile manufacturing industry (Colombo, Ehrenberg, and Sabavala 1994). There exists a two-way classification of car purchases wherein the rows of the data matrix describe the owners’ previous car makes, and the columns of the matrix describe the makes of the currently owned automobiles, a standard format for brand switching contingency tables. Each cell entry is the number of consumers who had owned the row-brand car and who now own the column-brand car. These data exist for four consecutive years, 1986 through 1989, for two countries, France and Great Britain; as an example, the matrix for the French 1989 market is found in Table 1. [The analyses for the remaining markets (French 1986, 1987, 1988, and U.K. 1986-1989) are available from the authors.]

Row and column percentages are also found in Table 1, and these represent basic market share information (for times t and t+1, respectively). These marginal indices afford an opportunity to make some initial observations on these data. The market contains a small number of highly active players (the three largest players are ordered: Renault, Peugeot, Citroen), suggesting national loyalty or distribution availability. At the other extreme, the market also contains several players who capture only nominal markets (1-2% market share, including: Alfa Romeo, BMW, Lada, Mercedes, Rover, Seat, Volvo). The remaining manufacturers serve approximately 5% shares (including: Fiat, Ford, GM, VW/Audi). [These relative market share patterns are robust in that they are highly consistent across all four time periods, whether examining row (t) or column (t+1) indices (for both the French and UK markets). The mean Pearson product-moment correlation among the four years' row percentages is .99, as is the mean correlation among the four years' column percentages, and for the row to column comparisons within respective years. However, this apparent stability does not mean there are no interesting dynamic phenomena in these data (as we shall show.)]

Marginal information, by definition, does not represent the associations internal to the body of a contingency table. For example, several automobile manufacturers in both markets have large row margins (e.g., Peugeot), suggesting the loss of many customers, but their respective column margins are also large, indicating that the customers are being replaced in nearly equal quantities. These large share competitors simply see much incoming and outgoing action. One might hypothesize that the action of the Citroen, Peugeot, and Renault brands is reciprocalCthat the French purchase primarily French cars, so former owners of Citroen will next buy Peugeot, for example. When we examine the associations within the body of the table, shortly, we shall see that this exchange hypothesis is somewhat too simple to describe the transactions.

We also know that the differential margins must be considered before examining the data for indicators of loyalty or brand-switching. There is no single agreed upon definition of loyalty (e.g., 100% brand purchases, majority of household brand purchases, sequences of similar brand purchases, etc.), but one naive view would be to examine the diagonal elements in the matrix in Table 1 because these values represent the number of consumers who purchased the same car at time t+1 as they had previously owned at time t. However, we have seen the variability across manufacturers in the margins, and we must tease apart market size information from market repeat or change behavior. It is nearly a truism that manufacturers that capture larger portions of markets also do tend to attract greater loyalty, but the phenomena are confounded in the raw frequencies, as in Table 1. The models of independence and quasi-independence described in the next section will statistically control for the heterogeneity in the margins, and proceed to reflect cleanly the associations internal to the body of the table.

LOG LINEAR MODELS OF INDEPENDENCE AND QUASI-INDEPENDENCE

The fundamental statistic an analyst examines in a two-way cross-tabulation is the Pearson X^{2} of fit for the model of independence. In order to facilitate later generalization, we refer to this model as the log linear model:

(1) ln { E(x_{ij}) }=u + u_{1(i)} + u_{2(j)},

terms that represent a grand total effect, a row effect, and a column effect, respectively. Even cursory inspection of the matrices in Table 1 might lead us to suspect that the model of independence will not fit these data. The hypothesis of independence states that consumer purchases are random, and would yield no apparent pattern in the table; that is, knowing what brand a consumer purchased at time t would be of no help in predicting what brand the consumer purchases subsequently (time t+1). This hypothesis of randomness appears implausible given the apparent structure in the data, but to officially test for randomness, we compute the likelihood ratio test statistic, G^{2}=S_{i}S_{j}x_{ij}ln(x_{ij}/E(x)_{ij}) which is distributed c^{2} on (I-1)(J-1) degrees of freedom for an I by J table X={x_{ij}}, with subscripts i denoting rows (i=1,2,...I) and j denoting columns (j=1,2,...J; I=J).

FREQUENCIES OF CONSUMERS WHO HAD OWNED ROW CARS AND NOW OWN COLUMN CARS

The fit statistic indicates that the model of independence does not fit; G^{2}=22043.749, df=169, p=.000. A substantively interesting plausible explanation for the statistically nonrandom purchase behavior may indeed be loyalty; most of the diagonal values are large, and this suggests that consumers are repeat purchasing (thus purchases at time t+1 are not independent of those at time t). As stated previously, the diagonal values in the matrices in Table 1 are not corrected for differential market shares. However, once the model is fit and the row and column parameters estimated (the u_{1(i)}’s and u_{2(j)}’s in equation 1), the "standardized residuals" can be computed with the result that the data have been essentially corrected for the margins.

The standardized residual is computed as: z_{ij}=(x_{ij}-E(x)_{ij})/sqrt(E(x)_{ij}), and is distributed as a z-score (Haberman 1973), so any standardized residual exceeding 1.96 in magnitude is significant. [This relationship is not difficult to see. A X^{2} is the sum of z-scores squared, and the Pearson X^{2} statistic is computed as the sum of terms, (x_{ij}-E(x)_{ij})^{2}/E(x)_{ij}); i.e., the square of the z_{ij} standardized residual terms.] So for example, a large positive standardized residual value on the diagonal would indicate far more consumers repeated their purchase of the focal brand than would be predicted by chance (after correcting for apparent share differences). [Similarly, a large negative value on the diagonal would indicate extreme customer dissatisfaction.]

Each of these diagonal z-scores exceeded 20, clearly indicating strong loyalty, even after correcting for differential market shares. Therefore, we might wish to address the question, "Even though the model of independence does not fit, in large part due to the loyal purchase behavior reflected in the large diagonal values, is the brand-switching random, or are there also predictable, structural patterns in the off-diagonal elements in this matrix?"

This question requires that we fit the model of independence to the off-diagonal frequencies temporarily ignoring the diagonal (loyalty) cells. We can do so by fitting the model of "quasi-independence" as described in Fienberg (1980), whereby the model of independence is fit to a contingency table in which one or more cells have been marked with structural zeros (cf. Colombo and Morrison 1989). [Structural zeros are distinguished from sampling zeros in that the former are combinations that will not occur in data (e.g., a cell reflecting the number of consumers purchasing a focal car brand in a market where the car is not available), whereas the latter zeros appear due to sampling or baseline rarity but presumable are possible combinations (e.g., number of consumers purchasing some expensive, exotic, tailored-made car). The typical application of quasi-independence is to fit the model of independence while being sensible about constraining cells corresponding to structural zeros to have expected cell counts of zero, since the observed cell count will necessarily equal zero. In the current paper, structural zeros are imputed into the diagonal cells to focus the modeling efforts to the off-diagonals. This use is still relatively novel (e.g., see Colombo and Morrison 1989), but it is defensible to the extent that: a) we have already analyzed and characterized the entire table, and b) we selected the diagonal cells as focal due to substantive theoretical reasons, not arbitrarily, and not even empirically (a la "data-fishing").] (For instruction in fitting these models, see the Appendix.)

CODED TABLE OF STANDARDIZED RESIDUALS FROM QUASI-INDEPENDENCE MODEL

The G^{2} for the model of quasi-independence is 640.659 on 155 df [The model of quasi-independence is fit on I(=J) fewer degrees of freedom than the model of independence due to our constraining the I(=J) zeros on the diagonal cells where the loyalty behavior is reflected.] (p=.000), indicating it does not fit. Nevertheless, the improvement infit from the model of independence to quasi-independence is significant (DG^{2}=21403.105, Ddf=14, p<.001). Thus, the DG^{2} indicates that correcting for loyalty is important and necessary, and the G^{2} indicates that we must conclude the consumer brand-switching purchases are non-random over time, even after adjusting for loyalty.

In the spirit of exploratory data analysis (cf. Tukey 1977; Velleman and Hoaglin 1981, and others), rather than presenting the table of 182 standardized residual values, we present in Table 2 an easier visual summary of the data. To study the structural patterns, we created a "coded table"; coding the standardized residuals cell-by-cell upon fitting the model of quasi-independence to these data: "#" and "+" represent significantly positive residuals (z˜4.00, and z˜2.58, respectively), and "-" and "=" represent significantly negative residuals (z--2.58 and z--4.00, respectively). [We used the more conservative cutoffs of 2.58 and an even higher value of 4.00 rather than 1.96 because we are studying the patterns in so many cells (184=196-14) simultaneously, and we wish to control somewhat for chance, particularly because the cells themselves are presumable interdependent, given that the model of independence did not fit.] Thus, the first two codes represent those elements in the matrix where there are more consumers purchasing the column car after having owned the row car than would be expected if quasi-independence held, and the second two codes indicate very infrequent transactions.

There are many phenomena in this table that give us a sense of the richness of the data, far beyond the previously reported simple row and column percentages. For example, note that BMW and Mercedes do more inter-trading than the model (hypothesizing quasi-randomness) would predict (the BMW-Mercedes and Mercedes-BMW cells contain "#"s). Consumers considering these car makes are also somewhat more likely to be trading-in or trading-to VW/Audi or Volvos. These consideration sets tell the manufacturers who their most likely competitors are, as a subset of all possible competitors.

Earlier, we had briefly entertained the hypothesis that the cars within the trio of {Citroen-Peugeot-Renault} were somehow interchangeable. There is some tendency for Peugeot owners to continue to purchase French brands (the "+" in the Peugeot row corresponding to the Renault column), but for example, Renault buyers are literally less predictable in their subsequent purchases (the Renault row is comprised entirely of "."s, meaning few significant departures from random behavior for Renault owners). Citroen buyers show more structural predictability (there are more clear patterns of especially high or low purchase frequency patterns; i.e., more "#+-=" codes in the Citroen row), and national loyalty.

There are very few combinations that are particularly rare (i.e., the "="s, or even "-"s). The French 1989 market shows two (Citroen to VW/Audi, and VW/Audi to Renault); that is, it was highly unusual (less often than chance) for Citroen owners to purchase VW/Audis, and for VW/Audi owners to purchase Renaults. It would be interesting to understand why these purchase patterns are so rare. One might argue that one car in each pair is in a different sort of "market" from the other in the pair, but that could be true of many of the pairs within these markets, and there were no a priori expectations that the data would flag these purchases as particularly unusual temporal combinations.

Finally, these structural patterns could be used by any one of the manufacturers’ marketing managers to study their brand in the marketplace compared with others. For example, the Citroen row in the matrix has a fair number of marks ("#+-="s), indicating this competitor is more likely (than randomness) to lose consumers to Peugeot and Renault ("#" and "+"), and not likely to lose consumers to Ford, GM, or VW/Audi ("-" and "="). While that list may seem like a good number of competitors with which to contend in managing the loss of consumers to other manufacturers, at least the Citroen manager’s picture is clear: Citroen knows the players in the market to whom it is most likely to lose owners.

Contrast Citroen’s picture with say that of Alfa Romeo, Lada, or Renault. For these car makes, the rows (if we are still focusing on where lost consumers are ging), show no strong indications of pattern (all cells are "."s). Alfa Romeo, Lada and Renault are likely to lose consumers to any other maker with equal likelihood (in proportion to the market shares which have been factored out). Given this observation, their managements are more challenged because they are just as likely to be losing their consumers to all the other manufacturers. Citroen’s competitive picture was more crystalized compared with the near-random behavior of say, the Alfa Romeo purchaser (according to the abundance of non-significant residualsCthe "."s).

We can do a similar analysis within a competitor comparing its row to its column. For example, Citroen has more "codes" in its row compared to its column. Thus, the data on Citroen indicate more clearly where its customers are going than from which competitors its customers have come. Such a market situation actually suggests that Citroen is not executing (or doing so successfully) a targeted marketing plan, but rather is attracting consumers from all sorts of previous car makes. Surely a car manufacturer could be happy in such a situation, as long as incoming volumes were high, but should the volume drop off, or should the manager wish to understand the consumers s/he is likely to attract, this information picture is not clear. On the other hand, the outgoing (row) information is more clearly structured, so Citroen would know where to focus to stop the leak of customers (i.e., Peugeuot and Renault, and not Ford, GM, or VW/Audi). That is, Citroen has a clearer means of identifying how to retain customers than how to attract them.

As a final class of observations, we note the asymmetries in the codes in Table 2. There exist two symmetric dyads, in which the behavior from row i to column j resembles that from row j to column i, including BMW-Mercedes, and Ford-GM. The BMW-VW/Audi pair is also nearly symmetric. For these three pairs, the manufacturers involved are essentially switching a portion of their customers back and forth. All the remaining pairs, however, demonstrate asymmetric switching. For example, more consumers switch from Fiat to Alfa Romeos than the reverse. The same is true for Ford or GM to Citroen, Citroen to Peugeot or Renault, Fiat to Peugeot or Rover, Rover to Ford, VW/Audi to Citroen, Renault to VW/Audi, etc. [One can imagine a walk through a graph theoretic path structure as descriptive of aspirative car purchases; e.g., perhaps the GM owner moves onto purchase a Citroen, and then a Peugeot, and then a Rover, and Ford, etc. Most manufacturers would desire to be a "sink" in a graph; retaining customers fed from other manufacturers, or "sources/"]

In the next section of this paper, we explore symmetry systematically. We have pointed to many pairs of car makes with different codes (e.g., one "." and one "="), but we must acknowledge that the majority of pairs in this matrix are symmetric if only because they are insignficant (".",".")’s. Thus, an objective measure of (a)symmetry is sought.

LOG LINEAR MODELS OF SYMMETRY AND QUASI-SYMMETRY

In this section, we explore a means of testing for symmetry and asymmetry in these brand-switching data. We apply the methods of Bishop, Fienberg, and Holland (1975) who present a log linear model for a two-way table with elements x_{ij} in which a three-way table is created:

(2) x_{ij1}=x_{ij} i>j, 0 elsewhere, and

x_{ij2}=x_{ji} i>j, 0 elsewhere.

The first model fit to this three-dimensional array represents symmetry:

(3) Ln { E(x)_{ijk} }=u +u_{1(i)} +u_{2(j)} +u_{3(k)} +u_{12(ij)}.

The partner model is that of "quasi-symmetry" which is like the above, but with a relaxation on the assumption of homogeneous margins (cf. Novak 1993, p.270) [The model of quasi-symmetry shares with the model of quasi-independence a relaxation of certain assumptions, but while the latter involves structural zeros, the former involves allowing margins to not be perfectly equal (if a matrix were symmetric, it would follow that its row and column margins would be exactly equal).]:

(4) Ln { E(x)_{ijk} }=u +u_{1(i)} +u_{2(j)} +u_{3(k)} +u_{12(ij)} +u_{13(ik)} +u_{23(jk)}.

The fit statistic for the model of symmetry is large, indicating the data do indeed display sufficient asymmetries that the model does not fit (G^{2}=845.723 df=286, p<.001). While the relaxation of the assumption of marginal homogeneity in the quasi-symmetry model improves the fits significantly (DG^{2}=752.450, df=26, p<.001), the quasi-symmetry model nevertheless still does not fit the data (G^{2}=93.273, df=260, p<.050).

Thus, while it may be the case that some pairs or small groups of car manufacturers symmetrically trade consumers back and forth (e.g., BMW and Mercedes), these results suggest that asymmetry is a property that describes this market better, on average. It is conceivable that these asymmetries are a result of consumer car purchases being hierarchical, if some cars are more desirable to trade-up to than others, for example.

CONCLUSIONS

In this paper, we analyzed brand switching data using methods created for contingency tables. Specifically, we fit the models of independence and quasi-independence to the two-way contingency table describing consumer car purchases at times t and t+1. The modeling is straightforward, and the results are rich and telling. The model of independence did not fit, and the first hypothesis was that this result was driven by likely high loyalties. The model of quasi-independence fit significantly better, confirming the importance of controlling for the loyalties, but it also did not fit, indicating that the brand-switching patterns were also nonrandom and structured.

The analyses also suggested asymmetries in the brand-switching behavior that was not obvious by the simple marginal percentages. The asymmetric structure was confirmed via the fitting of models of symmetry and quasi-symmetry.

The results of inspecting the standardized residuals were especially enlightening. Managerial implications were obvious; cross-competitor comparisons were possible, as were within-competitor analyses of customers lost versus customers gained.

The methods presented in this paper are easy to execute, understand, and communicate, and they yield clear insights into competitive structure and manufacturer properties. We encourage their more frequent utilitization.

HOW TO FIT THE MODELS (IN SPSS).

REFERENCES

Bishop, Yvonne M.M., Stephen E. Fienberg, and Paul W. Holland (1975), Discrete Multivariate Analysis: Theory and Practice, Cambridge, MA: The MIT Press.

Carpenter, Gregory S. and Donald R. Lehmann (1985), "A Model of Marketing Mix, Brand Switching, and Competition," Journal of Marketing Research, 22(3), 318-329.

Colombo, Richard, Andrew Ehrenberg, and Darius Sabavala (1994) "The Car Challenge: Diversity in Analyzing Brand Switching Tables," Unpublished manuscript.

Colombo, Richard A., and Donald G. Morrison (1989), "A Brand Switching Model with Implications for Marketing Strategies," Marketing Science, 8, 89-99.

Deighton, John, Caroline M. Henderson, and Scott A. Neslin (1994), "The Effects of Advertising on Brand Switching and Repeat Purchasing," Journal of Marketing Research, 31(1), 28-43.

Fienberg, Stephen E. (1980), The Analysis of Cross-Classified Categorical Data, (2nd ed.), Cambridge, MA: The MIT Press.

Grover, Rajiv, and V. Srinivasan (1992), "Evaluating the Multiple Effects of Retail Promotion on Brand Loyal and Brand Switching Segments," Journal of Marketing Research, 29(1), 76-89.

Haberman, Shelby J. (1973), "The Analysis of Residuals in Cross-Classified Tables," Biometrika, 29, 205-220.

Novak, Thomas P. (1993), "Log-Linear Trees: Models of Market Structure in Brand Switching Data," Journal of Marketing Research, 30, 267-287.

Stephan, S. Kent, and Barry L. Tannenholz (1994), "The Real Reason for Brand Switching," Advertising Age, 65(25), 31.

Tukey, John W. (1977), Exploratory Data Analysis, Reading, MA: Addison-Wesley.

Velleman, Paul F. and David C. Hoaglin (1981), Applications, Basics, and Computing of Exploratory Data Analysis. Boston: Duxbury Press.

----------------------------------------

Tweet
window.twttr = (function (d, s, id) { var js, fjs = d.getElementsByTagName(s)[0], t = window.twttr || {}; if (d.getElementById(id)) return; js = d.createElement(s); js.id = id; js.src = "https://platform.twitter.com/widgets.js"; fjs.parentNode.insertBefore(js, fjs); t._e = []; t.ready = function (f) { t._e.push(f); }; return t; } (document, "script", "twitter-wjs"));