A Management Science Assessment of a Behavioral Measure of Brand Loyalty

Terry Elrod, Vanderbilt University
ABSTRACT - This paper offers a new behavioral measure of brand loyalty, denoted Scaled Probability of Purchase (SPOP), explores its properties, and then examines its performance relative to Jacoby and Chestnut's (1978) nine criticisms of behavioral measures of brand loyalty. This author contends that the increase in availability, accuracy, and affordability of behavioral (scanner panel) data warrants a reexamination of Jacoby and Chestnut's conclusion that brand loyalty measures should be a composite of attitudinal and behavioral measures. Since attitudinal measures of brand loyalty are not normally available from scanner panel households, the gains in behavioral data collection do not apply to composite measures as well. This author concludes that behavioral measures such as SPOP are sufficiently informative to warrant their continued use, at least for frequently-bought consumer goods.
[ to cite ]:
Terry Elrod (1988) ,"A Management Science Assessment of a Behavioral Measure of Brand Loyalty", in NA - Advances in Consumer Research Volume 15, eds. Micheal J. Houston, Provo, UT : Association for Consumer Research, Pages: 481-486.

Advances in Consumer Research Volume 15, 1988      Pages 481-486

A MANAGEMENT SCIENCE ASSESSMENT OF A BEHAVIORAL MEASURE OF BRAND LOYALTY

Terry Elrod, Vanderbilt University

ABSTRACT -

This paper offers a new behavioral measure of brand loyalty, denoted Scaled Probability of Purchase (SPOP), explores its properties, and then examines its performance relative to Jacoby and Chestnut's (1978) nine criticisms of behavioral measures of brand loyalty. This author contends that the increase in availability, accuracy, and affordability of behavioral (scanner panel) data warrants a reexamination of Jacoby and Chestnut's conclusion that brand loyalty measures should be a composite of attitudinal and behavioral measures. Since attitudinal measures of brand loyalty are not normally available from scanner panel households, the gains in behavioral data collection do not apply to composite measures as well. This author concludes that behavioral measures such as SPOP are sufficiently informative to warrant their continued use, at least for frequently-bought consumer goods.

INTRODUCTION

In their definitive study of brand loyalty, Jacoby and Chestnut (1978) conclude that measurement of brand loyalty should be composite, i.e. based upon both behavioral and attitudinal data. They see behavioral and attitudinal data guarding against each other's deficiencies.

Since 1978, marketing has seen great strides in the routine collection of behavioral data. In particular scanner data at the store level and at the household level are now widely available for low-priced frequently purchased goods. While advantages of diary panels remain (such as complete coverage of purchases), now entire cities have most stores equipped to record the purchases of scanner panel members in an unobtrusive manner.

Attitudinal measures from these scanner panel consumers are not normally available, however. Economies of scale in data collection mean that third parties (market research suppliers) will continue to collect these data and then sell them piecemeal to companies on a product by product basis. Market research suppliers must see to it that their panel remains as representative of the general population as possible. Obtaining new panel members for scanner panels is more difficult than for diary panels because of the requirement that panelists reside in one of the cities equipped to record scanned panel purchases. This means that attitudinal measures of brand loyalty from these same households are likely to remain unavailable for the foreseeable future.

Does the unavailability of attitudinal data from scanner panels imply that the scanner data should not be used to study brand loyalty? The attractiveness of these data in other respects prompted me to reexamine the issue of brand loyalty. This paper will not argue that scanner panel data are a panacea. Rather, it asks whether meaningful measures of loyalty, at least for low-priced frequently-purchased brands, can be obtained from scanner panel data.

Behavioral measures of brand loyalty are most likely to be meaningful for frequently purchased low-priced branded goods. In this case there are few barriers to purchase (such as price and availability) that would force behavior to differ from attitude, and so behavioral and the more expensive composite measures of brand loyalty are likely to agree [I am indebted to an anonymous reviewer at ACR for this insight among others.].

USES OF THE BRAND LOYALTY CONCEPT

Brand loyalty is a simple concept. Drawing upon the-Webster's Third New International Dictionary definition of loyalty, we have brand loyalty as being "fidelity or tenacious adherence" to a brand on the part of buyers. A loyal friend is someone who will stick by you pretty much no matter what, and a loyal buyer will stick by your brand.

The purpose of identifying brand loyal consumers can be any of several:

To Exploit the Brand Loyal

It can be argued that if consumers are delighted with a brand and its terms of purchase, then the brand manager is doing something wrong. At a minimum the manager is charging too low a price, or is overspending on promotion or some other element of the marketing mix. Identifying brand loyal consumers is a starting point for determining a differentiated marketing mix to that group (one that is more profitable). "Exploitation" of loyal buyers, so defined, leads to all buyers benefiting equally from the exchange: enough to retain their patronage, but not much more.

To Identify the Loyal Buyer in Case of Hard Times

Marketers may wish to sustain the surplus goodwill that is brand loyalty as a defense or deterrent -against aggressive marketing-by a competitor or as a hedge against adverse economic conditions. The brand loyal buyer is the consumer to retreat to in case of a bad turn of fortune, and it pays to know in advance who the loyal buyers are.

To Better Market to Those Who Can Be Made Brand Loyal

Knowing which customers are very pleased with a brand indicates that psycho- or sociodemographically similar consumers in the general population are also good prospects. This use of brand loyalty presumes further penetration of the market is feasible and that similar consumers can be identified, which is more likely to be true for growing product categories and for brands that are new and truly differentiated.

A MEASURE OF BRAND LOYALTY BASED ON PURCHASE PROBABILITY

After locating 33 different behavioral measures of brand loyalty in the literature, Jacoby and Chestnut (1978) criticize these measures on nine grounds. This paper begins by proposing yet a 34th measure of brand loyalty, and the section following will evaluate this measure against Jacoby and Chestnut's criticisms. Jacoby and Chestnut will hereafter be referred to as J&C.

The measure proposed, which I call Scaled Probability Of Purchase (SPOP), is not offered as the measure of brand loyalty, in part because there are too many other contenders for the title, and in part because the best measure may depend upon the data available and the purpose. SPOP is defined as follows. Presuming for the moment a good estimate of a household's purchase probabilities for a set of J brands (P1,---,PJ), the household's loyalty towards the ith brand in the set is defined as:

Li = J(Pi- 1/J)

    = J Pi - 1          (1)

Thus SPOP is a linear transformation of the household's purchase probabilities for the brands. Many of SPOP's properties will be explained in the context of J&C's criticisms, but the rationale underlying the measure and its essential properties warrant immediate explanation.

Translating Probability of Purchase to a Meaningful Origin

Construction of the SPOP measure begins with subtracting 1/J from all purchase probabilities to give a meaningful origin to the measure. A household that buys -a brand with probability exceeding 1/J is buying the brand more often than the average brand and therefore shows some degree of loyalty to the brand. This household receives a positive SPOP score. A household buying a brand with probability less than 1/1 is disloyal and receives a negative SPOP score for that brand.

Scaling Translated Probabilities Of Purchase

(Pi - 1/J) has a maximum value of a-1)/J and a minimum of -1/J. It seems that a measure of brand loyalty should be somewhat robust to the number of brands included in the analysis. If a household never buys a brand, then the maximum disloyalty cannot be less than this, regardless of what other brands are bought Multiplying (Pi - 1/J) by J yields a maximum disloyalty score of -1 regardless of the number of brands and leaves the origin unchanged. However, multiplying (Pi - 1/J) by J also changes the maximum loyalty measure to J-1. Thus maximum attainable loyalty increases with the number of brands in the analysis. This is conceptually pleasing for two reasons. First, since always buying a brand out of a larger set of competing brands is a stronger (and rarer) indication of loyalty to the brand, this ought to be reflected in a larger maximum attainable score for the brand loyalty measure. Adding brands with very low purchase probabilities to a choice set makes the loyalty scores for the initial brands increase, even from being disloyal to being loyal. For example, suppose that a consumer's probabilities of purchase for the two brands constituting a market are .6 and .4. Then we would have to say that the consumer is not very committed to the second brand. In fact, the SPOP scores would be .2 and -.2, respectively, indicating that the consumer is slightly disloyal to the second brand. Suppose next that 4 brands are added to the product category, but that none of these brands attract purchases from this buyer. The buyer's purchase probabilities for the 6 brands become .6, .4, 0, 0, 0, 0. Now we would be inclined to say that the consumer is somewhat loyal to the first two brands, and disloyal to the other four. This intuition is reflected in the SPOP scores for the brands: on a scale now of 5 to -1, the SPOP scores for the 6 brands have become 2.6, 1.4, -1, -1, -1, -1.

Second, adding a brand to the choice set that attains an average probability of purchase (and hence a SPOP score of zero) by drawing share proportionately from existing brands leaves the SPOP scores for these brands unchanged. For example, suppose a household's probabilities of purchase for three brands are given by .5,.3,.2. The corresponding SPOP scores are 0.5, -0.1, -0.4. Adding the fourth brand changes the household's purchase probabilities to: .375, .225, .150, .250 but the SPOP scores are still: 0.5, -0.1, -0.4, 0.0.

Note that SPOP yields a separate measure of a household's loyalty towards each of the brands in the choice set, and that the measure is continuous and not discrete. In addition, the average SPOP brand loyalty score over all brands is always zero.

Obtaining Good Estimates of a Household's Probabilities Of Purchase

The SPOP measure of brand loyalty presumes a good measure of a household's probability of buying each brand. Observed purchase frequency is not a good measure. It is unstable for short periods of observation and yet over longer periods of observation many households are likely to change their behavior.

One resolution of this dilemma is to use a Bayes estimate of a household's purchase probability for each brand. A Bayes estimate recognizes that the household is sampled from a population and uses information about the population to yield an improved estimate of the household's purchase probability.

If the purchase probabilities are distributed Dirichlet (with parameters a1,..., aJ), then the Bayes estimates of a household's purchase probabilities are easily derived. The distribution of the purchase probability for brand i for households observed to buy brand i n i times on N purchase occasions is Beta with parameters (ai + ni, A-ai + N-ni), where a i is the parameter of the Dirichlet distribution corresponding to brand i and A = a1 +...+aJ. The mean of this distribution is the Bayes estimate of purchase probability. It is the minimum variance estimator and it is given by:

PBi = ai + ni / (A + N)      (2)

Notice that the estimate works even for households that were never observed to buy any of the brands. In the absence of any information about a household, our best estimate of its purchase probabilities are the average purchase probabilities in the population. On the other hand, as a household is observed to make more and more purchases, the estimates of the household's purchase probabilities approaches that of its observed purchase frequency (n i /N).

Basing SPOP on Bayes estimates of purchase probabilities avoids misattributing brand loyalty to households that bought the same brand repeatedly due to chance. To see this, consider a hypothetical market consisting of two brands: 1 and 2. Everyone in that market buys whichever brand is on sale that week, and let's say that each brand is cheaper half the time. Therefore DO buyer is brand loyal to either brand, and every buyer has a probability of purchasing each brand of one-half. If a large sample of such buyers are observed for two purchase occasions, then we would observe the buying behavior portrayed in Table 1. In truth all consumers are identical, neither loyal nor disloyal to either brand (with SPOP scores of zero for both).

TABLE 1

HYPOTHETICAL BUYING BY NONLOYAL HOUSEHOLDS

Fitting the Dirichlet-multinomial model (known as the Beta-binomial in this two-brand case) to these data yields parameter estimates a1 = a2 = infinity. These strange parameter estimates reflect the fact that all buyers have identical true purchase probabilities of one half, a limiting case of the Beta distribution. Nevertheless, substituting these parameter estimates into the formula for P Bi yields true purchase probability estimates of .5 for all households, regardless of their observed purchase behavior, and fitted SPOP scores of zero for all brands and consumers. Thus even in this extreme case, the SPOP measure is not "fooled" into misattributing brand loyalty, provided that Bayes estimates of purchase Probabilities are used.

CRITIQUE OF BEHAVIORAL MEASURES OF BRAND LOYALTY

I now turn to a consideration of the J&C criticisms of behavioral measures of brand loyalty and discuss how well SPOP measures up.

"BEHAVIORAL MEASURES LACK A LOGICAL-CONCEPTUAL BASIS"

Based as it is on purchase probability, SPOP has the economic theory of random utility maximization as its basis. On a randomly selected choice occasion, P i is the probability that brand i has the highest utility of all brands:

Pi = Prob [ Uit > Ujt, for all j not equal to i]       (3)

where Ukt = Vk + ekt, Vk is the nonstochastic utility for the brand, and ekt is the stochastic component of utility that varies from one choice occasion to the next. In practice, ekt will be capturing the effects of changes in consumption situation and in marketing mix variables.

For every Pi there is an implied Vi and vice versa (Daganzo 1979). The simplest specification of Vi is ln(Pi), but widely used (e.g. Nakanishi and Cooper 1974) is a transformation that standardizes Vi to have zero mean across brands (just at SPOP does):

vi = ln(Pi/P*)        (4)

where P* is the geometric mean of the purchase probabilities.

Vi so defined would make an attractive measure of brand loyalty because it shows the household's underlying enduring preference for the brand. I chose to work instead with a linear transformation of purchase probability for two reasons. First, Vi has values of minus infinity and of plus infinity for brands with purchase probabilities of zero and of one, respectively. These will arise when observed frequencies must be used, and infinities do not work well as data. Second, as was shown above, there exists for purchase probabilities a simple formula for estimating a household's purchase probabilities based upon only a few purchases. Using SPOP with these estimates avoids the misattribution of brand loyalty to repeat buying that is driven by stochastic errors.

Since purchase probability (and hence SPOP) has a monotonic relationship to the enduring random utility preferences (the Vi's), purchase probability is justifiably referred to as a measure of brand preference as well.

"Behavioral Measures Use Arbitrary Criteria to Distinguish Between Loyal and Disloyal Buyers"

SPOP is scaled so that zero is the dividing point between loyalty and disloyalty, that point corresponding to the average purchase probability across brands (1/J). Returning to the random utility origins, it corresponds to the average brand preference--consumers are described as disloyal when they like a brand less than the average brand in the category. This measure of brand loyalty is relative but nonarbitrary.

"Behavioral Measures Are Overly Simplistic"

This objection is hard for a management scientist to deal with because, ceteris paribus, simplicity in a model or measure is a virtue. On the other hand, too v much of anything is bad, and so the charge as stated is unanswerable.

The substance of J&C's objection seems to be that brand loyalty is complex, therefore any measure of it must be complex. My view is that there are measures of constructs and then there are models that capture the relationships between constructs. It is a matter of choice whether complexity is embodied within measures or within models. Management scientists tend to prefer simplicity in measures, relying on models to explain essential relationships.

"Behavioral Measures Fail to Account for Disloyalty"

The SPOP measure distinguishes between loyalty and disloyalty. Any positive value for SPOP represents loyalty to the brand, any negative value disloyalty, and a value of zero neutrality. Thus the SPOP measure can be used to investigate consumer behavior along a loyalty-disloyalty continuum.

"Behavioral Measures Fail to Specify the Appropriate Unit of Measurement

SPOP uses as data the number of units purchased by a household of each brand. Some argue that the Achilles' heel of panel data is that one does not observe the buying behavior of individuals and therefore the household-level data constitute a misaggregation of behavior by heterogeneous individuals.

Several points can be made in defense of household-level data. First, for some products the household is the consuming unit. Regular ground coffee is thought to be an example of this--the product is perishable once opened and is typically prepared in batches, and so most families consume at home a single brand of regular ground coffee at any one time.

Second, and for a much wider range of products, it may be better to study household-level behavior because it captures the interdependence of family members on each other's consumption behavior. For example, are the breakfast cereals bought by a household really the summation of individually-determined brand choices? Studies focusing on individual preferences frequently ignore the interdependencies within families of attitude, behavior and choice.

Third, while it is true that aggregating individual behavior into household behavior will disguise some of the heterogeneity in preferences in the population if preferences are truly individual, the extent of this misaggregation is an empirical issue that is not resolved. It is known for most frequently-bought goods that considerable heterogeneity in buying behavior is retained in the household-level data. Whether this reflects all or even most of the heterogeneity in behavior is hard to say, largely because individual-level behavior is not at present well monitored.

Finally, even when we have measures of individual-level behavior and/or attitude, the problem of misaggregation remains. To the extent that preference is determined by consumption situation, any measure of brand loyalty that is not conditional on situation is misaggregated (Srivastava, Alpert and Shocker 1984). There is a tendency to reject household-level data based upon a suspicion of misaggregation while blithely risking misaggregation across consumption situations. This is not to say that loyalty should always be measured conditional on both individual and situation, nor that we should always resort to household-level data because "misaggregation is unavoidable." Rather, a level of aggregation should be selected after consideration of the risk of misaggregation for that product category, recognizing that less aggregate data lower this risk while being more costly and perhaps less reliable in other respects.

Interestingly, J&C do not argue the misaggregation case against household-level data. Instead, they argue that any unit of measurement defined at the household level cannot be tied very well to individual-level data collected from the same households (such as personality scores) because we do not observe which family members are doing the purchasing. Yet the noncombinability of observed household behavior with attitudinal measures of individuals is a problem for composite measures of brand loyalty, and not with either type of measurement taken singly. If forced to choose between behavioral or attitudinal measures, J&C confess to a bias towards attitudinal measures. I suspect that most management scientists have a bias towards the behavioral because the data are objective and because they are closely Lied to the behavior of interest, i.e. to what people buy and not to what they say they like.

TABLE 2

J&C'S FOUR HYPOTHETICAL HOUSEHOLDS

"Behavioral Measures Explain Only the Outcome of the Decision-Making Process, and Not the Process Itself"

This assertion also makes a management scientist pause, because a measure that tells you what you want to know and nothing else is, ceteris paribas, a good measure. However, it is incorrect to suggest that purchase probability cannot be used to yield insight into the causes of loyalty to a brand. This role can be served by models such as logit that relate SPOP to marketing variables and household characteristics using a formulation such as:

ln[(SPOPi+1)/(J-SPOPi-1)] = b ln(Xi) + ei       (5)

where b is a parameter (or vector of parameters) and Xi is a positive exogenous variable (or vector of such variables). Variables not logically positive are first made so by exponentiation. The SPOP scores are more reliably estimated for some households (such as those that bought many times) than others, which means that ei is heteroscedastic. But the error variance is known to be

Var(SPOPi | ni) = J2 (ai + ni) (A - ai + N - ni) / [(A + N)2 (A + N + 1)]      (6)

and this can be used to estimate the logit model by weighted least squares (Jones and Zufryden 1980).

"Most Behavioral Measures Fail to Account For Multibrand Loyalty"

SPOP accounts for multibrand loyalty. A household that buys two brands half the time out of four available would be registered as being (equally) loyal to the two brands and disloyal to the others (the SPOP scores would be: +1, +1, -1, -1). The loyalty to either brand would not be as high as for households that bought only a single brand (the SPOP scores are then: +3, -1, -1, -1), but then neither should it be.

"Relations Among the Various Behavioral Measures Have Not Been Studied Empirically"

Here, I&C provide an example of purchasing by four households and show how different measures would reach different conclusions about who is loyal and who is not. Disagreement among measures is not the fault of any single measure, nor is the absence of research showing agreement. Reproduced here from I&C (p. 44) are 10 hypothetical purchases by 4 households towards the 8 brands A through H. Remember that these examples were invented by the authors to make their point. Their examples are given in Table 2, along with the implied SPOP scores.

The SPOP measures of brand loyalty in this case are the purchase shares times 8 less 1. (For panel data there is a better estimate of purchase probabilities, as was discussed above.) Households 3 and 4 have the same SPOP measures for the brands. J&C invented households 3 and 4 to show that a "three purchases in a row" criterion for brand loyalty would create an unwanted distinction between the two households in their loyalty to brand A.

That household 1 should have the same measured loyalty to brand A as households 3 and 4 makes one pause. Household 1 concentrates its .3 non-A purchase probability on brand B, but households 3 and 4 spread their .3 non-A purchase probability evenly among three other brands. Should the degree of-concentration of non-A purchases affect the loyalty measure for brand A as does their frequency? The best answer is no for three reasons. First, sales lost are sales lost, and whether to one other brand or to several seems to belong to the realm of second order effects.

Second, analysis of the logit market attraction model of purchase probability supports ignoring degree of concentration of purchasing among other brands. The market attraction model appears in its simplest form as:

Pi = sib / (Sj sjb)     (7)

where Xi and b are as defined previously. If, as is usually assumed, revenues are linear in purchase probability and costs are linear in the marketing variable, then the optimal level of the marketing variable for brand i (ignoring competitive response) is determined by the derivative of purchase probability with respect to the marketing variable:

dPi / dXi = b Pi (1 - Pi) / Xi       (8)

Note that the distribution of purchases among non-i brands is irrelevant.

Finally, SPOP is a vector of a household's loyalties to all brands in the market, and so information about the distribution of loyalty over non-A brands is retained. Attempting to incorporate loyalty towards non-A brands into the measure of loyalty towards A (and doing this simultaneously for all brands A-H) would be redundant and unnecessarily confusing.

"Behavioral Measures Lack Demonstrated Reliability and Validity"

The reliability for SPOP is readily computed. The reliability coefficient for an estimate is given by the ratio of the variance in true scores to the variance in observed scores (cf. Nunnally 1978, p.215). In the case of SPOP, this is the variance in true purchase probability for a brand over the sample of households, relative to the variance in the within-household market share (n i /N) over households. The former is given by (Greene 1982, p. 155):

Var(Pi) = ai (A - ai) / [A2 (A+1)]      (9)

The variance in ni is given by (Greene 1982, p. 157):

Var(ni) = Nai (A - ai) (A + N) / [A2 (A+1)]      (10)

Hence the reliability coefficient is given by Var(Pi)/Var(ni/N):

r = N/ (A + N)      (11)

Since SPOP is a linear function of purchase probability, its reliability is N / (A + N) as well.

Interestingly, psychometricians (Nunnally 1978, p. 217) recommend that deviations in true scores (from their mean) be estimated from the deviations in observed scores by multiplying the latter by the reliability coefficient. In our case, the estimate of a household's purchase probability for brand i (Pri) is:

Pri - ai / A = r (ni / N - ai / A), or

            Pri = (ni + ai) / (A + N)      (12)

This is the same as the Bayes estimator PBi recommended above.

AN EMPIRICAL EXAMPLE

SAMI/Burke, Inc. made available the records of purchases of six liquid laundry detergents by a panel for a one year period. 1268 households bought at least one _, nit, and the average number bought by them was 7.6. Observed for each household is the number of units bought of each brand. A fit of the Dirichlet-Multinomial model by maximum likelihood yielded parameter estimates a 1 ,...,a 6 equal to (.198, .512, .193, .150, .179, .116). A = a 1 +...+a 6 is therefore estimated as 1.348. Then the reliability of SPOP for a household that bought a total of N units (all brands combined) is equal to N / (N + 1.348), regardless of which brands were bought. Households bought anywhere between 1 and 70 units, but the average reliability over all households is .73. The means and standard deviations of the SPOP scores over all households for the six brands are shown in Table 3.

TABLE 3

SCORES FOR SIX LIQUID LAUNDRY DETERGENTS

From Table 3 it is apparent that the highest average loyalty is to Wisk. This brand is in fact the "prototypical" brand--the one that founded the liquid detergent category. It's having the highest average loyalty measure reflects its having the largest market share. However, the households differ more in their loyalty to Wisk than to any other brand. The standard deviations for all brands are high, indicating that considerable heterogeneity in consumer behavior is retained by the household-level data. This is not surprising for laundry detergent if you believe that the household, and not individual family members, is the consuming unit. Were demographic data available for these households, then brand loyalty could be related to these characteristics by logit, as was shown previously.

CONCLUSION

At the time of the Jacoby and Chestnut (1978) study, the benefit I cost = value of several methods of measuring brand loyalty were such that composite (behavioral plus attitudinal) measures were the best value overall. But in the past decade the widespread availability of low cost behavioral data has raised the value of behavioral data without facilitating composite measurement. A comparison of the different measures of brand loyalty in terms of value may be as shown in Table 4. It appears that, in the case of frequently-bought consumer goods, behavioral measures of brand loyalty such as SPOP are here to stay for the foreseeable future.

TABLE 4

VALUE OF DIFFERENT MEASURES OF BRAND LOYALTY FOR FREQUENTLY-BOUGHT CONSUMER GOODS

REFERENCES

Daganzo, Carlos (1979), Multinomial Probit, New York: Academic.

Greene, Jerome D. (1982), Consumer Behavior Models for NonStatisticians, New York: Praeger.

Jacoby, Jacob and Robert W. Chestnut (1978), Brand Loyalty Measurement and Management, New York: Wiley.

Jones, J. Morgan and Fred S. Zufryden (1980), "Adding Explanatory Variables to a Consumer Purchase Behavior Model: An Exploratory Study," Journal of Marketing Research, 17 (August), pp. 323-344.

Nakanishi, Masao and Lee G. Cooper (1974), "Parameter Estimation for a Multiplicative Competitive Interaction Model-Least Squares Approach," Journal of Marketing Research, 11 (August), pp. 303-311.

Nunnally, Jum C. (1978), Psychometric Theory, New York: McGraw-Hill.

Srivastava, Rajendra K., Mark I. Alpert and Allan D. Shocker (1984), "A Customer-oriented Approach for Determining Market Structure," 48 (Spring), pp. 32-45.

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