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A maximum likelihood methodology for clusterwise linear regression

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Abstract

This paper presents a conditional mixture, maximum likelihood methodology for performing clusterwise linear regression. This new methodology simultaneously estimates separate regression functions and membership inK clusters or groups. A review of related procedures is discussed with an associated critique. The conditional mixture, maximum likelihood methodology is introduced together with the E-M algorithm utilized for parameter estimation. A Monte Carlo analysis is performed via a fractional factorial design to examine the performance of the procedure. Next, a marketing application is presented concerning the evaluations of trade show performance by senior marketing executives. Finally, other potential applications and directions for future research are identified.

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References

  • ADDELMAN, S. (1962), “Orthogonal Main Effects Plans for Asymmetrical Factorial Experiments,”Technometrics, 4, 21–46.

    Google Scholar 

  • AKAIKE, H. (1974), “A New Look at Statistical Model Identification,”IEEE Transactions on Automatic Control, AC-19, 716–723.

    Google Scholar 

  • BASFORD, K.E., and MCLACHLAN, G.J. (1985), “The Mixture Method of Clustering Applied to Three-Way Data,”Journal of Classification, 2, 109–125.

    Google Scholar 

  • BINDER, D.A. (1978), “Bayesian Cluster Analysis,”Biometrika, 65, 31–38.

    Google Scholar 

  • BONOMA, T.V. (1983), “Get More Out of Your Trade Shows,”Harvard Business Review, 61, 75–83.

    Google Scholar 

  • BOZDOGAN, H. (1983), “Determining the Number of Component Clusters in Standard Multivariate Normal Mixture Models Using Model-Selection Criterion,”Technical Report VIC/DOM/A83-1, Army Research Office.

  • CAVANAUGH, S. (1976), “Setting Objectives and Evaluating the Effectiveness of Trade Show Exhibits,”Journal of Marketing, 40, 100–103.

    Google Scholar 

  • CHARLIER, C.V.L., and WICKSELL, S.D. (1924), “On the Dissection of Frequency Functions,”Arkiv för Matematik, Astronomi Och Fysik., Bd.18, No.6, 85–98.

    Google Scholar 

  • CLEAVER, J. (1982), “You Don't Have to be a Star in this Show,”Advertising Age, 53, 9.

    Google Scholar 

  • COHEN, A.C. (1967), “Estimation in Mixtures of Two Normal Distributions,”Technometrics, 9, 15–28.

    Google Scholar 

  • COOPER, P. W. (1964), “Non Supervised Adaptive Signal Detection and Pattern Recognition,”Information and Control, 7, 416–444.

    Google Scholar 

  • DAY, N.E. (1969), “Estimating the Components of a Mixture of Normal Distributions,”Biometrika, 56, 463–474.

    Google Scholar 

  • DEMPSTER, A.P., LAIRD, N.M., and RUBIN, D.B. (1977), “Maximum likelihood from Incomplete Data Via the E-M Algorithm,”Journal of the Royal Statistical Society, B39, 1–38.

    Google Scholar 

  • DESARBO, W.S. (1982), “GENNCLUS: New Models for General Nonhierarchical Clustering Analysis,”Psychometrika, 47, 449–476.

    Google Scholar 

  • DESARBO, W.S., and CARROLL, J.D. (1985), “Three Way Metric Unfolding Via Alternating Weighted Least Squares,”Psychometrika, 50, 275–300.

    Google Scholar 

  • DESARBO, W.S., CARROLL, J.D., CLARK, L.A., and GREEN, P.E. (1984), “Synthesized Clustering: A Method for Amalgamating Alternative Clustering Bases with Differential Weighting of Variables,”Psychometrika, 49, 57–78.

    Google Scholar 

  • DESARBO, W.S., and MAHAJAN, V. (1984), “Constrained Classification: The Use of a Priori Information in Cluster Analysis,”Psychometrika, 49, 187–215.

    Google Scholar 

  • DESARBO, W.S., OLIVER, R., and RANGASWAMY, A. (1988), “A Simulated Annealing Methodology for Clusterwise Linear Regression,”Working Paper, University of Michigan, Ann Arbor, MI.

    Google Scholar 

  • De SOETE, G., DESARBO, W.S., FURNAS, G.W., and CARROLL, J.D. (1984), “The Representation of Nonsymmetric Rectangular Proximity Data by Ultrametric and Path Length Tree Structure,”Psychometrika, 49, 289–310.

    Google Scholar 

  • De SOETE, G., DESARBO, W.S., and CARROLL, J.D. (1985), “Optimal Variable Weighting for Hierarchical Clustering: An Alternating Least Squares Algorithm,”Journal of Classification, 2, 173–192.

    Google Scholar 

  • DUNN, S.W., and BARBAN, A.M. (1986),Advertising, 6th ed., Hinsdale, IL: Dryden.

    Google Scholar 

  • DYNKIN, E.B. (1961), “Necessary and Sufficient Statistics for a Family of Probability Distributions,”Selected Translations in Mathematical Statistics and Probability, Providence, RI: American Mathematical Society, 17–40.

  • EVERITT, B.S., and HAND, D.J. (1981), Finite Mixture Distribution, New York: Chapman and Hall.

    Google Scholar 

  • GANESALINGAM, S., and MCLACHLAN, G.J. (1981), “Some Efficiency Results for the Estimation of the Mixing Proportion in a Mixture of Two Normal Distributions,”Biometrics, 37, 23–33.

    Google Scholar 

  • GREEN, P.E., and RAO, V.R. (1971), “Conjoint Measurement for Quantifying Judgmental Data,”Journal of Marketing Research, 8, 355–363.

    Google Scholar 

  • GREEN, P.E., and TULL, D.S. (1978),Research for Marketing Decisions, 4th ed., Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • GUILFORD, J.A. (1954),Psychometric Methods, 2nd ed., New York: McGraw-Hill.

    Google Scholar 

  • HAAS, R.W. (1982),Industrial Marketing Management, 2nd ed., Boston: Kent.

    Google Scholar 

  • HARTIGAN, J.A. (1975),Clustering Algorithms, New York: Wiley.

    Google Scholar 

  • HARTIGAN, J.A. (1977), “Distribution Problems in Clustering,” inClassification and Clustering, ed. J. Van Ryzin, New York: Academic Press, 45–71.

    Google Scholar 

  • HASSELBLAD, V. (1966), “Estimation of Parameters for a Mixture of Normal Distributions,”Technometrics, 8, 431–444.

    Google Scholar 

  • HATCH, M. (1981),How to Improve Sales Success at Trade Shows, New Canaan, CT: Trade Show Bureau.

    Google Scholar 

  • HOSMER, D.W. (1974), “Maximum Likelihood Estimates of the Parameters of a Mixture of Two Regression Lines,”Communications in Statistics, 3, 995–1006.

    Google Scholar 

  • HUTT, M.D., and SPEH, T.W. (1985),Industrial Marketing Management, 2nd ed., Hinsdale, IL: Dryden.

    Google Scholar 

  • JOHNSTON, J. (1984),Econometric Methods, 3rd ed., New York: McGraw-Hill.

    Google Scholar 

  • JUDGE, G.C., GRIFFITHS, W.E., HILL, R.C., LUTKEPOHL, H., and LEE, T.C. (1985),Theory and Practice of Econometrics, New York: Wiley.

    Google Scholar 

  • KERIN, R.A., and CRON, W.L. (1987), “Assessing Trade Show Functions and Performance: An Exploratory Study,”Journal of Marketing, 51, 87–94.

    Google Scholar 

  • KONIKOW, R.B. (1983),How to Participate Profitably in Trade Shows, rev. ed., Chicago: Dartnell.

    Google Scholar 

  • KRAYENBUEHL, T.E. (1985),Country Risk: Assessment and Monitoring, Lexington, MA: Lexington.

    Google Scholar 

  • LACHENBRUCH, P.A. (1967), “An Almost Unbiased Method of Obtaining Confidence Intervals for the Probability of Misclassification in Discriminant Analysis,”Biometrics, 23, 639–645.

    PubMed  Google Scholar 

  • LACHENBRUCH, P.A. (1975),Discriminant Analysis, New York: Hafner Press.

    Google Scholar 

  • LILIEN, G.L. (1983), “A Descriptive Model of the Trade Show Budgeting Decision Process,”Industrial Marketing Management, 12, 25–29.

    Google Scholar 

  • LOUIS, T.A. (1982), “Finding the Observed Information Matrix When Using the E-M Algorithm,”Journal of the Royal Statistical Society, B44, 226–233.

    Google Scholar 

  • MACQUEEN, J. (1967), “Some Methods for Classification and Analysis of Multivariate Observations,” in the5th Berkeley Symposium of Mathematics, Statistics and Probability, Vol. 1, eds. L.M. LeCam and J. Neyman, Los Angeles, CA: University of California Press, 281–298.

    Google Scholar 

  • MADDALA, G.S. (1976),Econometrics, New York: McGraw-Hill.

    Google Scholar 

  • MARRIOTT, F.H.C. (1975), “Separating Mixtures of Normal Distributions,”Biometrics, 31, 767–769.

    Google Scholar 

  • MCLACHLAN, G.J. (1982), “The Classification and Mixture Maximum Likelihood Approaches to Cluster Analysis,” inHandbook of Statistics, Vol. 2, eds. P.R. Krishnaiah and L.N. Kanal, Amsterdam: North-Holland, 199–208.

    Google Scholar 

  • MCLACHLAN, G.J. (1987), “On Bootstrapping the Likelihood Ratio Test Statistic for the Number of Components in a Normal Mixture,”Applied Statistics, 36, 318–324.

    Google Scholar 

  • MCLACHLAN, G.J. and BASFORD, K.E. (1988),Mixture Models: Inference and Applications to Clustering, New York: Marcel Dekker.

    Google Scholar 

  • MEE, W.W. (1982),Trade Show Exhibit Cost Analysis, East Orleans, MA: Trade Show Bureau.

    Google Scholar 

  • MEE, W.W. (1983a),Trade Show Industry Growth 1972–1981, Projected Growth 1981–1991, East Orleans, MA: Trade Show Bureau.

    Google Scholar 

  • MEE, W.W. (1983b),The Exhibitors: Their Trade Show Practices, East Orleans, MA: Trade Show Bureau.

    Google Scholar 

  • MEE, W.W. (1984),Audience Characteristics — Regional and National Trade Shows, East Orleans, MA: Trade Show Bureau.

    Google Scholar 

  • MORRISON, D. (1969), “On the Interpretation of Discriminant Analysis,”Journal of Marketing Research, 6, 156–163.

    Google Scholar 

  • PEARSON, K. (1894), “Contribution to the Mathematical Theory of Evolution,”Philosophical Transactions of the Royal Society of London, A, 185, 71–110.

    Google Scholar 

  • PETERS, B.C., and WALKER, H.F. (1978), “An Iterative Procedure for Obtaining Maximum Likelihood Estimates of the Parameters for a Mixture of Normal Distributions,”SIAM Journal on Applied Mathematics, 35, 362–378.

    Google Scholar 

  • PORTER, M.E. (1980),Competitive Strategy, New York: Free Press.

    Google Scholar 

  • QUANDT, R.E. (1972), “A New Approach to Estimating Switching Regressions,”Journal of the American Statistical Association, 67, 306–310.

    Google Scholar 

  • QUANDT, R.E., and RAMSEY, J.B. (1978), “Estimating Mixtures of Normal Distributions and Switching Regressions,”Journal of the American Statistical Association, 73, 730–738.

    Google Scholar 

  • REDNER, R.A., and WALKER, H.F. (1984), “Mixture Densities, Maximum Likelihood, and the E-M Algorithm,”SIAM Review, 2, 195–239.

    Google Scholar 

  • RICH, M. (1985), “Regional Shows Give Small Marketers an Even Break,”Marketing News, May 27, 19, p. 15.

    Google Scholar 

  • SCLOVE, S.C. (1977), “Population Mixture Models and Clustering Algorithms,”Communication in Statistics, A6, 417–434.

    Google Scholar 

  • SCLOVE, S.L. (1983), “Application of the Conditional Population Mixture Model to Image Segmentation,”IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-5, 428–433.

    Google Scholar 

  • SCLOVE, S.L. (1987), “Application of Model-Selection Criteria to Some Problems in Multivariate Analysis,”Psychometrika, 52, 333–343.

    Google Scholar 

  • SCOTT, A.J., and SYMONS, M.J. (1971), “Clustering Methods Based on Likelihood Ratio Criteria,”Biometrics, 27, 238–397.

    Google Scholar 

  • SPÄTH, H. (1979), “Algorithm 39: Clusterwise Linear Regression,”Computing, 22, 367–373.

    Google Scholar 

  • SPÄTH, H. (1981), “Correction to Algorithm 39: Clusterwise Linear Regression,”Computing, 26, 275.

    Google Scholar 

  • SPÄTH, H. (1982), “Algorithm 48: A Fast Algorithm for Clusterwise Linear Regression,”Computing, 29, 175–181.

    Google Scholar 

  • SPÄTH, H. (1985),Cluster Dissection and Analysis, New York: Wiley.

    Google Scholar 

  • SYMONS, M.J. (1981), “Clustering Criteria and Multivariate Normal Mixtures,”Biometrics, 37, 35–43.

    Google Scholar 

  • TEICHER, H. (1961), “Identifiability of Mixtures,”Annals of Mathematical Statistics, 32, 244–248.

    Google Scholar 

  • TEICHER, H. (1963), “Identifiability of Finite Mixtures,”Annals of Mathematical Statistics, 34, 1265–1269.

    Google Scholar 

  • THEIL, H. (1971),Principles of Econometrics, New York: Wiley.

    Google Scholar 

  • TITTERINGTON, D.M., SMITH, A.F.M., and MAKOV, U.E. (1985),Statistical Analysis of Finite Mixture Distributions, New York: Wiley.

    Google Scholar 

  • VEAUX, R.D. (1986), “Parameter Estimation for a Mixture of Linear Regressions,”Technical Report No.247, Department of Statistics, Stanford University, Stanford, CA.

    Google Scholar 

  • WILSON, D.L., and SARGENT, R.G. (1979), “Some Results of Monte Carlo Experiments in Estimating the Parameters of the Finite Mixed Exponential Distribution,” in theProceedings of the Twelfth Annual Symposium Interface, ed. J.F. Gentleman, University of Waterloo, Ontario, Canada, 461–465.

  • WOLFE, J.H. (1965), “A Computer Programs for the Maximum Likelihood Analysis of Types.” Technical Bulletin, 65-15, U.S. Naval Personnel Research Activity, San Diego, CA.

    Google Scholar 

  • WOLFE, J.H. (1967), “NORMIX: Computational Methods for Estimating the Parameters of Multivariate Normal Mixtures of Distributions,”Research Memorandum SRM 68-2, U.S. Naval Personnel Research Activity, San Diego, CA.

    Google Scholar 

  • WOLFE, J.H. (1970), “Pattern Clustering by Multivariate Mixture Analysis,”Multivariate Behavioral Research, 5, 329–350.

    Google Scholar 

  • WOLFE, J.H. (1971), “A Monte Carlo Study of the Sampling Distribution of the Likelihood Ratio for Mixture of Multinormal Distributions,”Technical Bulletin: Naval Personnel and Training Research Laboratory, STB 72-2, San Diego, CA.

  • YAKOWITZ, S.J. (1970), “Unsupervised Learning and the Identification of Finite Mixtures,”IEEE Transactions on Information Theory and Control, IT-16, 330–338.

    Google Scholar 

  • YAKOWITZ, S.J., and SPRAGINS, J.D. (1968), “On the Identifiability of Finite Mixtures,”Annals of Mathematical Statistics, 39, 209–214.

    Google Scholar 

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Authors and Affiliations

  1. Departments of Marketing and Statistics, Business School of the University of Michigan, 48104, Ann Arbor, MI, USA

    Wayne S. DeSarbo

  2. Department of Marketing, Edwin L. Cox School of Business, Southern Methodist University, 75275, Dallas, TX, USA

    William L. Cron

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  1. Wayne S. DeSarbo
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  2. William L. Cron
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DeSarbo, W.S., Cron, W.L. A maximum likelihood methodology for clusterwise linear regression. Journal of Classification 5, 249–282 (1988). https://doi.org/10.1007/BF01897167

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