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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
ADDELMAN, S. (1962), “Orthogonal Main Effects Plans for Asymmetrical Factorial Experiments,”Technometrics, 4, 21–46.
AKAIKE, H. (1974), “A New Look at Statistical Model Identification,”IEEE Transactions on Automatic Control, AC-19, 716–723.
BASFORD, K.E., and MCLACHLAN, G.J. (1985), “The Mixture Method of Clustering Applied to Three-Way Data,”Journal of Classification, 2, 109–125.
BINDER, D.A. (1978), “Bayesian Cluster Analysis,”Biometrika, 65, 31–38.
BONOMA, T.V. (1983), “Get More Out of Your Trade Shows,”Harvard Business Review, 61, 75–83.
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.
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.
CLEAVER, J. (1982), “You Don't Have to be a Star in this Show,”Advertising Age, 53, 9.
COHEN, A.C. (1967), “Estimation in Mixtures of Two Normal Distributions,”Technometrics, 9, 15–28.
COOPER, P. W. (1964), “Non Supervised Adaptive Signal Detection and Pattern Recognition,”Information and Control, 7, 416–444.
DAY, N.E. (1969), “Estimating the Components of a Mixture of Normal Distributions,”Biometrika, 56, 463–474.
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.
DESARBO, W.S. (1982), “GENNCLUS: New Models for General Nonhierarchical Clustering Analysis,”Psychometrika, 47, 449–476.
DESARBO, W.S., and CARROLL, J.D. (1985), “Three Way Metric Unfolding Via Alternating Weighted Least Squares,”Psychometrika, 50, 275–300.
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.
DESARBO, W.S., and MAHAJAN, V. (1984), “Constrained Classification: The Use of a Priori Information in Cluster Analysis,”Psychometrika, 49, 187–215.
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.
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.
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.
DUNN, S.W., and BARBAN, A.M. (1986),Advertising, 6th ed., Hinsdale, IL: Dryden.
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.
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.
GREEN, P.E., and RAO, V.R. (1971), “Conjoint Measurement for Quantifying Judgmental Data,”Journal of Marketing Research, 8, 355–363.
GREEN, P.E., and TULL, D.S. (1978),Research for Marketing Decisions, 4th ed., Englewood Cliffs, NJ: Prentice-Hall.
GUILFORD, J.A. (1954),Psychometric Methods, 2nd ed., New York: McGraw-Hill.
HAAS, R.W. (1982),Industrial Marketing Management, 2nd ed., Boston: Kent.
HARTIGAN, J.A. (1975),Clustering Algorithms, New York: Wiley.
HARTIGAN, J.A. (1977), “Distribution Problems in Clustering,” inClassification and Clustering, ed. J. Van Ryzin, New York: Academic Press, 45–71.
HASSELBLAD, V. (1966), “Estimation of Parameters for a Mixture of Normal Distributions,”Technometrics, 8, 431–444.
HATCH, M. (1981),How to Improve Sales Success at Trade Shows, New Canaan, CT: Trade Show Bureau.
HOSMER, D.W. (1974), “Maximum Likelihood Estimates of the Parameters of a Mixture of Two Regression Lines,”Communications in Statistics, 3, 995–1006.
HUTT, M.D., and SPEH, T.W. (1985),Industrial Marketing Management, 2nd ed., Hinsdale, IL: Dryden.
JOHNSTON, J. (1984),Econometric Methods, 3rd ed., New York: McGraw-Hill.
JUDGE, G.C., GRIFFITHS, W.E., HILL, R.C., LUTKEPOHL, H., and LEE, T.C. (1985),Theory and Practice of Econometrics, New York: Wiley.
KERIN, R.A., and CRON, W.L. (1987), “Assessing Trade Show Functions and Performance: An Exploratory Study,”Journal of Marketing, 51, 87–94.
KONIKOW, R.B. (1983),How to Participate Profitably in Trade Shows, rev. ed., Chicago: Dartnell.
KRAYENBUEHL, T.E. (1985),Country Risk: Assessment and Monitoring, Lexington, MA: Lexington.
LACHENBRUCH, P.A. (1967), “An Almost Unbiased Method of Obtaining Confidence Intervals for the Probability of Misclassification in Discriminant Analysis,”Biometrics, 23, 639–645.
LACHENBRUCH, P.A. (1975),Discriminant Analysis, New York: Hafner Press.
LILIEN, G.L. (1983), “A Descriptive Model of the Trade Show Budgeting Decision Process,”Industrial Marketing Management, 12, 25–29.
LOUIS, T.A. (1982), “Finding the Observed Information Matrix When Using the E-M Algorithm,”Journal of the Royal Statistical Society, B44, 226–233.
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.
MADDALA, G.S. (1976),Econometrics, New York: McGraw-Hill.
MARRIOTT, F.H.C. (1975), “Separating Mixtures of Normal Distributions,”Biometrics, 31, 767–769.
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.
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.
MCLACHLAN, G.J. and BASFORD, K.E. (1988),Mixture Models: Inference and Applications to Clustering, New York: Marcel Dekker.
MEE, W.W. (1982),Trade Show Exhibit Cost Analysis, East Orleans, MA: Trade Show Bureau.
MEE, W.W. (1983a),Trade Show Industry Growth 1972–1981, Projected Growth 1981–1991, East Orleans, MA: Trade Show Bureau.
MEE, W.W. (1983b),The Exhibitors: Their Trade Show Practices, East Orleans, MA: Trade Show Bureau.
MEE, W.W. (1984),Audience Characteristics — Regional and National Trade Shows, East Orleans, MA: Trade Show Bureau.
MORRISON, D. (1969), “On the Interpretation of Discriminant Analysis,”Journal of Marketing Research, 6, 156–163.
PEARSON, K. (1894), “Contribution to the Mathematical Theory of Evolution,”Philosophical Transactions of the Royal Society of London, A, 185, 71–110.
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.
PORTER, M.E. (1980),Competitive Strategy, New York: Free Press.
QUANDT, R.E. (1972), “A New Approach to Estimating Switching Regressions,”Journal of the American Statistical Association, 67, 306–310.
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.
REDNER, R.A., and WALKER, H.F. (1984), “Mixture Densities, Maximum Likelihood, and the E-M Algorithm,”SIAM Review, 2, 195–239.
RICH, M. (1985), “Regional Shows Give Small Marketers an Even Break,”Marketing News, May 27, 19, p. 15.
SCLOVE, S.C. (1977), “Population Mixture Models and Clustering Algorithms,”Communication in Statistics, A6, 417–434.
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.
SCLOVE, S.L. (1987), “Application of Model-Selection Criteria to Some Problems in Multivariate Analysis,”Psychometrika, 52, 333–343.
SCOTT, A.J., and SYMONS, M.J. (1971), “Clustering Methods Based on Likelihood Ratio Criteria,”Biometrics, 27, 238–397.
SPÄTH, H. (1979), “Algorithm 39: Clusterwise Linear Regression,”Computing, 22, 367–373.
SPÄTH, H. (1981), “Correction to Algorithm 39: Clusterwise Linear Regression,”Computing, 26, 275.
SPÄTH, H. (1982), “Algorithm 48: A Fast Algorithm for Clusterwise Linear Regression,”Computing, 29, 175–181.
SPÄTH, H. (1985),Cluster Dissection and Analysis, New York: Wiley.
SYMONS, M.J. (1981), “Clustering Criteria and Multivariate Normal Mixtures,”Biometrics, 37, 35–43.
TEICHER, H. (1961), “Identifiability of Mixtures,”Annals of Mathematical Statistics, 32, 244–248.
TEICHER, H. (1963), “Identifiability of Finite Mixtures,”Annals of Mathematical Statistics, 34, 1265–1269.
THEIL, H. (1971),Principles of Econometrics, New York: Wiley.
TITTERINGTON, D.M., SMITH, A.F.M., and MAKOV, U.E. (1985),Statistical Analysis of Finite Mixture Distributions, New York: Wiley.
VEAUX, R.D. (1986), “Parameter Estimation for a Mixture of Linear Regressions,”Technical Report No.247, Department of Statistics, Stanford University, Stanford, CA.
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.
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.
WOLFE, J.H. (1970), “Pattern Clustering by Multivariate Mixture Analysis,”Multivariate Behavioral Research, 5, 329–350.
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.
YAKOWITZ, S.J., and SPRAGINS, J.D. (1968), “On the Identifiability of Finite Mixtures,”Annals of Mathematical Statistics, 39, 209–214.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Issue Date:
DOI: https://doi.org/10.1007/BF01897167