An inter-models distance for clustering utility functions

Statistica Applicata Vol. 18, n. 3, 2006 521

AN INTER-MODELS DISTANCE FOR CLUSTERINGUTILITY FUNCTIONS1

Elvira Romano, Carlo Lauro

Dipartimento di Matematica e Statistica, Università di Napoli Federico II,Complesso Universitario di Monte S. Angelo, Via Cinthia, I - 80126 Napoli [email protected]; [email protected]

Giuseppe Giordano

Dipartimento di Scienze Economiche e Statistiche Università di SalernoVia Ponte Don Melillo, 84084 Fisciano (Sa), [email protected]

Abstract

Conjoint Analysis is one of the most widely used techniques in the assessment of theconsumer’s behaviors. This method allows to estimate the partial utility coefficientsaccording to a statistical model linking the overall note of preference with the attributelevels describing the stimuli. Conjoint analysis results are useful in new-product positioningand market segmentation. In this paper a cluster-based segmentation strategy based on anew metric has been proposed. The introduced distance is based on a convex linearcombination of two Euclidean distances em bedding information both on the estimatedparameters and on the model fitting. Market segments can be then defined according to theproximity of the part-worth coecients and to the explicative power of the estimated models.

Key words: Multiattribute Preference Data, Conjoint Analysis, Cluster Analysis, MarketSegmentation.

1 This paper was financially supported by MIUR grants: “Multivariate Statistical and VisualizationMethods to Analyze, to Summarize, to Evaluate Performance Indicators”, coordinated by prof.M.R. D’Esposito and “Models for Designing and Measuring Customer Satisfaction”, coordinatedby prof. C.N. Lauro.

522 Romano E., Lauro C., Giordano G.

1. INTRODUCTION

2. THE DATA STRUCTURE AND THE INTER-MODELS DISTANCE

An inter-models distance for clustering utility functions 523

Tab.1: The data collection.

Utility models Coefficients Model fitting

Model 1 w11, . . . , w

1k, . . . , w

1K – 1 w1

K

… … …Model j w j1, . . . , w jk, . . . ,w j K – 1 w j K

… … …Model J wJ

1, . . . , w Jk, . . . ,w JK – 1 w JK



3. CLUSTERING UTILITY FUNCTIONS


4. SIMULATIONS STUDY


Tab. 2: Simulation plan for the three classes of models with dierent coecients and similar fittingvalues.

In order to generate three sets of models with a quite good approximation, webuild the global preference ratings according to the model (10) and with coecientsgiven in Table 2. They are used as dependent variables in a multivariate multipleregression model with dummy explicative variables defined by the orthogonalexperimental design in Table 3.

Tab. 3: Experimental design.

Intercept x1 x2 x3 x4

1 1 1 -1 -1 12 1 -1 -1 -1 13 1 1 1 0 14 1 -1 1 0 15 1 1 0 1 16 1 -1 0 1 17 1 1 -1 -1 -18 1 -1 -1 -1 -19 1 1 1 0 -1

10 1 -1 1 0 -111 1 1 0 1 -112 1 -1 0 1 -1


Fig: 1. Box-plot of the adj - R2, the intercept and the four estimated coecients.


Fig. 2. The tree structure of the simulated models.

Fig. 3. Distribution of the optimum λ -values.


Tab. 4: Simulation plan involving a hidden tree structure.

N. w0 w1 w2 w3 w4 ei

Class A 20 6.50 -1.33 1.00 1.25 -1.83 N(0, 1)

Class B 20 6.50 -1.33 1.00 1.25 -1.83 N(0, 3)

Class C 20 6.50 -0.50 -0.25 3.00 0.00 N(0, 1)

5. CONCLUDING REMARKS


Fig. 4: Distribution of the λ -values corresponding to the maximum cophenetic coecient in 100replications of the clustering.

Fig. 5: Dendrogram of the models. λ = 1:the bad fitted models are hidden in the two clusterstructure.


Fig. 6: Dendrogram of the models. λ = 0.2:the cophenetic coecient is maximum, a three clusterstructure is more evident.


REFERENCES

GREEN P.E., SRINIVASAN V. (1990), Conjoint Analysis in marketing: New Developments WithImplication for Research and Practice, Journal of Marketing, 25: 3–19.

GUSTAFSSON A., HERMANN A., HUBER F. (2000), Conjoint Measurement. Methods andApplication. Berlin, Heidelberg: Springer Verlag.

HENNING C. (2000), Identifiability of Models for Clusterwise Linear Regression, Journal ofClassification, 17: 273–296.

LAURO C., SCEPI G., GIORDANO G., (2002) Cluster Based Conjoint Analysis, in Proceedings ofSixth International Conference on Social Science Methodology, RC Logic & 33 Methodology,August 17-20, 2004 Amsterdam.

PLAIA A. (2003), Constrained Clusterwise Linear Regression, in New Developments in Classificationand Data Analysis, M. Vichi P. Monari, S. Mignani, S. Montanarini eds., Springer, Bologna79–86.

SPAETH H. (1979), Clusterwise Linear Regression, Computing, 22: 367–373.

TAKANE Y., DE LEW J., YOUNG F.W. (1990), Regression with Qualitative and QuantitativeVariables: An Alternating Least Squares Method with Optimal Scaling Features, Psycometrika,41(4): 505–529.

VRIENS M., WEDEL M. and WILMS T. (1996), Metric Conjoint Segmentation Methods: a MonteCarlo comparison, Journal of Marketing Research, 33: 73–85.

CLASSIFICAZIONE DI FUNZIONI DI UTILITÀ ATTRAVERSOUNA DISTANZA TRA MODELLI

Riassunto

La Conjoint Analysis è una delle tecniche maggiormente utilizzate nella valutazionedel comportamento dei consumatori. Questa metodologia consente di stimare i coefficientidi utilità parziale in base ad un modello statistico che lega la valutazione globale dipreferenza alle caratteristiche descrittive degli stimoli (prodotti o servizi). I risultati dellaConjoint Analysis trovano vasta applicazione nella segmentazione del mercato.

In questo lavoro viene proposta una strategia di classificazione basata su una nuovametrica. La distanza introdotta è definita come combinazione convessa di due distanze.Essa consente di tener conto di una duplice qualità dell’informazione relativa al modello:il valore dei coefficienti stimati e la bontà di adattamento. Di conseguenza, la differenzia-zione tra segmenti di mercato è ottenuta considerando la prossimità dei modelli di utilitàindividuali stimati e la capacità predittiva degli stessi.

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An inter-models distance for clustering utility functions