2006 54 2 445 459 c 2006 2006 2 10 2006 6 14 intensity Cox Cox Cox 1. one-to-one 2004 2000 intensity Cox Hughes 1994 RFM Recency Frequency and Monetary value 186 8603 2 1; konishi@ier.hit-u.ac.jp
446 54 2 2006 1. 2. 3. Jonker et al. 2004 Tsai and Chiu 2004 3 Cox 4 5 2. 2.1 2.2 3 2.3 2.1 2003 7 2004 9 416 4639 42 22 14 80 4200 6300 6700 90 12 78 27 15 km 5km
447 2.2 Korgaonkar et al. 1985 Akaah et al. 1995 3 5km 15 km 30 km 5.5 1 2 1 Sheather and Jones 1991 1. : 7909 4.5 : 3911 6.1 : 1693 10.1
448 54 2 2006 30 35 4 20 5 10 5 4 5 2.3 1. 30 km 1 2. 3. 1 2 3 4 5 Wedel and Kamakura 1998 2 homogeneity 416 1 2 1 2 2 3 KM T T S(t)=Pr{T t i} S(t) KM S(t)KM =1 S(t)KM t i <t t i+1 S(t) KM (2.1) S(t) KM = S(t i) KM 1 Ri n i. S(t i) KM t>t i 1 R i t i n i t i 3 0
449 2. 3. KM
450 54 2 2006 Cox Kalbfleisch and Prentice 1980 1 1001 189 24 3. Cox 3.2 3.3 3.1 Cox 1972 Cox intensity T S(t) λ(t) S(t) t t + (3.1) λ i(t x)=λ 0(t)exp(β 1x 1i + β 2x 2i + + β mx mi), (i =1,...,k). (3.2) S i(t x) Cox = S 0(t) exp(β 1x 1i +β 2 x 2i + +β mx mi ). β =(β 1,...,β m) x =(x 1,...,x m) λ 0(t) S 0(t) 3.1 λ 0(t) exp(β 1x 1i + β 2x 2i + + β mx mi) 3.3 R t 1,...,t i,...,t R K(t i)={j : t j t i} t i t j = t i + (3.3) L(β)= R i=1 exp(β x i) Èj K(t i ) exp(β x. j) Newton-Raphson ˆβ
451 1. (3.4) LL(β)= R β x i R log ¾ i=1 i=1 j K(t i ) exp(β x j). 3.3 Breslow 1974 3.2 416 1 3 3 2 3 1 9 3 3.3 2 Cox AIC 2 3 a 5 b
452 54 2 2006 1 Cox 0.006 20 21 Cox exp(0.006 21)/exp(0.006 20) = 1.006 1 1.006 0.6 10 6.1 intensity 10 19.7 1 1000 3.8 0.728 1km 2.07 2km 3km 2.07 62 19 3 =0 =1 1cm =0 =1 3 40 3
453 2. Cox n 4 3. Cox n 5
454 54 2 2006 Cox 4. 4.2 4.1 Hughes 1994 RFM Recency Frequency, and Monetary value R F M 3 RFM Fader et al. 2005 F M R 40 2004 8 2004 9 10 100 100 100 3.2 S i(t x) Cox 2.1 KM Cox KM t i <t t i+1 t i 1 R i t>ti n i S i(t x) Cox S 0(t) 1 R i R n i 1 i (1/R i ) n R i i t i R i =exp(β x i) n i n i = È K k = È exp(β x k ) t i S 0(t)= S 0(t i) 1 ˆR i (1/ ˆR i ) 3.2 ˆn i (4.1) ¼ ½ β S i(t x) Cox = S exp( β x i ) 0(t) = S 0(t i) 1 ˆR (1/ i ˆR i ) exp( x i ). ˆn i 40 l 1 0 4.2 4.1 t>100 4.3 S(t) S(t) SE(t) 95 (4.2) (4.3) E{1 {Ti t i }} = P {1 {Ti t i }} =1 S i(t x) Cox. = 1 l l i=1 1 S 0(t i) exp( β x i ).
455 (4.4) = SE(t) 1 l 1 S 0(t i) exp( β x i ) S 0(t i) exp( β x i ). 4 Cox KM KM t>100 2.1 È 1/2 R S(t j) i i n i (n i Ri) Cox 100 95 261 295 262 Cox KM KM 4 KM Cox S 0(t) KM Cox 2.1 S 0(t) Cox S 0(t) 4.1 KM exp( β x i) 0 2005 1 0.4 mm 1 1.25 cm 2 2005 2 3 3 2 3 4.
456 54 2 2006 4. KM Cox : S 0 (t) PWP Prentice et al., 1981 Cox KM 4.2 1 3 1 5 100 689 27 184 5 10 15
457 35 40 18 10 70 60 1 3 5. Cox Cox 1. 2. 3. KM Cox 2004 11 12 International Conference on Recent Development of Statistical Modeling in Marketing Latent Variable and Latent Structure Approach
458 54 2 2006 Akaah, I. P., Korgaonkar, P. K. and Lund, D. 1995. Direct marketing attitudes, Journal of Business Research, 34, 211 219. Breslow, N. 1974. Covariance analysis of censored survival data, Biometrics, 30, 89 99. Cox, D. R. 1972. Regression models and life tables with discussion Journal of the Royal Statistical Society Series B, 34, 187 220. Fader, P. S., Hardie, Bruce G. S. and Lee, K. L. 2005. RFM and CLV: Using Iso-value curves for customer base analysis, Journal of Marketing Research, XLII, 415 430. Hughes, A. M. 1994. Strategic Database Marketing: The Master Plan for Starting and Managinga Profitable, Customer-Based Marketing Program, Probus Pub Co., Chicago. Jonker, J. J., Piersma, N. and Van den Poel, D. 2004. Joint optimization of customer segmentation and marketing policy to maximize long-term profitability, Expert Systems with Applications, 27, 159 168. Kalbfleisch, J. D. and Prentice, R. L. 1980. The Statistical Analysis of Failure Time Data, John Wiley and Sons, New York. Korgaonkar, P. K., Lund, D. and Price, B. 1985. A structural equations approach toward examination of store attitude and store patronage behavior, Journal of Retailing, 61, 39 60. 2005. http://www.jhcia.org/haircolor/coloring/ index.html. 2005. http://www. perm. or. jp/txt/index. html. Prentice, R. L., Williams, B. J. and Peterson, A. V. 1981. On the regression analysis of multivariate failure time data, Biometrika, 68, 373 379. Sheather, S. J. and Jones, M. C. 1991. A reliable data-based bandwidth selection method for kernel density estimation, Journal of the Royal Statistical Society Series B, 53, 683 690. Tsai,C.Y.andChiu,C.C. 2004. A purchase-based market segmentation methodology, Expert Systems with Applications, 27, 265 276. Wedel, M. and Kamakura, W. A. 1998. Market Segmentation: Conceptual and Methodological Foundations, Kluwer Academic Publishers, Boston, Dordrecht, London. 2000. bit 169 178. 2004. 2004 5 316 324.
Proceedings of the Institute of Statistical Mathematics Vol. 54, No. 2, 445 459 (2006) 459 A Duration Analysis of Hair Salon Consumers Behavior and Prediction of Revisit Rates Yoko Konishi Institute of Economic Research, Hitotsubashi University In this study, we apply duration analysis to specify a model of hair salon customers behavior regarding their visits and then predict their next revisit rates. We estimate the intensity function by adopting the Cox model for the hair salon data to examine which demographic, geographic, and/or behavioral variables influence each customer s attitude. Our target customers use three hair services; haircut, color, and permanent wave. These customers were divided into non-loyal and loyal customers using the RFM method. Estimation results showed that the intensity functions of three items have been specified by different models, and we found differences in intensity between non-loyal and loyal customers. As a next step, we apply the Cox model estimation results to calculate the revisit rates for each customer within 100 days since the last visit from the interval prediction of his/her next revisit at 95 confidence interval level. We then compare the predicted rates with the actual rates of the customers revisits. For haircut, loyal and non-loyal customers actual revisit rates were within the predicted confidence interval based on the Cox model s results. However, hair color and permanent wave results based on the Cox model cannot predict the revisit rates well, but the predicted results based on the Kaplan-Meier estimation, which do not consider differences among individuals, yielded actual revisit rates within the prescribe confidence intervals. The prediction of revisit rates tell us when and how many customers will visit a salon and which services they will receive. The use of this information can help achieve an effective direct mailing strategy. Key words: Duration analysis, Cox model, hair salon.