A GENERALIZED ADDITIVE LOGIT MODEL OF BRAND CHOICE * SUNIL K SAPRA California State University Los Angeles, United States

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1 Journal of Contemorary Issues n Busness Research Volume, Issue No. 3, 22 Journal of Contemorary Issues n Busness Research ISSN (Onlne), 22, Vol., No. 3, Coyrght of the Academc Journals JCIBR All rghts reserved. A GENERALIZED ADDITIVE LOGIT MODEL OF BRAND CHOICE * SUNIL K SAPRA Calforna State Unversty Los Angeles, Unted States ABSTRACT The aer resents an econometrc alcaton of generalzed addtve Logt regresson models (GALRMs) for brand choce. Our sem-arametrc models are flexble and robust extensons of the Logt model. The GALRMs are ft to bnary resonse data by maxmzng a enalzed log lkelhood or a enalzed log artal-lkelhood. The GAMs allow us to buld a regresson surface as a sum of lower-dmensonal nonarametrc terms crcumventng the curse of dmensonalty: the slow convergence of an estmator to the true value n hgh dmensons. Four GALRMs are comared wth a Logt model for brand choce and the best model s selected usng varous model selecton crtera. Keywords: Logt Model; Generalzed Addtve Model; Interactons; Backfttng Algorthm; Penalzed Regresson Slnes. INTRODUCTION Logt models are ervasve n busness and economcs. The most common alcatons of Logt models have been to the analyss of brand choce data n marketng (Baltas 997 and Guadagn and Lttle 983) and transortaton choce data n economcs (Greene 28 and Mansk and McFadden 983). Logt models belong to the class of generalzed lnear models, whch relax the assumton that the resonse s normally dstrbuted by allowng t to follow any dstrbuton from the exonental famly (for examle, normal, Posson, bnomal, gamma etc.). Inference for GLMs s based on lkelhood theory. McCullagh and Nelder (989) rovde an authortatve account of GLMs and Cameron and Trved (25) and Greene (28) rovde econometrc alcatons. In recent years, sem-arametrc extensons of lnear regressons have been emloyed n economcs and busness. Nevertheless, there have been relatvely few alcatons of semarametrc extensons of generalzed lnear models. Ths aer s an attemt n ths drecton. We resent an econometrc alcaton of a generalzed addtve model (GAM) to bnary choce data, whch s a sem-arametrc extenson of GLMs. A GAM s a sem-arametrc GLM n whch art of the lnear redctor s secfed n terms of a sum of unknown smooth functons of exlanatory varables. Generalzed addtve models (GAMs) are a owerful generalzaton of lnear, logstc, and Posson regresson models. GAMs are very flexble, and can rovde an excellent ft n the resence of nonlnear relatonshs. GAMs and GLMs can be aled n smlar stuatons, but they serve dfferent analytc uroses. GLMs emhasze estmaton and nference for the arameters of the model, whle GAMs focus on exlorng * The vews or onons exressed n ths manuscrt are those of the author(s) and do not necessarly reflect the oston, vews or onons of the edtor(s), the edtoral board or the ublsher. Corresondng author Emal: ssara@exchange.calstatela.edu 69

2 Journal of Contemorary Issues n Busness Research Volume, Issue No. 3, 22 data nonarametrcally. The GAM aroach offers more flexblty n model form than the GLM aroach does. These models estmate an addtve aroxmaton to the multvarate lnk functon. The key beneft of ths aroach s that each of the ndvdual addtve terms s estmated usng a unvarate smoother nstead of a multvarate smoother for a hghdmensonal non-arametrc term crcumventng the curse of dmensonalty: the slow convergence of an estmator to the true value n hgh dmensons. The GAM formulaton of the Logt regresson model allows us to buld a regresson surface as a sum of lowerdmensonal nonarametrc terms. The two man aroaches to estmaton of GAMs are backfttng wth local scorng algorthm (Haste and Tbshran (99)) and enalzed regresson slnes (Wood (26)). Backfttng has the advantage that t can be used wth any scatterlot smoother whle the latter method has the advantage that estmaton of the smoothng arameter usng generalzed cross valdaton s ntegrated nto estmaton. Ths aer resents an econometrc alcaton of the GAM extenson of the Logt model and demonstrates that t can overcome a serous weakness of the Logt model n falng to dentfy the nonlneartes n the lnk functon. The aer s organzed as follows. Secton 2 ntroduces the generalzed addtve Logt regresson model (GALRM). Secton 3 resents the enalzed regresson method for the estmaton of GALRMs. Secton 4 resents an econometrc alcaton of GALRM to Coke data on choce between Coke and Pes. Secton 5 rovdes some concludng remarks. GENERALIZED ADDITIVE MODELS Generalzed addtve models are nonarametrc generalzed lnear models. Generalzed addtve models extend tradtonal lnear models n another way, namely by allowng for a lnk between the nonlnear redctor f(x,...,x) and the exected value of y. Ths amounts to allowng for an alternatve dstrbuton for the underlyng random varaton besdes ust the normal dstrbuton. Whle Gaussan models can be used n many statstcal alcatons, for several tyes of roblems they are not arorate. For examle, the normal dstrbuton may not be adequate for modelng dscrete resonses such as counts, or bounded resonses such as roortons. Generalzed addtve models consst of a random comonent, an addtve comonent, and a lnk functon relatng these two comonents. The resonse y, the random comonent, s assumed to have a densty n the exonental famly. f Y y b( ) ( y;, ) ex{ c( y. ) a( ) Where s called the natural arameter and Φ s the scale arameter. The normal, bnomal, and Posson dstrbutons are all n ths famly. Unlke the generalzed lnear models, the mean μ = E(y x, x2,, x) s not lnked to the lnear redctor xβ, but to the nonlnear nonarametrc redctor. η = g( ) ( ), s x Where s( ),..., s( ) are smooth nonarametrc functons defnes the addtve comonent. Fnally, the relatonsh between the mean μ of the resonse varable and η s defned by a lnk functon g(μ) = η. The most commonly used lnk functon s the canoncal lnk, for whch η = θ. A combnaton of backfttng and local scorng algorthms are used n the actual fttng of the model. In order to ft GAMs to the data, followng Wood (26), we use bass exansons of smooth functons and enalzed lkelhood maxmzaton for model estmaton 7

3 Journal of Contemorary Issues n Busness Research Volume, Issue No. 3, 22 n whch wggly models are enalzed more heavly than smooth models n a controllable manner, and degree of smoothness s chosen based on cross valdaton, or AIC or Mallows crteron. The generalzed addtve models are ft to count data or bnary resonse data by maxmzng a enalzed log lkelhood or a enalzed log artal-lkelhood. To maxmze t, the backfttng rocedure s used n conuncton wth a maxmum lkelhood or maxmum artal lkelhood algorthm (Haste and Tbshran (99) and Wood (26)). The Newton- Rahson method for maxmzng log-lkelhoods n these models can be resented n an IRLS (teratvely reweghted least squares) form. It nvolves a reeated weghted lnear regresson of a constructed resonse varable on the covarates: each regresson yelds a new value of the arameter estmates whch gve a new constructed varable, and the rocess s terated. In the generalzed addtve model, the weghted lnear regresson s smly relaced by a weghted backfttng algorthm (Haste and Tbshran (99)). ESTIMATION OF GENERALIZED ADDITIVE LOGIT REGRESSION MODEL USING PENALIZED REGRESSION METHOD Algorthm for Penalzed Iteratvely Re-weghted Least Squares (PIRLS) (Wood (26) The GALRMs are ft to bnary resonse data by maxmzng a enalzed log lkelhood or a enalzed log artal-lkelhood. The followng algorthm s used to mlement these methods. The R-ackage mgcv (Wood (22) ) was used for comutatons. Ste : Intalze ˆ g(/ N N y ), s,,2,...,. Ste 2 :Construct an adusted deendent varable z as z ( y )( / g( ) s ( x Ste 3:Comuteweghts w ( / ) ( V ), ) ) and g where V sthe varance of y at, Ste 4: Penalzed Slne Re gresson Mnmze smoothng arameter.comutes of 2 ( ). the matrx of data on bass functons used to reresent the regresson functon, W s a dagonal matrx wth - th dagonal element matrx of known coeffcentsn the enalty functon ' S and s a s,, and. W ( z X ) ' S wth resect to, where X s convergenc e s obtaned. Ste 5:Reeat stes2-4 relacng,, and, the second stage estmates w, S s a by untl the dfference between twosuccessve values of s less than a small resecfed number and The Generalzed Addtve Logt Model The generalzed addtve Logt model assumes that 7

4 Journal of Contemorary Issues n Busness Research Volume, Issue No. 3, 22 g( ) logt( ) ln where E( y x ) ex{ s ( x ), s ( x )}/ ex{ s ( x )}. The adusted deendent varable z and the weghts w used n the algorthm above are z ( y ) / ( ), w ( ) where g ( ), s ( x ) The functons s, s 2,, s are estmated by an algorthm lke the one descrbed earler. AN EMPIRICAL APPLICATION OF GAM LOGIT MODEL TO BRAND CHOICE GAM Logt Aled to Coke Data The dataset s from ERIM ublc data base, James M. Klts Center, Unversty of Chcago Booth School of Busness and conssts of data on 4 ndvduals who urchased Coke or Pes. Data Descrton The deendent varable COKE s defned as follows: COKE = f Coke s chosen, = f Pes s chosen PR_PEPSI = Prce of 2 lter bottle of Pes PR_COKE = Prce of 2 lter bottle of Coke DISP_PEPSI = f Pes s dslayed at tme of urchase, otherwse = DISP_COKE = f Coke s dslayed at tme of urchase, otherwse = PRATIO = Prce of Coke relatve to rce of Pes TABLE Summary Statstcs Varable Obs. Mean Std. Dev. Mn Max COKE PR_PEPSI PR_COKE DISP_PEPSI DISP_COKE PRATIO Source. ERIM ublc data base, James M. Klts Center, Unversty of Chcago Booth School of Busness Models The followng models were ft to the data. Model s a Logt model. Gven the nonlnearty of the Logt lnk functon n PRATIO as dslayed n the artal resdual lot of PRATIO n Fg., Model 2, a Generalzed Addtve Logt model ntroduces a nonarametrc smooth term s(pratio). The ossblty of nteracton between PRATIO and DISP_PEPSI and between PRATIO and DISP_COKE was nvestgated n models 3 and 4 resectvely by ntroducng a nonarametrc nteracton term n each model. Fnally, Model 5 s a GLM Logt 72

5 Journal of Contemorary Issues n Busness Research Volume, Issue No. 3, 22 model, whch ntroduces arametrc nteractons between PRATIO and DISP_PEPSI as well as between PRATIO and DISP_COKE. FIGURE Partal Resdual Plot of V6 Model : Logt Re gresson Model g( ) DISP _ PEPSI DISP _ COKE PRATIO Model 2 : Generalzed Addtve Logt Re gresson Models g( ) DISP _ PEPSI DISP _ COKE s( PRATIO) 4 5 Model 3 : Generalzed Addtve Logt Re gresson Models wth a Nonarametrc Interacto nterm g( ) DISP _ COKE s( PRATIO * DISP _ PEPSI ) 5 Model 4 : Generalzed Addtve Logt Re gresson Models wth a Nonarametrc Interacto nterm g( ) DISP _ PEPSI s( PRATIO * DISP _ COKE) 4 Model 5 : Logt Re gresson Model wth Interacto ns between Prato and DISP _ PEPSI and between Prato and DISP _ COKE g( ) DISP _ PEPSI DISP _ COKE PRATIO PRATIO * DISP _ PEPSI PRATIO * DISP _ COKE

6 Journal of Contemorary Issues n Busness Research Volume, Issue No. 3, 22 TABLE 2 Model : GLM Logt: Coke Data Varable Coeffcent Estmate Std. Error t-rato -value Intercet x -9*** DISP_PEPSI x -5*** DISP_COKE * PRATIO x -*** Sgnf. codes: ***. **. *.5.. (Dserson arameter for bnomal famly taken to be ) Null devance: on 39 degrees of freedom Resdual devance: 48.9 on 36 degrees of freedom AIC = Number of Fsher Scorng teratons: 3 TABLE 3 Model 2: GAM Logt: Coke Data Varable Coeffcent Estmate Std. Error t-rato -value Intercet DISP_PEPSI *** DISP_COKE * Aroxmate sgnfcance of smooth terms: Varable Estmated df Refned df Ch-squared -value s(pratio) x - ** Sgnf. codes: ***. **. *.5.. R-sq.(ad) =.35 Devance exlaned =.% UBRE score =.2483 Scale est. = n = 4 AIC = Null Devance: on 39 degrees of freedom Resdual Devance: on 33 degrees of freedom Number of Local Scorng Iteratons: 8 74

7 Journal of Contemorary Issues n Busness Research Volume, Issue No. 3, 22 TABLE 4 Model 3: GAM Logt Model wth a Smooth Nonarametrc Functon and Interacton between V4 and V6: Coke Data Varable Coeffcent Estmate Std. Error t-rato -value Intercet DISP_PEPSI DISP_COKE ** Aroxmate sgnfcance of smooth terms: Varable Estmated df Refned df Ch-squared -value s(pratio) x -2 *** s(disp_pepsi*pratio) *** Sgnf. codes: ***. **. *.5.. R-sq.(ad) =.42 Devance exlaned =.5% UBRE score = Scale est. = n = 4 AIC = Null Devance: on 39 degrees of freedom Resdual Devance: on 29 degrees of freedom Number of Local Scorng Iteratons: 2 TABLE 5 Model 4: GAM Logt Model wth a Smooth Nonarametrc Functon and Interacton between V5 and V6: Coke Data Varable Coeffcent Estmate Std. Error t-rato -value Intercet DISP_PEPSI * DISP_COKE Aroxmate sgnfcance of smooth terms: Varable Estmated df Refned df Ch-squared -value s(pratio) x -6 *** s(disp_coke*pratio) Sgnf. codes: ***. **. *.5.. R-sq.(ad) =.42 Devance exlaned = 2% UBRE score =.2375 Scale est. = n = 4 AIC = Null Devance: on 39 degrees of freedom Resdual Devance: on 29 degrees of freedom Number of Local Scorng Iteratons: 9 75

8 Journal of Contemorary Issues n Busness Research Volume, Issue No. 3, 22 TABLE 6 Model 5: GLM Logt wth Interacton: Coke Data Varable Coeffcent Estmate Std. Error t-rato -value Intercet x -8*** DISP_PEPSI x -5*** DISP_COKE * PRATIO x -9*** DISP_PEPSI*PRATIO ** DISP_COKE*PRATIO Sgnf. codes: ***. **. *.5.. (Dserson arameter for bnomal famly taken to be ) Null devance: on 39 degrees of freedom Resdual devance: 4.7 on 34 degrees of freedom AIC: 42.7 Number of Fsher Scorng teratons: 4 Comarng the Models The estmaton and results are resented n tables 2 through 6. A comarson of models usng the AIC s resented n Table 7. TABLE 7 Models and the AICs MODEL AIC Model 3, a modfed GAM, whch allows for nonarametrc nteracton between a dscrete varable DISP_PEPSI and a contnuous varable PRATIO, has the lowest AIC and UBRE score among the fve models studed and s therefore the best model. The Logt regresson model (Model ) has the hghest AIC. Ths s not surrsng snce the Logt model msses the nonlnearty n PRATIO n the lnk functon. Model 3 also has the lowest devance of on 29 degrees of freedom, whle Model has the hghest devance of 48.9 on 36 degrees of freedom. The statstcal sgnfcance of PRATIO dffers markedly between the Logt regresson model and the varous GAM Logt models emloyed here. The sgns of the coeffcent estmates are all exected n all of the models but Model 4. For nstance, the sgn of DISP_PEPSI s negatve and the sgn of DISP_COKE s ostve across models through 3 and model 5 ndcatng that the odds of choosng Coke over Pes ncrease f Pes s dslayed at the tme of urchase and decrease f Coke s dslayed at the tme of urchase. The analyss of devance n Tables 2, 3, and 4 ndcates sgnfcant nonlnear contrbuton from the varable PRATIO. The nonarametrc nteracton term n Model 3 for nonlnear nteracton between DISP_PEPSI and PRATIO turns out to be hghly sgnfcant ndcatng another nonlnearty n the lnk functon mssed by the Logt model. The arametrc lnear nteracton term for nteracton between DISP_PEPSI and PRATIO n Model 5 also turns out to be hghly sgnfcant. Nevertheless, Table 5 ndcates a weak nteracton between nonlnear effects of PRATIO and the varable DISP_COKE. The hgh degree of nonlnearty n PRATIO s also seen n the artal resdual smoothng lot of PRATIO n Fgure. The dotted curves around the sold curve reresent +-2 standard errors 76

9 Journal of Contemorary Issues n Busness Research Volume, Issue No. 3, 22 around the sold curve. The only surrsng result s the negatve sgn of the varable DISP_COKE n Table 4. CONCLUSIONS The aer has studed four generalzed addtve Logt regresson models (GALRMs) as alternatves to the Logt regresson model. In all of the emrcal alcatons, each of the GALRMs rovdes a much better ft than the Logt regresson model as reflected n lower AICs and lower devances. The econometrc technques used n the aer are wdely alcable to the analyss of count, bnary resonse and duraton tyes of data occurrng frequently n economcs and busness. Nevertheless, the GALRMs are not wthout drawbacks. The comutatonal algorthms are comlex and nterretatons can be dffcult. These models are useful manly when smle models for the lnear redctor rovde an nadequate ft for the data. REFERENCES Baltas, G. (997). Determnants of store brand choce: a behavoral analyss. Journal of Product & Brand Management, 6 (5), Cameron, A. C., and Trved, P. (998). Mcroeconometrcs. Cambrdge Unversty Press. Greene, W. H. (28). Econometrc Analyss, Pearson/Prentce Hall, New York. Guadagn, P. M., and Lttle, John, D.C. (983). A Logt Model of Brand Choce Calbrated on Scanner Data. Marketng Scence Summer Vol. 2 (3), Haste, T., & Tbshran, R. (99). Generalzed Addtve Models. Chaman and Hall, London. McCullagh, P., and Nelder, J. (989). Generalzed Lnear Models. Chaman and Hall, London. Mansk, C., and Danel McFadden (98). Structural Analyss of Dscrete Data and Econometrc Alcatons. MIT Press, Cambrdge. Wood, S. N. (22), R-ackage mgcv. Download From htt://cran.rroect.org/web/ackages/mgcv/mgcv.df Wood, S. N. (26). Generalzed Addtve Models: an ntroducton wth R. CRC, Boca Raton. 77