Estimation of the Probability of Success Based on Communication History

Transcription

1 Workng paper, presented at 7-th Valenca Meetng n Bayesan Statstcs, June 22 Estmaton of the Probablty of Success Based on Communcaton Hstory Arkady E Shemyakn Unversty of St Thomas, Sant Paul, Mnnesota, USA Abstract The paper presents a study of how the detaled hstory of communcaton between a sales agent and a customer can be used to estmate the probablty of success (sale) It utlzes the data obtaned from a Mnnesota based company that uses telesales n a busness-to-busness envronment Etensve telesales database was nvestgated and prncpal varables nfluencng the sale event were determned The number of decson-maker contacts X s analyzed as an eplanatory varable A comparson s made between logstc regresson technque and dscrmnant analyss based on negatve bnomal or zero-modfed Posson dstrbuton of the number of contacts There seems to be a good agreement between the results obtaned by both methods Dscrmnant analyss as demonstrated by Efron (975) can be more effcent f some model assumptons are made regardng the dstrbuton of X However, logstc regresson may be used n order to valdate the choce n a certan class of models A Introducton A Introducton and Methods Suppose we observe communcaton hstory of a seller wth a number of customers The goal of communcaton s to acheve a sales event Each complete communcaton record, correspondng to th k j customer, s a vector X ( X, X ; S ), where X are some observable varables regardng communcaton patterns (frequency, length, and other characterstcs of contacts) j For the sake of smplcty, let us assume that all X are countng (non-negatve dscrete nteger) varables Varable S s a bnary success varable, whch s assumed to equal n case of success (communcaton leadng to a sale) and n case of falure (no sale, communcaton termnated) We obtan N complete records Incomplete records nclude some or all of the values of j X, and no values for S (ongong communcaton) Our objectve s to use the nformaton contaned n complete records to study the relatonshp between the k vector X ( X, X ) of eplanatory varables and S as the response varable, and then to predct the probablty of success for ncomplete records based on ths relatonshp We wll consder two conventonal approaches to statstcal problems nvolvng bnary data: lnear dscrmnant analyss and logstc regresson A2Dscrmnant Analyss Dscrmnant analyss approach suggests that the customers represent two dstnct populatons: success populaton and falure populaton Before the communcaton hstory s accumulated, the pror probablty of a customer to belong to the success populaton s π P( S ), whle the probablty of a customer to

2 2 belong to the falure populaton s supples us wth addtonal nformaton regardng the posteror: where P( S k π π Communcaton hstory represented by X ( X, X ) p r πp( ) s X ) π p ( ) + π p, ( ) ( ) P( X S r), r, Mamum lkelhood estmaton n ths case requres the mamzaton of the uncondtonal lkelhood p( s ) p( ) N N + N N N p( ) π p( ) N + π, () where N and N N N are the numbers of complete success and falure records respectvely Ths mamzaton s usually equvalent to separate mamum lkelhood estmaton of dstrbuton parameters for success and falure populatons Pror probabltes can be emprcally estmated by the relatve frequency of successes and falures n the sample (Clogg et al, 99) A3 Logstc Regresson Logstc regresson suggests that there ests a stable statstcal relatonshp, whch allows one to determne the k probablty of success ( S ) P gven the values of X ( X, X ) : P( S X ) + ep( α β) j No assumpton s made concernng the dstrbuton of X Coeffcents α and β can be found drectly by mamum lkelhood estmaton Essentally, we may look for the values of α and β, mamzng the condtonal lkelhood p( s ) N P( S s X ) N ( + ep( α β )) (2) Two models (dscrmnant analyss and logstc regresson) may be consdered equvalent f we assume 2

3 3 π p( ) α log + log π p ( ) β (3) Nevertheless, as we wll show, correspondng estmaton procedures may lead to dfferent results due to dfferent mamzaton procedures n () and (2) Efron (975) demonstrated that dscrmnant analyss s more effcent for the case of X havng a multvarate normal dstrbuton It can be epected to be true n general, f there ests a model for X, n whch we trust However, f we do not have a good model for the dstrbuton of X to start wth, the dstrbuton-free logstc regresson, optmzng condtonally on the observed values of X, seems to be a logcal choce Logstc regresson va formula (3) can even be used for dagnostcs and valdaton of parametrc models for X B Data B Data Analyss The data set we wll use for demonstraton purposes was obtaned from a Mnnesota based company that uses telesales n a busness-to-busness envronment It ncluded 782 complete records of communcaton ncludng 62 successful sales (S ) and 62 falures (S ) By a falure we mean a customer contact, whch dd not lead to a sale The data were presented for analyss as the prntouts from the company database In the most of 782 cases the data ncluded the followng: dates of the calls, status of the calls (conversaton, answerng machne, no answer), name and status of the contact (decson-maker or not), names of the company representatves Most of the records contaned well-defned date of the frst decson-maker contact, dates of producton sample kts sent, and dates of prce quotes sent In many cases addtonal data were avalable on the echanges of faes and prntng materals, closng dates, and follow-up actvtes The goals of data analyss were to dentfy the varables, study correlatons and buld models estmatng probablty of a fnal success based on communcaton patterns The company s nterested to know n partcular, what are the chances of a fnal success wth a customer after 4 and 6 months of communcaton actvtes These are the ponts when manageral decsons typcally have to be made whether to proceed wth communcaton or drop the customer to save costs The pror probablty of success may be evaluated at appromately 2, based on the eperence of the prevous years B2 Varables The followng varables were determned as potentally mportant for the decson-makng: - tme from the frst contact to the frst producton sample kt sent (KtTme), - tme between the frst kt sent to the frst prce quote sent (QuoteTme), - total number of contacts wth the decson-makers (DMT), - total number of calls n the frst two-month-perod ( to 6 days) after the frst contact (C), - total number of decson-maker contacts n the frst two-month-perod ( to 6 days) after the frst contact (DM), - total number of calls n the second two-month-perod (6 to 2 days) after the frst contact (C2), - total number of calls n the second two-month-perod (6 to 2 days) after the frst contact (DM2), - total number of calls n the thrd two-months-perod (2 to 8 days) after the frst contact (C3), - total number of calls n the thrd two-month-perod (2 to 8 days) after the frst contact (DM3), - success varable, assumng values for sale and for non-sale (S) The values of these varables were calculated for each of 62 sales and 62 non-sales Such varables as the value or the volume of the contract, or eact closng date were unavalable Accordng to the prelmnary analyss, they were not lkely to be mportant for the fnal conclusons 3

4 4 On the bass of the prncpal varables the followng ratos were derved: - percentage of effectve calls n the frst two-month perod R DM / C, - percentage of effectve calls n the second two-month perod R2 DM2/C2, - percentage of effectve calls n the thrd two-month perod R3 DM3/C3 B3 Correlaton Analyss We observed the Pearson s sample correlaton coeffcents for S wth all potental predctors, takng the mssng values nto account A very weak correlaton wth S was detected for KtTme and QuoteTme (actual value depended on the method of treatng mssng values) It was also dffcult to determne whether KtTme or QuoteTme be observable at any gven tme after the frst decson-maker contact A satsfactory correlaton wth S was observed for DMT: corr(s,dmt) 46 However, the value of DMT (total number of decson-maker contacts) cannot be determned before the fnal sale, whch makes ths varable not a good predctor Moderate correlaton wth S, ncreasng wth tme, was observed for DM, DM2, and DM3: corr(s,dm) 25, corr(s,dm2) 3, corr(s,dm3) 42 The values of DM, DM2, and DM3 can be determned fully n two, four, and s months respectvely after the frst decson-maker contact Ths encourages the practcal use of these varables as predctors A lower correlaton wth S, though also ncreasng wth tme, was observed for C, C2, and C3: corr(s,c) 22, corr(s,c2) 26, corr(s,c3) 34 The values of C, C2, and C3 can be determned fully n two, four, and s months respectvely after the frst decson-maker contact However, the resdual analyss shows that these varables do not carry any sgnfcant nformaton, whch s not contaned n DM, DM2, and DM3 Contrary to pror epectatons, the ratos R, R2, and R3 proved not to be correlated wth S : corr(s, R) 7, corr(s,r2) 5, corr(s,r3) 7 Other eperments wth ratos or lnear functons of the varables dd not gve any postve results Therefore, we suggest the use of DM, DM2, and DM3 as sngle predctors for S n two, four, and s months after the frst decson-maker contact C Logstc Regresson C Implementaton of the Methods In order to determne the condtonal probablty of a sale gven a value of the chosen predctor for an ncomplete record (DM, DM2, DM3 dependng on the term of observaton), we apply the logstc regresson model P( S X ) + ep( α β) Implementaton of the logstc regresson algorthm yelded: () Two months after the frst decson-maker contact: α 26, β 96, the value of DM; (2) Four months after the frst decson-maker contact: α 9, β 247, the value of DM2; (3) S months after the frst decson-maker contact: α 24, β 4, the value of DM3 In all three cases t-values for α and β were above 6, whch gves the evdence of a good regresson ft 4

5 5 C2 Dscrmnant Analyss Three parametrc models were chosen for testng Keepng n mnd that X DM s a countng random varable, we suggest the followng: P( X S l) pl ( ), l, negatve bnomal or zero-modfed Posson Method of mamum lkelhood was used to evaluate the parameters of the dstrbutons by the avalable samples of szes, l, The results obtaned are presented below: N l Negatve Bnomal + rl rl pl ( ) ql ( ql ),,,2, rl For a reason clarfed later, we wll dstngush two possbltes: ) three-parameter model ( q, q, r), r r r (embeddng geometrc dstrbuton for r ) 2) four-parameter model ( q, q, r, r ), r r Table Negatve Bnomal Three-parameter ˆq ˆq rˆ Four-parameter ˆq ˆq ˆr ˆr st Perod nd Perod rd Perod Zero-Modfed Posson Table 2 ql λl λl pl ( ) ql ; pl ( ) e,,2, λl e! Zero-Modfed Posson ˆq ˆλ ˆq λ st Perod nd Perod rd Perod C3 Probablty of Success The followng table contans the posteror probabltes of success calculated accordng to the models above For the sake of an llustraton, we took the 2 nd and the 3 rd Perods (6 to 9 days and 9 to 2 days snce prmary contact) The frst column contans the number of decson-maker contacts per perod The second column (hghlghted) gves success probabltes accordng to logstc regresson model wth the coeffcents found above The fourth column (also hghlghted) represents the results of applyng zero-modfed Posson model The thrd and ffth columns show results correspondng to three- and four- parameter negatve bnomal model 5

6 6 Table 3 Perod 2 Logstc Regresson Zero-modfed Posson X NegBn(3) NegBn(4) Table 4 Perod 3 Logstc Regresson Zero-modfed Posson X NegBn(3) NegBn(4)

7 7 The entres hghlghted by n blue n the tables above correspond to the success probablty whch are hgher than the pror π 2, that can be obtaned emprcally from the aggregate data C4 Model Choce and Comparson Formula (3) clearly states the condton on the dstrbutons of X, for whch the dscrmnant analyss approach s equvalent to logstc regresson: log p( ) π β α + log p ( ) π Log-lkelhood rato on the left-hand sde must be lnear n For the three-parameter negatve bnomal model, ths s satsfed wth q β log and q Ths s also true for the zero-modfed Posson model: λ β log and λ q π α log + log r q π λ q π e α log + log + log λ q π e However, the four-parameter negatve bnomal model does not satsfy (3) because of the non-lnear second term on the rght-hand sde of p ( ) q log log ( ) p q Γ( + ) r + Γ( + r ) + One can observe n Tables 3 and 4 that the zero-modfed Posson model brngs about results, whch are drastcally dfferent from the other methods It seems that the results of condtonal lkelhood mamzaton by logstc regresson, are n a good agreement wth the negatve bnomal models (both three- and four-parameter), but not wth the zero-modfed Posson D Conclusons Therefore, logstc regresson can be suggested as a dagnostc tool n order to compare the parametrc models above The choce of negatve bnomal model s supported by the dstrbuton-free condtonal lkelhood mamzaton If we use the estmates from Table n formula (3), we can see that ˆ -for the 2 nd Perod β log[( qˆ ) /( qˆ )] 24 and ˆ α 85, -for the 3 rd Perod ˆ β log[( qˆ ) /( qˆ )] 43 and α ˆ 2 8 These numbers are farly close to respectvely, β 247, α 9 for the 2 nd Perod, and β 4, α 24 for the 3 rd Perod, obtaned by condtonal lkelhood mamzaton n the logstc regresson model 7

8 8 In the meantme, the values ˆ ˆ -for the 2 nd Perod β log( λ / λ ) 64 and α ˆ 3 93, -for the 3 rd Perod ˆ β log( ˆ λ / ˆ λ ) 85 and α ˆ 4 67, ˆ obtaned from Table 2 and formula (3), are sgnfcantly dfferent In order to make the pont, why we should be nterested n parametrc models at all, and not stay satsfed wth logstc regresson, let us ntroduce an elementary formula for the computaton of success probablty n the embedded case of geometrc dstrbuton: ˆ( /( ) N m + m S X ) N m /( + m + N /( + m ) + ) + m P, + where N () are total number of successes (falures) n the database, m () are sample mean values of decson-maker contacts for the sample of successes (falures) References CLOGG, Clfford C et al, Multple Imputaton of Industry and Occupaton Codes n Census Publc-use Samples Usng Bayesan Logstc Regresson, 99, J Amer Statst Assoc, March 99, Vol 86, No 43, EFRON, Bradley, The Effcency of Logstc Regresson Compared to Normal Dscrmnant Analyss, J Amer Statst Assoc, December 975, Vol 7, No 352, PREGIBON, Daryl, Logstc Regresson Dagnostcs, Ann Statst, 98, Vol 3, No 4, RUIZ-VELASCO, S, Asymptotc Effcency of Logstc Regresson Relatve to Lnear Dscrmnant Analyss, Bometrka, 99, Vol 78, No 2,