# The Goal-Gradient Hypothesis Resurrected: Purchase Acceleration, Illusionary Goal Progress, and Customer Retention

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5 The Goal-Gradient Hypothesis Resurrected 43 were marked on the back so that they could not be redeemed for a reward. The control customers were randomly sampled from the population of program members. Research assistants (posing as café employees) intercepted customers who requested a regular program card and offered instead to enroll them in a purchase-habit study designed to help the café management better understand its customers. Participants were asked to carry a transparent card and have it stamped every time they made a qualifying purchase at the café. They received \$5 when they agreed to participate in the study, and they were told that they would receive \$15 more when they surrendered their cards six weeks later, regardless of how many coffee purchases they made during that time. We verified that control participants did not use the regular RP cards during the study. Results A plot of the raw mean interpurchase times, aggregated across all redeemed cards (excluding transparent and buyback cards), demonstrates purchase acceleration as a function of smaller goal distance (see Figure 3). 3 Consistent with H 1, as members accumulated more stamps on their cards, the average length of time before their next coffee purchase decreased. The mean difference between the first and the last observed interpurchase times was.7 days (t = 2.6, p <.05), representing an average acceleration of 20% from the first to the last interpurchase time. As an estimate of the overall effect of acceleration on the average card, it is possible to compare the mean observed time to complete a card, which was 24.6 days, with the number of days it would have taken to complete a card at the rate of the first observed interpurchase time, which was 29.4 days. This yields a difference of nearly 5 days (16%) in card completion time. 3We excluded from the analysis days on which the café was closed. Figure 3 PURCHASE ACCELERATION AS A FUNCTION OF SMALLER Mean Interpurchase Time GOAL DISTANCE Progress toward goal Number of Stamps on Card d t = Although the analysis of raw data provides preliminary support for purchase acceleration (H 1 ), it does not account for various important factors. Accordingly, we used more sophisticated data analysis, namely, a discrete-time proportional hazard rate model. This modeling approach incorporates time-varying covariates and controls (e.g., weekly number of issued stamps) intended to rule out alternative explanations, such as time-trend effects. Our modeling approach also enables us to account for unobserved heterogeneity (i.e., individual differences in base purchase rates and acceleration tendencies). Hazard rate modeling methodology. In the main data set, each row represented one day per customer on a card on which the customer could have made a purchase; we included the days after the first stamp was received up to the day of the last stamp. There are variables in the data set at the customer level, day level, and card level. Overall, from 949 completed (i.e., redeemed) ten-stamp cards that captured nearly 10,000 coffee purchases, this data set yielded 29,076 rows of data. Hazard rate models (Cox 1972) are an important method to model interpurchase times. In these models, the instantaneous probability of purchase (called the hazard function, h[t]) is estimated, conditional on the amount of time since the prior purchase. In the discrete-time model (Gupta 1991; Helsen and Schmittlein 1993), the hazard model likelihood is decomposed into probabilities of purchase within given time intervals. In a hazard rate model, the baseline continuous survival function S(t) represents the probability that no purchase will occur after time t has elapsed since the previous purchase: t () 1 St () = exp hudu ( ). 0 We used a discrete-time proportional hazard model parameterized as the discretized survival function. In line with Seetharaman and Chintagunta s (2003) derivation, the full discretized survival function can be expressed as a function of the baseline hazard function h(t), time-varying covariates X t (including the proportional distance to the goal), and estimated covariate coefficients β (including a constant term): t u ( 2) St (, Xt) = exp ex u β hwdw ( ). u = 1 u 1 In our application, we decompose the survival function into day-specific components, and our dependent variable is the probability of purchase on a given day, conditional on no purchase having yet occurred: tu St (, Xt) ()Pr(, 3 t Xt) = 1 = 1 exp St ( 1, Xt 1) ex tβ h( u) du tl = St ( u) 1 St ( l ) extβ.

6 44 JOURNAL OF MARKETING RESEARCH, FEBRUARY 2006 In conducting our analysis at the day level, we assume that each day is a potential purchase occasion, except for days on which the café is closed. Because we estimate each day s probability as the difference in survival probability from the start of the day to the end of the day, it was necessary to code each day t as a range of continuous times between the lower bound t l and the upper bound t u when applying Equation 3. Purchases made on the same day were coded as occurring between time t l = 0 and time t u =.5, purchases on the subsequent day were coded as occurring between t l =.5 and t u = 1.5, and so forth. In the following likelihood function for an observed purchase, we denote the observed number of days elapsed at time of purchase by T, and we code an indicator function δ v, which represents whether a purchase occurred on day v (δ v = 1) or did not occur on day v (δ v = 0): T ( 4) L = Pr( v, ) v[ 1 Pr( v, )] 1 v v X δ X δ v. v = 0 The full likelihood is the product of all the purchasespecific likelihoods across cards and customers (Seetharaman and Chintagunta 2003). We determined the best-fitting base hazard function with Schwarz s Bayesian information criterion (BIC) measure. The BIC measure trades off improvements in the loglikelihood for increases in the number of parameters. We used latent classes to account for unobserved heterogeneity in the hazard rate parameters (Kamakura and Russell 1989), which is important to rule out heterogeneity bias. Because the unit of analysis in the latent-class model is the customer, we took into account the common error variance when multiple cards belonged to a single individual, and we specifically accounted for cross-customer unobserved heterogeneity. 4 The latent-class modeling can be considered a nonparametric multivariate distribution on the hazard rate parameters across participants; each latent class represents a support point in the distribution. Although we found significant unobserved heterogeneity in the hazard rate parameters, when we ran the models without latent-class segmentation, all of the results still held. In all models, we used GAUSS software to implement Newton Raphson maximum likelihood estimation, and we determined the number of latent classes using the BIC criterion. Analyses of acceleration with common parameters across consumers. In this subsection, the primary focus of our modeling is the effect of goal distance on interpurchase times. Recall that the distance to the reward goal is captured with the measure d t = (r n t )/r. In the café RP, n t is the number of stamps accumulated on the card at time t, and r is the total number of required stamps. Given that in the main data set we model probability of purchase when there are between 1 and 9 stamps accumulated on the card (and r = 10), the measure d t ranges between.9 and.1. To test the goal-gradient hypothesis (H 1 ) in this and the subsequent empirical applications, we constructed the GDM, which includes linear and quadratic parameters that 4All the results are replicated when we exclude from the analyses the subsequent cards of members who redeemed more than one card. capture the effect of goal distance on observed behavior. 5 Here, we added the GDM as a covariate in the proportional hazard model. We parameterized the model by defining the probability of purchase for customer i on a given day t as follows: St ( u) () 5 Pr(, i t X it) = 1, St ( l ) where g = exp[β 0 + β 1 d it + β 2 (d it d i ) 2 + γx it ]; S(t) = the baseline survival function; d it = the proportion of total distance remaining to the goal for individual i at time t; d it d i = the mean-centered proportion of total distance remaining to the goal; β 1 and β 2 = the linear and quadratic goal-distance parameters, respectively; X it = the vector of covariates (i.e., control variables); and γ = the corresponding vector of coefficients. The parameters for estimation in the GDM are the intercept β 0, the goal-distance parameters β 1 and β 2, and the vector of coefficients γ. Consistent with H 1, we expect the parameter β 1 to be less than zero, capturing the predicted increase in the probability of purchase (hazard rate) as a function of smaller goal distance (d it ). Note that if β 1 is greater than zero, we observe effort deceleration (i.e., lower probability of purchase as a function of smaller goal distance). Among the time-varying covariates, we included both the weekly number of issued stamps and a code for whether a given day was after the end of the spring classes to control for alternative explanations based on time trend. 6 We accounted for unobserved heterogeneity in the base hazard rate parameters (but not in the goal-distance or the covariate parameters) using the methodology we described previously. Table 1 displays the estimated parameters for the log-logistic hazard rate function and the GDM in Equation 5. 7 The linear goal-distance parameter b 1 was less than zero (p <.01). This result supports H 1 and demonstrates that members accelerated their coffee purchases as they got closer to earning a free coffee. Consistent with the goalgradient curve in Figure 3, the negative quadratic parameter b 2 (p <.01) implies a diminishing rate of acceleration. A nested likelihood ratio test indicated that the GDM provided an improvement in fit over a naive model in which β 1 and β 2 were restricted to zero (χ 2 = 18.8, d.f. = 2, p <.01). 5In this and subsequent empirical applications, we examined higherorder parameters using orthogonal contrast codes (Fisher and Yates 1957), but we found that these were not significant. 6The control variables also included card type (indicating whether the member purchased American or Italian coffees) and additional timevarying covariates, such as midterm break (a code for whether the day was during the midterm break), day of week (linear trend from Monday through Thursday), and dummy codes for Friday and Saturday Sunday. None of the covariates were allowed to vary across the latent classes. 7All covariates were normalized before model-fitting in this and the subsequent model estimations. g

12 50 JOURNAL OF MARKETING RESEARCH, FEBRUARY 2006 Figure 5 THE JABOOM MUSIC-RATING WEB INTERFACE site visits. Given that during most (i.e., 96%) of the Web site visits members rated multiple songs, we examined both the quantity of ratings in each visit and the intervisit times. We modeled intervisit times using the discrete-time proportional hazard rate model we described previously. The findings were similar to those of the café RP. We briefly describe them here and then report the analyses and results of the quantity and persistence operationalizations of the goal-gradient hypothesis. Tests of the Goal-Gradient Hypothesis with Intervisit Times In our data set, there were 371 attempted reward certificates (i.e., with at least 1 song rated toward the certificate) and 262 earned certificates (i.e., with 51 songs rated). Of the 262 earned certificates, 114 were completed in a single visit. Although such single-visit certificates arguably signify strong goal motivation, they must be excluded from the analyses of intervisit times because they do not permit testing for acceleration (or deceleration). Therefore, we calibrated the hazard rate version of the GDM on the data from the remaining 148 certificates, which were earned in two or more visits. The data set was constructed such that each row represented a unique day on which a particular member could have visited the Jaboom Web site; we included the days after the first visit had occurred up to the day of the last visit In this section and in the subsequent subsections, we obtained similar results when we excluded from the analyses the data from subsequent certificates of members who earned more than one certificate. We parameterized the survival function with the GDM shown in Equation 5 and used the likelihood function shown in Equation 4. Recall that the variable capturing the hypothesized acceleration is d t = (r n t )/r, that is, the proportion of total distance remaining before the goal. In this empirical application, r is equal to 51 songs (i.e., the original total distance to the goal), and n t is the number of song ratings accumulated at time t toward earning the reward certificate. Given that we modeled the probability of visiting when there were between 0 and 50 song ratings accumulated, the measure d t ranges between 1 (when no goal progress has yet been made) and.02 (when the goal is almost achieved), respectively. Table 3 displays the parameters for the GDM that we estimated with Weibull hazard parameterization. The linear goal-distance parameter was less than zero (p <.01). This result supports H 1 and demonstrates that members visited the Jaboom Web site more frequently as they got closer to earning the reward (i.e., intervisit times decreased as a function of goal proximity). Consistent with Hull s (1934) findings with rats and our results of the café RP, the rate of acceleration diminished, as indicated by the negative quadratic goal-distance parameter (p <.01). Note that the model includes two variables as controls: (1) the total number of visits to the Jaboom Web site at the day level, which was intended to control for time-trend effects, and (2) the number of visits it took to complete each certificate, which was intended to rule out aggregation bias. We obtained similar results when we excluded either or both of these controls from the model. In addition, the linear and quadratic goaldistance parameters remained significant and did not vary

13 The Goal-Gradient Hypothesis Resurrected 51 Table 3 THE HAZARD RATE GDM OF INTERVISIT TIMES (MUSIC- RATING PROGRAM) A: Latent Class-Level Parameters Class 1 Class 2 Segment size (%) α (hazard rate) 0.5** 2.1** β 0 1.4**.9** B: Parameters Not Varying by Latent Class Acceleration Parameters Estimate Linear effect of goal distance, b 1.25** Quadratic effect of goal distance, b 2.22** Covariate Parameters (i.e., Control Variables) Daily number of visits 0.10** Total visits to earn certificate.01** Saturday Sunday 0.03** *p <.05 (based on Wald test; two-tailed). **p <.01 (based on Wald test; two-tailed). in magnitude across models with different numbers of latent classes in the base hazard rate. We also calibrated the GDM by simultaneously estimating unobserved heterogeneity in the acceleration and hazard rate parameters. Although the BIC favors a one-class solution, we report the two-class solution to rule out heterogeneity bias. In the larger class (65%), we found linear acceleration (b 1 =.6, p <.1; b 2 =.1, p >.1), and in the smaller class (35%), we found nonsignificant linear acceleration (b 1 =.1, p >.1; b 2 =.3, p <.1). The finding that both segments demonstrate acceleration is inconsistent with the heterogeneity bias rival account. Finally, we recalibrated the same GDM on the data obtained from incomplete certificates (i.e., those with less than 51 songs rated toward the unattained certificate). There were 36 incomplete certificates with at least two Web site visits (i.e., at least one intervisit time that could be modeled as a function of goal distance). Consistent with the analysis of the incomplete cards in the café RP, we found deceleration (b 1 =.7, p <.1; b 2 =.01, p >.9). Furthermore, we calibrated the GDM on a data set that combined both complete and incomplete certificates, and we added an interaction term, β INT, between completion (yes versus no) and linear acceleration. We found significantly stronger linear acceleration for complete than for incomplete certificates (b INT =.5; Wald χ 2 = 16.2, p <.01). That is, whereas acceleration is related to retention and goal attainment, deceleration is associated with program defection and goal abandonment. Tests of the Goal-Gradient Hypothesis with Rating Quantity Thus far, we have examined how goal distance influences interpurchase and intervisit times. In many cases, however, customers can accelerate their efforts by increasing the quantity of credits earned in a given interaction with the RP. Accordingly, in this subsection, we test the goal-gradient hypothesis by examining whether the quantity of songs rated per visit increases with goal proximity. We report separate tests of pooled and segment-level quantity acceleration, behavior on incomplete certificates, and postreward resetting. Figure 6 presents the raw data obtained from the 148 complete certificates (with at least two visits) that we observed. Consistent with the goal-gradient hypothesis, as members approached the 51-song goal, they rated more songs in later visits. To model the quantity of songs rated in a visit, we used a Type I Tobit model (Tobin 1958). This approach enables us to capture the underlying motivation to rate songs, while taking into account the constraint of 51 songs per certificate. Specifically, in the final visit of each certificate, the number of song ratings is censored at 51 n t, where n t is the number of song ratings accumulated toward the certificate at the start of visit t. By definition, any additional songs rated in visit t beyond the 51 n t limit would be credited toward the next certificate rather than the goal of earning the present 51-song certificate. Relatedly, as 51 n t approaches zero, the inherent motivation to rate songs may increase (because d t 0), but the 51-song constraint (i.e., the reward threshold) would suppress such an effect. For uncensored data, the Tobit model is equivalent to maximum likelihood regression. However, because applying a regression to censored data leads to biased parameter estimates (Breen 1996), the Tobit models the censored data as the probability of rating 51 n t or more songs. Thus, we model the quantity Q it rated in each visit t by participant i, incorporating the GDM into a Type I Tobit model: g if g < 51 nit () 6 Qi(, t X it) =, 51 nit if g > 51 nit Songs Rated Figure 6 NUMBER OF SONGS RATED AS A FUNCTION OF GOAL PROGRESS Visit Number 2-visit certificates 3-visit certificates 4-visit certificates

16 54 JOURNAL OF MARKETING RESEARCH, FEBRUARY 2006 of postreward resetting is a key corollary of the goalgradient hypothesis; it demonstrates that effort expenditure is a function of goal distance and rules out learning and other time-trend effects as well as a self-selection (or survivor) rival account. Self-selection is also inconsistent with the analysis of participants entire sequence of ratings (including ratings that did not eventually lead to reward), which revealed significant goal gradients. 11 Next, we use individual differences to explore the relationship between the goal gradient and customer retention. IMPLICATIONS OF THE GOAL GRADIENT FOR CUSTOMER RETENTION Prior research with animals has shown that a steeper goal gradient was generated by an increased drive (e.g., hunger) to attain the reward (Hull 1934). This finding suggests that RP members who exhibit enhanced acceleration possess a stronger motivation to earn free rewards (e.g., due to a higher achievement motivation). If, indeed, the motivation to earn free rewards is related to the steepness of the goal gradient, individual differences in the tendency to accelerate (when we hold constant the overall program effort) should predict customer retention after attainment of the first reward. Specifically, we expect members who accelerate more strongly toward their first reward to be more likely to reengage in the program and earn a second reward. Relatedly, we expect stronger acceleration to lead to faster reengagement in the RP. To examine these predictions, we recalibrated the GDM with unobserved heterogeneity in both the hazard rate and the goal-distance parameters, using a subsample of the café 11Detailed analyses that rule out the self-selection (survivor) account are available on request. RP data that excluded subsequent cards. We used this model to obtain individual-level linear acceleration estimates based on the member s first coffee card. We calculated these estimates by multiplying the latent class parameters of goal distance (i.e., acceleration) by the individual-level probabilities of class membership. We then used the individual-level acceleration estimates as independent variables in predicting member reengagement in the café RP. Importantly, the tests we report subsequently included covariates that statistically controlled for the length of time it took participants to complete the first card (i.e., the participant s overall program effort and product liking) and the date of completion of the card. That is, we investigated the effect of individual differences in the slope of the goal gradient (the linear goaldistance parameter), holding constant the base hazard rate (i.e., the average interpurchase time) and possible seasonality (or right-censoring) effects. Retention Probability We used a logistic regression to test the prediction that members who accelerate more strongly toward their first reward will be more likely to earn a second reward. The (dummy) dependent variable received a value of 1 if the member earned a second reward and 0 if otherwise. The effect of the first-card individual-level acceleration estimate for the member was in the hypothesized direction (b =.5, Wald χ 2 = 5.3; p <.05). In particular, members who accelerated more strongly toward their first reward were more likely to earn a second reward. To demonstrate this effect visually, we also split the sample on the basis of first-card estimated linear acceleration into three equally sized groups: decelerators (mean b =.1), accelerators (mean b =.05), and strong accelerators (mean b =.1). Figure 8 (left panel) depicts the probability of completing a second card (based on the raw data) for each of these three groups. Figure 8 EFFECT OF FIRST-CARD ACCELERATION ON REENGAGEMENT PROBABILITY AND TIMING Probability of Earning Second Reward 40% 35% 30% 25% 20% 15% 10% 5% 0% Average Reengagement Time (Days) Decelerator Accelerator Strong Accelerator Decelerator Accelerator Strong Accelerator

20 58 JOURNAL OF MARKETING RESEARCH, FEBRUARY 2006 Thaler, Richard (1980), Toward a Positive Theory of Consumer Choice, Journal of Economic Behavior and Organization, 1 (March), Tobin, James (1958), Liquidity Preference as Behavior Towards Risk, The Review of Economic Studies, 25 (1), Trope, Yaacov and Nira Liberman (2003), Temporal Construal, Psychological Review, 110 (3), Van Heerde, Harald J., Peter S.H. Leeflang, and Dick R. Wittink (2000), The Estimation of Pre- and Postpromotion Dips with Store-Level Scanner Data, Journal of Marketing Research, 37 (August), Wertenbroch, Klaus (1998), Consumption Self-Control by Rationing Purchase Quantities of Virtue and Vice, Marketing Science, 17 (4), Winer, Russell S. (1986), A Reference Price Model of Demand for Frequently Purchased Products, Journal of Consumer Research, 13 (September), (1999), Experimentation in the 21st Century: The Importance of External Validity, Journal of the Academy of Marketing Science, 27 (3), Wittink, Dick R. (2004) Journal of Marketing Research: 2 Ps, Journal of Marketing Research, 41 (February), 1 6.

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