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New Chapter “99” Event Duration Models This chapter covers models of elapsed duration. Customer Relationship Duration Loyalty Program Membership Duration Customer Retention Metrics Mathematical Marketing This Section is 95% taken from Helsen, Kristiaan and David C. Schmittlein (1993), "Analyzing Duration Times in Marketing: Evidence for the Effectiveness of Hazard Rate Models," Marketing Science, 12 (4), 395-414. Slide 99d.1 Hazard Rate Models Module Sequence The sequence of coverage Mathematical Marketing Definitions The Hazard Function Truncation Censoring Non-Parametric Models Parametric Models Slide 99d.2 Hazard Rate Models Key Definitions Define Ti as a random variable representing the duration for individual i. Then F(t) = Pr(Ti < t) is the probability function of duration failure times. The density, or unconditional failure rate is f(t) = F′(t) = Mathematical Marketing dF( t ) dt Slide 99d.3 Hazard Rate Models More On Survivorship and Failureship The cumulative failure function can now be written as an integral t F(t) = Pr(Ti < t) f (u )du 0 The survivorship function is the complement of the Failureship distribution, S(t) = 1 – F(t) = Pr(Ti > t) = f (u )du t Mathematical Marketing Slide 99d.4 Hazard Rate Models What Is An Hazard Function? The hazard function, or conditional (age specific) failure rate is h(t) Mathematical Marketing f (t) f (t ) 1 F( t ) S( t ) Slide 99d.5 Hazard Rate Models Elaboration on the Hazard Function pr [failure at t] f (t) f (t ) h(t) 1 F( t ) S( t ) pr [there has not been a failure up to t] It is the instantaneous rate of failure given survival until now, or the imminent failure risk Mathematical Marketing Slide 99d.6 Hazard Rate Models The Shape of the Hazard Function h(t) f (t) f (t ) 1 F( t ) S( t ) The hazard function can take on any shape: Mathematical Marketing 1. h(t) increases – snowballing (product adoption) 2. h(t) constant – no dynamics or memory dh ( t ) 0 dt 3. h(t) decreases – inertia (interpurchase times) dh ( t ) 0 dt Slide 99d.7 Hazard Rate Models Constant Hazard – No Memory The exponential distribution f(t) = e-t implies h(t) = and we have situation 2. Mathematical Marketing Slide 99d.8 Hazard Rate Models The Two-Parameter Weibull The Weibull distribution 1 t f (t ) t e implies h(t) = t-1 and we can create any of the three situations. Mathematical Marketing Slide 99d.9 Hazard Rate Models The Hazard Rate Impacts Average Retention Since h(t) f (t) f (t ) 1 F( t ) S( t ) We can solve for f(t) and see that the hazard rate will have an impact on the mean of f(t). So can we add independent variables to the model? First, a digression on censoring. Mathematical Marketing Slide 99d.10 Hazard Rate Models Truncation and Censoring Left Truncation Censoring Mathematical Marketing Right Ti is observed only if Ti < a Ti is observed only if Ti > a If Ti a, then Ti = a All values below a are observed as a If Ti a, then Ti = a All values above a are observed as a Slide 99d.11 Hazard Rate Models A Typical Relationship Between x and y y x Mathematical Marketing Slide 99d.12 Hazard Rate Models Guest Survey at a Hospitality Establishment There are no hotels cheaper than $0 per night In a guest survey you get left truncation Mathematical Marketing Slide 99d.13 Hazard Rate Models Truncated Dependent Variables Assume yi is observed only if yi > a Detection of the observation is therefore subject to a selection process This is called truncation Mathematical Marketing Slide 99d.14 Hazard Rate Models The Truncation Line Is y > a y a y>a x Mathematical Marketing Slide 99d.15 Hazard Rate Models Here Is What We Observe y x Mathematical Marketing Slide 99d.16 Hazard Rate Models Note That a New Line Has a Different Slope y x Mathematical Marketing Slide 99d.17 Hazard Rate Models Another View y x Mathematical Marketing Slide 99d.18 Hazard Rate Models Here Is Our Selection Process y y>a x Mathematical Marketing Slide 99d.19 Hazard Rate Models Here Is the Part We Observe y y>a x Mathematical Marketing Slide 99d.20 Hazard Rate Models Here Is the (Wrong) Line We Estimate y y>a x Mathematical Marketing Slide 99d.21 Hazard Rate Models A General Survey Does Not Save You In a general survey you get left censoring Assume if yi a, then yi = a All values above a are observed as a Mathematical Marketing Slide 99d.22 Hazard Rate Models True Relationship of x and Duration duration Each dependent value above the horizontal line will be redefined as equal to the line, i. e. y = a. Ti=a Ti=0 Mathematical Marketing x Slide 99d.23 Hazard Rate Models True Relationship of x and Duration duration Each dependent value above the horizontal line will be redefined as equal to the line, i. e. y = a. Ti=a How will the bias work? Ti=0 Mathematical Marketing x Slide 99d.24 Hazard Rate Models Customer Relationship Duration Time Ongoing Relationships Are Right-Censored Time of Study Mathematical Marketing Slide 99d.25 Hazard Rate Models Relationship Duration Is Generally Right Censored For right-censored individuals, we know only that Ti a. As we did the study at a certain time, all our current customers are right censored. Mathematical Marketing Slide 99d.26 Hazard Rate Models Proportional Hazards h(t) = h0(t) hx(t) This part is a function of individual x values It adjusts h0 up or down as a function of marketing instruments This part is constant for all individuals Mathematical Marketing Slide 99d.27 Hazard Rate Models Parametric Hazard Models We combine this model with “partial” maximum likelihood estimation. The partial likelihood is the probability that individual i had duration Ti given that someone out of the group had duration T. This model gets us two benefits: 1. 2. This partial likelihood is a ratio of individual likelihoods, so h0(t) cancels. Information from censored observations is appropriately taken into account. We generally use h x (t ) e Mathematical Marketing βxi Slide 99d.28 Hazard Rate Models Two Parametric Funtional Forms h(t) = h0(t) hx(t) e βx i 1 βx i t e Exponential distribution Weibull distribution Can we make the Exponential a special case of the Weibull? Mathematical Marketing Slide 99d.29 Hazard Rate Models ML Estimation Density function Survivorship function Pr(Ti > t) ln l i ln f (Ti | β) (1 i ) ln S(Ti | β) i i with Mathematical Marketing 1 i 0 for uncensored observations for censored observations Slide 99d.30 Hazard Rate Models SAS PROC LIFEREG ; proc lifereg data=input-data-set; model y *flag-var (1) = iv1 iv2 / distribution = weibull ; class nominal-var ; This var tracks whether the observation is right censored or not If flag-var is equal to this value, the observation is censored. Mathematical Marketing Slide 99d.31 Hazard Rate Models