1. Slide 16.1 Hazard Rate Models MathematicalMarketing Chapter 16 -- Event Duration Models This chapter covers models of elapsed duration.  Customer Relationship Duration  Loyalty Program Membership Duration  Customer Retention Metrics 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.

2. Slide 16.2 Hazard Rate Models MathematicalMarketing Module Sequence The sequence of coverage  Definitions  The Hazard Function  Truncation  Censoring  Parametric Models

3. Slide 16.3 Hazard Rate Models MathematicalMarketing Key Definitions Define T i as a random variable representing the duration for individual i. Then F(t) = Pr(T i < t) is the probability function of duration failure times. The density, or unconditional failure rate is f(t) = F′(t) =

4. Slide 16.4 Hazard Rate Models MathematicalMarketing More On Survivorship and Failureship The survivorship function is the complement of the Failureship distribution, S(t) = 1 – F(t) = Pr(T i > t) = The cumulative failure function can now be written as an integral F(t) = Pr(T i < t)

5. Slide 16.5 Hazard Rate Models MathematicalMarketing What Is a Hazard Function? The hazard function, or conditional (age specific) failure rate is

6. Slide 16.6 Hazard Rate Models MathematicalMarketing Elaboration on the Hazard Function pr [failure at 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

7. Slide 16.7 Hazard Rate Models MathematicalMarketing The Shape of the Hazard Function The hazard function can take on any shape: 1. h(t) increases – snowballing (product adoption) 2. h(t) constant – no dynamics or memory 3. h(t) decreases – inertia (interpurchase times)

8. Slide 16.8 Hazard Rate Models MathematicalMarketing Constant Hazard – No Memory The exponential distribution f(t) = e - t implies h(t) = and we have situation 2.

9. Slide 16.9 Hazard Rate Models MathematicalMarketing The Two-Parameter Weibull The Weibull distribution implies h(t) =  t  -1 and we can create any of the three situations.

10. Slide 16.10 Hazard Rate Models MathematicalMarketing The Hazard Rate Impacts Average Retention Since then So can we add independent variables to the model? First, a digression on censoring.

11. Slide 16.11 Hazard Rate Models MathematicalMarketing Customer Relationship Duration Ongoing Relationships Are Right-Censored Time Time of Study

12. Slide 16.12 Hazard Rate Models MathematicalMarketing Truncation and Censoring LeftRight TruncationT i is observed only if T i < a i T i is observed only if T i > a i Censoring If T i  a i, then T i = a i All values below a are observed as a If T i  a i, then T i = a i All values above a are observed as a

13. Slide 16.13 Hazard Rate Models MathematicalMarketing True Relationship of x and Duration duration x T i =0 TiTi Some dependent variable values will be reduced due to censoring.

14. Slide 16.14 Hazard Rate Models MathematicalMarketing True Relationship of x and Duration duration x T i =0 TiTi Some dependent variable values will be reduced due to censoring.

15. Slide 16.15 Hazard Rate Models MathematicalMarketing True Relationship of x and Duration duration x How will the bias work? Each dependent value above the horizontal line will be redefined as equal to the line, i. e. y = a. T i =0 T i =a

16. Slide 16.16 Hazard Rate Models MathematicalMarketing Proportional Hazards h(t) = h 0 (t) h x (t) This part is constant for all individuals This part is a function of individual x values It adjusts h 0 up or down as a function of marketing instruments

17. Slide 16.17 Hazard Rate Models MathematicalMarketing Proportional Hazard Models We generally use this parametric approach:

18. Slide 16.18 Hazard Rate Models MathematicalMarketing Two Parametric Functional Forms h(t) = h 0 (t) h x (t) Exponential distribution Weibull distribution Can you make the Exponential a special case of the Weibull?

19. Slide 16.19 Hazard Rate Models MathematicalMarketing ML Estimation with for uncensored observations for censored observations Survivorship function Pr(Ti > t) Density function

20. Slide 16.20 Hazard Rate Models MathematicalMarketing 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.