Duration Models

Question 1

Why duration models?

Answer 1

Because they take into account time and duration. Timing is important, question is often not “if” it is going to happen. But “when” it is going to happen.

Question 2

Two relating, interesting questions:

Answer 2

1) Analysis and prediction of when an event will happen? Or whether it happens at all.
2) Analysis of the effects of covariates on whether and when the event happens.

Question 3

Duration Data

Answer 3

Time it takes for the event of interest to happen

Question 4

Hazard rate

Answer 4

The probability that an event occurs at time period t, conditional that it did not happen yet.

Question 5

When is a observation censored?

Answer 5

If the event did not take place in the observed time period, it means that that observation is censored.

Question 6

What do we know about censored observations:

Answer 6

• DO KNOW: the event did not happen within the observation period
• DON’T KNOW: if and when the event will happen

Question 7

Why use censored data?

Answer 7

Models need to use as much information as possible, censored observations are not missing at random and they contain essential information.

Question 8

Hazard model

Answer 8

A model for the hazard rate, which is the probability that an event happens in a time interval given that it has not happend yet.

Question 9

Building blocks of the hazard model:

Answer 9

• Probability density function
• Cumulative distribution function
• Survival function
• Hazard rate

Question 10

Probability density function

Answer 10

Prob. that event happens in the time interval t

Question 11

Cumulative distribution function

Answer 11

Prob. that the even takes place AT or BEFORE time t.

Question 12

Survival function

Answer 12

Probability that event not happens before time t.

Question 13

Hazard rate

Answer 13

Conditional prob. that even occurs at t, given that it has not occurred until t.

Question 14

Kaplan Meier Survival Function

Answer 14

Uses the survival function (S)t: probability that the event did not happen till time period t.

Question 15

Kaplan-Meier estimator

Answer 15

Non-parametric estimator directly computed from the observed proportions of surviving cases (over-time periods)

Question 16

Cox Proportional Hazard model

Answer 16

Model for the Hazard rate, allows for including:

• fixed covariates
• time-varying covariates

Question 17

Cox Proportional Hazard model, formula:

Answer 17

h(t) = h0(t)*exp(B1X1 + ….. + BkXk)

Question 18

Why is Cox Proportional Hazard model popular:

Answer 18

1) Partial likelihood approach, no need to specify hazard baseline function
2) No need to specify the probability distribution
3) Very easy & fast to estimate

Question 19

Parameter interpretation via…

Answer 19

The hazard ratio, which is the ratio between two hazard rates.

Question 20

Example of a hazard ratio, in words and formula..

Answer 20

A model for churn and X is a dummy variable for gender ( 1 = female | 0 = male):

Hazard rate = Hfemale(t) / Hmale(t) = e^Bgender

Question 21

If parameter is corresponding to B, then the hazard rate increases by:

Answer 21

100*(exp(B)-1) %

Question 22

Key assumption of the proportional hazard model

Answer 22

Proportionally assumption:

• assumption that the hazards for different levels of covariates is constant overtime.
• in other words, the hazard ratio of males vs. females is constant overtime.

Question 23

Model fit and selection, via:

Answer 23

• Likelihood ratio test
• Wald-test
• Score test
• Pseudo R2

Question 24

Index of Concordance

Answer 24

The fraction of pairs in your data, where the observation with the higher survival time, has the higher probability of survival predicted by your model.

Question 25

Pseudo R2

Answer 25

Percentage of improvement in log-likelihood of the full model, relative to the null model.