Lecture 4: Likelihoods and Inference Likelihood function for censored data

Likelihood Function Start simple – All times are observed (i.e. NO censoring) – What does the likelihood look like? Assumptions: – Sample size is N – pdf denoted by:

Exponential…

That was Easy… So how do we handle censoring? What do we know if the actual time is not observed? Right censored data – Some patients have observed times – Some patients have censored times Only know that the haven’t failed by time t Include partial information

First Some Notation… Exact lifetimes: Right-censored: Left-censored: Interval censored:

Likelihood for Right-Censored Data From our previous slide – Exact lifetime – Right censored The likelihood

Other Censoring Generalized form of the likelihood What about truncation? – Left: – Right:

Left-Truncated Right Censored Data

Type I Right-Censoring Up to this point we have been working with event and censoring times X and C r However, when we sample from a population we observe either the event or censoring time What we actually observe is a random variable T and a censoring indicator, , yielding the r.v. pair {T,  } Thus within a dataset, we have two possibilities…

Type I Right Censoring Scenario 1:  = 0

Type I Right Censoring Scenario 2:  = 1

Back to our Exponential Example With right-censoring

What if X and C r are random variables… Assume we have a random censoring process So now each person has a lifetime X and a censoring time C r that are random variables How does this effect the likelihood? We still observe the r.v. pair {T,  } Again we have two possible scenarios – Observe the subjects censoring time – Observe the subjects event time

X and C r are random Scenario 1:  = 0

X and C r are random Scenario 2:  = 1

X and C r are random Likelihood:

What If X and C r are Not Independent These likelihoods are invalid Instead assume there is some joint survival distribution, S(X, C r ) that describes these event times The resulting likelihood: Results may be very different from the independent likelihood

MLEs Recall the MLE is found by maximizing the likelihood Recall likelihood setup under right censoring

MLE Example Consider our exponential example What is the MLE for ?

MLE Example

More on MLEs? What else might we want to know? – MLE variance? – Confidence Intervals? – Hypothesis testing?

MLE Variance Recall, I(  ) denotes the Fisher’s information matrix with elements The MLE has large sample propertied

Confidence Intervals for  The (1-  )*100% CI for 

Examples Data x 1, x 2,…, x n ~Exp( ) (iid)

Test Statistics Testing for fixed  0 – Wald Statistic – Score Statistic – LRT (Neyman-Pearson/Wilks)

Examples: Weibull, no censoring Data x 1, x 2,…, x n ~Weib( , ) (iid)

Fisher Information

Wald Test for Weibull From this we can construct the Wald Test:

Next Time We begin discussing nonparametric methods Homework 1: – Chapter 2: 2.2, 2.3, 2.4, 2.11 – Chapter 3: 3.2 – Additional: Find the pdf of the cure rate distribution assuming S*(t) ~ Weib(,  )