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: f x

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

,i r ii D i R

L t f t S C

if t

,r iS C

Other Censoring

• Generalized form of the likelihood

• What about truncation?– Left:

– Right:

1i r l i ii D i R i L i I

L f t S C S C S L S R

Left-Truncated Right Censored Data

Type I Right-Censoring• Up to this point we have been working with event

and censoring times X and Cr

• 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, d, yielding the r.v. pair {T, d}

• Thus within a dataset, we have two possibilities…

Type I Right Censoring• Scenario 1: d = 0

Type I Right Censoring• Scenario 2: d = 1

Back to our Exponential Example• With right-censoring

What if X and Cr are random variables…

• Assume we have a random censoring process

• So now each person has a lifetime X and a censoring time Cr that are random variables

• How does this effect the likelihood? We still observe the r.v. pair {T, d}

• Again we have two possible scenarios– Observe the subjects censoring time– Observe the subjects event time

X and Cr are random

• Scenario 1: d = 0

X and Cr are random

• Scenario 2: d = 1

X and Cr are random

• Likelihood:

What If X and Cr are Not Independent

• These likelihoods are invalid• Instead assume there is some joint survival

distribution, S(X, Cr ) that describes these event times• The resulting likelihood:

• Results may be very different from the independent likelihood

1

1

, ,i i

i i

n i i

ix t c t

dS x t dS t cL

dx dc

MLEs

• Recall the MLE is found by maximizing the likelihood

• Recall likelihood setup under right censoring

1

1

log log log

i in

i ii

i iuncensored censored

L f t S t

L f t S t

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

MLE Example

More on MLEs?

• What else might we want to know?– MLE variance?– Confidence Intervals?– Hypothesis testing?

MLE Variance

• Recall, I(q) denotes the Fisher’s information matrix with elements

• The MLE has large sample propertied

Confidence Intervals for q

• The (1-a)*100% CI for q

Examples

• Data x1, x2,…, xn ~Exp(l) (iid)

Test Statistics

• Testing for fixed q0 – Wald Statistic

– Score Statistic

– LRT (Neyman-Pearson/Wilks)

'2

0 0 0ˆ ˆ ~obs dI

1 20 0 0log log ~obs dL I L

0 22log ~ˆobs d

L

L

Examples: Weibull, no censoring• Data x1, x2,…, xn ~Weib(a, l) (iid)

0 : 1 . : 1AH vs H

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(l, a)