Module 3: (Cox Reg) Semi-Para Models

Question 1

Why Cox regression?

Answer 1

So we can model separate factors, eg weight, height, age etc

Question 2

What does vector Z hold;

Answer 2

the data about the person, eg weight, age, smoking status

Question 3

What does λ₀(t) represent?

Answer 3

The baseline hazad function.

HF when all the Z numbers are 0

Question 4

Expression for the hazard function under the cox model

Answer 4

λ₀(t) * exp( ∑_j=1 ^p β_jZ_ij )

Question 5

If β is +ve then….

Answer 5

Hazard rate (instant chance of death) increases with the j^th covariate.

Question 6

Does magnitude of β matter?

Answer 6

No, because of scaling.

Question 7

What does proportionality of hazard rates mean broadly;

Answer 7

The hazard rates for different individuals are proportional for any t. Ie they’ve curved but paralell lines.

Question 8

What is partial likelihood estimation used for

Answer 8

Estimating the β values

Question 9

What 2 assumptions does partial likelihood estimation rely on?

Answer 9

1. Non-informative censoring
2. Lives are independent

Question 10

What does the function R(t_j) tell us?

Answer 10

The set of the label of all lives who are still alive at t_j^-

Question 11

What are the two cases of partial likelihood regression we consider?

Answer 11

1. Deaths cannot occur simaltaneously
2. They can

Question 12

Properties of β under MLE estimation (2)

Answer 12

1. βhat is asymptotically unbiased.
2. It asymptotically follows a multivariate normat distribution.

Question 13

What 2 tests can we use to determine significance of covariates?

Answer 13

1. Likelihood ratio test
2. Wald test

Question 14

What H₀ will we set in our covariate significance testing?

Answer 14

We will introduce an extra set of q β’s and test if they all =0

Question 15

Likelihood ratio test statistic

Answer 15

-2 [log L_p - log L_p+q]

Question 16

How is the likelihood ratio test statistic distributed?

Answer 16

Chi squ dist with q degrees of freedom.

Question 17

Wald statistic is

Answer 17

Complicated, see p6 from notes.

Question 18

Wald statistic is distributed by

Answer 18

Chi squ distribution with q degrees of freedom

Question 19

How to find the Cov() term in the wald statistic?

Answer 19

See notes p7.

Uses information matrix.

Then partitions it up.

Question 20

Other than estimating the β terms, to estimate the hazard function we need to…

Answer 20

Estimate the cumulative baseline hazard rate.

Question 21

How do we estimate the baseline cumulative hazard rate?

Answer 21

Using Breslow’s estimator.

See notes for eqn.

Question 22

What are Cox-Snell residuals used for?

Answer 22

Seeing if the estimated b values are close to the true β