Ch 3. The Cox PH Model and Its Characteristics

Question 1

Tests statistics typically used with ML estimates (2)

Answer 1

• Wald statistic
• likelihood ratio (LR statistic): makes use of the log likelihood statistic.

Question 2

Testing the significance of interaction using Likelihood Ratio (LR) statistic

Answer 2

We need to compute the difference between the log likelihood statistic of the reduced model which does not contain the interaction term and the log likelihood statistic of the full model containing the interaction term.

In general, the LR statistic can be written in the form -2lnL_R minus -2lnL_F, where R denotes the reduced model and F denotes the full model.

LR = -2 ln L_R - (-2 ln L_F)

Question 3

Z Wald Statistics: How to obtain p-value in model

Answer 3

p-value is obtained by dividing the coefficient of the product term by its standard error and then assuming that this quantity is approximately a standard normal or Z variable.

Question 4

Statistical objectives of model (3)

Answer 4

• test for significance of effect
• point estimate of effect
• confidence interval for effect

Question 5

Point estimate of effect

Answer 5

A point estimate of the effect of the treatment is provided in the HR column. This value gives the estimated hazard ratio (HR) for the effect of the treatment;

Question 6

Calculation of confidence interval for effect

Answer 6

1. Compute a 95% confidence interval for the regression coefficient of the Rx variable (‘beta’1).
2. Exponentiate the two limits obtained for the confidence interval for the regression coefficient of Rx

exp[ ‘beta’₁ +/- 1.96 sqrt (Var • ‘beta’₁)

Question 7

Confounding (HR)

Answer 7

Crude versus adjusted HR are meaningfully different.

Question 8

Adjusted survival curves vs KM curves

Answer 8

Adjusted survival curves are mathe- matically different from Kaplan–Meier (KM) curves. KM curves do not adjust for covariates and, therefore, are not computed using results from a fitted Cox PH model. Adjusted: Adjusted for covariates Use fitted Cox model KM: No covariates No Cox model fitted

Question 9

The formula for the Cox PH Model

Answer 9

The Cox model formula says that the hazard at time t is the product of two quantities.

The first of these, h₀(t), is called the baseline hazard function.

The second quantity is the exponential expression e to the linear sum of ß_iX_i, where the sum is over the p explanatory X variables.

Question 10

What is a time-independent variable?

Answer 10

A time-independent variable is defined to be any variable whose value for a given individual does not change over time

Question 11

Property that makes the Cox model a semiparametric model

Answer 11

The baseline hazard, h₀(t), is an unspecified fuction

Question 12

What is a parametric model?

Answer 12

A parametric model is one whose functional form is completely specified, except for the values of the unknown parameters.

For example, the Weibull hazard model is a parametric model where the unknown parameters are lambda, p, and the ß_i’s. Note that for the Weibull model, h₀(t) is given by lambda*p*t^(p-1)

Question 13

Why is the Cox PH model is popular?

Answer 13

Cox PH model is ‘robust’: will closely approximate correct parametric model. (prefer parametric model if sure o correct model; e.g. use goodness-of-fit test.

Measure of effect: hazard ratio (HR) involves only ß’s, without estimating h₀(t).

Can estimate h(t,X) and S(t,X) for cox model using a minimum of assumptions.

Cox model prefereed to logistic model:

• uses more information: survival times and censoring
• uses (0, 1) outcome and ignores survival times and censoring.

Question 14

Why L is a partial likelihood (rather than a complete likelihood)

Answer 14

The term“partial” likelihood is used because the likelihood formula considers probabilities only for those subjects who fail, and d_oes not explicitly consider probabilities for those subjects who are censored._

Thus the likelihood for the Cox model does not consider probabilities for all subjects, and so it is called a “partial” likelihood.

Question 15

Eq. partial likelihood

Answer 15

The partial likelihood can be written as the product of several likelihoods, one for each of, say, k failure times. Thus, at the f-th failure time, L_f denotes the likelihood of failing at this time, given survival up to this time.

L = L₁ x L₂ x L₃ x … L_k = ΠL_j (j = 1 , k)

Cox likelihood based on order of events rather than their distribution

Question 16

Definition of ‘risk set’, R(t_(f))

Answer 16

The set of individuals at risk at the jth failure

Question 17

Solution by iteration

Answer 17

The solutioni sobtained in a stepwise manner, which starts with a guessed value for the solution, and then successively modifies the guessed value until a solution is finally obtained.

Question 18

General definition of hazard ratio (HR)

Answer 18

In general, a hazard ratio (HR) is defined as the hazard for one individual divided by the hazard for a different individual.

The two individuals being compared can be distinguished by their values for the set of predictors, that is, the X’s

Question 19

General rule for (0, 1) exposure variable when there are product terms:

Answer 19

Question 20

Formula of Cox model hazard function and Cox model survival function

Answer 20

Question 21

Meaning of the PH Assumption

Answer 21

The PH assumption requires that the HR is constant over time, or equivalently, that the hazard for one individual is proportional to the hazard for any other individual, where the proportionality constant is independent of time.

PH: HR is constant over time, i.e., h-‘hat’- (t, X*) = constant x h-‘hat’- (t, X)

Question 22

Obtaining maximum likelihood estimates (score equations)

Answer 22

Once the likelihood is formulated, the question becomes:

which values of the regression parameters would maximize L?

The process of maximizing the likelihood is typically carried out by setting the partial derivative of the natural log of L to zero and then solving the system of equations (called the score equations).

Question 23

Defintion of time 0 (t₀)

Answer 23

Starting time of the true survival time (starting point)

Question 24

Defining the starting point: Possible choices for time 0:

Answer 24

• Study entry
• Beginning of treatment
• Disease onset
• Disease diagnosis
• Surgery
• Point in calendar time
• Birth
• Conception

Question 25

Concept of left truncation at time t₀

Answer 25

The subject is not observed from time 0 to t₀.

If the subject has the event before time t₀, then that subject is not included in the study.

If the subject has the event after time t₀, the subject is included in the study but with the caveat that the subject was not at risk to be an observed event until time t₀

Question 26

Types of left truncation (2)

Answer 26

• Type 1 occurs if the subject has the event before t₀ and thus is not included in the study. (e.g. exposure (E) under study causes individuals to die before they could enter the study, this could lead to a (selective) survival bias that would underestimate the effect of exposure)
• Type 2 occurs if the subject survives beyond time t₀ (i.e., t > t₀). This is required in order for the subject to have his/ her survival time observed.

(e.g. developed the disease (time 0) prior to being diagnosed with the disease at time t₀ and was observed after t₀. Subject included in the study)

Question 27

Differences between left truncation / left censorship (4)

Answer 27

If a subject is left censored at time t, then that subject is:

• (i) included in the study,
• (ii) known to be event free at time 0,
• (iii) known to be at risk for the event after time 0, and
• (iv) known to have had the event before time t but with the exact time of event being unknown

Question 28

Approaches of measuring survival time (2)

Answer 28

• Time-on-study
• Age-at-follow-up until either an event or censorhip

Question 29

Time-on-Study vs. Age as Time Scale: (closed cohort vs open cohort)

Which one to use?

Answer 29

Key issue: Did all subjects first become at risk at their study entry?

Clinical trial:

• Subjects start to be followed for the outcome after random allocation.
• Reasonable to assume subjects start to be at risk upon study entry.

⇒ time-on-study as the time scale is typically appropriate.

Observational study:

• Subjects already at risk prior to study entry
• Unknown time or age when first at risk

Reasonable to assume that T = t_r + t; where:
T = true survival time; t_r = time at risk prior to study entry; t = observed time-on-study.

The individual’s true (i.e., total) survival time istherefore underestimated by the time-on-study information, i.e. the true survival time is left-truncated.

⇒ time-on-study follow-up times that ignores unkown delayed entry time may be questioned.

Question 30

Modified approach to use time-on-study vs. age as time scale

Answer 30