Ch 3. The Cox PH Model and Its Characteristics Flashcards
Tests statistics typically used with ML estimates (2)
- Wald statistic
- likelihood ratio (LR statistic): makes use of the log likelihood statistic.
Testing the significance of interaction using Likelihood Ratio (LR) statistic
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 -2lnLR minus -2lnLF, where R denotes the reduced model and F denotes the full model.
LR = -2 ln LR - (-2 ln LF)
Z Wald Statistics: How to obtain p-value in model
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.
Statistical objectives of model (3)
- test for significance of effect
- point estimate of effect
- confidence interval for effect
Point estimate of effect
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;
Calculation of confidence interval for effect
- Compute a 95% confidence interval for the regression coefficient of the Rx variable (‘beta’1).
- Exponentiate the two limits obtained for the confidence interval for the regression coefficient of Rx
exp[ ‘beta’1 +/- 1.96 sqrt (Var • ‘beta’1)
Confounding (HR)
Crude versus adjusted HR are meaningfully different.
Adjusted survival curves vs KM curves
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
The formula for the Cox PH Model
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 ßiXi, where the sum is over the p explanatory X variables.
What is a time-independent variable?
A time-independent variable is defined to be any variable whose value for a given individual does not change over time
Property that makes the Cox model a semiparametric model
The baseline hazard, h0(t), is an unspecified fuction
What is a parametric model?
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 ßi’s. Note that for the Weibull model, h0(t) is given by lambda*p*t^(p-1)
Why is the Cox PH model is popular?
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.
Why L is a partial likelihood (rather than a complete likelihood)
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.
Eq. partial likelihood
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, Lf denotes the likelihood of failing at this time, given survival up to this time.
L = L1 x L2 x L3 x … Lk = ΠLj (j = 1 , k)
Cox likelihood based on order of events rather than their distribution
Definition of ‘risk set’, R(t(f))
The set of individuals at risk at the jth failure
Solution by iteration
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.
General definition of hazard ratio (HR)
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
General rule for (0, 1) exposure variable when there are product terms:
Formula of Cox model hazard function and Cox model survival function
Meaning of the PH Assumption
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)
Obtaining maximum likelihood estimates (score equations)
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).
Defintion of time 0 (t0)
Starting time of the true survival time (starting point)
Defining the starting point: Possible choices for time 0:
- Study entry
- Beginning of treatment
- Disease onset
- Disease diagnosis
- Surgery
- Point in calendar time
- Birth
- Conception
Concept of left truncation at time t0
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
Types of left truncation (2)
- 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)
Differences between left truncation / left censorship (4)
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
Approaches of measuring survival time (2)
- Time-on-study
- Age-at-follow-up until either an event or censorhip
Time-on-Study vs. Age as Time Scale: (closed cohort vs open cohort)
Which one to use?
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 = tr + t; where:
T = true survival time; tr = 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.
Modified approach to use time-on-study vs. age as time scale
