The routine may be called by the names g12baf or nagf_surviv_coxmodel.
3Description
The proportional hazard model relates the time to an event, usually death or failure, to a number of explanatory variables known as covariates. Some of the observations may be right-censored, that is the exact time to failure is not known, only that it is greater than a known time.
Let , for , be the failure time or censored time for the th observation with the vector of covariates . It is assumed that censoring and failure mechanisms are independent. The hazard function, , is the probability that an individual with covariates fails at time given that the individual survived up to time . In the Cox proportional hazards model (see Cox (1972))
is of the form:
where is the base-line hazard function, an unspecified function of time, is a vector of unknown parameters and is a known offset.
Assuming there are ties in the failure times giving distinct failure times, such that individuals fail at , it follows that the marginal likelihood for is well approximated (see Kalbfleisch and Prentice (1980)) by:
(1)
where is the sum of the covariates of individuals observed to fail at and is the set of individuals at risk just prior to , that is, it is all individuals that fail or are censored at time along with all individuals that survive beyond time . The maximum likelihood estimates (MLEs) of , given by , are obtained by maximizing (1) using a Newton–Raphson iteration technique that includes step halving and utilizes the first and second partial derivatives of (1) which are given by equations (2) and (3) below:
(2)
for , where is the th element in the vector and
Similarly,
(3)
where
is the th component of a score vector and is the element of the observed information matrix whose inverse gives the variance-covariance matrix of .
It should be noted that if a covariate or a linear combination of covariates is monotonically increasing or decreasing with time then one or more of the 's will be infinite.
If varies across strata, where the number of individuals in the th stratum is , for with , then rather than maximizing (1) to obtain , the following marginal likelihood is maximized:
(4)
where is the contribution to likelihood for the observations in the th stratum treated as a single sample in (1). When strata are included the covariate coefficients are constant across strata but there is a different base-line hazard function .
The base-line survivor function associated with a failure time , is estimated as , where
(5)
where is the number of failures at time . The residual for the th observation is computed as:
where . The deviance is defined as (logarithm of marginal likelihood). There are two ways to test whether individual covariates are significant: the differences between the deviances of nested models can be compared with the appropriate -distribution; or, the asymptotic normality of the parameter estimates can be used to form tests by dividing the estimates by their standard errors or the score function for the model under the null hypothesis can be used to form tests.
4References
Cox D R (1972) Regression models in life tables (with discussion) J. Roy. Statist. Soc. Ser. B34 187–220
Gross A J and Clark V A (1975) Survival Distributions: Reliability Applications in the Biomedical Sciences Wiley
Kalbfleisch J D and Prentice R L (1980) The Statistical Analysis of Failure Time Data Wiley
Constraint:
if ,
and more than ip values of , for .
13: – Real (Kind=nag_wp)Output
On exit: the deviance, that is (maximized log marginal likelihood).
14: – Real (Kind=nag_wp) arrayInput/Output
On entry: initial estimates of the covariate coefficient parameters . must contain the initial estimate of the coefficient of the covariate in z corresponding to the th nonzero value of isz.
Suggested value:
in many cases an initial value of zero for may be used. For other suggestions see Section 9.
On exit: contains the estimate , the coefficient of the covariate stored in the th column of z where is the th nonzero value in the array isz.
15: – Real (Kind=nag_wp) arrayOutput
On exit: is the asymptotic standard error of the estimate contained in and score function in , for .
16: – Real (Kind=nag_wp) arrayOutput
On exit: is the value of the score function, , for the estimate contained in .
17: – Real (Kind=nag_wp) arrayOutput
On exit: the variance-covariance matrix of the parameter estimates in b stored in packed form by column, i.e., the covariance between the parameter estimates given in and , , is stored in .
18: – Real (Kind=nag_wp) arrayOutput
On exit: the residuals,
, for .
19: – IntegerOutput
On exit: the number of distinct failure times.
20: – Real (Kind=nag_wp) arrayOutput
On exit: contains the th distinct failure time, for .
21: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array sur
must be at least
.
On exit: if , contains the estimated survival function for the th distinct failure time.
If , contains the estimated survival function for the th distinct failure time in the th stratum.
22: – IntegerInput
On entry: the dimension of the array tp and the first dimension of the array sur as declared in the (sub)program from which g12baf is called.
Constraint:
.
23: – Real (Kind=nag_wp)Input
On entry: indicates the accuracy required for the estimation. Convergence is assumed when the decrease in deviance is less than . This corresponds approximately to an absolute precision if the deviance is small and a relative precision if the deviance is large.
Constraint:
.
24: – IntegerInput
On entry: the maximum number of iterations to be used for computing the estimates. If maxit is set to then the standard errors, score functions, variance-covariance matrix and the survival function are computed for the input value of in b but is not updated.
Constraint:
.
25: – IntegerInput
On entry: indicates if the printing of information on the iterations is required.
No printing.
The deviance and the current estimates are printed every iprint iterations. When printing occurs the output is directed to the current advisory message unit (see x04abf).
26: – Real (Kind=nag_wp) arrayWorkspace
27: – Integer arrayWorkspace
28: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: or .
On entry, .
Constraint: .
All observations are censored.
On entry, , and .
Constraint: .
On entry, and .
Constraint: or .
On entry, and .
Constraint: .
On entry, and minimum value for .
Constraint: of distinct failure times.
The matrix of second partial derivative is singular. Try different starting values or include fewer covariates.
Overflow has been detected in the calculations. Try using different starting values.
Convergence not achieved in iterations. The progress toward convergence can be examined by using a nonzero value of iprint. Any non-convergence may be due to a linear combination of covariates being monotonic with time. The full results are returned.
Too many step halvings required. In the current iteration step halvings have been preformed without decreasing the deviance from the previous iteration. The process is assumed to have converged.
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
Background information to multithreading can be found in the Multithreading documentation.
g12baf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g12baf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
g12baf uses mean centering which involves subtracting the means from the covariables prior to computation of any statistics. This helps to minimize the effect of outlying observations and accelerates convergence.
If the initial estimates are poor then there may be a problem with overflow in calculating or there may be non-convergence. Reasonable estimates can often be obtained by fitting an exponential model using g02gcf.
10Example
The data are the remission times for two groups of leukemia patients (see page 242 of Gross and Clark (1975)). A dummy variable indicates which group they come from. An initial estimate is computed using the exponential model and then the Cox proportional hazard model is fitted and parameter estimates and the survival function are printed.