NAG Library Routine Document

G12BAF

INTEGER	N, M, NS, LDZ, ISZ(M), IP, IC(N), ISI(), ND, NDMAX, MAXIT, IPRINT, IWK(2N), IFAIL
REAL (KIND=nag_wp)	Z(LDZ,M), T(N), OMEGA(), DEV, B(IP), SE(IP), SC(IP), COV(IP(IP+1)/2), RES(N), TP(NDMAX), SUR(NDMAX,), TOL, WK(IP(IP+9)/2+N)
CHARACTER(1)	OFFSET

3 Description

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

t_{i}

, for

i = 1, 2, \dots, n

, be the failure time or censored time for the

i

th observation with the vector of

p

covariates

z_{i}

. It is assumed that censoring and failure mechanisms are independent. The hazard function,

λ (t, z)

, is the probability that an individual with covariates

z

fails at time

t

given that the individual survived up to time

t

. In the Cox proportional hazards model (see Cox (1972))

λ (t, z)

is of the form:

λ (t, z) = λ_{0} (t) \exp (z^{T} β + ω)

where

λ_{0}

is the base-line hazard function, an unspecified function of time,

β

is a vector of unknown parameters and

ω

is a known offset.

Assuming there are ties in the failure times giving

n_{d} < n

distinct failure times,

t_{(1)} < \dots < t_{(n_{d})}

such that

d_{i}

individuals fail at

t_{(i)}

, it follows that the marginal likelihood for

β

is well approximated (see Kalbfleisch and Prentice (1980)) by:

L = \prod_{i = 1}^{n_{d}} \frac{\exp (s_{i}^{T} β + ω_{i})}{{[\sum_{l \in R (t_{(i)})} \exp (z_{l}^{T} β + ω_{l})]}^{d_{i}}}

(1)

where

s_{i}

is the sum of the covariates of individuals observed to fail at

t_{(i)}

and

R (t_{(i)})

is the set of individuals at risk just prior to

t_{(i)}

, that is, it is all individuals that fail or are censored at time

t_{(i)}

along with all individuals that survive beyond time

t_{(i)}

. The maximum likelihood estimates (MLEs) of

β

, given by

\hat{β}

, 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:

U_{j} (β) = \frac{\partial \ln L}{\partial β_{j}} = \sum_{i = 1}^{n_{d}} [s_{j i} - d_{i} α_{j i} (β)] = 0

(2)

for

j = 1, 2, \dots, p

, where

s_{j i}

is the

j

th element in the vector

s_{i}

and

α_{j i} (β) = \frac{\sum_{l \in R (t_{(i)})} z_{j l} \exp (z_{l}^{T} β + ω_{l})}{\sum_{l \in R (t_{(i)})} \exp (z_{l}^{T} β + ω_{l})} .

Similarly,

I_{h j} (β) = - \frac{\partial^{2} \ln L}{\partial β_{h} \partial β_{j}} = \sum_{i = 1}^{n_{d}} d_{i} γ_{h j i}

(3)

where

γ_{h j i} = \frac{\sum_{l \in R (t_{(i)})} z_{h l} z_{j l} \exp (z_{l}^{T} β + ω_{l})}{\sum_{l \in R (t_{(i)})} \exp (z_{l}^{T} β + ω_{l})} - α_{hi} (β) α_{j i} (β), h, j = 1, \dots, p .

U_{j} (β)

is the

j

th component of a score vector and

I_{h j} (β)

is the

(h, j)

element of the observed information matrix

I (β)

whose inverse

I {(β)}^{- 1} = {[I_{h j} (β)]}^{- 1}

gives the variance-covariance matrix of

β

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

β_{j}

's will be infinite.

λ_{0} (t)

varies across

ν

strata, where the number of individuals in the

k

th stratum is

n_{k}

, for

k = 1, 2, \dots, ν

with

n = \sum_{k = 1}^{ν} n_{k}

, then rather than maximizing (1) to obtain

\hat{β}

, the following marginal likelihood is maximized:

L = \prod_{k = 1}^{ν} L_{k},

(4)

where

L_{k}

is the contribution to likelihood for the

n_{k}

observations in the

k

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

λ_{0}

The base-line survivor function associated with a failure time

t_{(i)}

, is estimated as

\exp (- \hat{H} (t_{(i)}))

, where

\hat{H} (t_{(i)}) = \sum_{t_{(j)} \leq t_{(i)}} (\frac{d_{i}}{\sum_{l \in R (t_{(j)})} \exp (z_{l}^{T} \hat{β} + ω_{l})}),

(5)

where

d_{i}

is the number of failures at time

t_{(i)}

. The residual for the

l

th observation is computed as:

r (t_{l}) = \hat{H} (t_{l}) \exp (z_{l}^{T} \hat{β} + ω_{l})

where

\hat{H} (t_{l}) = \hat{H} (t_{(i)}), t_{(i)} \leq t_{l} < t_{(i + 1)}

. The deviance is defined as

- 2 \times

(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

χ^{2}

-distribution; or, the asymptotic normality of the parameter estimates can be used to form

z

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

z

tests.

4 References

Cox D R (1972) Regression models in life tables (with discussion) J. Roy. Statist. Soc. Ser. B 34 187–220

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

5 Parameters

1: $OFFSET$ – CHARACTER(1)Input

On entry: indicates if an offset is to be used.

$OFFSET ='Y'$: An offset must be included in OMEGA.
$OFFSET ='N'$: No offset is included in the model.

Constraint:

OFFSET ='Y'

'N'

2: $N$ – INTEGERInput

On entry:

n

, the number of data points.

Constraint:

N \geq 2

3: $M$ – INTEGERInput

On entry: the number of covariates in array Z.

Constraint:

M \geq 1

4: $NS$ – INTEGERInput

On entry: the number of strata. If

NS > 0

then the stratum for each observation must be supplied in ISI.

Constraint:

NS \geq 0

5: $Z (LDZ, M)$ – REAL (KIND=nag_wp) arrayInput

On entry: the

i

th row must contain the covariates which are associated with the

i

th failure time given in T.

6: $LDZ$ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which G12BAF is called.

Constraint:

LDZ \geq N

7: $ISZ (M)$ – INTEGER arrayInput

On entry: indicates which subset of covariates is to be included in the model.

$ISZ (j) \geq 1$: The $j$ th covariate is included in the model.
$ISZ (j) = 0$: The $j$ th covariate is excluded from the model and not referenced.

Constraint:

ISZ (j) \geq 0

and at least one and at most

n_{0} - 1

elements of ISZ must be nonzero where

n_{0}

is the number of observations excluding any with zero value of ISI.

8: $IP$ – INTEGERInput

On entry: the number of covariates included in the model as indicated by ISZ.

Constraints:

$IP \geq 1$ ;
$IP = number of nonzero values of ISZ$ .

9: $T (N)$ – REAL (KIND=nag_wp) arrayInput

On entry: the vector of

n

failure censoring times.

10: $IC (N)$ – INTEGER arrayInput

On entry: the status of the individual at time

t

given in T.

$IC (i) = 0$: The $i$ th individual has failed at time $T (i)$ .
$IC (i) = 1$: The $i$ th individual has been censored at time $T (i)$ .

Constraint:

IC (i) = 0

1

, for

i = 1, 2, \dots, N

11: $OMEGA (*)$ – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array OMEGA must be at least

N

OFFSET ='Y'

, and at least

1

otherwise.

On entry: if

OFFSET ='Y'

, the offset,

ω_{i}

, for

i = 1, 2, \dots, N

. Otherwise OMEGA is not referenced.

12: $ISI (*)$ – INTEGER arrayInput

Note: the dimension of the array ISI must be at least

N

NS > 0

, and at least

1

otherwise.

On entry: if

NS > 0

, the stratum indicators which also allow data points to be excluded from the analysis.

NS = 0

, ISI is not referenced.

$ISI (i) = k$: The $i$ th data point is in the $k$ th stratum, where $k = 1, 2, \dots, NS$ .
$ISI (i) = 0$: The $i$ th data point is omitted from the analysis.

Constraint: if

NS > 0

0 \leq ISI (i) \leq NS

and more than IP values of

ISI (i) > 0

, for

i = 1, 2, \dots, N

13: $DEV$ – REAL (KIND=nag_wp)Output

On exit: the deviance, that is

- 2 \times

(maximized log marginal likelihood).

14: $B (IP)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: initial estimates of the covariate coefficient parameters

β

B (j)

must contain the initial estimate of the coefficient of the covariate in Z corresponding to the

j

th nonzero value of ISZ.

Suggested value: in many cases an initial value of zero for

B (j)

may be used. For other suggestions see Section 9.

On exit:

B (j)

contains the estimate

{\hat{β}}_{i}

, the coefficient of the covariate stored in the

i

th column of Z where

i

is the

j

th nonzero value in the array ISZ.

15: $SE (IP)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

SE (j)

is the asymptotic standard error of the estimate contained in

B (j)

and score function in

SC (j)

, for

j = 1, 2, \dots, IP

16: $SC (IP)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

SC (j)

is the value of the score function,

U_{j} (β)

, for the estimate contained in

B (j)

17: $COV (IP \times (IP + 1) / 2)$ – 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

B (i)

and

B (j)

j \geq i

, is stored in

COV (j (j - 1) / 2 + i)

18: $RES (N)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the residuals,

r (t_{l})

, for

l = 1, 2, \dots, N

19: $ND$ – INTEGEROutput

On exit: the number of distinct failure times.

20: $TP (NDMAX)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

TP (i)

contains the

i

th distinct failure time, for

i = 1, 2, \dots, ND

21: $SUR (NDMAX *)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array SUR must be at least

\max (NS, 1)

On exit: if

NS = 0

SUR (i, 1)

contains the estimated survival function for the

i

th distinct failure time.

NS > 0

SUR (i, k)

contains the estimated survival function for the

i

th distinct failure time in the

k

th stratum.

22: $NDMAX$ – 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:

NDMAX \geq the number of distinct failure times. This is returned in ​ ND

23: $TOL$ – 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

TOL \times (1.0 + CurrentDeviance)

. This corresponds approximately to an absolute precision if the deviance is small and a relative precision if the deviance is large.

Constraint:

TOL \geq 10 \times machine precision

24: $MAXIT$ – INTEGERInput

On entry: the maximum number of iterations to be used for computing the estimates. If MAXIT is set to

0

then the standard errors, score functions, variance-covariance matrix and the survival function are computed for the input value of

β

in B but

β

is not updated.

Constraint:

MAXIT \geq 0

25: $IPRINT$ – INTEGERInput

On entry: indicates if the printing of information on the iterations is required.

$IPRINT \leq 0$: No printing.
$IPRINT \geq 1$: 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: $WK (IP \times (IP + 9) / 2 + N)$ – REAL (KIND=nag_wp) arrayWorkspace

27: $IWK (2 \times N)$ – INTEGER arrayWorkspace

28: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$OFFSET \neq'Y'$ or $'N'$ ,
or	$M < 1$ ,
or	$N < 2$ ,
or	$NS < 0$ ,
or	$IP < 1$ ,
or	$LDZ < N$ ,
or	$TOL < 10 \times machine precision$ ,
or	$MAXIT < 0$ .

$IFAIL = 2$

On entry,	$ISZ (i) < 0$ for some $i$ ,
or	the value of IP is incompatible with ISZ,
or	$IC (i) \neq 1$ or $0$ .
or	$ISI (i) < 0$ or $ISI (i) > NS$ ,
or	number of values of $ISZ (i) > 0$ is greater than or equal to $n_{0}$ , the number of observations excluding any with $ISI (i) = 0$ ,
or	all observations are censored, i.e., $IC (i) = 1$ for all $i$ ,
or	NDMAX is too small.

$IFAIL = 3$: The matrix of second partial derivatives is singular. Try different starting values or include fewer covariates.

$IFAIL = 4$: Overflow has been detected. Try using different starting values.

$IFAIL = 5$: Convergence has not been achieved in MAXIT 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.

Full results are returned.

$IFAIL = 6$: In the current iteration $10$ step halvings have been performed without decreasing the deviance from the previous iteration. Convergence is assumed.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

The accuracy is specified by TOL.

8 Parallelism and Performance

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.

9 Further 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

\exp (β^{T} z_{i})

or there may be non-convergence. Reasonable estimates can often be obtained by fitting an exponential model using G02GCF.

10 Example

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.

NAG Library Routine DocumentG12BAF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

G12BAF