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NAG Toolbox: nag_surviv_coxmodel_risksets (g12za)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_surviv_coxmodel_risksets (g12za) creates the risk sets associated with the Cox proportional hazards model for fixed covariates.

Syntax

[num, ixs, nxs, x, id, nd, tp, irs, ifail] = g12za(ns, z, isz, ip, t, ic, isi, mxn, 'n', n, 'm', m)
[num, ixs, nxs, x, id, nd, tp, irs, ifail] = nag_surviv_coxmodel_risksets(ns, z, isz, ip, t, ic, isi, mxn, 'n', n, 'm', m)

Description

The Cox proportional hazards model (see Cox (1972)) 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 ti, for i=1,2,,n, be the failure time or censored time for the ith observation with the vector of p covariates zi. It is assumed that censoring and failure mechanisms are independent. The hazard function, λt,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, λt,z is of the form
λt,z=λ0texpzTβ,  
where λ0 is the base-line hazard function, an unspecified function of time, and β is a vector of unknown arguments. As λ0 is unknown, the arguments β are estimated using the conditional or marginal likelihood. This involves considering the covariate values of all subjects that are at risk at the time when a failure occurs. The probability that the subject that failed had their observed set of covariate values is computed.
The risk set at a failure time consists of those subjects that fail or are censored at that time and those who survive beyond that time. As risk sets are computed for every distinct failure time, it should be noted that the combined risk sets may be considerably larger than the original data. If the data can be considered as coming from different strata such that λ0 varies from strata to strata but β remains constant, then nag_surviv_coxmodel_risksets (g12za) will return a factor that indicates to which risk set/strata each member of the risk sets belongs rather than just to which risk set.
Given the risk sets the Cox proportional hazards model can then be fitted using a Poisson generalized linear model (nag_correg_glm_poisson (g02gc) with nag_anova_dummyvars (g04ea) to compute dummy variables) using Breslow's approximation for ties (see Breslow (1974)). This will give the same fit as nag_surviv_coxmodel (g12ba). If the exact treatment of ties in discrete time is required, as given by Cox (1972), then the model is fitted as a conditional logistic model using nag_contab_condl_logistic (g11ca).

References

Breslow N E (1974) Covariate analysis of censored survival data Biometrics 30 89–99
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

Parameters

Compulsory Input Parameters

1:     ns int64int32nag_int scalar
The number of strata. If ns>0 then the stratum for each observation must be supplied in isi.
Constraint: ns0.
2:     zldzm – double array
ldz, the first dimension of the array, must satisfy the constraint ldzn.
The ith row must contain the covariates which are associated with the ith failure time given in t.
3:     iszm int64int32nag_int array
Indicates which subset of covariates are to be included in the model.
iszj1
The jth covariate is included in the model.
iszj=0
The jth covariate is excluded from the model and not referenced.
Constraint: iszj0 and at least one value must be nonzero.
4:     ip int64int32nag_int scalar
p, the number of covariates included in the model as indicated by isz.
Constraint: ip= the number of nonzero values of isz.
5:     tn – double array
The vector of n failure censoring times.
6:     icn int64int32nag_int array
The status of the individual at time t given in t.
ici=0
Indicates that the ith individual has failed at time ti.
ici=1
Indicates that the ith individual has been censored at time ti.
Constraint: ici=0 or 1, for i=1,2,,n.
7:     isi: int64int32nag_int array
The dimension of the array isi must be at least n if ns>0, and at least 1 otherwise
If ns>0, the stratum indicators which also allow data points to be excluded from the analysis.
If ns=0, isi is not referenced.
isii=k
Indicates that the ith data point is in the kth stratum, where k=1,2,,ns.
isii=0
Indicates that the ith data point is omitted from the analysis.
Constraint: if ns>0, 0isiins, for i=1,2,,n.
8:     mxn int64int32nag_int scalar
The first dimension of the array x and the dimension of the arrays ixs and id.
Constraint: mxn must be sufficiently large for the arrays to contain the expanded risk sets. The size will depend on the pattern of failures times and censored times. The minimum value will be returned in num unless the function exits with ifail=1 or 2.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the arrays t, ic and the first dimension of the array z. (An error is raised if these dimensions are not equal.)
n, the number of data points.
Constraint: n2.
2:     m int64int32nag_int scalar
Default: the dimension of the array isz and the second dimension of the array z. (An error is raised if these dimensions are not equal.)
The number of covariates in array z.
Constraint: m1.

Output Parameters

1:     num int64int32nag_int scalar
The number of values in the combined risk sets.
2:     ixsmxn int64int32nag_int array
The factor giving the risk sets/strata for the data in x and id.
If ns=0 or 1, ixsi=l for members of the lth risk set.
If ns>1, ixsi=j-1×nd+l for the observations in the lth risk set for the jth strata.
3:     nxs int64int32nag_int scalar
The number of levels for the risk sets/strata factor given in ixs.
4:     xmxnip – double array
The dimension of the array x will be num
The first num rows contain the values of the covariates for the members of the risk sets.
5:     idmxn int64int32nag_int array
Indicates if the member of the risk set given in x failed.
idi=1 if the member of the risk set failed at the time defining the risk set and idi=0 otherwise.
6:     nd int64int32nag_int scalar
The number of distinct failure times, i.e., the number of risk sets.
7:     tpn – double array
tpi contains the ith distinct failure time, for i=1,2,,nd.
8:     irsn int64int32nag_int array
Indicates rows in x and elements in ixs and id corresponding to the risk sets. The first risk set corresponding to failure time tp1 is given by rows 1 to irs1. The lth risk set is given by rows irsl-1+1 to irsl, for l=1,2,,nd.
9:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
On entry,m<1,
orn<2,
orns<0,
orldz<n.
   ifail=2
On entry,iszi<0 for some i,
orthe value of ip is incompatible with isz,
orici1 or 0.
orns>0 and isii<0,
orns>1 and isii>ns.
   ifail=3
mxn is too small, the minimum value is returned in num.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

When there are strata present, i.e., ns>1, not all the nxs groups may be present.

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. The risk sets are computed using nag_surviv_coxmodel_risksets (g12za) and the Cox's proportional hazard model is fitted using nag_contab_condl_logistic (g11ca).
function g12za_example


fprintf('g12za example results\n\n');

% Data
n = 42;
m = 1;
z          = zeros(n,m);
ic         = zeros(n,1,'int64');
ic(31:end) = 1;
z(22:end)  = 1;
isz        = [int64(1)];

t = [ 1;      1;      2;      2;      3;      4;      4;
      5;      5;      8;      8;      8;      8;     11;
     11;     12;     12;     15;     17;     22;     23;
      6;      6;      6;      7;     10;     13;     16;
     22;     23;      6;      9;     10;     11;     17;
     19;     20;     25;     32;     32;     34;     35];

% Parameters
ns  = int64(0);
ip  = int64(m);
isi = [int64(0)];
mxn = int64(1000);

% Create risk set
[num, ixs, nxs, x, id, nd, tp, irs, ifail] = ...
  g12za(...
        ns, z, isz, ip, t, ic, isi, mxn);

% Parameters for conditional logistic analysis
tol = 1e-05;
maxit = int64(10);

% initial estimate
b   = zeros(m,1);
% Calculate parameter estimates
[dev, b, se, sc, covar, nca, nct, ifail] = ...
g11ca( ...
       nxs, x, isz, id, ixs, b, tol, maxit, 'n', num);

% Display results
ns = max(ns,1);
fprintf(' Parameter      Estimate       Standard Error\n\n');
ivar = [1:m]';
fprintf('%6d%18.4f%18.4f\n',[ivar b se]');


g12za example results

 Parameter      Estimate       Standard Error

     1            1.6282            0.4331

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Chapter Introduction
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