nag_rank_regsn_censored (g08rbc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_rank_regsn_censored (g08rbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_rank_regsn_censored (g08rbc) calculates the parameter estimates, score statistics and their variance-covariance matrices for the linear model using a likelihood based on the ranks of the observations when some of the observations may be right-censored.

2  Specification

#include <nag.h>
#include <nagg08.h>
void  nag_rank_regsn_censored (Nag_OrderType order, Integer ns, const Integer nv[], const double y[], Integer p, const double x[], Integer pdx, const Integer icen[], double gamma, Integer nmax, double tol, double prvr[], Integer pdprvr, Integer irank[], double zin[], double eta[], double vapvec[], double parest[], NagError *fail)

3  Description

Analysis of data can be made by replacing observations by their ranks. The analysis produces inference for the regression model where the location parameters of the observations, θi, for i=1,2,,n, are related by θ=Xβ. Here X is an n by p matrix of explanatory variables and β is a vector of p unknown regression parameters. The observations are replaced by their ranks and an approximation, based on Taylor's series expansion, made to the rank marginal likelihood. For details of the approximation see Pettitt (1982).
An observation is said to be right-censored if we can only observe Yj* with Yj*Yj. We rank censored and uncensored observations as follows. Suppose we can observe Yj, for j=1,2,,n, directly but Yj*, for j=n+1,,q and nq, are censored on the right. We define the rank rj of Yj, for j=1,2,,n, in the usual way; rj equals i if and only if Yj is the ith smallest amongst the Y1,Y2,,Yn. The right-censored Yj*, for j=n+1,n+2,,q, has rank rj if and only if Yj* lies in the interval Yrj,Yrj+1, with Y0=-, Yn+1=+ and Y1<<Yn the ordered Yj, for j=1,2,,n.
The distribution of the Y is assumed to be of the following form. Let FL y=ey/1+ey, the logistic distribution function, and consider the distribution function Fγy defined by 1-Fγ=1-FLy 1/γ . This distribution function can be thought of as either the distribution function of the minimum, X1,γ, of a random sample of size γ-1 from the logistic distribution, or as the Fγy-logγ being the distribution function of a random variable having the F-distribution with 2 and 2γ-1 degrees of freedom. This family of generalized logistic distribution functions Fγ.;0γ< naturally links the symmetric logistic distribution γ=1 with the skew extreme value distribution (limγ0) and with the limiting negative exponential distribution (limγ). For this family explicit results are available for right-censored data. See Pettitt (1983) for details.
Let lR denote the logarithm of the rank marginal likelihood of the observations and define the q×1 vector a by a=lRθ=0, and let the q by q diagonal matrix B and q by q symmetric matrix A be given by B-A=-lRθ=0. Then various statistics can be found from the analysis.
(a) The score statistic XTa. This statistic is used to test the hypothesis H0:β=0 (see (e)).
(b) The estimated variance-covariance matrix of the score statistic in (a).
(c) The estimate β^R=MXTa.
(d) The estimated variance-covariance matrix M=XTB-AX -1 of the estimate β^R.
(e) The χ2 statistic Q=β^RM-1​ ​β^r=aTXXTB-AX -1XTa, used to test H0:β=0. Under H0, Q has an approximate χ2-distribution with p degrees of freedom.
(f) The standard errors Mii 1/2 of the estimates given in (c).
(g) Approximate z-statistics, i.e., Zi=β^Ri/seβ^Ri for testing H0:βi=0. For i=1,2,,n, Zi has an approximate N0,1 distribution.
In many situations, more than one sample of observations will be available. In this case we assume the model,
hk Yk = XkT β+ek ,   k=1,2,,ns ,
where ns is the number of samples. In an obvious manner, Yk and Xk are the vector of observations and the design matrix for the kth sample respectively. Note that the arbitrary transformation hk can be assumed different for each sample since observations are ranked within the sample.
The earlier analysis can be extended to give a combined estimate of β as β^=Dd, where
d=k= 1ns XkT ak ,
with ak, Bk and Ak defined as a, B and A above but for the kth sample.
The remaining statistics are calculated as for the one sample case.

4  References

Kalbfleisch J D and Prentice R L (1980) The Statistical Analysis of Failure Time Data Wiley
Pettitt A N (1982) Inference for the linear model using a likelihood based on ranks J. Roy. Statist. Soc. Ser. B 44 234–243
Pettitt A N (1983) Approximate methods using ranks for regression with censored data Biometrika 70 121–132

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     nsIntegerInput
On entry: the number of samples.
Constraint: ns1.
3:     nv[ns]const IntegerInput
On entry: the number of observations in the ith sample, for i=1,2,,ns.
Constraint: nv[i-1]1, for i=1,2,,ns.
4:     y[dim]const doubleInput
Note: the dimension, dim, of the array y must be at least i=1 ns nv[i-1].
On entry: the observations in each sample. Specifically, y[ k=1 i-1 nv[k-1]+j-1]  must contain the jth observation in the ith sample.
5:     pIntegerInput
On entry: the number of parameters to be fitted.
Constraint: p1.
6:     x[dim]const doubleInput
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×p when order=Nag_ColMajor;
  • max1, i=1 ns nv[i-1]×pdx when order=Nag_RowMajor.
Where Xi,j appears in this document, it refers to the array element
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=Nag_RowMajor.
On entry: the design matrices for each sample. Specifically, X k=1 i-1 nv[k-1] + j ,l  must contain the value of the lth explanatory variable for the jth observations in the ith sample.
Constraint: x must not contain a column with all elements equal.
7:     pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
  • if order=Nag_ColMajor, pdx i=1 ns nv[i-1];
  • if order=Nag_RowMajor, pdxp.
8:     icen[dim]const IntegerInput
Note: the dimension, dim, of the array icen must be at least i=1 ns nv[i-1].
On entry: defines the censoring variable for the observations in y.
If y[i-1] is uncensored.
If y[i-1] is censored.
Constraint: icen[i-1]=0 or 1, for i=1,2,, i=1 ns nv[i-1].
9:     gammadoubleInput
On entry: the value of the parameter defining the generalized logistic distribution. For gamma0.0001, the limiting extreme value distribution is assumed.
Constraint: gamma0.0.
10:   nmaxIntegerInput
On entry: the value of the largest sample size.
Constraint: nmax=max1insnv[i-1] and nmax>p.
11:   toldoubleInput
On entry: the tolerance for judging whether two observations are tied. Thus, observations Yi and Yj are adjudged to be tied if Yi-Yj<tol.
Constraint: tol>0.0.
12:   prvr[dim]doubleOutput
Note: the dimension, dim, of the array prvr must be at least
  • max1,pdprvr×p when order=Nag_ColMajor;
  • max1,p+1×pdprvr when order=Nag_RowMajor.
Where PRVRi,j appears in this document, it refers to the array element
  • prvr[j-1×pdprvr+i-1] when order=Nag_ColMajor;
  • prvr[i-1×pdprvr+j-1] when order=Nag_RowMajor.
On exit: the variance-covariance matrices of the score statistics and the parameter estimates, the former being stored in the upper triangle and the latter in the lower triangle. Thus for 1ijp, PRVRi,j contains an estimate of the covariance between the ith and jth score statistics. For 1jip-1, PRVRi+1,j contains an estimate of the covariance between the ith and jth parameter estimates.
13:   pdprvrIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array prvr.
  • if order=Nag_ColMajor, pdprvrp+1;
  • if order=Nag_RowMajor, pdprvrp.
14:   irank[nmax]IntegerOutput
On exit: for the one sample case, irank contains the ranks of the observations.
15:   zin[nmax]doubleOutput
On exit: for the one sample case, zin contains the expected values of the function g. of the order statistics.
16:   eta[nmax]doubleOutput
On exit: for the one sample case, eta contains the expected values of the function g. of the order statistics.
17:   vapvec[nmax×nmax+1/2]doubleOutput
On exit: for the one sample case, vapvec contains the upper triangle of the variance-covariance matrix of the function g. of the order statistics stored column-wise.
18:   parest[4×p+1]doubleOutput
On exit: the statistics calculated by the function.
The first p components of parest contain the score statistics.
The next p elements contain the parameter estimates.
parest[2×p] contains the value of the χ2 statistic.
The next p elements of parest contain the standard errors of the parameter estimates.
Finally, the remaining p elements of parest contain the z-statistics.
19:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, ns=value.
Constraint: ns1.
On entry, p=value.
Constraint: p1.
On entry, pdprvr=value.
Constraint: pdprvr>0.
On entry, pdx=value.
Constraint: pdx>0.
On entry, nmax=value and p=value.
Constraint: nmax>p.
On entry, pdprvr=value and p=value.
Constraint: pdprvrp.
On entry, pdprvr=value and p=value.
Constraint: pdprvrp+1.
On entry, pdx=value and p=value.
Constraint: pdxp.
On entry, pdx=value and sum nv[i-1]=value.
Constraint: pdx the sum of nv[i-1].
On entry, ns=value and nv[i-1]=value.
Constraint: nv[i-1]1, for i=1,2,,ns.
On entry M=value.
Constraint: M elements of array icen=0 or 1.
On entry M=value.
Constraint: M elements of array nv>0.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
The matrix XTB-AX is either singular or non positive definite.
All the observations were adjudged to be tied.
On entry, gamma=value.
Constraint: gamma0.0.
On entry, tol=value.
Constraint: tol>0.0.
On entry, all elements in column value of x are equal to value.
The largest sample size is value which is not equal to nmax, nmax=value.

7  Accuracy

The computations are believed to be stable.

8  Further Comments

The time taken by nag_rank_regsn_censored (g08rbc) depends on the number of samples, the total number of observations and the number of parameters fitted.
In extreme cases the parameter estimates for certain models can be infinite, although this is unlikely to occur in practice. See Pettitt (1982) for further details.

9  Example

This example fits a regression model to a single sample of 40 observations using just one explanatory variable.

9.1  Program Text

Program Text (g08rbce.c)

9.2  Program Data

Program Data (g08rbce.d)

9.3  Program Results

Program Results (g08rbce.r)

nag_rank_regsn_censored (g08rbc) (PDF version)
g08 Chapter Contents
g08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012