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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_nonpar_rank_regsn_censored (g08rb)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_nonpar_rank_regsn_censored (g08rb) 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.


[prvr, irank, zin, eta, vapvec, parest, ifail] = g08rb(nv, y, x, icen, gamma, nmax, tol, 'ns', ns, 'ip', ip)
[prvr, irank, zin, eta, vapvec, parest, ifail] = nag_nonpar_rank_regsn_censored(nv, y, x, icen, gamma, nmax, tol, 'ns', ns, 'ip', ip)


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.


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


Compulsory Input Parameters

1:     nvns int64int32nag_int array
The number of observations in the ith sample, for i=1,2,,ns.
Constraint: nvi1, for i=1,2,,ns.
2:     ynsum – double array
nsum, the dimension of the array, must satisfy the constraint nsum= i=1 ns nvi .
The observations in each sample. Specifically, y k=1 i-1 nvk+j  must contain the jth observation in the ith sample.
3:     xldxip – double array
ldx, the first dimension of the array, must satisfy the constraint ldxnsum.
The design matrices for each sample. Specifically, x k=1 i-1 nvk + 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.
4:     icennsum int64int32nag_int array
nsum, the dimension of the array, must satisfy the constraint nsum= i=1 ns nvi .
Defines the censoring variable for the observations in y.
If yi is uncensored.
If yi is censored.
Constraint: iceni=0 or 1, for i=1,2,,nsum.
5:     gamma – double scalar
The value of the parameter defining the generalized logistic distribution. For gamma0.0001, the limiting extreme value distribution is assumed.
Constraint: gamma0.0.
6:     nmax int64int32nag_int scalar
The value of the largest sample size.
Constraint: nmax=max1insnvi and nmax>ip.
7:     tol – double scalar
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.

Optional Input Parameters

1:     ns int64int32nag_int scalar
Default: the dimension of the array nv.
The number of samples.
Constraint: ns1.
2:     ip int64int32nag_int scalar
Default: the second dimension of the array x.
The number of parameters to be fitted.
Constraint: ip1.

Output Parameters

1:     prvrldprvrip – double array
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 1ijip, prvrij contains an estimate of the covariance between the ith and jth score statistics. For 1jiip-1, prvri+1j contains an estimate of the covariance between the ith and jth parameter estimates.
2:     iranknmax int64int32nag_int array
For the one sample case, irank contains the ranks of the observations.
3:     zinnmax – double array
For the one sample case, zin contains the expected values of the function g. of the order statistics.
4:     etanmax – double array
For the one sample case, eta contains the expected values of the function g. of the order statistics.
5:     vapvecnmax×nmax+1/2 – double array
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.
6:     parest4×ip+1 – double array
The statistics calculated by the function.
The first ip components of parest contain the score statistics.
The next ip elements contain the parameter estimates.
parest2×ip+1 contains the value of the χ2 statistic.
The next ip elements of parest contain the standard errors of the parameter estimates.
Finally, the remaining ip elements of parest contain the z-statistics.
7:     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:
On entry,ns<1,
ornmaxmax1ins nvi,
ornvi0 for some i, i=1,2,,ns,
On entry,iceni0 or 1, for some 1insum.
On entry, all the observations are adjudged to be tied. You are advised to check the value supplied for tol.
The matrix XTB-AX is either ill-conditioned or not positive definite. This error should only occur with extreme rankings of the data.
On entry,at least one column of the matrix X has all its elements equal.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The computations are believed to be stable.

Further Comments

The time taken by nag_nonpar_rank_regsn_censored (g08rb) 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.


This example fits a regression model to a single sample of 40 observations using just one explanatory variable.
function g08rb_example

fprintf('g08rb example results\n\n');

y = [143; 164; 188; 188; 190; 192; 206; 209; 213; 216;
     220; 227; 230; 234; 246; 265; 304; 216; 244; 142;
     156; 163; 198; 205; 232; 232; 233; 233; 233; 233;
     239; 240; 261; 280; 280; 296; 296; 323; 204; 344];
nv = [int64(numel(y))];
x  = zeros(nv,1);
x(20:end) = 1;
icen = zeros(nv,1,'int64');
icen(18:19) = 1;
icen(39:40) = 1;

gamma = 1e-05;
nmax  = int64(nv);
tol   = 1e-05;

ns    = size(y,2);
ip    = size(x,2);
fprintf('Number of samples            = %3d\n', ns);
fprintf('Number of parameters fitted  = %3d\n', ip);
fprintf('Distribution power parameter = %8.1e\n', gamma);
fprintf('Tolerance for ties           = %8.1e\n', tol);

[parvar, irank, zin, eta, vapvec, parest, ifail] = ...
  g08rb( ...
         nv, y, x, icen, gamma, nmax, tol);

% Display results
fprintf('\nScore statistic\n');
fprintf('%9.3f\n', parest(1:ip));
fprintf('\nCovariance matrix of score statistic\n');
for j = 1:ip
  fprintf('%9.3f', parvar(1:j,j));
fprintf('\nParameter estimates\n');
fprintf('%9.3f', parest(ip+1:ip+ip));
fprintf('\n\nCovariance matrix of parameter estimates\n');
for j = 1:ip
  fprintf('%9.3f', parvar(j+1,1:j));

chisq = parest(2*ip+1);
fprintf('\nChi-squared statistic = %8.3f with %2d d.f.\n\n', chisq, ip);

sterr = reshape(parest(2*ip+2:end),[ip,2]);
fprintf('Standard errors of estimates and approximate z-statistics\n');

g08rb example results

Number of samples            =   1
Number of parameters fitted  =   1
Distribution power parameter =  1.0e-05
Tolerance for ties           =  1.0e-05

Score statistic

Covariance matrix of score statistic

Parameter estimates

Covariance matrix of parameter estimates

Chi-squared statistic =    2.746 with  1 d.f.

Standard errors of estimates and approximate z-statistics
    0.3615    1.6571

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Chapter Contents
Chapter Introduction
NAG Toolbox

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