NAG Library Routine Document

G08RBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G08RBF 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

SUBROUTINE G08RBF (

NS, NV, NSUM, Y, IP, X, LDX, ICEN, GAMMA, NMAX, TOL, PRVR, LDPRVR, IRANK, ZIN, ETA, VAPVEC, PAREST, WORK, LWORK, IWA, IFAIL)

INTEGER	NS, NV(NS), NSUM, IP, LDX, ICEN(NSUM), NMAX, LDPRVR, IRANK(NMAX), LWORK, IWA(4*NMAX), IFAIL
REAL (KIND=nag_wp)	Y(NSUM), X(LDX,IP), GAMMA, TOL, PRVR(LDPRVR,IP), ZIN(NMAX), ETA(NMAX), VAPVEC(NMAX(NMAX+1)/2), PAREST(4IP+1), WORK(LWORK)

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,

θ_{i}

, for

i = 1, 2, \dots, n

, are related by

θ = X β

. Here

X

is an

n

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

Y_{j}^{*}

with

Y_{j}^{*} \leq Y_{j}

. We rank censored and uncensored observations as follows. Suppose we can observe

Y_{j}

, for

j = 1, 2, \dots, n

, directly but

Y_{j}^{*}

, for

j = n + 1, \dots, q

and

n \leq q

, are censored on the right. We define the rank

r_{j}

Y_{j}

, for

j = 1, 2, \dots, n

, in the usual way;

r_{j}

equals

i

if and only if

Y_{j}

is the

i

th smallest amongst the

Y_{1}, Y_{2}, \dots, Y_{n}

. The right-censored

Y_{j}^{*}

, for

j = n + 1, n + 2, \dots, q

, has rank

r_{j}

if and only if

Y_{j}^{*}

lies in the interval

[Y_{(r_{j})}, Y_{(r_{j} + 1)}]

, with

Y_{0} = - \infty

Y_{(n + 1)} = + \infty

and

Y_{(1)} < \dots < Y_{(n)}

the ordered

Y_{j}

, for

j = 1, 2, \dots, n

The distribution of the

Y

is assumed to be of the following form. Let

F_{L} (y) = e^{y} / (1 + e^{y})

, the logistic distribution function, and consider the distribution function

F_{γ} (y)

defined by

1 - F_{γ} = {[1 - F_{L} (y)]}^{1 / γ}

. This distribution function can be thought of as either the distribution function of the minimum,

X_{1, γ}

, of a random sample of size

γ^{- 1}

from the logistic distribution, or as the

F_{γ} (y - \log γ)

being the distribution function of a random variable having the

F

-distribution with

2

and

2 γ^{- 1}

degrees of freedom. This family of generalized logistic distribution functions

[F_{γ} (.); 0 \leq γ < \infty]

naturally links the symmetric logistic distribution

(γ = 1)

with the skew extreme value distribution (

\lim γ \to 0

) and with the limiting negative exponential distribution (

\lim γ \to \infty

). For this family explicit results are available for right-censored data. See Pettitt (1983) for details.

Let

l_{R}

denote the logarithm of the rank marginal likelihood of the observations and define the

q \times 1

vector

a

a = l_{R}^{'} (θ = 0)

, and let the

q

q

diagonal matrix

B

and

q

q

symmetric matrix

A

be given by

B - A = - l_{R}^{''} (θ = 0)

. Then various statistics can be found from the analysis.

(a)	The score statistic $X^{T} a$ . This statistic is used to test the hypothesis $H_{0} : β = 0$ (see (e)).
(b)	The estimated variance-covariance matrix of the score statistic in (a).
(c)	The estimate ${\hat{β}}_{R} = M X^{T} a$ .
(d)	The estimated variance-covariance matrix $M = {(X^{T} (B - A) X)}^{- 1}$ of the estimate ${\hat{β}}_{R}$ .
(e)	The $χ^{2}$ statistic $Q = {\hat{β}}_{R} M^{- 1} {\hat{β}}_{r} = a^{T} X {(X^{T} (B - A) X)}^{- 1} X^{T} a$ , used to test $H_{0} : β = 0$ . Under $H_{0}$ , $Q$ has an approximate $χ^{2}$ -distribution with $p$ degrees of freedom.
(f)	The standard errors $M_{i i}^{1 / 2}$ of the estimates given in (c).
(g)	Approximate $z$ -statistics, i.e., $Z_{i} = {\hat{β}}_{R_{i}} / s e ({\hat{β}}_{R_{i}})$ for testing $H_{0} : β_{i} = 0$ . For $i = 1, 2, \dots, n$ , $Z_{i}$ has an approximate $N (0, 1)$ distribution.

In many situations, more than one sample of observations will be available. In this case we assume the model,

h_{k} (Y_{k}) = X_{k}^{T} β + e_{k}, k = 1, 2, \dots, NS,

where NS is the number of samples. In an obvious manner,

Y_{k}

and

X_{k}

are the vector of observations and the design matrix for the

k

th sample respectively. Note that the arbitrary transformation

h_{k}

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

\hat{β} = D d

, where

D^{- 1} = \sum_{k = 1}^{NS} X^{T} (B_{k} - A_{k}) X_{k}

and

d = \sum_{k = 1}^{NS} X_{k}^{T} a_{k},

with

a_{k}

B_{k}

and

A_{k}

defined as

a

B

and

A

above but for the

k

th 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 Parameters

1: NS – INTEGERInput

On entry: the number of samples.

Constraint:

NS \geq 1

2: NV(NS) – INTEGER arrayInput

On entry: the number of observations in the

i

th sample, for

i = 1, 2, \dots, NS

Constraint:

NV (i) \geq 1

, for

i = 1, 2, \dots, NS

3: NSUM – INTEGERInput

On entry: the total number of observations.

Constraint:

NSUM = \sum_{i = 1}^{NS} NV (i)

4: Y(NSUM) – REAL (KIND=nag_wp) arrayInput

On entry: the observations in each sample. Specifically,

Y (\sum_{k = 1}^{i - 1} NV (k) + j)

must contain the

j

th observation in the

i

th sample.

5: IP – INTEGERInput

On entry: the number of parameters to be fitted.

Constraint:

IP \geq 1

6: X(LDX,IP) – REAL (KIND=nag_wp) arrayInput

On entry: the design matrices for each sample. Specifically,

X (\sum_{k = 1}^{i - 1} NV (k) + j, l)

must contain the value of the

l

th explanatory variable for the

j

th observations in the

i

th sample.

Constraint:

X

must not contain a column with all elements equal.

7: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G08RBF is called.

Constraint:

LDX \geq NSUM

8: ICEN(NSUM) – INTEGER arrayInput

On entry: defines the censoring variable for the observations in Y.

$ICEN (i) = 0$: If $Y (i)$ is uncensored.
$ICEN (i) = 1$: If $Y (i)$ is censored.

Constraint:

ICEN (i) = 0

1

, for

i = 1, 2, \dots, NSUM

9: GAMMA – REAL (KIND=nag_wp)Input

On entry: the value of the parameter defining the generalized logistic distribution. For

GAMMA \leq 0.0001

, the limiting extreme value distribution is assumed.

Constraint:

GAMMA \geq 0.0

10: NMAX – INTEGERInput

On entry: the value of the largest sample size.

Constraint:

NMAX = \max_{1 \leq i \leq NS} (NV (i))

and

NMAX > IP

11: TOL – REAL (KIND=nag_wp)Input

On entry: the tolerance for judging whether two observations are tied. Thus, observations

Y_{i}

and

Y_{j}

are adjudged to be tied if

|Y_{i} - Y_{j}| < TOL

Constraint:

TOL > 0.0

12: PRVR(LDPRVR,IP) – REAL (KIND=nag_wp) arrayOutput

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

1 \leq i \leq j \leq IP

PRVR (i, j)

contains an estimate of the covariance between the

i

th and

j

th score statistics. For

1 \leq j \leq i \leq IP - 1

PRVR (i + 1, j)

contains an estimate of the covariance between the

i

th and

j

th parameter estimates.

13: LDPRVR – INTEGERInput

On entry: the first dimension of the array PRVR as declared in the (sub)program from which G08RBF is called.

Constraint:

LDPRVR \geq IP + 1

14: IRANK(NMAX) – INTEGER arrayOutput

On exit: for the one sample case, IRANK contains the ranks of the observations.

15: ZIN(NMAX) – REAL (KIND=nag_wp) arrayOutput

On exit: for the one sample case, ZIN contains the expected values of the function

g (.)

of the order statistics.

16: ETA(NMAX) – REAL (KIND=nag_wp) arrayOutput

On exit: for the one sample case, ETA contains the expected values of the function

g' (.)

of the order statistics.

17: VAPVEC( $NMAX \times (NMAX + 1) / 2$ ) – REAL (KIND=nag_wp) arrayOutput

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( $4 \times IP + 1$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: the statistics calculated by the routine.

The first IP components of PAREST contain the score statistics.

The next IP elements contain the parameter estimates.

PAREST (2 \times IP + 1)

contains the value of the

χ^{2}

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.

19: WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace

20: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which G08RBF is called.

Constraint:

LWORK \geq NMAX \times (IP + 1)

21: IWA( $4 \times NMAX$ ) – INTEGER arrayWorkspace

22: 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,	$NS < 1$ ,
or	$TOL \leq 0.0$ ,
or	$NMAX \leq IP$ ,
or	$LDPRVR < IP + 1$ ,
or	$LDX < NSUM$ ,
or	$NMAX \neq \max_{1 \leq i \leq NS} (NV (i))$ ,
or	$NV (i) \leq 0$ for some $i$ , $i = 1, 2, \dots, NS$ ,
or	$NSUM \neq \sum_{i = 1}^{NS} NV (i)$ ,
or	$IP < 1$ ,
or	$GAMMA < 0.0$ ,
or	$LWORK < NMAX \times (IP + 1)$ .

$IFAIL = 2$

On entry,

ICEN (i) \neq 0

1

, for some

1 \leq i \leq NSUM

$IFAIL = 3$: On entry, all the observations are adjudged to be tied. You are advised to check the value supplied for TOL.

$IFAIL = 4$: The matrix $X^{T} (B - A) X$ is either ill-conditioned or not positive definite. This error should only occur with extreme rankings of the data.

$IFAIL = 5$

On entry,

at least one column of the matrix

X

has all its elements equal.

7 Accuracy

The computations are believed to be stable.

8 Further Comments

The time taken by G08RBF 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.

NAG Library Routine DocumentG08RBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G08RBF