g08raf 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.
The routine may be called by the names g08raf or nagf_nonpar_rank_regsn.
3Description
Analysis of data can be made by replacing observations by their ranks. The analysis produces inference for regression parameters arising from the following model.
For random variables we assume that, after an arbitrary monotone increasing differentiable transformation, , the model
(1)
holds, where is a known vector of explanatory variables and is a vector of unknown regression coefficients. The are random variables assumed to be independent and identically distributed with a completely known distribution which can be one of the following: Normal, logistic, extreme value or double-exponential. In Pettitt (1982) an estimate for is proposed as with estimated variance-covariance matrix . The statistics and depend on the ranks of the observations and the density chosen for .
The matrix is the matrix of explanatory variables. It is assumed that is of rank and that a column or a linear combination of columns of is not equal to the column vector of or a multiple of it. This means that a constant term cannot be included in the model (1). The statistics and are found as follows. Let have pdf and let . Let be order statistics for a random sample of size with the density . Define , then . To define we need , where is an diagonal matrix with and is a symmetric matrix with . In the case of the Normal distribution, the are standard Normal order statistics and , for .
The analysis can also deal with ties in the data. Two observations are adjudged to be tied if , where tol is a user-supplied tolerance level.
Various statistics can be found from the analysis:
(a)The score statistic . This statistic is used to test the hypothesis , see (e).
(b)The estimated variance-covariance matrix of the score statistic in (a).
(c)The estimate .
(d)The estimated variance-covariance matrix of the estimate .
(e)The statistic used to test . Under , has an approximate -distribution with degrees of freedom.
(f)The standard errors of the estimates given in (c).
(g)Approximate -statistics, i.e., for testing . For , has an approximate distribution.
In many situations, more than one sample of observations will be available. In this case we assume the model
where ns is the number of samples. In an obvious manner, and are the vector of observations and the design matrix for the th sample respectively. Note that the arbitrary transformation 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 as , where
and
with , and defined as , and above but for the th sample.
The remaining statistics are calculated as for the one sample case.
4References
Pettitt A N (1982) Inference for the linear model using a likelihood based on ranks J. Roy. Statist. Soc. Ser. B44 234–243
5Arguments
1: – IntegerInput
On entry: the number of samples.
Constraint:
.
2: – Integer arrayInput
On entry: the number of observations in the
th sample, for .
Constraint:
, for .
3: – IntegerInput
On entry: the total number of observations.
Constraint:
.
4: – Real (Kind=nag_wp) arrayInput
On entry: the observations in each sample. Specifically, must contain the th observation in the th sample.
5: – IntegerInput
On entry: the number of parameters to be fitted.
Constraint:
.
6: – Real (Kind=nag_wp) arrayInput
On entry: the design matrices for each sample. Specifically, must contain the value of the th explanatory variable for the th observation in the th sample.
Constraint:
must not contain a column with all elements equal.
7: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g08raf is called.
Constraint:
.
8: – IntegerInput
On entry: the error distribution to be used in the analysis.
Normal.
Logistic.
Extreme value.
Double-exponential.
Constraint:
.
9: – IntegerInput
On entry: the value of the largest sample size.
Constraint:
and .
10: – Real (Kind=nag_wp)Input
On entry: the tolerance for judging whether two observations are tied. Thus, observations and are adjudged to be tied if .
Constraint:
.
11: – 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 , contains an estimate of the covariance between the th and th score statistics. For , contains an estimate of the covariance between the th and th parameter estimates.
12: – IntegerInput
On entry: the first dimension of the array prvr as declared in the (sub)program from which g08raf is called.
Constraint:
.
13: – Integer arrayOutput
On exit: for the one sample case, irank contains the ranks of the observations.
14: – Real (Kind=nag_wp) arrayOutput
On exit: for the one sample case, zin contains the expected values of the function of the order statistics.
15: – Real (Kind=nag_wp) arrayOutput
On exit: for the one sample case, eta contains the expected values of the function of the order statistics.
16: – 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 of the order statistics stored column-wise.
17: – 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.
contains the value of the statistic.
The next ip elements of parest contain the standard errors of the parameter estimates.
Finally, the remaining ip elements of parest contain the -statistics.
18: – Real (Kind=nag_wp) arrayOutput
19: – IntegerInput
20: – Integer arrayOutput
On entry: are no longer required by g08raf but is retained for backwards compatibility.
21: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, elements of .
Constraint: .
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, .
On entry, , , or .
On entry, all the observations were adjudged to be tied. You are advised to check the value supplied for tol.
The matrix is either ill-conditioned or not positive definite. This error should only occur with extreme rankings of the data.
On entry, for , for all .
Constraint: for at least one .
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The computations are believed to be stable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g08raf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g08raf 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.
9Further Comments
The time taken by g08raf 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.
10Example
A program to fit a regression model to a single sample of observations using two explanatory variables. The error distribution will be taken to be logistic.