naginterfaces.library.nonpar.rank_​regsn

naginterfaces.library.nonpar.rank_regsn(nv, y, x, idist, nmax, tol)

rank_regsn 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.

For full information please refer to the NAG Library document for g08ra

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g08/g08raf.html

Parameters

nvint, array-like, shape $\left(\text{ns}\right)$

The number of observations in the $\text{i}$th sample, for $\text{i}=1,2,\dots ,\text{ns}$.

yfloat, array-like, shape $\left(sum\left(nv\right)\right)$

The observations in each sample. Specifically, $y[{\sum }_{k=1}^{i-1}nv[k-1]+j-1]$ must contain the $j$th observation in the $i$th sample.

xfloat, array-like, shape $(\text{nsum},\text{ip})$

The design matrices for each sample. Specifically, $x\left[{\sum }_{k=1}^{i-1}nv\right[k-1]+j-1,l-1]$ must contain the value of the $l$th explanatory variable for the $j$th observation in the $i$th sample.

idistint

The error distribution to be used in the analysis.

$idist=1$

Normal.

$idist=2$

Logistic.

$idist=3$

Extreme value.

$idist=4$

Double-exponential.

nmaxint

The value of the largest sample size.

tolfloat

The tolerance for judging whether two observations are tied. Thus, observations $Y_i$ and $Y_j$ are adjudged to be tied if $|Y_i-Y_j|<tol$.

Returns

prvrfloat, ndarray, shape $(\text{ip}+1,\text{ip})$

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 $1\le i\le j\le \text{ip}$, $prvr[i-1,j-1]$ contains an estimate of the covariance between the $i$th and $j$th score statistics. For $1\le j\le i\le \text{ip}-1$, $prvr[i,j-1]$ contains an estimate of the covariance between the $i$th and $j$th parameter estimates.

irankint, ndarray, shape $\left(nmax\right)$

For the one sample case, $irank$ contains the ranks of the observations.

zinfloat, ndarray, shape $\left(nmax\right)$

For the one sample case, $zin$ contains the expected values of the function $g(.)$ of the order statistics.

etafloat, ndarray, shape $\left(nmax\right)$

For the one sample case, $eta$ contains the expected values of the function $g\prime (.)$ of the order statistics.

vapvecfloat, ndarray, shape $(nmax\times (nmax+1)/2)$

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.

parestfloat, ndarray, shape $(4\times \text{ip}+1)$

The statistics calculated by the function.

The first $\text{ip}$ components of $parest$ contain the score statistics.

The next $\text{ip}$ elements contain the parameter estimates.

$parest[2\times \text{ip}]$ contains the value of the ${\chi }^2$ statistic.

The next $\text{ip}$ elements of $parest$ contain the standard errors of the parameter estimates.

Finally, the remaining $\text{ip}$ elements of $parest$ contain the $z$-statistics.

Raises

NagValueError

(errno $1$)

On entry, ${\text{max}}_i\left(nv\right[i\left]\right)=⟨value⟩$ and $nmax=⟨value⟩$.

Constraint: ${\text{max}}_i\left(nv\right[i\left]\right)=nmax$.

(errno $1$)

On entry, ${\sum }_i\left(nv\right[i\left]\right)=⟨value⟩$ and $\text{nsum}=⟨value⟩$.

Constraint: ${\sum }_i\left(nv\right[i\left]\right)=\text{nsum}$.

(errno $1$)

On entry, $⟨value⟩$ elements of $nv\text{are}<1$.

Constraint: $nv\left[i\right]\ge 1$.

(errno $1$)

On entry, $nmax=⟨value⟩$ and $\text{ip}=⟨value⟩$.

Constraint: $nmax>\text{ip}$.

(errno $1$)

On entry, $\text{ip}=⟨value⟩$.

Constraint: $\text{ip}\ge 1$.

(errno $1$)

On entry, $tol=⟨value⟩$.

Constraint: $tol>0.0$.

(errno $1$)

On entry, $\text{ns}=⟨value⟩$.

Constraint: $\text{ns}\ge 1$.

(errno $2$)

On entry, $idist=⟨value⟩$.

On entry, $idist=1$, $2$, $3$ or $4$.

(errno $3$)

On entry, all the observations were adjudged to be tied.

(errno $4$)

The matrix $X^T(B-A)X$ is either ill-conditioned or not positive definite.

(errno $5$)

On entry, for $j=⟨value⟩$, $x[i,j-1]=⟨value⟩$ for all $i$.

Constraint: $x[i,j-1]\ne x[i+1,j-1]$ for at least one $i$.

Notes

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 $Y_1,Y_2,\dots ,Y_n$ we assume that, after an arbitrary monotone increasing differentiable transformation, $h(.)$, the model

$$h\left(Y_i\right)=x_i^T\beta +{ϵ}_i$$

holds, where $x_i$ is a known vector of explanatory variables and $\beta$ is a vector of $p$ unknown regression coefficients. The ${ϵ}_i$ 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 $\beta$ is proposed as $\hat{\beta }=M{X}^{T}a$ with estimated variance-covariance matrix $M$. The statistics $a$ and $M$ depend on the ranks $r_i$ of the observations $Y_i$ and the density chosen for ${ϵ}_i$.

The matrix $X$ is the $n\times p$ matrix of explanatory variables. It is assumed that $X$ is of rank $p$ and that a column or a linear combination of columns of $X$ is not equal to the column vector of $1$ or a multiple of it. This means that a constant term cannot be included in the model (1). The statistics $a$ and $M$ are found as follows. Let ${ϵ}_i$ have pdf $f\left(ϵ\right)$ and let $g=-f^{\prime }/f$. Let $W_1,W_2,\dots ,W_n$ be order statistics for a random sample of size $n$ with the density $f(.)$. Define $Z_i=g\left(W_i\right)$, then $a_i=E\left(Z_{r_i}\right)$. To define $M$ we need $M^{-1}=X^T(B-A)X$, where $B$ is an $n\times n$ diagonal matrix with $B_{ii}=E\left(g^{\prime }\left(W_{r_i}\right)\right)$ and $A$ is a symmetric matrix with $A_{ij}=cov(Z_{r_i},Z_{r_j})$. In the case of the Normal distribution, the $Z_1<\cdots <Z_n$ are standard Normal order statistics and $E\left(g^{\prime }\left(W_i\right)\right)=1$, for $i=1,2,\dots ,n$.

The analysis can also deal with ties in the data. Two observations are adjudged to be tied if $|Y_i-Y_j|<tol$, where $tol$ is a user-supplied tolerance level.

Various statistics can be found from the analysis:

1. The score statistic $X^{T}a$. This statistic is used to test the hypothesis $H_0:\beta =0$, see (e).

2. The estimated variance-covariance matrix $X^T(B-A)X$ of the score statistic in (a).

3. The estimate $\hat{\beta }=M{X}^{T}a$.

4. The estimated variance-covariance matrix $M={\left(X^T(B-A)X\right)}^{-1}$ of the estimate $\hat{\beta }$.

5. The ${\chi }^2$ statistic $Q={\hat{\beta }}^T{M}^{-1}\hat{\beta }=a^{T}X{\left(X^T(B-A)X\right)}^{-1}{X}^{T}a$ used to test $H_0:\beta =0$. Under $H_0$, $Q$ has an approximate ${\chi }^2$-distribution with $p$ degrees of freedom.

6. The standard errors $M_{ii}^{1/2}$ of the estimates given in (c).

7. Approximate $z$-statistics, i.e., $Z_i={\hat{\beta }}_i/se\left({\hat{\beta }}_i\right)$ for testing $H_0:{\beta }_i=0$. 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

$$h_k\left(Y_k\right)=X_k^T\beta +e_k\text{,}k=1,2,\dots ,\text{ns}\text{,}$$

where $\text{ns}$ 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 $\beta$ as $\hat{\beta }=Dd$, where

$$D^{-1}=\sum _{k=1}^{\text{ns}}{X}_k^T(B_k-A_k)X_k$$

and

$$d=\sum _{k=1}^{\text{ns}}{X}_k^T{a}_k\text{,}$$

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.

References

Pettitt, A N, 1982, Inference for the linear model using a likelihood based on ranks, J. Roy. Statist. Soc. Ser. B (44), 234–243