NAG Library Routine Document

1Purpose

g02cdf performs a simple linear regression with no constant, with dependent variable $y$ and independent variable $x$, omitting cases involving missing values.

2Specification

Fortran Interface
 Subroutine g02cdf ( n, x, y,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(n), y(n), xmiss, ymiss Real (Kind=nag_wp), Intent (Out) :: result(21)
C Header Interface
#include <nagmk26.h>
 void g02cdf_ (const Integer *n, const double x[], const double y[], const double *xmiss, const double *ymiss, double result[], Integer *ifail)

3Description

g02cdf fits a straight line of the form
 $y=bx$
to those of the data points
 $x1,y1,x2,y2,…,xn,yn$
that do not include missing values, such that
 $yi=bxi+ei$
for those $\left({x}_{i},{y}_{i}\right)$, for $i=1,2,\dots ,n\text{ }\left(n\ge 2\right)$ which do not include missing values.
The routine eliminates all pairs of observations $\left({x}_{i},{y}_{i}\right)$ which contain a missing value for either $x$ or $y$, and then calculates the regression coefficient, $b$, and various other statistical quantities by minimizing the sum of the ${e}_{i}^{2}$ over those cases remaining in the calculations.
The input data consists of the $n$ pairs of observations $\left({x}_{1},{y}_{1}\right),\left({x}_{2},{y}_{2}\right),\dots ,\left({x}_{n},{y}_{n}\right)$ on the independent variable $x$ and the dependent variable $y$.
In addition two values, $\mathit{xm}$ and $\mathit{ym}$, are given which are considered to represent missing observations for $x$ and $y$ respectively. (See Section 7).
Let ${w}_{\mathit{i}}=0$, if the $\mathit{i}$th observation of either $x$ or $y$ is missing, i.e., if ${x}_{\mathit{i}}=\mathit{xm}$ and/or ${y}_{\mathit{i}}=\mathit{ym}$; and ${w}_{\mathit{i}}=1$ otherwise, for $\mathit{i}=1,2,\dots ,n$.
The quantities calculated are:
(a) Means:
 $x-=∑i=1nwixi ∑i=1nwi ; y-=∑i=1nwiyi ∑i=1nwi .$
(b) Standard deviations:
 $sx=∑i= 1nwi xi-x- 2 ∑i= 1nwi- 1 ; sy=∑i= 1nwi yi-y- 2 ∑i= 1nwi- 1 .$
(c) Pearson product-moment correlation coefficient:
 $r=∑i=1nwixi-x-yi-y- ∑i=1nwi xi-x- 2∑i=1nwi yi-y- 2 .$
(d) The regression coefficient, $b$:
 $b=∑i=1nwixiyi ∑i=1nwixi2 .$
(e) The sum of squares attributable to the regression, $SSR$, the sum of squares of deviations about the regression, $SSD$, and the total sum of squares, $SST$:
 $SST=∑i=1nwiyi2; SSD=∑i=1nwi yi-bxi 2; SSR=SST-SSD.$
(f) The degrees of freedom attributable to the regression, $DFR$, the degrees of freedom of deviations about the regression, $DFD$, and the total degrees of freedom, $DFT$:
 $DFT=∑i=1nwi; DFD=∑i=1nwi-1; DFR=1.$
(g) The mean square attributable to the regression, $MSR$, and the mean square of deviations about the regression, $MSD$:
 $MSR=SSR/DFR; MSD=SSD/DFD.$
(h) The $F$ value for the analysis of variance:
 $F=MSR/MSD.$
(i) The standard error of the regression coefficient:
 $seb=MSD ∑i= 1nwixi2 .$
(j) The $t$ value for the regression coefficient:
 $tb=bseb .$
(k) The number of observations used in the calculations:
 $nc=∑i= 1nwi.$

4References

Draper N R and Smith H (1985) Applied Regression Analysis (2nd Edition) Wiley

5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of pairs of observations.
Constraint: ${\mathbf{n}}\ge 2$.
2:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}\left(\mathit{i}\right)$ must contain ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
3:     $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{y}}\left(\mathit{i}\right)$ must contain ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
4:     $\mathbf{xmiss}$ – Real (Kind=nag_wp)Input
On entry: the value $xm$, which is to be taken as the missing value for the variable $x$ (see Section 7).
5:     $\mathbf{ymiss}$ – Real (Kind=nag_wp)Input
On entry: the value $ym$, which is to be taken as the missing value for the variable $y$ (see Section 7).
6:     $\mathbf{result}\left(21\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the following information:
 ${\mathbf{result}}\left(1\right)$ $\stackrel{-}{x}$, the mean value of the independent variable, $x$; ${\mathbf{result}}\left(2\right)$ $\stackrel{-}{y}$, the mean value of the dependent variable, $y$; ${\mathbf{result}}\left(3\right)$ ${s}_{x}$, the standard deviation of the independent variable, $x$; ${\mathbf{result}}\left(4\right)$ ${s}_{y}$, the standard deviation of the dependent variable, $y$; ${\mathbf{result}}\left(5\right)$ $r$, the Pearson product-moment correlation between the independent variable $x$ and the dependent variable, $y$; ${\mathbf{result}}\left(6\right)$ $b$, the regression coefficient; ${\mathbf{result}}\left(7\right)$ the value $0.0$; ${\mathbf{result}}\left(8\right)$ $se\left(b\right)$, the standard error of the regression coefficient; ${\mathbf{result}}\left(9\right)$ the value $0.0$; ${\mathbf{result}}\left(10\right)$ $t\left(b\right)$, the $t$ value for the regression coefficient; ${\mathbf{result}}\left(11\right)$ the value $0.0$; ${\mathbf{result}}\left(12\right)$ $SSR$, the sum of squares attributable to the regression; ${\mathbf{result}}\left(13\right)$ $DFR$, the degrees of freedom attributable to the regression; ${\mathbf{result}}\left(14\right)$ $MSR$, the mean square attributable to the regression; ${\mathbf{result}}\left(15\right)$ $F$, the $F$ value for the analysis of variance; ${\mathbf{result}}\left(16\right)$ $SSD$, the sum of squares of deviations about the regression; ${\mathbf{result}}\left(17\right)$ $DFD$, the degrees of freedom of deviations about the regression; ${\mathbf{result}}\left(18\right)$ $MSD$, the mean square of deviations about the regression; ${\mathbf{result}}\left(19\right)$ $SST$, the total sum of squares ${\mathbf{result}}\left(20\right)$ $DFT$, the total degrees of freedom; ${\mathbf{result}}\left(21\right)$ ${n}_{c}$, the number of observations used in the calculations.
7:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
After observations with missing values were omitted, fewer than two cases remained.
${\mathbf{ifail}}=3$
After observations with missing values were omitted, all remaining values of at least one of x and y were identical.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

g02cdf does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large $n$.
You are warned of the need to exercise extreme care in your selection of missing values. g02cdf treats all values in the inclusive range $\left(1±{0.1}^{\left({\mathbf{x02bef}}-2\right)}\right)×{xm}_{j}$, where ${\mathit{xm}}_{j}$ is the missing value for variable $j$ specified in xmiss.
You must therefore ensure that the missing value chosen for each variable is sufficiently different from all valid values for that variable so that none of the valid values fall within the range indicated above.
If, in calculating $F$ or $t\left(b\right)$  (see Section 3), the numbers involved are such that the result would be outside the range of numbers which can be stored by the machine, then the answer is set to the largest quantity which can be stored as a real variable, by means of a call to x02alf.

8Parallelism and Performance

g02cdf is not threaded in any implementation.

9Further Comments

The time taken by g02cdf depends on $n$ and the number of missing observations.
The routine uses a two-pass algorithm.

10Example

This example reads in eight observations on each of two variables, and then performs a simple linear regression with no constant, with the first variable as the independent variable, and the second variable as the dependent variable, omitting cases involving missing values ($0.0$ for the first variable, $99.0$ for the second). Finally the results are printed.

10.1Program Text

Program Text (g02cdfe.f90)

10.2Program Data

Program Data (g02cdfe.d)

10.3Program Results

Program Results (g02cdfe.r)