g02caf (linregs_const) : NAG Library, Mark 27

2 Specification

Fortran Interface

Subroutine g02caf (

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(n), y(n)
Real (Kind=nag_wp), Intent (Out)	::	result(20)

C Header Interface

#include <nag.h>

void	g02caf_ (const Integer n, const double x[], const double y[], double result[], Integer ifail)

The routine may be called by the names g02caf or nagf_correg_linregs_const.

3 Description

g02caf fits a straight line of the form

y = a + b x

to the data points

(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n}),

such that

y_{i} = a + b x_{i} + e_{i}, i = 1, 2, \dots, n (n > 2) .

The routine calculates the regression coefficient,

b

, the regression constant,

a

(and various other statistical quantities) by minimizing

\sum_{i = 1}^{n} e_{i}^{2} .

The input data consist of the

n

pairs of observations

(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})

on the independent variable

x

and the dependent variable

y

The quantities calculated are:

(a)Means:
$\bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}; \bar{y} = \frac{1}{n} \sum_{i = 1}^{n} y_{i} .$
(b)Standard deviations:
$s_{x} = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}; s_{y} = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}} .$
(c)Pearson product-moment correlation coefficient:
$r = \frac{\sum_{i = 1}^{n} (x_{i} - \bar{x}) (y_{i} - \bar{y})}{\sqrt{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} \sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}}} .$
(d)The regression coefficient, $b$ , and the regression constant, $a$ :
$b = \frac{\sum_{i = 1}^{n} (x_{i} - \bar{x}) (y_{i} - \bar{y})}{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}; a = \bar{y} - b \bar{x} .$
(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, $SST$ :
$SST = \sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}; SSD = \sum_{i = 1}^{n} {(y_{i} - a - b x_{i})}^{2}; SSR = SST - SSD .$
(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, $DFT$ :
$DFT = n - 1; DFD = n - 2; DFR = 1 .$
(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, $s e (b)$ , and the standard error of the regression constant, $s e (a)$ :
$s e (b) = \sqrt{\frac{MSD}{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}}; s e (a) = \sqrt{MSD (\frac{1}{n} + \frac{{\bar{x}}^{2}}{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}})} .$
(j)The $t$ value for the regression coefficient, $t (b)$ , and the $t$ value for the regression constant, $t (a)$ :
$t (b) = \frac{b}{s e (b)}; t (a) = \frac{a}{s e (a)} .$

5 Arguments

n

– Integer Input

On entry:

n

, the number of pairs of observations.

Constraint:

n > 2

x (n)

– Real (Kind=nag_wp) array Input

On entry:

x (i)

must contain

x_{i}

, for

i = 1, 2, \dots, n

y (n)

– Real (Kind=nag_wp) array Input

On entry:

y (i)

must contain

y_{i}

, for

i = 1, 2, \dots, n

result (20)

– Real (Kind=nag_wp) array Output

On exit: the following information:

$result (1)$	$\bar{x}$ , the mean value of the independent variable, $x$ ;
$result (2)$	$\bar{y}$ , the mean value of the dependent variable, $y$ ;
$result (3)$	$s_{x}$ the standard deviation of the independent variable, $x$ ;
$result (4)$	$s_{y}$ the standard deviation of the dependent variable, $y$ ;
$result (5)$	$r$ , the Pearson product-moment correlation between the independent variable $x$ and the dependent variable $y$ ;
$result (6)$	$b$ , the regression coefficient;
$result (7)$	$a$ , the regression constant;
$result (8)$	$s e (b)$ , the standard error of the regression coefficient;
$result (9)$	$s e (a)$ , the standard error of the regression constant;
$result (10)$	$t (b)$ , the $t$ value for the regression coefficient;
$result (11)$	$t (a)$ , the $t$ value for the regression constant;
$result (12)$	$SSR$ , the sum of squares attributable to the regression;
$result (13)$	$DFR$ , the degrees of freedom attributable to the regression;
$result (14)$	$MSR$ , the mean square attributable to the regression;
$result (15)$	$F$ , the $F$ value for the analysis of variance;
$result (16)$	$SSD$ , the sum of squares of deviations about the regression;
$result (17)$	$DFD$ , the degrees of freedom of deviations about the regression
$result (18)$	$MSD$ , the mean square of deviations about the regression;
$result (19)$	$SST$ , the total sum of squares;
$result (20)$	DFT, the total degrees of freedom.

ifail

– Integer Input/Output

On entry: ifail must be set to

0

- 1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

- 1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

- 1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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,

n = 〈value〉

.
Constraint:

n > 2

ifail = 2

On entry, all n values of at least one of x and y are identical.

ifail = - 99

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.

ifail = - 399

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.

ifail = - 999

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

g02caf does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large

n

If, in calculating

F

t (a)

t (b)

(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.

NAG FL Interface
g02caf (linregs_const)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interfaceg02caf (linregs_​const)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g02caf (linregs_const)