G02CBF : NAG Library, Mark 25

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

G02CBF performs a simple linear regression with no constant, with dependent variable

y

and independent variable

x

G02CBF fits a straight line of the form

y = b x

to the data points

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

such that

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

The routine calculates the regression coefficient,

b

, and the various other statistical quantities by minimizing

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

The input data consists 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

b = \frac{\sum_{i = 1}^{n} x_{i} y_{i}}{\sum_{i = 1}^{n} x_{i}^{2}} .

(e)

The sum of squares attributable to the regression,

S S R

, the sum of squares of deviations about the regression,

S S D

, and the total sum of squares,

S S T

S S T = \sum_{i = 1}^{n} y_{i}^{2}; S S D = \sum_{i = 1}^{n} {(y_{i} - b x_{i})}^{2}, S S R = S S T - S S D .

(f)

The degrees of freedom attributable to the regression,

D F R

, the degrees of freedom of deviations about the regression,

D F D

, and the total degrees of freedom,

D F T

D F T = n; D F D = n - 1, D F R = 1 .

(g)

The mean square attributable to the regression,

M S R

, and the mean square of deviations about the regression,

M S D .

M S R = S S R / D F R; M S D = S S D / D F D .

(h)

The

F

value for the analysis of variance:

F = M S R / M S D .

(i)

The standard error of the regression coefficient:

s e (b) = \sqrt{\frac{M S D}{\sum_{i = 1}^{n} x_{i}^{2}}} .

(j)

The

t

value for the regression coefficient:

t (b) = \frac{b}{s e (b)} .

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

On entry:

n

, the number of pairs of observations.

Constraint:

N > 2

On entry:

X (i)

must contain

x_{i}

, for

i = 1, 2, \dots, n

On entry:

Y (i)

must contain

y_{i}

, for

i = 1, 2, \dots, n

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;
$RESULT (7)$	the value $0.0$ ;
$RESULT (8)$	$s e (b)$ , the standard error of the regression coefficient;
$RESULT (9)$	the value $0.0$ ;
$RESULT (10)$	$t (b)$ , the $t$ value for the regression coefficient;
$RESULT (11)$	the value $0.0$ ;
$RESULT (12)$	$S S R$ , the sum of squares attributable to the regression;
$RESULT (13)$	$D F R$ , the degrees of freedom attributable to the regression;
$RESULT (14)$	$M S R$ , the mean square attributable to the regression;
$RESULT (15)$	$F$ , the $F$ value for the analysis of variance;
$RESULT (16)$	$S S D$ , the sum of squares of deviations about the regression;
$RESULT (17)$	$D F D$ , the degrees of freedom of deviations about the regression;
$RESULT (18)$	$M S D$ , the mean square of deviations about the regression;
$RESULT (19)$	$S S T$ , the total sum of squares;
$RESULT (20)$	$D F T$ , the total degrees of freedom.

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).

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:

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.8 in the Essential Introduction for further information.

Your licence key may have expired or may not have been installed correctly.

See Section 3.7 in the Essential Introduction for further information.

Dynamic memory allocation failed.

See Section 3.6 in the Essential Introduction for further information.

G02CBF 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 (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.

Computation time depends on

n

G02CBF uses a two-pass algorithm.

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. Finally the results are printed.

NAG Library Routine Document

G02CBF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine DocumentG02CBF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine Document

G02CBF