G02CAF : NAG Library, Mark 24

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 Parameters

1: N – INTEGERInput

On entry:

n

, the number of pairs of observations.

Constraint:

N > 2

2: X(N) – REAL (KIND=nag_wp) arrayInput

On entry:

X (i)

must contain

x_{i}

, for

i = 1, 2, \dots, n

3: Y(N) – REAL (KIND=nag_wp) arrayInput

On entry:

Y (i)

must contain

y_{i}

, for

i = 1, 2, \dots, n

4: RESULT(

20

) – REAL (KIND=nag_wp) arrayOutput

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.

5: IFAIL – INTEGERInput/Output

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).

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 Library Routine Document

G02CAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG02CAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G02CAF