(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:

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

4 References

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

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of pairs of observations.

Constraint:

n > 2

2: $x (n)$ – Real (Kind=nag_wp) array Input

On entry:

x (i)

must contain

x_{i}

, for

i = 1, 2, \dots, n

3: $y (n)$ – Real (Kind=nag_wp) array Input

On entry:

y (i)

must contain

y_{i}

, for

i = 1, 2, \dots, n

4: $result (20)$ – 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;
$result (7)$	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.

5: $ifail$ – 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

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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g02cbf is not threaded in any implementation.

9 Further Comments

Computation time depends on

n

g02cbf uses a two-pass algorithm.

10 Example

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.

g02cb: FL CL CPP AD PY MB

NAG FL Interfaceg02cbf (linregs_​noconst)

▸▿ 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
g02cbf (linregs_noconst)