g02ca:: Correlation and Regression Analysis (NAG Toolbox)

nag_correg_linregs_const (g02ca) 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 function 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)} .

References

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

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Errors or warnings detected by the function:

Accuracy

nag_correg_linregs_const (g02ca) 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 Description), 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 double variable, by means of a call to nag_machine_real_largest (x02al).

Further Comments

The time taken by nag_correg_linregs_const (g02ca) depends on

n

The function uses a two-pass algorithm.

Example

This example reads in eight observations on each of two variables, and then performs a simple linear regression with the first variable as the independent variable, and the second variable as the dependent variable. Finally the results are printed.

function g02ca_example


fprintf('g02ca example results\n\n');

x = [ 1.0   0.0    4.0    7.5   2.5   0.0  10.0   5.0];
y = [20.0  15.5   28.3   45.0  24.5  10.0  99.0  31.2];

n = numel(x);
fprintf('  i    independent(x)   dependent(y)\n');
fprintf('%3d%14.4f%14.4f\n',[1:n; x; y]);

[result, ifail] = g02ca(x, y);

fprintf('\n');
fprintf('Mean of independent variable               = %8.4f\n', result(1));
fprintf('Mean of   dependent variable               = %8.4f\n', result(2));
fprintf('Standard deviation of independent variable = %8.4f\n', result(3));
fprintf('Standard deviation of   dependent variable = %8.4f\n', result(4));
fprintf('Correlation coefficient                    = %8.4f\n', result(5));
fprintf('\n');
fprintf('Regression coefficient                     = %8.4f\n', result(6));
fprintf('Standard error of coefficient              = %8.4f\n', result(8));
fprintf('t-value for coefficient                    = %8.4f\n', result(10));
fprintf('\n');
fprintf('Regression constant                        = %8.4f\n', result(7));
fprintf('Standard error of constant                 = %8.4f\n', result(9));
fprintf('t-value for constant                       = %8.4f\n', result(11));

fprintf('\nAnalysis of regression table :-\n\n');

fprintf('     Source       Sum of squares  D.F.    Mean square     F-value\n');
fprintf('Due to regression %11.3f%8d%14.3f%14.3f\n', result(12:15));
fprintf('About  regression %11.3f%8d%14.3f\n', result(16:18));
fprintf('Total             %11.3f%8d\n', result(19:20));

g02ca example results

  i    independent(x)   dependent(y)
  1        1.0000       20.0000
  2        0.0000       15.5000
  3        4.0000       28.3000
  4        7.5000       45.0000
  5        2.5000       24.5000
  6        0.0000       10.0000
  7       10.0000       99.0000
  8        5.0000       31.2000

Mean of independent variable               =   3.7500
Mean of   dependent variable               =  34.1875
Standard deviation of independent variable =   3.6253
Standard deviation of   dependent variable =  28.2604
Correlation coefficient                    =   0.9096

Regression coefficient                     =   7.0905
Standard error of coefficient              =   1.3224
t-value for coefficient                    =   5.3620

Regression constant                        =   7.5982
Standard error of constant                 =   6.6858
t-value for constant                       =   1.1365

Analysis of regression table :-

     Source       Sum of squares  D.F.    Mean square     F-value
Due to regression    4625.303       1      4625.303        28.751
About  regression     965.245       6       160.874
Total                5590.549       7

$result (1)$	$\bar{x}$ , 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.

NAG Toolbox: nag_correg_linregs_const (g02ca)

▸▿ Contents

Purpose

Syntax

Description