naginterfaces.library.correg.linregs_​const_​miss

naginterfaces.library.correg.linregs_const_miss(x, y, xmiss, ymiss)

linregs_const_miss performs a simple linear regression with dependent variable $y$ and independent variable $x$, omitting cases involving missing values.

For full information please refer to the NAG Library document for g02cc

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g02/g02ccf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

$x[\text{i}-1]$ must contain $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

yfloat, array-like, shape $\left(n\right)$

$y[\text{i}-1]$ must contain $y_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

xmissfloat

The value $xm$ which is to be taken as the missing value for the variable $x$. See Accuracy.

ymissfloat

The value $ym$ which is to be taken as the missing value for the variable $y$. See Accuracy.

Returns

resultfloat, ndarray, shape $\left(21\right)$

The following information:

$result\left[0\right]$	$\overline{x}$, the mean value of the independent variable, $x$;
$result\left[1\right]$	$\overline{y}$, the mean value of the dependent variable, $y$;
$result\left[2\right]$	$s_x$, the standard deviation of the independent variable, $x$;
$result\left[3\right]$	$s_y$, the standard deviation of the dependent variable, $y$;
$result\left[4\right]$	$r$, the Pearson product-moment correlation between the independent variable $x$ and the dependent variable $y$
$result\left[5\right]$	$b$, the regression coefficient;
$result\left[6\right]$	$a$, the regression constant;
$result\left[7\right]$	$se\left(b\right)$, the standard error of the regression coefficient;
$result\left[8\right]$	$se\left(a\right)$, the standard error of the regression constant;
$result\left[9\right]$	$t\left(b\right)$, the $t$ value for the regression coefficient;
$result\left[10\right]$	$t\left(a\right)$, the $t$ value for the regression constant;
$result\left[11\right]$	$SSR$, the sum of squares attributable to the regression;
$result\left[12\right]$	$DFR$, the degrees of freedom attributable to the regression;
$result\left[13\right]$	$MSR$, the mean square attributable to the regression;
$result\left[14\right]$	$F$, the $F$ value for the analysis of variance;
$result\left[15\right]$	$SSD$, the sum of squares of deviations about the regression;
$result\left[16\right]$	$DFD$, the degrees of freedom of deviations about the regression;
$result\left[17\right]$	$MSD$, the mean square of deviations about the regression;
$result\left[18\right]$	$SST$, the total sum of squares;
$result\left[19\right]$	$DFT$, the total degrees of freedom;
$result\left[20\right]$	$n_c$, the number of observations used in the calculations.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>2$.

(errno $2$)

After observations with missing values were omitted, two or fewer cases remained.

(errno $3$)

After observations with missing values were omitted, all remaining values of at least one of $x$ and $y$ were identical.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

linregs_const_miss fits a straight line of the form

$$y=a+bx$$

to those of the data points

$$(x_1,y_1),(x_2,y_2),\dots ,(x_n,y_n)$$

that do not include missing values, such that

$$y_i=a+b{x}_i+e_i$$

for those $(x_i,y_i)$, $i=1,2,\dots ,n(n>2)$ which do not include missing values.

The function eliminates all pairs of observations $(x_i,y_i)$ which contain a missing value for either $x$ or $y$, and then calculates the regression coefficient, $b$, the regression constant, $a$, and various other statistical quantities, by minimizing the sum of the $e_i^2$ over those cases remaining in the calculations.

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 $y$.

In addition two values, $\text{xm}$ and $\text{ym}$, are given which are considered to represent missing observations for $x$ and $y$ respectively. (See Accuracy).

Let $w_i=0$ if the $i$th observation of either $x$ or $y$ is missing, i.e., if $x_i=\text{xm}$ and/or $y_i=\text{ym}$; and $w_i=1$ otherwise, for $\text{i}=1,2,\dots ,n$.

The quantities calculated are:

1. Means:

   $$\overline{x}=\frac{{\sum }_{i=1}^n{w}_i{x}_i}{{\sum }_{i=1}^n{w}_i}\text{;}\overline{y}=\frac{{\sum }_{i=1}^n{w}_i{y}_i}{{\sum }_{i=1}^n{w}_i}\text{.}$$

2. Standard deviations:

   $$s_x=\sqrt{\frac{{\sum }_{i=1}^n{w}_i{(x_i-\overline{x})}^2}{{\sum }_{i=1}^n{w}_i-1}}\text{;}{s}_y=\sqrt{\frac{{\sum }_{i=1}^n{w}_i{(y_i-\overline{y})}^2}{{\sum }_{i=1}^n{w}_i-1}}\text{.}$$

3. Pearson product-moment correlation coefficient:

   $$r=\frac{{\sum }_{i=1}^n{w}_i(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\sum }_{i=1}^n{w}_i{(x_i-\overline{x})}^2{\sum }_{i=1}^n{w}_i{(y_i-\overline{y})}^2}}\text{.}$$

4. The regression coefficient, $b$, and the regression constant, $a$:

   $$b=\frac{{\sum }_{i=1}^n{w}_i(x_i-\overline{x})(y_i-\overline{y})}{{\sum }_{i=1}^n{w}_i{(x_i-\overline{x})}^2}\text{,}a=\overline{y}-b\overline{x}\text{.}$$

5. 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, $SST$:

   $$SST=\sum _{i=1}^n{w}_i{(y_i-\overline{y})}^2\text{;}SSD=\sum _{i=1}^n{w}_i{(y_i-a-b{x}_i)}^2\text{;}SSR=SST-SSD\text{.}$$

6. 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, $DFT$:

   $$DFT=\sum _{i=1}^n{w}_i-1\text{;}DFD=\sum _{i=1}^n{w}_i-2\text{;}DFR=1\text{.}$$

7. The mean square attributable to the regression, $MSR$, and the mean square of deviations about the regression, $MSD$:

   $$MSR=SSR/DFR\text{;}MSD=SSD/DFD\text{.}$$

8. The $F$ value for the analysis of variance:

   $$F=MSR/MSD\text{.}$$

9. The standard error of the regression coefficient, $se\left(b\right)$, and the standard error of the regression constant, $se\left(a\right)$:

   $$se\left(b\right)=\sqrt{\frac{MSD}{{\sum }_{i=1}^n{w}_i{(x_i-\overline{x})}^2}}\text{;}se\left(a\right)=\sqrt{MSD(\frac{1}{{\sum }_{i=1}^n{w}_i}+\frac{{\overline{x}}^2}{{\sum }_{i=1}^n{w}_i{(x_i-\overline{x})}^2})}\text{.}$$

10. The $t$ value for the regression coefficient, $t\left(b\right)$, and the $t$ value for the regression constant, $t\left(a\right)$:

    $$t\left(b\right)=\frac{b}{se\left(b\right)}\text{;}t\left(a\right)=\frac{a}{se\left(a\right)}\text{.}$$

11. The number of observations used in the calculations:

    $$n_c=\sum _{i=1}^n{w}_i\text{.}$$

References

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