naginterfaces.library.correg.linregm_​stat_​resinf

naginterfaces.library.correg.linregm_stat_resinf(n, ip, res, h, rms)

linregm_stat_resinf calculates two types of standardized residuals and two measures of influence for a linear regression.

For full information please refer to the NAG Library document for g02fa

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02faf.html

Parameters

nint

$n$, the number of observations included in the regression.

ipint

$p$, the number of linear parameters estimated in the regression model.

resfloat, array-like, shape $\left(\text{nres}\right)$

The residuals, $r_i$.

hfloat, array-like, shape $\left(\text{nres}\right)$

The diagonal elements of $H$, $h_i$, corresponding to the residuals in $res$.

rmsfloat

The estimate of ${\sigma }^2$ based on all $n$ observations, $s^2$, i.e., the residual mean square.

Returns

sresfloat, ndarray, shape $(\text{nres},4)$

The standardized residuals and influence statistics.

For the observation with residual, $r_i$, given in $res[i-1]$.

$sres[i-1,0]$

Is the internally standardized residual, ${RI}_i$.

$sres[i-1,1]$

Is the externally standardized residual, ${RE}_i$.

$sres[i-1,2]$

Is Cook’s $D$ statistic, $D_i$.

$sres[i-1,3]$

Is Atkinson’s $T$ statistic, $T_i$.

Raises

NagValueError

(errno $1$)

On entry, $\text{nres}=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $\text{nres}\le n$.

(errno $1$)

On entry, $rms=⟨value⟩$.

Constraint: $rms>0.0$.

(errno $1$)

On entry, $ip=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $n-1>ip$.

(errno $1$)

On entry, $ip=⟨value⟩$.

Constraint: $ip\ge 1$.

(errno $1$)

On entry, $\text{nres}=⟨value⟩$.

Constraint: $\text{nres}\ge 1$.

(errno $2$)

On entry, $h[⟨value⟩]=⟨value⟩$.

Constraint: $0.0<h\left[i\right]<1.0$, for all $i$.

(errno $3$)

On entry, a value in $res$ is too large for given $rms$. $res[⟨value⟩]=⟨value⟩$ and $rms=⟨value⟩$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

For the general linear regression model

$$y=X\beta +ϵ\text{,}$$

where	$y$ is a vector of length $n$ of the dependent variable,
	$X$ is an $n\times p$ matrix of the independent variables,
	$\beta$ is a vector of length $p$ of unknown parameters,
and	$ϵ$ is a vector of length $n$ of unknown random errors such that $var\left(ϵ\right)={\sigma }^{2}I$.

The residuals are given by

$$r=y-\hat{y}=y-X\hat{\beta }$$

and the fitted values, $\hat{y}=X\hat{\beta }$, can be written as $Hy$ for an $n\times n$ matrix $H$. The $i$th diagonal elements of $H$, $h_i$, give a measure of the influence of the $i$th values of the independent variables on the fitted regression model. The values of $r$ and the $h_i$ are returned by linregm_fit().

linregm_stat_resinf calculates statistics which help to indicate if an observation is extreme and having an undue influence on the fit of the regression model. Two types of standardized residual are calculated:

1. The $i$th residual is standardized by its variance when the estimate of ${\sigma }^2$, $s^2$, is calculated from all the data; this is known as internal Studentization.

   $$R{I}_i=\frac{r_i}{s\sqrt{1-h_i}}\text{.}$$

2. The $i$th residual is standardized by its variance when the estimate of ${\sigma }^2$, $s_{-i}^2$ is calculated from the data excluding the $i$th observation; this is known as external Studentization.

   $$R{E}_i=\frac{r_i}{s_{-i}\sqrt{1-h_i}}=r_i\sqrt{\frac{n-p-1}{n-p-R{I}_i^2}}\text{.}$$

The two measures of influence are:

1. Cook’s $D$

   $$D_i=\frac{1}{p}R{E}_i^2\frac{h_i}{1-h_i}\text{.}$$

2. Atkinson’s $T$

   $$T_i=\left|R{E}_i\right|\sqrt{\left(\frac{n-p}{p}\right)\left(\frac{h_i}{1-h_i}\right)}\text{.}$$

References

Atkinson, A C, 1981, Two graphical displays for outlying and influential observations in regression, Biometrika (68), 13–20

Cook, R D and Weisberg, S, 1982, Residuals and Influence in Regression, Chapman and Hall