naginterfaces.library.correg.glm_​gamma

naginterfaces.library.correg.glm_gamma(x, isx, y, link='R', mean='M', wt=None, s=0, a=0.0, v=None, tol=0.0, maxit=10, iprint=0, eps=0.0, io_manager=None)

glm_gamma fits a generalized linear model with gamma errors.

For full information please refer to the NAG Library document for g02gd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g02/g02gdf.html

Parameters

xfloat, array-like, shape $(n,m)$

$x[\text{i}-1,\text{j}-1]$ must contain the $\text{i}$th observation for the $\text{j}$th independent variable, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

isxint, array-like, shape $\left(m\right)$

Indicates which independent variables are to be included in the model.

If $isx[j-1]>0$, the variable contained in the $j$th column of $x$ is included in the regression model.

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

$y$, the dependent variable.

linkstr, length 1, optional

Indicates which link function is to be used.

$link=\text{'E'}$

An exponential link is used.

$link=\text{'I'}$

An identity link is used.

$link=\text{'L'}$

A log link is used.

$link=\text{'S'}$

A square root link is used.

$link=\text{'R'}$

A reciprocal link is used.

meanstr, length 1, optional

Indicates if a mean term is to be included.

$mean=\text{'M'}$

A mean term, intercept, will be included in the model.

$mean=\text{'Z'}$

The model will pass through the origin, zero-point.

wtNone or float, array-like, shape $\left(n\right)$, optional

If provided $wt$ must contain the weights to be used with the model.

If $wt[i-1]=0.0$, the $i$th observation is not included in the model, in which case the effective number of observations is the number of observations with nonzero weights.

If $wt$ is not provided the effective number of observations is $n$.

sfloat, optional

The scale parameter for the gamma model, ${\nu }^{-1}$.

$s=0.0$

The scale parameter is estimated with the function using the formula described in Notes.

afloat, optional

If $link=\text{'E'}$, $a$ must contain the power of the exponential.

If $link\ne \text{'E'}$, $a$ is not referenced.

vNone or float, array-like, shape $(n,\text{ip}+7)$, optional

The offset values $o_{\text{i}}$, otherwise $v$ need not be set and no offset is used.

tolfloat, optional

Indicates the accuracy required for the fit of the model.

The iterative weighted least squares procedure is deemed to have converged if the absolute change in deviance between iterations is less than $tol\times (1.0+\text{Current Deviance})$.

This is approximately an absolute precision if the deviance is small and a relative precision if the deviance is large.

If $0.0\le tol<\text{machine precision}$ then the function will use $10\times \text{machine precision}$ instead.

maxitint, optional

The maximum number of iterations for the iterative weighted least squares.

$maxit=0$

A default value of $10$ is used.

iprintint, optional

Indicates if the printing of information on the iterations is required.

$iprint\le 0$

There is no printing.

$iprint>0$

Every $iprint$ iteration, the following are printed:

the deviance;

the current estimates;

and if the weighted least squares equations are singular, then this is indicated.

When printing occurs the output is directed to the file object associated with the advisory I/O unit (see FileObjManager).

epsfloat, optional

The value of $eps$ is used to decide if the independent variables are of full rank and, if not, what is the rank of the independent variables. The smaller the value of $eps$ the stricter the criterion for selecting the singular value decomposition.

If $0.0\le eps<\text{machine precision}$, the function will use machine precision instead.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

sfloat

If on input $s=0.0$, $s$ contains the estimated value of the scale parameter, ${\hat{\nu }}^{-1}$.

If on input $s\ne 0.0$, $s$ is unchanged on exit.

devfloat

The adjusted deviance for the fitted model.

idfint

The degrees of freedom asociated with the deviance for the fitted model.

bfloat, ndarray, shape $\left(\text{ip}\right)$

The estimates of the parameters of the generalized linear model, $\hat{\beta }$.

If $mean=\text{'M'}$, the first element of $b$ will contain the estimate of the mean parameter and $b\left[i\right]$ will contain the coefficient of the variable contained in column $j$ of $x$, where $isx[j-1]$ is the $i$th positive value in the array $isx$.

If $mean=\text{'Z'}$, $b[i-1]$ will contain the coefficient of the variable contained in column $j$ of $x$, where $isx[j-1]$ is the $i$th positive value in the array $isx$.

irankint

The rank of the independent variables.

If the model is of full rank, $irank=\text{ip}$.

If the model is not of full rank, $irank$ is an estimate of the rank of the independent variables. $irank$ is calculated as the number of singular values greater that $eps\times$(largest singular value).

It is possible for the SVD to be carried out but for $irank$ to be returned as $\text{ip}$.

sefloat, ndarray, shape $\left(\text{ip}\right)$

The standard errors of the linear parameters.

$se[\text{i}-1]$ contains the standard error of the parameter estimate in $b[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{ip}$.

covfloat, ndarray, shape $(\text{ip}\times (\text{ip}+1)/2)$

The upper triangular part of the variance-covariance matrix of the $\text{ip}$ parameter estimates given in $b$. They are stored in packed form by column, i.e., the covariance between the parameter estimate given in $b[i-1]$ and the parameter estimate given in $b[j-1]$, $j\ge i$, is stored in $cov[(j\times (j-1)/2+i)-1]$.

vfloat, ndarray, shape $(n,\text{ip}+7)$

Auxiliary information on the fitted model.

$v[i-1,0]$	contains the linear predictor value, ${\eta }_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,1]$	contains the fitted value, ${\hat{\mu }}_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,2]$	contains the variance standardization, $\frac{1}{{\tau }_{\text{i}}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,3]$	contains the square root of the working weight, $w_{\text{i}}^{\frac{1}{2}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,4]$	contains the Anscombe residual, $r_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,5]$	contains the leverage, $h_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.
$v[i-1,6]$	contains the offset, $o_{\text{i}}$, for $\text{i}=1,2,\dots ,n$. If $v\text{is}None$, all values will be zero.
$v[i-1,j-1]$	for $j=8,\dots ,\text{ip}+7$, contains the results of the $QR$ decomposition or the singular value decomposition.

If the model is not of full rank, i.e., $irank<\text{ip}$, the first $\text{ip}$ rows of columns $8$ to $\text{ip}+7$ contain the $P^{\ast }$ matrix.

Raises

NagValueError

(errno $1$)

On entry, $\text{ip}=⟨value⟩$.

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

(errno $1$)

On entry, $eps=⟨value⟩$.

Constraint: $eps\ge 0.0$.

(errno $1$)

On entry, $tol=⟨value⟩$.

Constraint: $tol\ge 0.0$.

(errno $1$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit\ge 0$.

(errno $1$)

On entry, $\text{offset}=⟨value⟩$.

Constraint: $\text{offset}=\text{'Y'}$ or $\text{'N'}$.

(errno $1$)

On entry, $link=⟨value⟩$.

Constraint: $link=\text{'E'}$, $\text{'I'}$, $\text{'L'}$, $\text{'S'}$ or $\text{'R'}$.

(errno $1$)

On entry, $a=0.0$ and $link=\text{'E'}$.

Constraint: if $link=\text{'E'}$, $a\ne 0.0$.

(errno $1$)

On entry, $s=⟨value⟩$.

Constraint: $s\ge 0.0$.

(errno $1$)

On entry, $\text{weight}=⟨value⟩$.

Constraint: $\text{weight}=\text{'W'}$ or $\text{'U'}$.

(errno $1$)

On entry, $mean=⟨value⟩$.

Constraint: $mean=\text{'M'}$ or $\text{'Z'}$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $2$)

On entry, $wt[⟨value⟩]<0.0$.

Constraint: $wt[i-1]\ge 0.0$, for $i=1,2,\dots ,n$.

(errno $3$)

On entry, $isx[⟨value⟩]<0$.

Constraint: $isx[j-1]\ge 0.0$, for $j=1,2,\dots ,m$.

(errno $3$)

On entry, $\text{ip}$ incompatible with number of nonzero values of $isx$: $\text{ip}=⟨value⟩$.

(errno $3$)

Number of requested x-variables greater than $\text{n}$.

(errno $4$)

On entry, $y[⟨value⟩]=⟨value⟩$.

Constraint: $y[i-1]\ge 0.0$, for $i=1,2,\dots ,n$.

(errno $5$)

A fitted value is at the boundary, i.e., $\hat{\mu }=0.0$. This may occur if there are $y$ values of $0.0$ and the model is too complex for the data. The model should be reformulated with, perhaps, some observations dropped.

(errno $6$)

SVD solution failed to converge.

(errno $7$)

The iterative weighted least squares has failed to converge in $maxit$ (or default $10$) iterations. The value of $maxit$ could be increased but it may be advantageous to examine the convergence using the $iprint$ option. This may indicate that the convergence is slow because the solution is at a boundary in which case it may be better to reformulate the model.

Warns

NagAlgorithmicWarning

(errno $8$)

The rank of the model has changed during the weighted least squares iterations. The estimate for $\beta$ returned may be reasonable, but you should check how the deviance has changed during iterations.

(errno $9$)

The degrees of freedom for error are $0$. A saturated model has been fitted.

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.

A generalized linear model with gamma errors consists of the following elements:

1. a set of $n$ observations, $y_i$, from a gamma distribution with probability density function:

   $$\frac{1}{\Gamma \left(\nu \right)}{\left(\frac{\nu y}{\mu }\right)}^{\nu }exp(-\frac{\nu y}{\mu })\frac{1}{y}$$

   $\nu$ being constant for the sample.

2. $X$, a set of $p$ independent variables for each observation, $x_1,x_2,\dots ,x_p$.

3. a linear model:

   $$\eta =\sum {\beta }_j{x}_j\text{.}$$

4. a link between the linear predictor, $\eta$, and the mean of the distribution, $\mu$, $\eta =g\left(\mu \right)$. The possible link functions are:

   1. exponent link: $\eta ={\mu }^a$, for a constant $a$,

   2. identity link: $\eta =\mu$,

   3. log link: $\eta =\log \left(\mu \right)$,

   4. square root link: $\eta =\sqrt{\mu }$,

   5. reciprocal link: $\eta =\frac{1}{\mu }$.

5. a measure of fit, an adjusted deviance. This is a function related to the deviance, but defined for $y=0$:

   $$\sum _{i=1}^n{dev}^{\ast }(y_i,{\hat{\mu }}_i)=\sum _{i=1}^{n}2(\log \left({\hat{\mu }}_i\right)+\left(\frac{y_i}{{\hat{\mu }}_i}\right))\text{.}$$

The linear parameters are estimated by iterative weighted least squares. An adjusted dependent variable, $z$, is formed:

$$z=\eta +(y-\mu )\frac{d\eta }{d\mu }$$

and a working weight, $w$,

$$w^{-1}={\left(\tau \frac{d\eta }{d\mu }\right)}^2\text{, where}\tau =\frac{1}{\mu }\text{.}$$

At each iteration an approximation to the estimate of $\beta$, $\hat{\beta }$ is found by the weighted least squares regression of $z$ on $X$ with weights $w$.

glm_gamma finds a $QR$ decomposition of $w^{\frac{1}{2}}X$, i.e.,

$w^{\frac{1}{2}}X=QR$ where $R$ is a $p\times p$ triangular matrix and $Q$ is an $n\times p$ column orthogonal matrix.

If $R$ is of full rank then $\hat{\beta }$ is the solution to:

$R\hat{\beta }=Q^T{w}^{\frac{1}{2}}z$

If $R$ is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of $R$.

$$\begin{array}{c}R=Q_{\ast }\left(\begin{array}{cc}D& 0\\ 0& 0\end{array}\right)P^T\text{.}\end{array}$$

where $D$ is a $k\times k$ diagonal matrix with nonzero diagonal elements, $k$ being the rank of $R$ and $w^{\frac{1}{2}}X$.

This gives the solution

$$\begin{array}{c}\hat{\beta }=P_1{D}^{-1}\left(\begin{array}{cc}{Q}_{\ast }& 0\\ 0& I\end{array}\right)Q^T{w}^{\frac{1}{2}}z\text{,}\end{array}$$

where $P_1$ is the first $k$ columns of $P$, i.e., $P=\left(P_1{P}_0\right)$.

The iterations are continued until there is only a small change in the deviance.

The initial values for the algorithm are obtained by taking

$$\hat{\eta }=g\left(y\right)\text{.}$$

The scale parameter, ${\nu }^{-1}$ is estimated by a moment estimator:

$${\hat{\nu }}^{-1}=\sum _{i=1}^n\frac{{\left[(y_i-{\hat{\mu }}_i)/{\hat{\mu }}_i\right]}^2}{(n-k)}\text{.}$$

The fit of the model can be assessed by examining and testing the deviance, in particular, by comparing the difference in deviance between nested models, i.e., when one model is a sub-model of the other. The difference in deviance or adjusted deviance between two nested models with known $\nu$ has, asymptotically, a ${\chi }^2$-distribution with degrees of freedom given by the difference in the degrees of freedom associated with the two deviances.

The parameters estimates, $\hat{\beta }$, are asymptotically Normally distributed with variance-covariance matrix:

$C=R^{-1}{R^{-1}}^T{\nu }^{-1}$ in the full rank case, otherwise

$C=P_1{D}^{-2}{P}_1^T{\nu }^{-1}$.

The residuals and influence statistics can also be examined.

The estimated linear predictor $\hat{\eta }=X\hat{\beta }$, can be written as $H{w}^{\frac{1}{2}}z$ 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. These are known as leverages.

The fitted values are given by $\hat{\mu }=g^{-1}\left(\hat{\eta }\right)$.

glm_gamma also computes the Anscombe residuals, $r$:

$$r_i=\frac{3(y_i^{\frac{1}{3}}-{\hat{\mu }}_i^{\frac{1}{3}})}{{\hat{\mu }}_i^{\frac{1}{3}}}\text{.}$$

An option allows the use of prior weights, ${\omega }_i$. This gives a model with:

$${\nu }_i=\nu {\omega }_i\text{.}$$

In many linear regression models the first term is taken as a mean term or an intercept, i.e., $x_{i,1}=1$, for $i=1,2,\dots ,n$. This is provided as an option.

Often only some of the possible independent variables are included in a model, the facility to select variables to be included in the model is provided.

If part of the linear predictor can be represented by a variables with a known coefficient then this can be included in the model by using an offset, $o$:

$$\eta =o+\sum {\beta }_j{x}_j\text{.}$$

If the model is not of full rank the solution given will be only one of the possible solutions. Other estimates may be obtained by applying constraints to the parameters. These solutions can be obtained by using glm_constrain() after using glm_gamma. Only certain linear combinations of the parameters will have unique estimates, these are known as estimable functions, and can be estimated and tested using glm_estfunc().

Details of the SVD are made available in the form of the matrix $P^{\ast }$:

$$\begin{array}{c}{P}^{\ast }=\left(\begin{array}{c}{D}^{-1}{P}_1^T\\ P_0^T\end{array}\right)\text{.}\end{array}$$

References

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

McCullagh, P and Nelder, J A, 1983, Generalized Linear Models, Chapman and Hall