g02efc calculates a full stepwise selection from $p$ variables by using Clarke's sweep algorithm on the correlation matrix of a design and data matrix, $Z$. The (weighted) variance-covariance, (weighted) means and sum of weights of $Z$ must be supplied.
The function may be called by the names: g02efc, nag_correg_linregm_fit_stepwise or nag_full_step_regsn.
3Description
The general multiple linear regression model is defined by
$$y={\beta}_{0}+X\beta +\epsilon \text{,}$$
where
$y$ is a vector of $n$ observations on the dependent variable,
${\beta}_{0}$ is an intercept coefficient,
$X$ is an $n\times p$ matrix of $p$ explanatory variables,
$\beta $ is a vector of $p$ unknown coefficients, and
$\epsilon $ is a vector of length $n$ of unknown, Normally distributed, random errors.
g02efc employs a full stepwise regression to select a subset of explanatory variables from the $p$ available variables (the intercept is included in the model) and computes regression coefficients and their standard errors, and various other statistical quantities, by minimizing the sum of squares of residuals. The method applies repeatedly a forward selection step followed by a backward elimination step and halts when neither step updates the current model.
The criterion used to update a current model is the variance ratio of residual sum of squares. Let ${s}_{1}$ and ${s}_{2}$ be the residual sum of squares of the current model and this model after undergoing a single update, with degrees of freedom ${q}_{1}$ and ${q}_{2}$, respectively. Then the condition:
must be satisfied if a variable $k$ will be considered for removal from the current model, where ${f}_{1}$ and ${f}_{2}$ are user-supplied values and ${f}_{2}\le {f}_{1}$.
In the entry step the entry statistic is computed for each variable not in the current model. If no variable is associated with a test value that exceeds ${f}_{1}$ then this step is terminated; otherwise the variable associated with the largest value for the entry statistic is entered into the model.
In the removal step the removal statistic is computed for each variable in the current model. If no variable is associated with a test value less than ${f}_{2}$ then this step is terminated; otherwise the variable associated with the smallest value for the removal statistic is removed from the model.
The data values $X$ and $y$ are not provided as input to the function. Instead, summary statistics of the design and data matrix $Z=(X\mid y)$ are required.
Explanatory variables are entered into and removed from the current model by using sweep operations on the correlation matrix $R$ of $Z$, given by:
where ${r}_{\mathit{i}\mathit{j}}$ is the correlation between the explanatory variables $\mathit{i}$ and $\mathit{j}$, for $\mathit{i}=1,2,\dots ,p$ and $\mathit{j}=1,2,\dots ,p$, and ${r}_{yi}$ (and ${r}_{iy}$) is the correlation between the response variable $y$ and the $\mathit{i}$th explanatory variable, for $\mathit{i}=1,2,\dots ,p$.
A sweep operation on the $k$th row and column ($k\le p$) of $R$ replaces:
for a user-supplied value ($>0$) of $\tau $ and where the index $i$ runs over explanatory variables in the current model. The sweep operation is its own inverse, therefore, pivoting on an explanatory variable $k$ in the current model has the effect of removing it from the model.
Once the stepwise model selection procedure is finished, the function calculates:
(a)the least squares estimate for the $i$th explanatory variable included in the fitted model;
(b)standard error estimates for each coefficient in the final model;
(c)the square root of the mean square of residuals and its degrees of freedom;
(d)the multiple correlation coefficient.
The function makes use of the symmetry of the sweep operations and correlation matrix which reduces by almost one half the storage and computation required by the sweep algorithm, see Clarke (1981) for details.
4References
Clarke M R B (1981) Algorithm AS 178: the Gauss–Jordan sweep operator with detection of collinearity Appl. Statist.31 166–169
Dempster A P (1969) Elements of Continuous Multivariate Analysis Addison–Wesley
Draper N R and Smith H (1985) Applied Regression Analysis (2nd Edition) Wiley
5Arguments
1: $\mathbf{m}$ – IntegerInput
On entry: the number of explanatory variables available in the design matrix, $Z$.
Constraint:
${\mathbf{m}}>1$.
2: $\mathbf{n}$ – IntegerInput
On entry: the number of observations used in the calculations.
Note: the dimension, dim, of the array c
must be at least
$({\mathbf{m}}+1)\times ({\mathbf{m}}+2)/2$.
On entry: the upper-triangular variance-covariance matrix packed by column for the design matrix, $Z$. Because the function computes the correlation matrix $R$ from c, the variance-covariance matrix need only be supplied up to a scaling factor.
5: $\mathbf{sw}$ – doubleInput
On entry: if weights were used to calculate c then sw is the sum of positive weight values; otherwise sw is the number of observations used to calculate c.
On entry: the value of
${\mathbf{isx}}\left[\mathit{j}-1\right]$ determines the set of variables used to perform full stepwise model selection, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
${\mathbf{isx}}\left[\mathit{j}-1\right]=-1$
To exclude the variable corresponding to the $j$th column of $X$ from the final model.
${\mathbf{isx}}\left[\mathit{j}-1\right]=1$
To consider the variable corresponding to the $j$th column of $X$ for selection in the final model.
${\mathbf{isx}}\left[\mathit{j}-1\right]=2$
To force the inclusion of the variable corresponding to the $j$th column of $X$ in the final model.
Constraint:
${\mathbf{isx}}\left[\mathit{j}-1\right]=-1,1\text{ or}2$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
On exit: the value of ${\mathbf{isx}}\left[\mathit{j}-1\right]$ indicates the status of the $j$th explanatory variable in the model.
${\mathbf{isx}}\left[\mathit{j}-1\right]=-1$
Forced exclusion.
${\mathbf{isx}}\left[\mathit{j}-1\right]=0$
Excluded.
${\mathbf{isx}}\left[\mathit{j}-1\right]=1$
Selected.
${\mathbf{isx}}\left[\mathit{j}-1\right]=2$
Forced selection.
7: $\mathbf{fin}$ – doubleInput
On entry: the value of the variance ratio which an explanatory variable must exceed to be included in a model.
Suggested value:
${\mathbf{fin}}=4.0$.
Constraint:
${\mathbf{fin}}>0.0$.
8: $\mathbf{fout}$ – doubleInput
On entry: the explanatory variable in a model with the lowest variance ratio value is removed from the model if its value is less than fout. fout is usually set equal to the value of fin; a value less than fin is occasionally preferred.
On entry: the tolerance, $\tau $, for detecting collinearities between variables when adding or removing an explanatory variable from a model. Explanatory variables deemed to be collinear are excluded from the final model.
On exit: ${\mathbf{b}}\left[0\right]$ contains the estimate for the intercept term in the fitted model. If ${\mathbf{isx}}\left[j-1\right]\ne 0$, then ${\mathbf{b}}\left[j\right]$ contains the estimate for the $j$th explanatory variable in the fitted model; otherwise ${\mathbf{b}}\left[j\right]=0$.
On exit: ${\mathbf{se}}\left[\mathit{j}-1\right]$ contains the standard error for the estimate of ${\mathbf{b}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}+1$.
12: $\mathbf{rsq}$ – double *Output
On exit: the ${R}^{2}$-statistic for the fitted regression model.
13: $\mathbf{rms}$ – double *Output
On exit: the mean square of residuals for the fitted regression model.
14: $\mathbf{df}$ – Integer *Output
On exit: the number of degrees of freedom for the sum of squares of residuals.
15: $\mathbf{monfun}$ – function, supplied by the userExternal Function
You may define your own function or specify the NAG defined default function g02efh. If this facility is not required then the NAG defined null function macro NULLFN can be substituted.
Backward elimination did not remove any variables from the current model.
${\mathbf{flag}}=\mathrm{Nag\_BeginStepwise}$
Starting stepwise selection procedure.
${\mathbf{flag}}=\mathrm{Nag\_VarianceRatio}$
The variance ratio for variable var takes the value val.
${\mathbf{flag}}=\mathrm{Nag\_FinishStepwise}$
Finished stepwise selection procedure.
2: $\mathbf{var}$ – IntegerInput
On entry: the index of the explanatory variable in the design matrix $Z$ to which flag pertains.
3: $\mathbf{val}$ – doubleInput
On entry: if ${\mathbf{flag}}=\mathrm{Nag\_VarianceRatio}$, val is the variance ratio value for the coefficient associated with explanatory variable index var.
4: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to monfun.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling g02efc you may allocate memory and initialize these pointers with various quantities for use by monfun when called from g02efc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
16: $\mathbf{comm}$ – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
17: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_FREE_VARS
On entry, ${\mathbf{isx}}\left[i-1\right]\ne 1$, for all $i=1,2,\dots ,{\mathbf{m}}$.
Constraint: there must be at least one free variable.
NE_INT
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}>1$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>1$.
NE_INT_ARRAY_ELEM_CONS
On entry, ${\mathbf{isx}}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{isx}}\left[\mathit{j}-1\right]=-1$, $1$ or $2$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MODEL_INFEASIBLE
All variables are collinear, no model to select.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_POS_DEF
The design and data matrix $Z$ is not positive definite, results may be inaccurate. All output is returned as documented.
NE_REAL
On entry, ${\mathbf{fin}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{fin}}>0.0$.
On entry, ${\mathbf{sw}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{sw}}>1.0$.
On entry, ${\mathbf{tau}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tau}}>0.0$.
NE_REAL_2
On entry, ${\mathbf{fout}}=\u27e8\mathit{\text{value}}\u27e9$; ${\mathbf{fin}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $0.0\le {\mathbf{fout}}\le {\mathbf{fin}}$.
NE_ZERO_DIAG
On entry at least one diagonal element of ${\mathbf{c}}\le 0.0$.
Constraint: c must be positive definite.
7Accuracy
g02efc returns a warning if the design and data matrix is not positive definite.
8Parallelism and Performance
g02efc is not threaded in any implementation.
9Further Comments
Although the condition for removing or adding a variable to the current model is based on a ratio of variances, these values should not be interpreted as $F$-statistics with the usual interpretation of significance unless the probability levels are adjusted to account for correlations between variables under consideration and the number of possible updates (see, e.g., Draper and Smith (1985)).
g02efc allocates internally $\mathcal{O}(4\times {\mathbf{m}}+({\mathbf{m}}+1)\times ({\mathbf{m}}+2)/2+2)$ of double storage.
10Example
This example calculates a full stepwise model selection for the Hald data described in Dempster (1969). Means, the upper-triangular variance-covariance matrix and the sum of weights are calculated by g02buc. The NAG defined default monitor function g02efh is used to print information at each step of the model selection process.