NAG Library Function Document

nag_full_step_regsn (g02efc)


    1  Purpose
    7  Accuracy


nag_full_step_regsn (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.


#include <nag.h>
#include <nagg02.h>
void  nag_full_step_regsn (Integer m, Integer n, const double wmean[], const double c[], double sw, Integer isx[], double fin, double fout, double tau, double b[], double se[], double *rsq, double *rms, Integer *df,
void (*monfun)(Nag_FullStepwise flag, Integer var, double val, Nag_Comm *comm),
Nag_Comm *comm, NagError *fail)


The general multiple linear regression model is defined by
y = β0 +Xβ+ε,  
nag_full_step_regsn (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 s1 and s2 be the residual sum of squares of the current model and this model after undergoing a single update, with degrees of freedom q1 and q2, respectively. Then the condition:
s2 - s1 / q2 - q1 s1 / q1 > f1 ,  
must be satisfied if a variable k will be considered for entry to the current model, and the condition:
s1 - s2 / q1 - q2 s1 / q1 < f2 ,  
must be satisfied if a variable k will be considered for removal from the current model, where f1 and f2 are user-supplied values and f2f1.
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 f1 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 f2 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=Xy 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:
R = 1 r1p r1y rp1 1 rpy ry1 ryp 1 ,  
where rij is the correlation between the explanatory variables i and j, for i=1,2,,p and j=1,2,,p, and ryi (and riy) is the correlation between the response variable y and the ith explanatory variable, for i=1,2,,p.
A sweep operation on the kth row and column (kp) of R replaces:
rkk ​ by ​ -1 / rkk ; rik ​ by ​ rik / rkk ,  i=1,2,,p+1 ​ ​ ik ; rkj ​ by ​ rkj / rkk ,  j=1,2,,p+1 ​ ​ jk ; rij ​ by ​ rij - rik rkj / rkk ,  ​ i=1,2,,p+1 ​ ​ ik ; ​ j=1,2,,p+1 ​ ​ jk .  
The kth explanatory variable is eligible for entry into the current model if it satisfies the collinearity tests: rkk>τ and
rii - rik rki rkk τ1 ,  
for a user-supplied value (>0) of τ 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 ith 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.


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


1:     m IntegerInput
On entry: the number of explanatory variables available in the design matrix, Z.
Constraint: m>1.
2:     n IntegerInput
On entry: the number of observations used in the calculations.
Constraint: n>1.
3:     wmean[m+1] const doubleInput
On entry: the mean of the design matrix, Z.
4:     c[dim] const doubleInput
Note: the dimension, dim, of the array c must be at least m+1×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:     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.
Constraint: sw>1.0.
6:     isx[m] IntegerInput/Output
On entry: the value of isx[j-1] determines the set of variables used to perform full stepwise model selection, for j=1,2,,m.
To exclude the variable corresponding to the jth column of X from the final model.
To consider the variable corresponding to the jth column of X for selection in the final model.
To force the inclusion of the variable corresponding to the jth column of X in the final model.
Constraint: isx[j-1]=-1,1​ or ​2, for j=1,2,,m.
On exit: the value of isx[j-1] indicates the status of the jth explanatory variable in the model.
Forced exclusion.
Forced selection.
7:     fin doubleInput
On entry: the value of the variance ratio which an explanatory variable must exceed to be included in a model.
Suggested value: fin=4.0.
Constraint: fin>0.0.
8:     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.
Suggested value: fout=fin.
Constraint: 0.0foutfin.
9:     tau doubleInput
On entry: the tolerance, τ, 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.
Suggested value: tau=1.0×10-6.
Constraint: tau>0.0.
10:   b[m+1] doubleOutput
On exit: b[0] contains the estimate for the intercept term in the fitted model. If isx[j-1]0 then b[j] contains the estimate for the jth explanatory variable in the fitted model; otherwise b[j]=0.
11:   se[m+1] doubleOutput
On exit: se[j-1] contains the standard error for the estimate of b[j-1], for j=1,2,,m+1.
12:   rsq double *Output
On exit: the R2-statistic for the fitted regression model.
13:   rms double *Output
On exit: the mean square of residuals for the fitted regression model.
14:   df Integer *Output
On exit: the number of degrees of freedom for the sum of squares of residuals.
15:   monfun function, supplied by the userExternal Function
You may define your own function or specify the NAG defined default function nag_full_step_regsn_monfun (g02efg). If this facility is not required then the NAG defined null function macro NULLFN can be substituted.
The specification of monfun is:
void  monfun (Nag_FullStepwise flag, Integer var, double val, Nag_Comm *comm)
1:     flag Nag_FullStepwiseInput
On entry: the value of flag indicates the stage of the stepwise selection of explanatory variables.
Variable var was added to the current model.
Beginning the backward elimination step.
Variable var failed the collinearity test and is excluded from the model.
Variable var was dropped from the current model.
Beginning the forward selection step
Backward elimination did not remove any variables from the current model.
Starting stepwise selection procedure.
The variance ratio for variable var takes the value val.
Finished stepwise selection procedure.
2:     var IntegerInput
On entry: the index of the explanatory variable in the design matrix Z to which flag pertains.
3:     val doubleInput
On entry: if flag=Nag_VarianceRatio, val is the variance ratio value for the coefficient associated with explanatory variable index var.
4:     comm Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to monfun.
userdouble *
iuserInteger *
The type Pointer will be void *. Before calling nag_full_step_regsn (g02efc) you may allocate memory and initialize these pointers with various quantities for use by monfun when called from nag_full_step_regsn (g02efc) (see Section in How to Use the NAG Library and its Documentation).
16:   comm Nag_Comm *
The NAG communication argument (see Section in How to Use the NAG Library and its Documentation).
17:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, isx[i-1]1, for all i=1,2,,m.
Constraint: there must be at least one free variable.
On entry, m=value.
Constraint: m > 1.
On entry, n=value.
Constraint: n>1.
On entry, isx[value]=value.
Constraint: isx[j-1]=-1,1​ or ​2, for j=1,2,,m.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
All variables are collinear, no model to select.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
The design and data matrix Z is not positive definite, results may be inaccurate. All output is returned as documented.
On entry, fin=value.
Constraint: fin > 0.0.
On entry, sw=value.
Constraint: sw>1.0.
On entry, tau=value.
Constraint: tau>0.0.
On entry, fout=value; fin=value.
Constraint: 0.0foutfin.
On entry at least one diagonal element of c 0.0.
Constraint: c must be positive definite.


nag_full_step_regsn (g02efc) returns a warning if the design and data matrix is not positive definite.

Parallelism and Performance

nag_full_step_regsn (g02efc) is not threaded in any implementation.

Further 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)).
nag_full_step_regsn (g02efc) allocates internally O 4×m+ m+1 × m+2 /2+2  of double storage.


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 nag_sum_sqs (g02buc). The NAG defined default monitor function nag_full_step_regsn_monfun (g02efg) is used to print information at each step of the model selection process.

Program Text

Program Text (g02efce.c)

Program Data

Program Data (g02efce.d)

Program Results

Program Results (g02efce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017