Fits a general linear regression model to new dependent variable

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NAG[g02dgc] NAG[nag_regsn_mult_linear_newyvar] - Fits a general linear regression model to new dependent variable

Calling Sequence

g02dgc(rss, rank, cov, q, svd, p, y, b, se, res, com_ar, 'n'=n, 'wt'=wt, 'ip'=ip, 'tdq'=tdq, 'fail'=fail)

nag_regsn_mult_linear_newyvar(. . .)

Parameters

rss - assignable;

Note: On exit the variable rss will have a value of type float.

On entry: the residual sum of squares for the original dependent variable.

On exit: the residual sum of squares for the new dependent variable.

rank - integer;

On entry: the rank of the independent variables, as given by g02dac (nag_regsn_mult_linear).

Constraint: and if , otherwise .

cov - Vector(1.., datatype=float[8]);

Note: the dimension, dim, of the array cov must be at least .

On entry: the covariance matrix of the argument estimates as given by g02dac (nag_regsn_mult_linear).

On exit: the upper triangular part of the variance-covariance matrix of the ip argument estimates given in b. They are stored packed by column, i.e., the covariance between the argument estimate given in and the argument estimate given in , , is stored in for and .

q - Matrix(1..n, 1..tdq, datatype=float[8], order=C_order);

On entry: the results of the decomposition as returned by g02dac (nag_regsn_mult_linear).

On exit: the first column of q contains the new values of , the remainder of q will be unchanged.

svd - boolean;

On entry: indicates if a singular value decomposition was used by g02dac (nag_regsn_mult_linear).

A singular value decomposition was used by g02dac (nag_regsn_mult_linear).

A singular value decomposition was not used by g02dac (nag_regsn_mult_linear).

p - Vector(1.., datatype=float[8]);

Note: the dimension, dim, of the array p must be at least .

On entry: details of the decomposition and SVD, if used, as returned in array p by g02dac (nag_regsn_mult_linear).

If , only the first ip elements of p are used, these will contain the zeta values for the decomposition (see f01qcc (nag_real_qr) for details).

If , the first ip elements of p will contain the zeta values for the decomposition (see f01qcc (nag_real_qr) for details) and the next ip elements of p contain singular values. The following ip by ip elements contain the matrix stored by rows.

y - Vector(1..n, datatype=float[8]);

On entry: the new dependent variable .

b - Vector(1..ip, datatype=float[8]);

On exit: , contain the least-squares estimates of the arguments of the regression model, .

se - Vector(1..ip, datatype=float[8]);

On exit: , contain the standard errors of the ip argument estimates given in b.

res - Vector(1..n, datatype=float[8]);

On exit: the residuals for the new regression model.

com_ar - Vector(1.., datatype=float[8]);

Note: the dimension, dim, of the array com_ar must be at least .

On entry: if , com_ar must be unaltered from the previous call to g02dac (nag_regsn_mult_linear).

'n'=n - integer; (optional)

Default value: the first dimension of the arrays wt, q, y, res.

On entry: the number of observations, .

Constraint: . .

'wt'=wt - Vector(1..n, datatype=float[8]); (optional)

On entry: if weighted estimates are required then wt must contain the weights to be used in the weighted regression. Otherwise wt need not be defined and may be set to the null pointer NULL, i.e., (double *)0.

If , then the th observation is not included in the model, in which case the effective number of observations is the number of observations with non-zero weights. The values of res and h will be set to zero for observations with zero weights. If wt is NULL, then the effective number of observations is .

Constraint: or , for . .

'ip'=ip - integer; (optional)

Default value: the first dimension of the arrays b, se and the second dimension of the arrays b, sethe array q.

On entry: the number of independent variables in the model (including the mean if fitted).

Constraint: . .

'tdq'=tdq - integer; (optional)

On entry: the second dimension of the array q as declared in the function from which nag_regsn_mult_linear_newyvar (g02dgc) is called.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_regsn_mult_linear_newyvar (g02dgc) calculates the estimates of the arguments of a general linear regression model for a new dependent variable after a call to g02dac (nag_regsn_mult_linear).

Description

nag_regsn_mult_linear_newyvar (g02dgc) uses the results given by g02dac (nag_regsn_mult_linear) to fit the same set of independent variables to a new dependent variable.

g02dac (nag_regsn_mult_linear) computes a decomposition of the matrix of independent variables and also, if the model is not of full rank, a singular value decomposition (SVD). These results can be used to compute estimates of the arguments for a general linear model with a new dependent variable. The decomposition leads to the formation of an upper triangular by matrix and an by orthogonal matrix . In addition the vector (or ) is computed. For a new dependent variable, , nag_regsn_mult_linear_newyvar (g02dgc) computes a new value of or .

If is of full rank, then the least-squares argument estimates, , are the solution to: , where is the first elements of .

If is not of full rank, then g02dac (nag_regsn_mult_linear) will have computed the SVD of ,

R=Q[*]([[D,0],[0,0]])P^T

where is a by diagonal matrix with non-zero diagonal elements, being the rank of , and and are by orthogonal matrices. This gives the solution

being the first columns of , i.e., and being the first columns of . Details of the SVD are made available by g02dac (nag_regsn_mult_linear) in the form of the matrix :

P^(*)=([[D^(-1)P[1]^T],[P[0]^T]])

The matrix is made available through the com_ar argument of g02dac (nag_regsn_mult_linear).

In addition to argument estimates, the new residuals are computed and the variance-covariance matrix of the argument estimates are found by scaling the variance-covariance matrix for the original regression.

Error Indicators and Warnings

"NE_INT_ARG_LE"

On entry, rank must not be less than or equal to 0: .

"NE_INT_ARG_LT"

On entry, ip must not be less than 1: .

"NE_REAL_ARG_LE"

On entry, rss must not be less than or equal to 0.0: .

"NE_REAL_ARG_LT"

On entry, must not be less than 0.0: .

"NE_SVD_RANK_GT_IP"

On entry, the Boolean variable, svd, is true and rank must not be greater than ip: rank = , .

"NE_SVD_RANK_NE_IP"

On entry, the Boolean variable, svd, is false and rank must be equal to ip: , .

	Accuracy
	The same accuracy as g02dac (nag_regsn_mult_linear) is obtained.

	Further Comments
	The values of the leverages, , are unaltered by a change in the dependent variable so a call to g02fac (nag_regsn_std_resid_influence) can be made using the value of h from g02dac (nag_regsn_mult_linear).

Examples

n := 12:
rss := 22.22679999999998:
ip := 5:
rank := 4:
tdq := 6:
svd := true:
wt := Vector([], datatype=float[8]):
cov := Vector([0.1481786666666665, 0.03704466666666668, 0.703848666666666, 0.0370446666666667, -0.222268, 0.7038486666666662, 0.03704466666666655, -0.2222679999999998, -0.2222679999999997, 0.7038486666666665, 0.03704466666666659, -0.2222679999999998, -0.2222679999999999, -0.2222680000000002, 0.7038486666666667], datatype=float[8]):
q := Matrix([[-132.3142479415325, -3.464101615137754, -0.8660254037844386, -0.8660254037844386, -0.8660254037844386, -0.8660254037844386], [-4.385000000000004, 0.2542949013547401, 1.5, -0.5000000000000001, -0.5000000000000001, -0.4999999999999998], [3.450681092190353, 0.2542949013547401, 0.2464408878569321, -1.414213562373095, 0.7071067811865477, 0.7071067811865478], [-4.564215887385989, 0.2542949013547401, 0.2464408878569321, -0.1493782079840454, -1.224744871391589, 1.224744871391589], [-2.03658664553691, 0.2542949013547401, 0.2464408878569321, -0.1493782079840454, -0.2512250416211345, -7.079982118054818e-16], [-2.146916163163576, 0.2542949013547401, 0.2464408878569321, 0.4236386677182777, 0.0475798492653138, 0.1965008449194822], [-2.301954099286771, 0.2542949013547401, 0.2464408878569321, -0.1493782079840454, -0.2512250416211345, 0.2776580774169763], [0.8127756607468626, 0.2542949013547401, -0.3431208422711535, -0.0579580547393013, -0.09747415577321773, 0.5078285895463452], [1.612755059311305, 0.2542949013547401, 0.2464408878569321, -0.1493782079840454, 0.4136728082784364, -0.07427328568675991], [2.032775660746862, 0.2542949013547401, -0.3431208422711535, -0.0579580547393013, -0.09747415577321773, 0.5078285895463452], [-0.8772449406886967, 0.2542949013547401, 0.2464408878569321, -0.1493782079840454, 0.4136728082784364, -0.07427328568675991], [-0.08691616316357387, 0.2542949013547401, 0.2464408878569321, 0.4236386677182777, 0.0475798492653138, 0.1965008449194822]], datatype=float[8], order='C_order'):
p := Vector([1.135198279858991, 1.13078348304906, 1.234006904805161, 1.228003039942231, 1.148419798367432, 3.872983346207417, 1.732050807568878, 1.732050807568877, 1.732050807568877, 3.166264259090703e-16, 0.2309401076758502, 0.05773502691896256, 0.05773502691896257, 0.0577350269189626, 0.05773502691896259, -8.986850838355147e-17, -0.2180297316413536, -0.3448196554213736, 0.3466016006011757, 0.2162477864615517, 3.472144456476812e-18, 0.4492089773025322, -0.3565253366776346, -0.03799964639604183, -0.05468399422885582, 4.280983223691876e-18, -0.02596788076090121, -0.06315923956148034, -0.3583620478433494, 0.4474891681657309, -0.447213595499958, 0.4472135954999579, 0.447213595499958, 0.4472135954999578, 0.4472135954999579], datatype=float[8]):
y := Vector([63, 69, 68, 71, 68, 65, 65, 66, 72, 67, 70, 67], datatype=float[8]):
b := Vector(5, datatype=float[8]):
se := Vector(5, datatype=float[8]):
res := Vector(12, datatype=float[8]):
com_ar := Vector([-1, 7.782841126037044e-17, -3.006965304918233e-18, -3.707440224892163e-18, 4.087629581716687e-17, -3.644677812977964e-17, -0.4360594632827073, 0.8984179546050645, -0.05193576152180249, 1.099662896404934e-16, 6.704435184107242e-17, 0.8856432516484704, 0.4386657356682224, 0.1523430443446021, -1.368371848221454e-16, -1.602297734032577e-17, -0.1596501653338369, -0.02043406944075239, 0.986962143911796, -2.565248701484727e-16, -4.087629581716687e-17, -1.32681057203528e-16, 3.235224838401304e-17, -3.191913700752707e-16, -0.9999999999999999, 0.2886751345948128, -0.3448196554213737, -0.3565253366776346, -0.06315923956148034, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3.397315261335401e-311, 7.476584967756268e-306, 3.755036198176092e-309, 3.270287702751045e-317, 0, 9.515376782229773e+150, 9.645643603758656e-320, 0, 0, 0, 0, 0, 0, 8.146899419624402e-316, 2.556463956594714e-261, 2.573660592467583e-261, 2.568434901423488e-261, 8.487983163861089e-314, 2.568357443336539e-261, 5.087599856778982e-05, -2.116067917594497e-208, 6.395607724371421e-46, 2.781534841469659e-309, 2.617317011560295e-310, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], datatype=float[8]):
NAG:-g02dgc(rss, rank, cov, q, svd, p, y, b, se, res, com_ar, 'n' = n, 'wt' = wt, 'ip' = ip, 'tdq' = tdq):

rss;

cov;

se;

res;

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20 (3) 2–25 Searle S R (1971) Linear Models Wiley g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.