void	g03ccc (Nag_FacScoreMethod method, Nag_FacRotation rotate, Integer nvar, Integer nfac, const double fl[], Integer tdfl, const double psi[], const double e[], const double r[], Integer tdr, double fs[], Integer tdfs, NagError *fail)

The function may be called by the names: g03ccc, nag_mv_factor_score or nag_mv_fac_score.

3 Description

A factor analysis model aims to account for the covariances among

p

variables, observed on

n

individuals, in terms of a smaller number,

k

, of unobserved variables or factors. The values of the factors for an individual are known as factor scores. g03cac fits the factor analysis model by maximum likelihood and returns the estimated factor loading matrix,

Λ

, and the diagonal matrix of variances of the unique components,

Ψ

. To obtain estimates of the factors, a

p \times k

matrix of factor score coefficients,

Φ

, is formed. The estimated vector of factor scores,

\hat{f}

, is then given by:

\hat{f} = x^{T} Φ,

where

x

is the vector of observed variables for an individual.

There are two commonly used methods of obtaining factor score coefficients.

The regression method:

Φ = Ψ^{−1} Λ {(I + Λ^{T} Ψ^{−1} Λ)}^{−1},

and Bartlett's method:

Φ = Ψ^{−1} Λ {(Λ^{T} Ψ^{−1} Λ)}^{−1} .

See Lawley and Maxwell (1971) for details of both methods. In the regression method as given above, it is assumed that the factors are not correlated and have unit variance; this is true for models fitted by g03cac. Further, for models fitted by g03cac,

Λ^{T} Ψ^{−1} Λ = Θ - I,

where

Θ

is the diagonal matrix of eigenvalues of the matrix

S^{*}

, as described in g03cac.

The factors may be orthogonally rotated using an orthogonal rotation matrix,

R

, as computed by g03bac. The factor scores for the rotated matrix are then given by

Λ R

4 References

Lawley D N and Maxwell A E (1971) Factor Analysis as a Statistical Method (2nd Edition) Butterworths

5 Arguments

1: $method$ – Nag_FacScoreMethod Input

On entry: indicates which method is to be used to compute the factor score coefficients.

$method = Nag_FacScoreRegsn$: The regression method is used.
$method = Nag_FacScoreBart$: Bartlett's method is used.

Constraint:

method = Nag_FacScoreRegsn

Nag_FacScoreBart

2: $rotate$ – Nag_FacRotation Input

On entry: indicates whether a rotation is to be applied.

$rotate = Nag_FacRotate$: A rotation will be applied to the coefficients and the rotation matrix, $R$ , must be given in r.
$rotate = Nag_FacNoRotate$: No rotation is applied.

Constraint:

rotate = Nag_FacRotate

Nag_FacNoRotate

3: $nvar$ – Integer Input

On entry: the number of observed variables in the factor analysis,

p

Constraint:

nvar \geq nfac

4: $nfac$ – Integer Input

On entry: the number of factors in the factor analysis,

k

Constraint:

nfac \geq 1

5: $fl [nvar \times tdfl]$ – const double Input

Note: the

(i, j)

th element of the matrix is stored in

fl [(i - 1) \times tdfl + j - 1]

On entry: the matrix of unrotated factor loadings,

Λ

, as returned by g03cac.

6: $tdfl$ – Integer Input

On entry: the stride separating matrix column elements in the array fl.

Constraint:

tdfl \geq nfac

7: $psi [nvar]$ – const double Input

On entry: the diagonal elements of

Ψ

, as returned by g03cac.

Constraint:

psi [i - 1] > 0.0

, for

i = 1, 2, \dots, p

8: $e [nvar]$ – const double Input

On entry: the eigenvalues of the matrix

S^{*}

, as returned by g03cac.

Constraint:

e [i - 1] > 1.0

, for

i = 1, 2, \dots, p

9: $r [nfac \times tdr]$ – const double Input

Note: the

(i, j)

th element of the matrix

R

is stored in

r [(i - 1) \times tdr + j - 1]

On entry: if

rotate = Nag_FacRotate

, then r must contain the orthogonal rotation matrix,

R

, as returned by g03bac.

rotate = Nag_FacNoRotate

then r need not be set.

10: $tdr$ – Integer Input

On entry: the stride separating matrix column elements in the array r.

Constraint: if

rotate = Nag_FacRotate

then

tdr \geq nfac

11: $fs [nvar \times tdfs]$ – double Output

On exit: the matrix of factor score coefficients,

Φ

fs [(i - 1) \times tdfs + j - 1]

contains the factor score coefficient for the

j

th factor and the

i

th observed variable, for

i = 1, 2, \dots, p

and

j = 1, 2, \dots, k

12: $tdfs$ – Integer Input

On entry: the stride separating matrix column elements in the array fs.

Constraint:

tdfs \geq nfac

13: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_ENUM_CONS: On entry, $tdr = ⟨ value ⟩$ while $nfac = ⟨ value ⟩$ and $rotate = Nag_FacRotate$ . These arguments must satisfy $tdr \geq nfac$ when $rotate = Nag_FacRotate$ .
NE_2_INT_ARG_LT: On entry, $nvar = ⟨ value ⟩$ while $nfac = ⟨ value ⟩$ . These arguments must satisfy $nvar \geq nfac$ .

On entry, $tdfl = ⟨ value ⟩$ while $nfac = ⟨ value ⟩$ . These arguments must satisfy $tdfl \geq nfac$ .

On entry, $tdfs = ⟨ value ⟩$ while $nfac = ⟨ value ⟩$ . These arguments must satisfy $tdfs \geq nfac$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument method had an illegal value.

On entry, argument rotate had an illegal value.
NE_INT_ARG_LT: On entry, $nfac = ⟨ value ⟩$ .
Constraint: $nfac \geq 1$ .
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.
NE_REAL_ARRAY_INPUT: On entry, $e [⟨ value ⟩] = ⟨ value ⟩$ .
Constraint: $e [⟨ value ⟩] > 1.0$ .

On entry, $psi [⟨ value ⟩] = ⟨ value ⟩$ .
Constraint: $psi [⟨ value ⟩] > 0.0$ .

7 Accuracy

Accuracy will depend on the accuracy requested when computing the estimated factor loadings using g03cac.

8 Parallelism and Performance

g03ccc is not threaded in any implementation.

9 Further Comments

To compute the factor scores using the factor score coefficients, the values for the observed variables first need to be standardized by subtracting the sample means and, if the factor analysis is based upon a correlation matrix, dividing by the sample standard deviations. This may be performed using g03zac. The standardized variables are then post-multiplied by the factor score coefficients. This may be performed using functions from the F16 Chapter Introduction, for example f16yac.

If principal component analysis is required, the function g03aac computes the principal component scores directly. Hence, the factor score coefficients are not needed.

10 Example

The example is taken from Lawley and Maxwell (1971). The correlation matrix for 220 observations on six school subjects is input and a factor analysis model with two factors fitted using g03cac. The factor score coefficients are computed using the regression method.

g03cc: FL CL CPP AD

NAG CL Interfaceg03ccc (factor_​score)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g03ccc (factor_score)