g03cac computes the maximum likelihood estimates of the arguments of a factor analysis model. Either the data matrix or a correlation/covariance matrix may be input. Factor loadings, communalities and residual correlations are returned.

2 Specification

#include <nag.h>

void	g03cac (Nag_FacMat matrix, Integer n, Integer m, const double x[], Integer tdx, Integer nvar, const Integer isx[], Integer nfac, const double wt[], double e[], double stat[], double com[], double psi[], double res[], double fl[], Integer tdfl, Nag_E04_Opt options, double eps, NagError fail)

The function may be called by the names: g03cac or nag_mv_factor.

3 Description

Let

p

variables,

x_{1}, x_{2}, \dots, x_{p}

, with variance-covariance matrix

Σ

be observed. The aim of factor analysis is to account for the covariances in these

p

variables in terms of a smaller number,

k

, of hypothetical variables, or factors,

f_{1}, f_{2}, \dots, f_{k}

. These are assumed to be independent and to have unit variance. The relationship between the observed variables and the factors is given by the model:

x_{i} = \sum_{j = 1}^{k} λ_{i j} f_{j} + e_{i}

λ_{i j}

, for

i = 1, 2, \dots, p

and

j = 1, 2, \dots, k

, are the factor loadings and

e_{i}

, for

i = 1, 2, \dots, p

, are independent random variables with variances

ψ_{i}

, for

i = 1, 2, \dots, p

. The

ψ_{i}

represent the unique component of the variation of each observed variable. The proportion of variation for each variable accounted for by the factors is known as the communality. For this function it is assumed that both the

k

factors and the

e_{i}

's follow independent Normal distributions.

The model for the variance-covariance matrix,

Σ

, can be written as:

Σ = Λ Λ^{T} + Ψ

(1)

where

Λ

is the matrix of the factor loadings,

λ_{i j}

, and

Ψ

is a diagonal matrix of unique variances,

ψ_{i}

, for

i = 1, 2, \dots, p

The estimation of the arguments of the model,

Λ

and

Ψ

, by maximum likelihood is described by Lawley and Maxwell (1971). The log-likelihood is:

- \frac{1}{2} (n - 1) \log (|Σ|) - \frac{1}{2} (n - 1) trace (S Σ^{- 1}) + constant,

where

n

is the number of observations,

S

is the sample variance-covariance matrix or, if weights are used,

S

is the weighted sample variance-covariance matrix and

n

is the effective number of observations, that is, the sum of the weights. The constant is independent of the arguments of the model. A two stage maximization is employed. It makes use of the function

F (Ψ)

, which is, up to a constant,

- 2 / (n - 1)

times the log-likelihood maximized over

Λ

. This is then minimized with respect to

Ψ

to give the estimates,

\hat{Ψ}

, of

Ψ

. The function

F (Ψ)

can be written as:

F (Ψ) = \sum_{j = k + 1}^{p} (θ_{j} - {\log θ}_{j}) - (p - k),

where values

θ_{j}

, for

j = 1, 2, \dots, p

are the eigenvalues of the matrix:

S^{*} = Ψ^{- 1 / 2} S Ψ^{- 1 / 2} .

The estimates

\hat{Λ}

, of

Λ

, are then given by scaling the eigenvectors of

S^{*}

, which are denoted by

V

\hat{Λ} = Ψ^{1 / 2} V {(Θ - I)}^{1 / 2} .

where

Θ

is the diagonal matrix with elements

θ_{i}

, and

I

is the identity matrix.

The minimization of

F (Ψ)

is performed using e04lbc which uses a modified Newton algorithm. The computation of the Hessian matrix is described by Clark (1970). However, instead of using the eigenvalue decomposition of the matrix

S^{*}

as described above, the singular value decomposition of the matrix

R Ψ^{- 1 / 2}

is used, where

R

is obtained either from the

Q R

decomposition of the (scaled) mean-centred data matrix or from the Cholesky decomposition of the correlation/covariance matrix. The function e04lbc ensures that the values of

ψ_{i}

are greater than a given small positive quantity,

δ

, so that the communality is always less than

1

. This avoids the so called Heywood cases.

In addition to the values of

Λ

Ψ

and the communalities, g03cac returns the residual correlations, i.e., the off-diagonal elements of

C - (Λ Λ^{T} + Ψ)

where

C

is the sample correlation matrix. g03cac also returns the test statistic:

χ^{2} = [n - 1 - (2 p + 5) / 6 - 2 k / 3] F (\hat{Ψ})

which can be used to test the goodness-of-fit of the model (1), see Lawley and Maxwell (1971) and Morrison (1967).

4 References

Clark M R B (1970) A rapidly convergent method for maximum likelihood factor analysis British J. Math. Statist. Psych.

Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20(3) 2–25

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

Morrison D F (1967) Multivariate Statistical Methods McGraw–Hill

5 Arguments

1: $matrix$ – Nag_FacMat Input

On entry: selects the type of matrix on which factor analysis is to be performed.

$matrix = Nag_DataCorr$ (Data input): The data matrix will be input in x and factor analysis will be computed for the correlation matrix.
$matrix = Nag_DataCovar$: The data matrix will be input in x and factor analysis will be computed for the covariance matrix, i.e., the results are scaled as described in Section 9.
$matrix = Nag_MatCorr_Covar$: The correlation/variance-covariance matrix will be input in x and factor analysis computed for this matrix.

Constraint:

matrix = Nag_DataCorr

Nag_DataCovar

Nag_MatCorr_Covar

2: $n$ – Integer Input

On entry: if

matrix = Nag_DataCorr

Nag_DataCovar

the number of observations in the data array x.

matrix = Nag_MatCorr_Covar

the (effective) number of observations used in computing the (possibly weighted) correlation/variance-covariance matrix input in x.

Constraint:

n > nvar

3: $m$ – Integer Input

On entry: the number of variables in the data/correlation/variance-covariance matrix.

Constraint:

m \geq nvar

4: $x [dim1 \times tdx]$ – const double Input

On entry: the input matrix.

$matrix = Nag_DataCorr$ or $Nag_DataCovar$: x must contain the data matrix, i.e., $x [(i - 1) \times tdx + j - 1]$ must contain the $i$ th observation for the $j$ th variable, for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m$ .
$matrix = Nag_MatCorr_Covar$: x must contain the correlation or variance-covariance matrix. Only the upper triangular part is required.

5: $tdx$ – Integer Input

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

Constraint:

tdx \geq m

6: $nvar$ – Integer Input

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

p

Constraint:

nvar \geq 2

7: $isx [m]$ – const Integer Input

On entry:

isx [j - 1]

indicates whether or not the

j

th variable is to be included in the factor analysis.

isx [j - 1] \geq 1

, then the variable represented by the

j

th column of x is included in the analysis; otherwise it is excluded, for

j = 1, 2, \dots, m

Constraint:

isx [j - 1] > 0

for nvar values of

j

8: $nfac$ – Integer Input

On entry: the number of factors,

k

Constraint:

1 \leq nfac \leq nvar

9: $wt [n]$ – const double Input

On entry: if

matrix = Nag_DataCorr

Nag_DataCovar

then the elements of wt must contain the weights to be used in the factor analysis. The effective number of observations is the sum of the weights. If

wt [i - 1] = 0.0

then the

i

th observation is not included in the analysis.

matrix = Nag_MatCorr_Covar

or wt is NULLthen wt is not referenced and the effective number of observations is

n

Constraint: if wt is referenced, then

wt [i - 1] \geq 0

for

i = 1, 2, \dots, n

, and the sum of the weights

> nvar

10: $e [nvar]$ – double Output

On exit: the eigenvalues

θ_{i}

, for

i = 1, 2, \dots, p

11: $stat [4]$ – double Output

On exit: the test statistics.

stat [0]

contains the value

F (\hat{Ψ})

stat [1]

contains the test statistic,

χ^{2}

stat [2]

contains the degrees of freedom associated with the test statistic.

stat [3]

contains the significance level.

12: $com [nvar]$ – double Output

On exit: the communalities.

13: $psi [nvar]$ – double Output

On exit: the estimates of

ψ_{i}

, for

i = 1, 2, \dots, p

14: $res [nvar \times (nvar - 1) / 2]$ – double Output

On exit: the residual correlations. The residual correlation for the

i

th and

j

th variables is stored in

res [(j - 1) (j - 2) / 2 + i - 1]

i < j

15: $fl [nvar \times tdfl]$ – double Output

On exit: the factor loadings.

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

contains

λ_{i j}

, for

i = 1, 2, \dots, p

and

j = 1, 2, \dots, k

16: $tdfl$ – Integer Input

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

Constraint:

tdfl \geq nfac

17: $options$ – Nag_E04_Opt * Input/Output

On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional parameters for e04lbc. These structure members offer the means of adjusting some of the argument values of the algorithm.

If the optional parameters are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call. See the document for e04lbc for further details.

18: $eps$ – double Input

On entry: a lower bound for the value of

Ψ_{i}

Constraint:

machine precision \leq eps < 1.0

19: $fail$ – 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_GT: On entry, $nfac = 〈value〉$ while $nvar = 〈value〉$ . These arguments must satisfy $nfac \leq nvar$ .
NE_2_INT_ARG_LE: On entry, $n = 〈value〉$ while $nvar = 〈value〉$ . These arguments must satisfy $n > nvar$ .
NE_2_INT_ARG_LT: On entry, $m = 〈value〉$ while $nvar = 〈value〉$ . These arguments must satisfy $m \geq nvar$ .

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

On entry, $tdx = 〈value〉$ while $m = 〈value〉$ . These arguments must satisfy $tdx \geq m$ .
NE_2_REAL_ARG_LT: On entry, $step_max = 〈value〉$ while $optim_tol = 〈value〉$ . These arguments must satisfy $step_max \geq optim_tol$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument matrix had an illegal value.

On entry, argument $print_level$ had an illegal value.
NE_INT_ARG_LT: On entry, $nfac = 〈value〉$ .
Constraint: $nfac \geq 1$ .

On entry, $nvar = 〈value〉$ .
Constraint: $nvar \geq 2$ .
NE_INTERNAL_ERROR: Additional error messages are output if the optimization fails to converge or if the options are set incorrectly. Details of these can be found in the e04lbc document.

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_INVALID_INT_RANGE_1: Value $〈value〉$ given to $max_iter$ is not valid. Correct range is $max_iter \geq 0$ .
NE_INVALID_REAL_RANGE_EF: Value $〈value〉$ given to eps is not valid. Correct range is machine precision $\leq optim_tol < 1.0$ .
NE_INVALID_REAL_RANGE_FF: Value $〈value〉$ given to $linesearch_tol$ is not valid. Correct range is $0.0 \leq linesearch_tol < 1.0$ .
NE_MAT_RANK: On entry, $matrix = Nag_DataCorr$ or $matrix = Nag_DataCovar$ and the data matrix is not of full column rank, or $matrix = Nag_MatCorr_Covar$ and the input correlation/variance-covariance matrix is not positive definite. This exit may also be caused by two of the eigenvalues of $S^{*}$ being equal; this is rare (see Lawley and Maxwell (1971)) and may be due to the data/correlation matrix being almost singular.
NE_NEG_WEIGHT_ELEMENT: On entry, $wt [〈value〉] = 〈value〉$ .
Constraint: when referenced, all elements of wt must be non-negative.
NE_NOT_APPEND_FILE: Cannot open file $〈string〉$ for appending.
NE_NOT_CLOSE_FILE: Cannot close file $〈string〉$ .
NE_OBSERV_LT_VAR: With weighted data, the effective number of observations given by the sum of weights $= 〈value〉$ , while the number of variables included in the analysis, $nvar = 〈value〉$ .
Constraint: effective number of observations $> nvar + 1$ .
NE_OPT_NOT_INIT: Options structure not initialized.
NE_SVD_NOT_CONV: A singular value decomposition has failed to converge. This is a very unlikely error exit.
NE_VAR_INCL_INDICATED: The number of variables, nvar in the analysis $= 〈value〉$ , while number of variables included in the analysis via array $isx = 〈value〉$ .
Constraint: these two numbers must be the same.
NW_COND_MIN: The conditions for a minimum have not all been satisfied but a lower point could not be found. Note that in this case all the results are computed. See e04lbc for further details.
NW_TOO_MANY_ITER: The maximum number of iterations, $〈value〉$ , have been performed.

7 Accuracy

The accuracy achieved is discussed in e04lbc.

8 Parallelism and Performance

g03cac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g03cac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The factor loadings may be orthogonally rotated by using g03bac and factor score coefficients can be computed using g03ccc. The maximum likelihood estimators are invariant to a change in scale. This means that the results obtained will be the same (up to a scaling factor) if either the correlation matrix or the variance-covariance matrix is used. As the correlation matrix ensures that all values of

ψ_{i}

are between 0 and 1 it will lead to a more efficient optimization. In the situation when the data matrix is input the results are always computed for the correlation matrix and then scaled if the results for the covariance matrix are required. When you input the covariance/correlation matrix the input matrix itself is used and so you are advised to input the correlation matrix rather than the covariance matrix.

10 Example

The example is taken from Lawley and Maxwell (1971). The correlation matrix for nine variables is input and the arguments of a factor analysis model with three factors are estimated and printed.

Interfaces: FL CL AD

NAG CL Interface Introduction

G03 (Mv) Chapter Contents

G03 (Mv) Chapter Introduction

g03ca: FL CL AD

NAG CL Interfaceg03cac (factor)

▸▿ 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
g03cac (factor)