nag_mv_factor (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.

Let  variables, , with variance-covariance matrix  be observed. The aim of factor analysis is to account for the covariances in these  variables in terms of a smaller number, , of hypothetical variables, or factors, . 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:

where , for ; , are the factor loadings and , for , are independent random variables with variances , for . The  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  factors and the 's follow independent Normal distributions.

The model for the variance-covariance matrix, , can be written as:

(1)

where  is the matrix of the factor loadings, , and  is a diagonal matrix of unique variances, , for .

The estimation of the arguments of the model,  and , by maximum likelihood is described by Lawley and Maxwell (1971). The log likelihood is:

where  is the number of observations,  is the sample variance-covariance matrix or, if weights are used,  is the weighted sample variance-covariance matrix and  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 , which is, up to a constant,  times the log likelihood maximized over . This is then minimized with respect to  to give the estimates, , of . The function  can be written as:

where values , for  are the eigenvalues of the matrix:

The estimates , of , are then given by scaling the eigenvectors of , which are denoted by :

where  is the diagonal matrix with elements , and  is the identity matrix.

The minimization of  is performed using e04lbc (nag_opt_bounds_2nd_deriv) 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  as described above, the singular value decomposition of the matrix  is used, where  is obtained either from the  decomposition of the (scaled) mean-centred data matrix or from the Cholesky decomposition of the correlation/covariance matrix. The function e04lbc (nag_opt_bounds_2nd_deriv) ensures that the values of  are greater than a given small positive quantity, , so that the communality is always less than one. This avoids the so called Heywood cases.

In addition to the values of ,  and the communalities, nag_mv_factor (g03cac) returns the residual correlations, i.e., the off-diagonal elements of  where  is the sample correlation matrix. nag_mv_factor (g03cac) also returns the test statistic:

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

"NE_2_INT_ARG_GT"

On entry,  while . These arguments must satisfy .

"NE_2_INT_ARG_LE"

On entry,  while . These arguments must satisfy .

"NE_2_INT_ARG_LT"

On entry,  while . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument matrix_type had an illegal value.

"NE_INT_ARG_LT"

On entry, nfac must not be less than 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 consult NAG for assistance.

"NE_INVALID_REAL_RANGE_EF"

Value  given to eps is not valid.Correct range is machine precision .

"NE_NEG_WEIGHT_ELEMENT"

On entry, . Constraint: When referenced, all elements of wt must be non-negative.

"NE_NOT_APPEND_FILE"

Cannot open file  for appending.

"NE_NOT_CLOSE_FILE"

Cannot close file .

"NE_OBSERV_LT_VAR"

With weighted data, the effective number of observations given by the sum of weights , while the number of variables included in the analysis, . Constraint: effective number of observations .

"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 , while number of variables included in the analysis via array . 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 (nag_opt_bounds_2nd_deriv) for further details.

"NW_TOO_MANY_ITER"

The maximum number of iterations, , have been performed.

The accuracy achieved is discussed in e04lbc (nag_opt_bounds_2nd_deriv).

The factor loadings may be orthogonally rotated by using g03bac (nag_mv_orthomax) and factor score coefficients can be computed using g03ccc (nag_mv_fac_score). 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  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 the user inputs the covariance/correlation matrix the input matrix itself is used and so the user is advised to input the correlation matrix rather than the covariance matrix.