void	nag_mv_orthomax (Nag_RotationLoading stand, double g, Integer nvar, Integer k, double fl[], Integer tdf, double flr[], double r[], Integer tdr, double acc, Integer maxit, Integer iter, NagError fail)

3 Description

Let

Λ

be the

p

k

matrix of loadings from a variable-directed multivariate method, e.g., canonical variate analysis or factor analysis. This matrix represents the relationship between the original

p

variables and the

k

orthogonal linear combinations of these variables, the canonical variates or factors. The latter are only unique up to a rotation in the

k

-dimensional space they define. A rotation can then be found that simplifies the structure of the matrix of loadings, and hence the relationship between the original and the derived variables. That is, the elements,

λ_{i j}^{*}

, of the rotated matrix,

Λ^{*}

, are either relatively large or small. The rotations may be found by minimizing the criterion:

V = \sum_{j = 1}^{k} \sum_{i = 1}^{p} {(λ_{i j}^{*})}^{4} - \frac{γ}{p} \sum_{j = 1}^{k} {[\sum_{i = 1}^{p} {(λ_{i j}^{*})}^{2}]}^{2}

where the constant

γ

gives a family of rotations with

γ = 1

giving varimax rotations and

γ = 0

giving quartimax rotations.

It is generally advised that factor loadings should be standardized, so that the sum of squared elements for each row is one, before computing the rotations.

The matrix of rotations,

R

, such that

Λ^{*} = Λ R

, is computed using first an algorithm based on that described by Cooley and Lohnes (1971), which involves the pairwise rotation of the factors. Then a final refinement is made using a method similar to that described by Lawley and Maxwell (1971), but instead of the eigenvalue decomposition, the algorithm has been adapted to incorporate a singular value decomposition.

4 References

Cooley W C and Lohnes P R (1971) Multivariate Data Analysis Wiley

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

5 Arguments

1: $stand$ – Nag_RotationLoadingInput

On entry: indicates if the matrix of loadings is to be row standardized before rotation.

$stand = Nag_RoLoadStand$: The loadings are row standardized.
$stand = Nag_RoLoadNotStand$: The loadings are left unstandardized.

Constraint:

stand = Nag_RoLoadStand

Nag_RoLoadNotStand

2: $g$ – doubleInput

On entry: the criterion constant,

γ

, with

γ = 1.0

giving varimax rotations and

γ = 0.0

giving quartimax rotations.

Constraint:

g \geq 0.0

3: $nvar$ – IntegerInput

On entry: the number of original variables,

p

Constraint:

nvar \geq k

4: $k$ – IntegerInput

On entry: the number of derived variates or factors,

k

Constraint:

k \geq 2

5: $fl [nvar \times tdf]$ – doubleInput/Output

On entry: the matrix of loadings,

Λ

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

must contain the loading for the

i

th variable on the

j

th factor, for

i = 1, 2, \dots, p

and

j = 1, 2, \dots, k

On exit: if

stand = Nag_RoLoadStand

the elements of fl are standardized so that the sum of squared elements for each row is 1.0 and then after, the computation of the rotations are rescaled; this may lead to slight differences between the input and output values of fl. If

stand = Nag_RoLoadNotStand

, fl will be unchanged on exit.

6: $tdf$ – IntegerInput

On entry: the stride separating matrix column elements in the arrays fl, flr.

Constraint:

tdf \geq k

7: $flr [nvar \times tdf]$ – doubleOutput

On exit: the rotated matrix of loadings,

Λ^{*}

flr [(i - 1) \times tdf + j - 1]

will contain the rotated loading for the

i

th variable on the

j

th factor, for

i = 1, 2, \dots, p

and

j = 1, 2, \dots, k

8: $r [k \times tdr]$ – doubleOutput

Note: the

(i, j)

th element of the matrix

R

is stored in

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

On exit: the matrix of rotations,

R

9: $tdr$ – IntegerInput

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

Constraint:

tdr \geq k

10: $acc$ – doubleInput

On entry: indicates the accuracy required. The iterative procedure of Cooley and Lohnes (1971) will be stopped and the final refinement computed when the change in

V

is less than

acc \times \max (1.0, V)

. If acc is greater than or equal to 0.0 but less than machine precision, or if acc is greater than 1.0, then machine precision will be used instead.

It is suggested that acc be set to 0.00001.

Constraint:

acc \geq 0.0

11: $maxit$ – IntegerInput

On entry: the maximum number of iterations. It is suggested that maxit be set to 30.

Constraint:

maxit \geq 1

12: $iter$ – Integer *Output

On exit: the number of iterations performed.

13: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $nvar = 〈value〉$ while $k = 〈value〉$ . These arguments must satisfy $nvar \geq k$ .

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

On entry, $tdr = 〈value〉$ while $k = 〈value〉$ . These arguments must satisfy $tdr \geq k$ .
NE_ACC_ITER: The algorithm to find R has failed to reach the required accuracy in the given number of iterations, $〈value〉$ . Try increasing acc or increasing maxit. The returned solution should be a reasonable approximation.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument stand had an illegal value.
NE_INT_ARG_LE: On entry, $maxit = 〈value〉$ .
Constraint: $maxit > 0$ .
NE_INT_ARG_LT: On entry, $k = 〈value〉$ .
Constraint: $k \geq 2$ .
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_ARG_LT: On entry, acc must not be less than $0.0$ : $acc = 〈value〉$ .

On entry, g must not be less than $0.0$ : $g = 〈value〉$ .
NE_SVD_NOT_CONV: The singular value decomposition has failed to converge. This is an unlikely error exit.

7 Accuracy

The accuracy is determined by the value of acc.

8 Parallelism and Performance

nag_mv_orthomax (g03bac) is not threaded in any implementation.

9 Further Comments

If the results of a principal component analysis as carried out by nag_mv_prin_comp (g03aac) are to be rotated, the loadings as returned in the array

p

by nag_mv_prin_comp (g03aac) can be supplied via the argument fl to nag_mv_orthomax (g03bac). The resulting rotation matrix can then be used to rotate the principal component scores as returned in the array

v

by nag_mv_prin_comp (g03aac). The function nag_dgemm (f16yac) may be used for this matrix multiplication.

10 Example

The example is taken from page 75 of Lawley and Maxwell (1971). The results from a factor analysis of ten variables using three factors are input and rotated using varimax rotations without standardizing rows.

NAG Library Function Documentnag_mv_orthomax (g03bac)

▸▿ 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 Library Function Document

nag_mv_orthomax (g03bac)