The above power transformation tends to increase the difference between high and low values of loadings and is intended to increase the interpretability of a solution.

In the second step a solution of:

X W = Y, X, Y \in ℝ^{n \times m}, ​ W \in ℝ^{m \times m},

is found for

W

in the least squares sense by use of singular value decomposition of the orthogonal loadings

X

. The ProMax rotation matrix

R

is then given by

R = O W \tilde{W}, O, ​ \tilde{W} \in ℝ^{m \times m},

where

O

is the rotation matrix from an orthogonal transformation, and

\tilde{W}

is a matrix with the square root of diagonal elements of

{(W^{T} W)}^{- 1}

on its diagonal and zeros elsewhere.

The ProMax factor pattern matrix

P

is given by

P = X W \tilde{W}, P \in ℝ^{n \times m};

the inter-factor correlations

Φ

are given by

Φ = {(Q^{T} Q)}^{- 1}, Φ \in ℝ^{m \times m};

where

Q = W \tilde{W}

; and the factor structure

S

is given by

S = P Φ, S \in ℝ^{n \times m} .

Optionally, the rows of target matrix

Y

can be scaled by the communalities of loadings.

References

None.

Parameters

Compulsory Input Parameters

1: $stand$ – string (length ≥ 1)

Indicates how loadings are normalized.

$stand ='S'$: Rows of $Y$ are (Kaiser) normalized by the communalities of the loadings.
$stand ='U'$: Rows are not normalized.

Constraint:

stand ='U'

'S'

2: $x (ldx, m)$ – double array

ldx, the first dimension of the array, must satisfy the constraint

ldx \geq n

The loadings matrix following an orthogonal rotation,

X

3: $ro (ldro, m)$ – double array

ldro, the first dimension of the array, must satisfy the constraint

ldro \geq m

The orthogonal rotation matrix,

O

4: $power$ – double scalar

p

, the value of exponent.

Constraint:

power > 1.0

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array x.
$n$ , the number of points.

Constraint: $n \geq m$ .
2: $m$ – int64int32nag_int scalar: Default: the first dimension of the array ro and the second dimension of the arrays x, ro. (An error is raised if these dimensions are not equal.)
$m$ , the number of dimensions.

Constraint: $m \geq 1$ .

Output Parameters

1: $fp (ldfp, m)$ – double array: The factor pattern matrix, $P$ .
2: $r (ldr, m)$ – double array: The ProMax rotation matrix, $R$ .
3: $phi (ldphi, m)$ – double array: The matrix of inter-factor correlations, $Φ$ .
4: $fs (ldfs, m)$ – double array: The factor structure matrix, $S$ .
5: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $m \geq 1$ .

Constraint: $power > 1.0$ .

Constraint: $stand ='U'$ or $'S'$ .

$ifail = 2$: Constraint: $ldfp \geq n$ .

Constraint: $ldfs \geq n$ .

Constraint: $ldphi \geq m$ .

Constraint: $ldro \geq m$ .

Constraint: $ldr \geq m$ .

Constraint: $ldx \geq n$ .

Constraint: $n \geq m$ .

$ifail = 20$: SVD failed to converge.

$ifail = 100$: 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The calculations are believed to be stable.

Further Comments

None.

Example

This example reads a loadings matrix and calculates a varimax transformation before calculating

P

R

and

σ

for a ProMax rotation.

Open in the MATLAB editor: g03bd_example

function g03bd_example


fprintf('g03bd example results\n\n');

fl = [0.74215 -0.57806;
      0.71370 -0.55515;
      0.87899 -0.15847;
      0.62533  0.76621;
      0.71447  0.67936];

% Calculate orthogonal rotation
stand = 's';
g     = 1;
[n,m] = size(fl);
nvar  = int64(n);

[~, x, ro, iter, ifail] = g03ba( ...
				 stand, g, nvar, fl);

% Calculate ProMax rotation

power = 3;
[fp, r, phi, fs, ifail] = g03bd( ...
				 stand, x, ro, power);

mtitle = 'Factor pattern';
matrix = 'General';
diag   = ' ';

[ifail] = x04ca( ...
                 matrix, diag, fp, mtitle);

fprintf('\n');
mtitle = 'Promax rotation';
[ifail] = x04ca( ...
                 matrix, diag, r, mtitle);

fprintf('\n');
mtitle = 'Inter-factor correlations';
[ifail] = x04ca( ...
                 matrix, diag, phi, mtitle);

fprintf('\n');
mtitle = 'Factor structure';
[ifail] = x04ca( ...
                 matrix, diag, fs, mtitle);

g03bd example results

 Factor pattern
             1          2
 1      0.9556    -0.0979
 2      0.9184    -0.0935
 3      0.7605     0.3393
 4     -0.0791     1.0019
 5      0.0480     0.9751

 Promax rotation
          1       2
 1   0.7380  0.5420
 2  -0.7055  0.8653

 Inter-factor correlations
          1       2
 1   1.0000  0.2019
 2   0.2019  1.0000

 Factor structure
          1       2
 1   0.9358  0.0950
 2   0.8995  0.0919
 3   0.8290  0.4928
 4   0.1232  0.9860
 5   0.2448  0.9848

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox