rotate_criteria_description.ihlp

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{* 05apr2005}{...}
{title:Rotation criteria}

{pstd}
In the descriptions below, the matrix to be rotated is denoted as {bf:A},
{it:p} denotes the number of rows of {bf:A}, and {it:f} denotes the number
of columns of {bf:A} (factors or components).  If {bf:A} is a loading matrix
from {cmd:factor} or {cmd:pca}, {it:p} is the number of variables, and {it:f}
the number of factors or components.

    {title:Criteria suitable only for orthogonal rotations}

{phang2}{opt varimax} and {opt vgpf}
apply the orthogonal varimax rotation.  {cmd:varimax} maximizes the variance
of the squared loadings within factors (columns of {bf:A}).  It is equivalent
to {cmd:cf(}{it:1/p}{cmd:)} and to {cmd:oblimin(1)}.  {cmd:varimax},
the most popular rotation, is implemented with a dedicated fast
algorithm and ignores all {it:optimize_options}.  Specify {cmd:vgpf} to switch
to the general GPF algorithm used for the other criteria.

{phang2}{opt quartimax}
uses the quartimax criterion.  {cmd:quartimax} maximizes the
variance of the squared loadings within the variables (rows of {bf:A}).  For
orthogonal rotations, {cmd:quartimax} is equivalent to {cmd:cf(0)} and to
{cmd:oblimax}.

{phang2}{opt equamax}
specifies the orthogonal equamax rotation.  {cmd:equamax} maximizes a weighted
sum of the {cmd:varimax} and {cmd:quartimax} criteria, reflecting a concern
for simple structure within variables (rows of {bf:A}) as well as within
factors (columns of {bf:A}).  {cmd:equamax} is equivalent to
{cmd:oblimin(}{it:p}{cmd:/2)} and {cmd:cf(}{it:f}{cmd:/(2}{it:p}{cmd:))}.

{phang2}{opt parsimax}
specifies the orthogonal parsimax rotation.  {cmd:parsimax} is equivalent
to {cmd:cf((}{it:f}{cmd:-1)/(}{it:p}{cmd:+}{it:f}{cmd:-2))}.

{phang2}{opt entropy}
applies the minimum entropy rotation criterion.

{phang2}{opt tandem1}
specifies that the first principle of Comrey's tandem is to be applied.
According to Comrey, this principle should be used to judge which "small"
factors be dropped.  Use {opt tandem2} for polishing.

{phang2}{opt tandem2}
specifies that the second principle of Comrey's tandem be applied.  According
to Comrey, {opt tandem2} should be used for "polishing", while {opt tandem1}
should be used to judge which "small" factors be dropped.

    {title:Criteria suitable only for oblique rotations}

{phang2}{cmd:promax}[{cmd:(}{it:#}{cmd:)}]
specifies the oblique promax rotation.  The optional argument specifies the
{cmd:promax} power.  Not specifying the argument is equivalent to specifying
{cmd:promax(3)}.  Values less than 4 are recommended, but the choice is yours.
Larger {cmd:promax} powers simplify the loadings (generate numbers closer to
zero and one) but at the cost of additional correlation between factors.
Choosing a value is a matter of trial-and-error, but most sources find values
in excess of 4 undesirable in practice.  The power must be greater than 1 but
is not restricted to integers.

{pmore2}
Promax rotation is an oblique rotation method that was developed before the
"analytical methods" (based on criterion optimization) became computationally
feasible.  Promax rotation is comprised of an oblique Procrustean rotation of
the original loadings {bf:A} towards the element-wise {it:#}-power of the
orthogonal varimax rotation of {bf:A}.

    {title:Criteria suitable for orthogonal and oblique rotations}

{phang2}{cmd:oblimin}[{cmd:(}{it:#}{cmd:)}]
specifies that the oblimin criterion with gamma = {it:#} be used.  When
restricted to orthogonal transformations, the {cmd:oblimin()} family is
equivalent to the orthomax criterion function.  Special cases of
{cmd:oblimin()} include

{space 16}{c TLC}{hline 36}{c TRC}
{space 16}{c |} gamma    special cases             {c |}
{space 16}{c LT}{hline 36}{c RT}
{space 16}{c |} 0        quartimax / quartimin     {c |}
{space 16}{c |} 1/2      biquartimax / biquartimin {c |}
{space 16}{c |} 1        varimax / covarimin       {c |}
{space 16}{c |} {it:p}/2      equamax                   {c |}
{space 16}{c BLC}{hline 36}{c BRC}
{space 16}  {it:p} = number of rows of {bf:A}

{pmore2}
gamma defaults to zero.
gamma {ul:<} 0 is recommended for oblique rotations.  For gamma > 0 it is
possible that optimal oblique rotations do not exist; the iterative procedure
used to compute the solution will wander off to a degenerate solution.

{phang2}{opt cf(#)}
specifies that a criterion from the Crawford-Ferguson family be used
with kappa = {it:#}.  {opt cf(kappa)} can be seen as
(1-{it:kappa})cf_1({bf:A}) + {it:kappa} cf_2({bf:A}), where cf_1({bf:A}) is a
measure of row parsimony, and cf_2({bf:A}) is a measure of column parsimony.
cf_1({bf:A}) attains its greatest lower bound when no row of {bf:A} has more
than one nonzero element, while cf_2({bf:A}) reaches zero if no column of
{bf:A} has more than one nonzero element.

{pmore2}
For orthogonal rotations the Crawford-Ferguson family is equivalent to the
{cmd:oblimin()} family.  For orthogonal rotations, special cases include

{space 16}{c TLC}{hline 40}{c TRC}
{space 16}{c |} kappa            description           {c |}
{space 16}{c LT}{hline 40}{c RT}
{space 16}{c |} 0                quartimax / quartimin {c |}
{space 16}{c |} 1/{it:p}              varimax / covarimin   {c |}
{space 16}{c |} {it:f}/(2{it:p})           equamax               {c |}
{space 16}{c |} ({it:f}-1)/({it:p}+{it:f}-2)    parsimax              {c |}
{space 16}{c |} 1                factor parsimony      {c |}
{space 16}{c BLC}{hline 40}{c BRC}
{space 16}  {it:p} = number of rows of {bf:A}
{space 16}  {it:f} = number of columns of {bf:A}

{phang2}{opt bentler}
specifies that Bentler's "invariant pattern simplicity" criterion be used.

{phang2}{opt oblimax}
specifies the oblimax criterion.  {cmd:oblimax}  maximizes the number of
high and low loadings.  {cmd:oblimax} is equivalent to {cmd:quartimax}
for orthogonal rotations.

{phang2}{opt quartimin}
specifies that the quartimin criterion be used.  For orthogonal rotations,
{cmd:quartimin} is equivalent to {cmd:quartimax}.

{phang2}{opt target(Tg)}
specifies that {bf:A} be rotated as near as possible to the conformable matrix
{it:Tg}.  Nearness is expressed by the Frobenius matrix norm.

{phang2}{opt partial(Tg W)}
specifies that {bf:A} be rotated as near as possible to the conformable matrix
{it:Tg}.  Nearness is expressed by a weighted (by {it:W}) Frobenius matrix
norm.  {it:W} should be nonnegative; and usually is zero-one valued, with ones
identifying the target values to be reproduced as closely as possible by the
factor loadings, while zeros identify loadings to remain unrestricted.
{p_end}