2nd order factor analysis

[DR VEENA Joshi]

Hi,

I have responses on Healthcare perceptions on 20 questions (likert scale of 5).
Using PCA Varimax rotation I got 5 factors. They are not correlated.
Is it correct to use factor analysis (second order) for the variables
in the first factor to get 2 factors? (I have read in a book that one
can use 2nd order factor analysis only if one has used oblic
rotation).
After getting 2 factors from the first factor, I now have 6 factors
and some of them are  correlated with these two new factors.

Is this corect procedure? If not what will be the correct procedure?

After getting 6 factors I used these factor scores as covariates and
used multinomial regression. I am getting expected results. But not
sure if I have followed correct procedure.

Factor scores should be correlated  or not ? When I did PAF varimax,
or obline, factor scores correlated.
Which will be the best model?

Veena

[Art Kendall]

How did you decide how many factors to retain?

Why do you want to split the first factor?

Are you trying to create scales from the items?

How many cases did you have?

Art Kendall
Social Research Consultants


DR VEENA Joshi wrote:

> Hi,
>
> I have responses on Healthcare perceptions on 20 questions (likert
> scale of 5).
> Using PCA Varimax rotation I got 5 factors. They are not correlated.
> Is it correct to use factor analysis (second order) for the variables
> in the first factor to get 2 factors? (I have read in a book that one
> can use 2nd order factor analysis only if one has used oblic
> rotation).
> After getting 2 factors from the first factor, I now have 6 factors
> and some of them are  correlated with these two new factors.
>
> Is this corect procedure? If not what will be the correct procedure?
>
> After getting 6 factors I used these factor scores as covariates and
> used multinomial regression. I am getting expected results. But not
> sure if I have followed correct procedure.
>
> Factor scores should be correlated  or not ? When I did PAF varimax,
> or obline, factor scores correlated.
> Which will be the best model?
>
> Veena
>
>

[Hector E. Maletta]

Joshi,

1. First a general idea worth recalling. Factors are
not objective things. They are artificial constructs
arising from a particular algorithm based on the
correlation of observed variables. There are several
such algorithms producing different sets of factors
out of the same data, and furthermore, you can rotate
your initial factors (in several ways) to obtain as
many rotated factor structures and factor scores. This
would mean (to some) that by carefully choosing your
factor extraction and rotation procedures you may be
able to "prove" or "disprove" any number of different
statements. In fact you cannot prove or disprove much
by means of factor analysis as such. This does not
mean it is useless, of course, but do not expect from
it more than it is capable of yielding. More on this
below.

2. If you had 20 questions and got 5 factors from them
(i.e., five factors with significant loadings; in fact
you could obtain as many as 20 factors out of 20
variables), and these 5 factors are not correlated
among themselves, it makes no sense to make any factor
analysis of the 5 factor scores derived from these 5
factors. As these factor scores would be 5
uncorrelated variables, with zero covariance, there
would be no commonality and of course no common factor
to explain it. If the 5 factors were obtained by
oblique rotation they would be inter-correlated, and
therefore they would share part of their variance. In
that case you could possibly use factor analysis on
them. This would be a "second order factor analysis"
(i.e. a factor analysis of correlated factor scores
resulting from a previous obliquely rotated factor
extraction").

3. If you select a subset of variables, e.g. those
variables which happened to have high loadings on the
first factor in the previous factor analysis, and
apply a factor analysis on them alone, it is perfectly
possible that you may extract 2 factors from them.
This is NOT a second order factor analysis. This is
another first order factor analysis performed on a
different set of variables. Again, if this subset is
composed of, say, 7 variables, you could obtain up to
7 factors (not all necessarily significant). But these
factors, whatever their number, obtained in this
second analysis of a subset of variables, cannot be
added to the factors obtained from the entire set of
20 variables. Notice that you could select any other
subset of 4, 5, 7 or any other number of variables
from your original set of 20, and perform a factor
analysis on them, each time extracting some factors,
but each of these exercises would be always a first
order factor analysis, separate and independent from
the others.

4. Whether factors SHOULD be correlated or not, is a
question that cannot be answered in general. It
depends on the nature of the problem. If you interpret
the factors as representing underlying traits that (in
your theory) can be correlated, then you should seek a
solution yielding correlated factors. If your theory
postulates that the factors are not correlated, then
do not try any oblique rotation. Again: factor
analysis cannot prove whether the underlying or latent
variables (represented by the extracted factors, which
are an observable linear function of the observed
variables) are correlated or not.,

5. One way of deciding (in a manner of speaking)
between correlated or uncorrelated factors is having
some external criterion against which you may contrast
your hypothesis. Suppose for instance your factors
represent different underlying traits contributing to
the prediction of an observable behaviour, for
instance school performance or criminal relapse after
jail release. You may try the correlated and
uncorrelated versions to assess which is the better
predictor of the behaviour in question. The conclusion
would be along these lines: "IF this kind of model,
with this functional form (e.g. linear or logistic or
whatever), with these and no other predictors, governs
this behaviour, THEN this particular form of factor
extraction and rotation fits the data best among the
various forms I have tried." This cautious conclusion
does not rule out that (a) some other functional form
is better, or (b) some other predictor should be
included, or some of those present be excluded or
modified, or (c) some other factor extraction or
rotation method should be used that you have not tried
yet.

Hope this helps.

Hector


----- Mensaje original -----
De: DR VEENA Joshi <[hidden email]>
Fecha: Sábado, Septiembre 23, 2006 12:03 pm
Asunto: 2nd order factor analysis

> Hi,
>
> I have responses on Healthcare perceptions on 20
questions (likert
> scale of 5).
> Using PCA Varimax rotation I got 5 factors. They are
not correlated.
> Is it correct to use factor analysis (second order)
for the variables
> in the first factor to get 2 factors? (I have read
in a book that one
> can use 2nd order factor analysis only if one has
used oblic
> rotation).
> After getting 2 factors from the first factor, I now
have 6 factors
> and some of them are  correlated with these two new
factors.
>
> Is this corect procedure? If not what will be the
correct procedure?
>
> After getting 6 factors I used these factor scores
as covariates and
> used multinomial regression. I am getting expected
results. But not
> sure if I have followed correct procedure.
>
> Factor scores should be correlated  or not ? When I
did PAF varimax,
> or obline, factor scores correlated.
> Which will be the best model?
>
> Veena
>

__________________________________________________
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Tired of spam?  Yahoo! Mail has the best spam protection around
http://mail.yahoo.com

[Hector E. Maletta]

Joshi,

1. First a general idea worth recalling. Factors are
not objective things. They are artificial constructs
arising from a particular algorithm based on the
correlation of observed variables. There are several
such algorithms producing different sets of factors
out of the same data, and furthermore, you can rotate
your initial factors (in several ways) to obtain as
many rotated factor structures and factor scores. This
would mean (to some) that by carefully choosing your
factor extraction and rotation procedures you may be
able to "prove" or "disprove" any number of different
statements. In fact you cannot prove or disprove much
by means of factor analysis as such. This does not
mean it is useless, of course, but do not expect from
it more than it is capable of yielding. More on this
below.

2. If you had 20 questions and got 5 factors from them
(i.e., five factors with significant loadings; in fact
you could obtain as many as 20 factors out of 20
variables), and these 5 factors are not correlated
among themselves, it makes no sense to make any factor
analysis of the 5 factor scores derived from these 5
factors. As these factor scores would be 5
uncorrelated variables, with zero covariance, there
would be no commonality and of course no common factor
to explain it. If the 5 factors were obtained by
oblique rotation they would be inter-correlated, and
therefore they would share part of their variance. In
that case you could possibly use factor analysis on
them. This would be a "second order factor analysis"
(i.e. a factor analysis of correlated factor scores
resulting from a previous obliquely rotated factor
extraction").

3. If you select a subset of variables, e.g. those
variables which happened to have high loadings on the
first factor in the previous factor analysis, and
apply a factor analysis on them alone, it is perfectly
possible that you may extract 2 factors from them.
This is NOT a second order factor analysis. This is
another first order factor analysis performed on a
different set of variables. Again, if this subset is
composed of, say, 7 variables, you could obtain up to
7 factors (not all necessarily significant). But these
factors, whatever their number, obtained in this
second analysis of a subset of variables, cannot be
added to the factors obtained from the entire set of
20 variables. Notice that you could select any other
subset of 4, 5, 7 or any other number of variables
from your original set of 20, and perform a factor
analysis on them, each time extracting some factors,
but each of these exercises would be always a first
order factor analysis, separate and independent from
the others.

4. Whether factors SHOULD be correlated or not, is a
question that cannot be answered in general. It
depends on the nature of the problem. If you interpret
the factors as representing underlying traits that (in
your theory) can be correlated, then you should seek a
solution yielding correlated factors. If your theory
postulates that the factors are not correlated, then
do not try any oblique rotation. Again: factor
analysis cannot prove whether the underlying or latent
variables (represented by the extracted factors, which
are an observable linear function of the observed
variables) are correlated or not.,

5. One way of deciding (in a manner of speaking)
between correlated or uncorrelated factors is having
some external criterion against which you may contrast
your hypothesis. Suppose for instance your factors
represent different underlying traits contributing to
the prediction of an observable behaviour, for
instance school performance or criminal relapse after
jail release. You may try the correlated and
uncorrelated versions to assess which is the better
predictor of the behaviour in question. The conclusion
would be along these lines: "IF this kind of model,
with this functional form (e.g. linear or logistic or
whatever), with these and no other predictors, governs
this behaviour, THEN this particular form of factor
extraction and rotation fits the data best among the
various forms I have tried." This cautious conclusion
does not rule out that (a) some other functional form
is better, or (b) some other predictor should be
included, or some of those present be excluded or
modified, or (c) some other factor extraction or
rotation method should be used that you have not tried
yet.

Hope this helps.

Hector

>
> ----- Mensaje original -----
> De: DR VEENA Joshi <[hidden email]>
> Fecha: Sábado, Septiembre 23, 2006 12:03 pm
> Asunto: 2nd order factor analysis
>
> > Hi,
> >
> > I have responses on Healthcare perceptions on 20
> questions (likert
> > scale of 5).
> > Using PCA Varimax rotation I got 5 factors. They
> are
> not correlated.
> > Is it correct to use factor analysis (second
> order)
> for the variables
> > in the first factor to get 2 factors? (I have read
> in a book that one
> > can use 2nd order factor analysis only if one has
> used oblic
> > rotation).
> > After getting 2 factors from the first factor, I
> now
> have 6 factors
> > and some of them are  correlated with these two
> new
> factors.
> >
> > Is this corect procedure? If not what will be the
> correct procedure?
> >
> > After getting 6 factors I used these factor scores
> as covariates and
> > used multinomial regression. I am getting expected
> results. But not
> > sure if I have followed correct procedure.
> >
> > Factor scores should be correlated  or not ? When
> I
> did PAF varimax,
> > or obline, factor scores correlated.
> > Which will be the best model?
> >
> > Veena
> >
>
> __________________________________________________
> Do You Yahoo!?
> Tired of spam?  Yahoo! Mail has the best spam
> protection around
> http://mail.yahoo.com
>

__________________________________________________
Do You Yahoo!?
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[statisticsdoc]

Stephen Brand
www.statisticsdoc.com

Dear Dr. Joshi,

What you describe is a potentially useful approach to exploratory analysis
for the purpose of scale construction.  However, additional analyses should
be performed in order to confirm the structure and internal consistency of
the scales.  First, as you state in your next e-mail to this list, AMOS (or
your favorite structural equation modeling package) can be utilized to
conduct a confirmatory factor analysis of the proposed five-dimensional
model for scoring the items into scales.  In addition, carry out reliability
analyses to examine the internal consistency of the scales.  With five
scales and 20 items, some scales might have very few items and hence have
relatively Cronbach alpha levels.

As an aside, the term "secondary factor analysis" is sometimes used in two
quite different - even opposite senses.  The most widely accepted use refers
to the analysis of oblique factors to determine the higher order dimensions
underlying the factor structure (e.g., consider a lengthy personality
inventory with 17 first-order oblique factors - the 17 oblique factors might
in turn reflect a smaller set of say 2 to 5 higher order secondary factors.
A quite different sense is when factor analysis is used to split apart a
dimension, as you have done, by seeing whether the items that load on the
same factor in the initial analysis can be differentiated into separate
factors when just those items are factor analzed.  This is sometimes a quite
useful analysis when a factor (often the first), combines items that are
correlated but seem to tap different sub-areas of a dimension.  The latter
is a useful exporatory approach, but the results stand in need of a
confirmatory analysis.

Best,

Stephen Brand

For personalized and professional consultation in statistics and research
design, visit
www.statisticsdoc.com


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]]On Behalf Of
DR VEENA Joshi
Sent: Saturday, September 23, 2006 11:03 AM
To: [hidden email]
Subject: 2nd order factor analysis


Hi,

I have responses on Healthcare perceptions on 20 questions (likert scale of
5).
Using PCA Varimax rotation I got 5 factors. They are not correlated.
Is it correct to use factor analysis (second order) for the variables
in the first factor to get 2 factors? (I have read in a book that one
can use 2nd order factor analysis only if one has used oblic
rotation).
After getting 2 factors from the first factor, I now have 6 factors
and some of them are  correlated with these two new factors.

Is this corect procedure? If not what will be the correct procedure?

After getting 6 factors I used these factor scores as covariates and
used multinomial regression. I am getting expected results. But not
sure if I have followed correct procedure.

Factor scores should be correlated  or not ? When I did PAF varimax,
or obline, factor scores correlated.
Which will be the best model?

Veena