factor analyzing the Tetrachoric correlations

[Dogan, Enis]

Dear list,

 

I am factor analyzing a set of 42 dichotomously scored test items (in
order to understand test dimensionality).

I calculated the Tetrachoric correlations (I can share the syntax if
anyone's interested); so my input file is a correlation matrix. My
question is:

Is there a particular type of rotation that works "better" with this
type of data? Or the same good-old rules apply here too?

 

Best

 

Enis

 

 

 

[Hector Maletta]

The best rotation does not depend on the type of correlation coefficient, or
the measurement level of variables, but on the underlying structure and the
purposes of your research.
Hector

-----Mensaje original-----
De: SPSSX(r) Discussion [mailto:[hidden email]] En nombre de
Dogan, Enis
Enviado el: Monday, July 31, 2006 7:46 PM
Para: [hidden email]
Asunto: factor analyzing the Tetrachoric correlations

Dear list,



I am factor analyzing a set of 42 dichotomously scored test items (in
order to understand test dimensionality).

I calculated the Tetrachoric correlations (I can share the syntax if
anyone's interested); so my input file is a correlation matrix. My
question is:

Is there a particular type of rotation that works "better" with this
type of data? Or the same good-old rules apply here too?



Best



Enis

[Richard Ristow]

At 08:45 PM 7/31/2006, Dogan, Enis wrote:

>I am factor analyzing a set of 42 dichotomously scored test items. I
>calculated the Tetrachoric correlations; so my input file is a
>correlation matrix.

Hector has replied very succinctly. Scanning, I found a more extended
note he'd written on the same subject. It seems excellent, and I'm
reproducing it below. Hector, I hope you don't mind.

I add one rough-and-ready point. It is not visible in the correlation
matrix, but a dichotomous observation always conveys less information
than does a continuous one. I can't think of a way to quantify how much
less. It depends on factors including the relative proportion of 0's
and 1's in the dichotomous variable, and the signal-to-noise ratio of
the continuous one. I'll throw out, off the cuff, that a dichotomous
variable might be about 1/5 as informative as a continous one.

Pending the opinions of those much better qualified (Hector, Marta,
...), be very careful of your sample size. I don't know the recommended
ratio of cases to variables for FA, but whatever it is, it must be
increased for FA of dichotomous variables. Quintuple? Maybe.


But, Hector's earlier response: At 01:24 PM 12/15/2004, Hector Maletta
wrote (subject "Re: factor analysis of tetrachloric correlations"):

>As I see it, the selection of a method for extraction or rotation is
>not related to the kind of correlation you have. In fact, so called
>tetrachoric correlations are equivalent to Pearson product-moment
>correlations for binary variables coded 0,1. The choice of extraction
>and rotation depends on the purpose of your analysis and the
>underlying theoretical model. For instance, the Principal Components
>extraction method would try to locate a major underlying factor
>explaining as much commolatity as possible, while the Principal Axis
>method would look for several possible independent (orthogonal)
>factors explaining different parts of the commonalities.
>
>Historically, the PC method was developed by Spearmann when looking
>for a General Intelligence factor underlying all intelligence tests,
>while the PA method was introduced by Thurstone to prove his point
>that there are several dimensions of intelligence and not just one. As
>it turns out, factors identified by factor analysis are but analytical
>constructs, not real objects, and by careful choice of method you can
>obtain a variety of solutions that reduce the dimensionality of your
>variable-space to a factor space defined by one or many factors,
>orthogonal or correlated, without proving anything about the validity
>of each solution: it only proves that various different mathematical
>models are consistent with the data.

[Dale Glaser]

..and to add to what Richard said, I don't have the Mplus manual in front of me, but there is a different WLS-type of estimator that Mplus uses for binary factor analysis that does not correspond to SPSS...I ran a binary EFA a few weeks ago in both Mplus and SPSS (importing the tetrachorics) and there was quite a difference in the magnitude of the loadings.....I am assuming that is due to the estimator involved.........

  Dale

Richard Ristow <[hidden email]> wrote:
  At 08:45 PM 7/31/2006, Dogan, Enis wrote:

>I am factor analyzing a set of 42 dichotomously scored test items. I
>calculated the Tetrachoric correlations; so my input file is a
>correlation matrix.

Hector has replied very succinctly. Scanning, I found a more extended
note he'd written on the same subject. It seems excellent, and I'm
reproducing it below. Hector, I hope you don't mind.

I add one rough-and-ready point. It is not visible in the correlation
matrix, but a dichotomous observation always conveys less information
than does a continuous one. I can't think of a way to quantify how much
less. It depends on factors including the relative proportion of 0's
and 1's in the dichotomous variable, and the signal-to-noise ratio of
the continuous one. I'll throw out, off the cuff, that a dichotomous
variable might be about 1/5 as informative as a continous one.

Pending the opinions of those much better qualified (Hector, Marta,
...), be very careful of your sample size. I don't know the recommended
ratio of cases to variables for FA, but whatever it is, it must be
increased for FA of dichotomous variables. Quintuple? Maybe.


But, Hector's earlier response: At 01:24 PM 12/15/2004, Hector Maletta
wrote (subject "Re: factor analysis of tetrachloric correlations"):

>As I see it, the selection of a method for extraction or rotation is
>not related to the kind of correlation you have. In fact, so called
>tetrachoric correlations are equivalent to Pearson product-moment
>correlations for binary variables coded 0,1. The choice of extraction
>and rotation depends on the purpose of your analysis and the
>underlying theoretical model. For instance, the Principal Components
>extraction method would try to locate a major underlying factor
>explaining as much commolatity as possible, while the Principal Axis
>method would look for several possible independent (orthogonal)
>factors explaining different parts of the commonalities.
>
>Historically, the PC method was developed by Spearmann when looking
>for a General Intelligence factor underlying all intelligence tests,
>while the PA method was introduced by Thurstone to prove his point
>that there are several dimensions of intelligence and not just one. As
>it turns out, factors identified by factor analysis are but analytical
>constructs, not real objects, and by careful choice of method you can
>obtain a variety of solutions that reduce the dimensionality of your
>variable-space to a factor space defined by one or many factors,
>orthogonal or correlated, without proving anything about the validity
>of each solution: it only proves that various different mathematical
>models are consistent with the data.

Dale Glaser, Ph.D.
Principal--Glaser Consulting
Lecturer--SDSU/USD/CSUSM/AIU
4003 Goldfinch St, Suite G
San Diego, CA 92103
phone: [phone]
fax: [phone]
email: [hidden email]
website: www.glaserconsult.com

[Anthony Babinec]

Here are a few points:

For dichotomous indicators, the tetrachoric correlation
is NOT equivalent to the pearson correlation computed on
the dichotomies.

The matrix of tetrachoric correlations is not necessarily
positive definite, so situations can arise in FACTOR where
you get those error messages about negative eigenvalues
and noninvertibility.

There's a good current paper by Sara Finney and Christine
DiStefano entitled "Nonnormal and Categorical Data
in Structural Equation Modeling" that reviews current
approaches in software and makes recommendations. Factor analysis
and regression are of course subsumed in SEM. The paper is in
"Structural Equation Modeling: A Second Course" edited
by Gregory Hancock and Ralph Mueller.

-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
Dale Glaser
Sent: Wednesday, August 02, 2006 9:23 PM
To: [hidden email]
Subject: Re: factor analyzing the Tetrachoric correlations

..and to add to what Richard said, I don't have the Mplus manual in front of
me, but there is a different WLS-type of estimator that Mplus uses for
binary factor analysis that does not correspond to SPSS...I ran a binary EFA
a few weeks ago in both Mplus and SPSS (importing the tetrachorics) and
there was quite a difference in the magnitude of the loadings.....I am
assuming that is due to the estimator involved.........

  Dale

Richard Ristow <[hidden email]> wrote:
  At 08:45 PM 7/31/2006, Dogan, Enis wrote:

>I am factor analyzing a set of 42 dichotomously scored test items. I
>calculated the Tetrachoric correlations; so my input file is a
>correlation matrix.

Hector has replied very succinctly. Scanning, I found a more extended
note he'd written on the same subject. It seems excellent, and I'm
reproducing it below. Hector, I hope you don't mind.

I add one rough-and-ready point. It is not visible in the correlation
matrix, but a dichotomous observation always conveys less information
than does a continuous one. I can't think of a way to quantify how much
less. It depends on factors including the relative proportion of 0's
and 1's in the dichotomous variable, and the signal-to-noise ratio of
the continuous one. I'll throw out, off the cuff, that a dichotomous
variable might be about 1/5 as informative as a continous one.

Pending the opinions of those much better qualified (Hector, Marta,
...), be very careful of your sample size. I don't know the recommended
ratio of cases to variables for FA, but whatever it is, it must be
increased for FA of dichotomous variables. Quintuple? Maybe.


But, Hector's earlier response: At 01:24 PM 12/15/2004, Hector Maletta
wrote (subject "Re: factor analysis of tetrachloric correlations"):

>As I see it, the selection of a method for extraction or rotation is
>not related to the kind of correlation you have. In fact, so called
>tetrachoric correlations are equivalent to Pearson product-moment
>correlations for binary variables coded 0,1. The choice of extraction
>and rotation depends on the purpose of your analysis and the
>underlying theoretical model. For instance, the Principal Components
>extraction method would try to locate a major underlying factor
>explaining as much commolatity as possible, while the Principal Axis
>method would look for several possible independent (orthogonal)
>factors explaining different parts of the commonalities.
>
>Historically, the PC method was developed by Spearmann when looking
>for a General Intelligence factor underlying all intelligence tests,
>while the PA method was introduced by Thurstone to prove his point
>that there are several dimensions of intelligence and not just one. As
>it turns out, factors identified by factor analysis are but analytical
>constructs, not real objects, and by careful choice of method you can
>obtain a variety of solutions that reduce the dimensionality of your
>variable-space to a factor space defined by one or many factors,
>orthogonal or correlated, without proving anything about the validity
>of each solution: it only proves that various different mathematical
>models are consistent with the data.

Dale Glaser, Ph.D.
Principal--Glaser Consulting
Lecturer--SDSU/USD/CSUSM/AIU
4003 Goldfinch St, Suite G
San Diego, CA 92103
phone: [phone]
fax: [phone]
email: [hidden email]
website: www.glaserconsult.com

[Hector Maletta]

I thought factor analysis can proceed if some of the correlations are
negative.
Hector
-----Mensaje original-----
De: SPSSX(r) Discussion [mailto:[hidden email]] En nombre de
Anthony Babinec
Enviado el: Thursday, August 03, 2006 11:24 AM
Para: [hidden email]
Asunto: Re: factor analyzing the Tetrachoric correlations

Here are a few points:

For dichotomous indicators, the tetrachoric correlation
is NOT equivalent to the pearson correlation computed on
the dichotomies.

The matrix of tetrachoric correlations is not necessarily
positive definite, so situations can arise in FACTOR where
you get those error messages about negative eigenvalues
and noninvertibility.

There's a good current paper by Sara Finney and Christine
DiStefano entitled "Nonnormal and Categorical Data
in Structural Equation Modeling" that reviews current
approaches in software and makes recommendations. Factor analysis
and regression are of course subsumed in SEM. The paper is in
"Structural Equation Modeling: A Second Course" edited
by Gregory Hancock and Ralph Mueller.

-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
Dale Glaser
Sent: Wednesday, August 02, 2006 9:23 PM
To: [hidden email]
Subject: Re: factor analyzing the Tetrachoric correlations

..and to add to what Richard said, I don't have the Mplus manual in front of
me, but there is a different WLS-type of estimator that Mplus uses for
binary factor analysis that does not correspond to SPSS...I ran a binary EFA
a few weeks ago in both Mplus and SPSS (importing the tetrachorics) and
there was quite a difference in the magnitude of the loadings.....I am
assuming that is due to the estimator involved.........

  Dale

Richard Ristow <[hidden email]> wrote:
  At 08:45 PM 7/31/2006, Dogan, Enis wrote:

>I am factor analyzing a set of 42 dichotomously scored test items. I
>calculated the Tetrachoric correlations; so my input file is a
>correlation matrix.

Hector has replied very succinctly. Scanning, I found a more extended
note he'd written on the same subject. It seems excellent, and I'm
reproducing it below. Hector, I hope you don't mind.

I add one rough-and-ready point. It is not visible in the correlation
matrix, but a dichotomous observation always conveys less information
than does a continuous one. I can't think of a way to quantify how much
less. It depends on factors including the relative proportion of 0's
and 1's in the dichotomous variable, and the signal-to-noise ratio of
the continuous one. I'll throw out, off the cuff, that a dichotomous
variable might be about 1/5 as informative as a continous one.

Pending the opinions of those much better qualified (Hector, Marta,
...), be very careful of your sample size. I don't know the recommended
ratio of cases to variables for FA, but whatever it is, it must be
increased for FA of dichotomous variables. Quintuple? Maybe.


But, Hector's earlier response: At 01:24 PM 12/15/2004, Hector Maletta
wrote (subject "Re: factor analysis of tetrachloric correlations"):

>As I see it, the selection of a method for extraction or rotation is
>not related to the kind of correlation you have. In fact, so called
>tetrachoric correlations are equivalent to Pearson product-moment
>correlations for binary variables coded 0,1. The choice of extraction
>and rotation depends on the purpose of your analysis and the
>underlying theoretical model. For instance, the Principal Components
>extraction method would try to locate a major underlying factor
>explaining as much commolatity as possible, while the Principal Axis
>method would look for several possible independent (orthogonal)
>factors explaining different parts of the commonalities.
>
>Historically, the PC method was developed by Spearmann when looking
>for a General Intelligence factor underlying all intelligence tests,
>while the PA method was introduced by Thurstone to prove his point
>that there are several dimensions of intelligence and not just one. As
>it turns out, factors identified by factor analysis are but analytical
>constructs, not real objects, and by careful choice of method you can
>obtain a variety of solutions that reduce the dimensionality of your
>variable-space to a factor space defined by one or many factors,
>orthogonal or correlated, without proving anything about the validity
>of each solution: it only proves that various different mathematical
>models are consistent with the data.

Dale Glaser, Ph.D.
Principal--Glaser Consulting
Lecturer--SDSU/USD/CSUSM/AIU
4003 Goldfinch St, Suite G
San Diego, CA 92103
phone: [phone]
fax: [phone]
email: [hidden email]
website: www.glaserconsult.com