Dr. Alan Reifman's SEM Course

2012-01-18T21:44:00.001-08:00

Welcome to the Spring 2012 semester. Links to my lecture notes, organized by topic, are available here.

2011-04-06T13:26:00.000-07:00

Here's the announcement for this year's musical!

We'll have a new song or two, plus we'll be singing some "oldies" from SEM the Musical 1, 2, 3, and 4 (just click directly on the numbers to access previous years' lyrics).

SEM Musical FIVE!
Lyrics by Alan Reifman (retread from last year)
(May be sung to the tune of “Let’s Get it Started,” Will Adams et al. for the Black Eyed Peas)

(Softly) The models keep runnin-runnin, and runnin-runnin, and runnin-runnin, and runnin-runnin, and runnin-runnin, and runnin-runnin, and runnin-runnin, and runnin-runnin, and...

We’re back again, to have some fun,
We’re gonna bust some rhyme, have a good time,
We’re gonna sing some songs, about SEM technique,
Access your inner geek, let your voices speak,
SEM is different, your measurement model’s explicit,
The whole model, gets tested for fit,
Is it identified? We know how hard you’ve tried,
Knowns and unknowns, side by side,
It takes you on a ride, finally you’re satisfied,
Your output’s now just fine, you’ve arrived, you can take pride…

NFI, TLI, CFI,
Calculate estimates, let it run, have some fun, yeah…
SEM Musical (FIVE!), SEM Musical (HERE!),
SEM Musical (FIVE!), SEM Musical (HERE!),
SEM Musical (FIVE!), SEM Musical (HERE!),
SEM Musical (FIVE!), SEM Musical (HERE!),
Yeah,

Build your constructs, get this straight,
Make sure the indicators, correlate,
Draw your pathways, residuals too,
Don’t leave out, the fixed 1 value,
Take your time, think it through,
Don’t worry if you’re new, we’ll walk with you,
Step by step, right up the pyramid,
For SEM, we’re really groovin,’
Hope you get an acceptable solution,
Submit your model and get it movin,’

NFI, TLI, CFI,
Calculate estimates, let it run, have some fun, yeah…
SEM Musical (FIVE!), SEM Musical (HERE!),
SEM Musical (FIVE!), SEM Musical (HERE!),
SEM Musical (FIVE!), SEM Musical (HERE!),
SEM Musical (FIVE!), SEM Musical (HERE!),
Yeah…

SEM Pyramid of Success (explanation)
Lyrics by Andrea Swenson
(May be sung to the tune of "Seasons of Love," Jonathan Larson, from the musical "Rent")

(Long opening on piano, about 40 seconds)

One hundred, thirty-nine thousand, two hundred seconds,
One hundred, thirty-nine thousand, moments to learn,
One hundred, thirty-nine thousand, two hundred seconds,
That is, how long, we sit in this class,*

It starts with, correlation,
Regression, and path an-al-y-sis,
E-F-A, builds into,
C-F-A, in time,

One hundred, thirty-nine thousand, two hundred seconds,
How, do you start? And, where do you go?

To get to, S.....E.....M.....
To get to, S.....E.....M.....
To get to, S.....E.....M.....
Measure it well....

Pyramid of.... (slow) success,
Pyramid of.... (slow) success,

One hundred, thirty-nine thousand, two hundred seconds,
One hundred, thirty-nine thousand, moments to learn,
One hundred, thirty-nine thousand, two hundred seconds,
That is, how long, we sit in this class,

Starting with, correlation,
Moving up to, regression,
In exploring, factors,
And confirming them,

It’s time now, to remember,
To bring it all together,
Let's, bring it all together, to do SEM,

Remember the pyramid (Oh you got to you got to remember the pyramid)
Remember the pyramid (You know that SEM is a starts from the r)
Remember the pyramid (regress, factor, SEM)
Assess the model (Learn, learn SEM)

Pyramid of success
Pyramid of success (that’s how we learn, learn SEM)

---
*Number of seconds in the class, based on 29 periods of 80 minutes each.

Hey, Hey, Heywood Cases
Lyrics by Nora "Felix" Phillips
(May be sung to the theme from "The Monkees," Boyce/Hart)

Let it run, the computations go through,
You get an error message, it leaves you feeling blue,

Hey Hey Heywood Cases!
Bringing, my AMOS, model down,
With your, negative variance,
You know that, isn't allowed,

Mis-specification,
Of the model, that you've drawn,
Or maybe, your own sample,
Was just, a tad bit, too small?

Hey Hey Heywood Cases!
I won't let you bring me down,
I can constrain, residuals,
To a, small positive, amount!

---

Nestedness
Lyrics by Alan Reifman
May be sung to the tune of “Yesterday” (Lennon/McCartney)

Nestedness,
It’s the way, models can be compared,
Should new paths be added in or spared?
The delta-test needs nestedness,

Can’t you see?
One model might have simplicity,
But more paths increase fidelity,
Which one to choose, the chi-square’s key,

Inside, the big one, the small one, is self-contained,
One has, extra paths, the other, does not maintain...

Nestedness,
To the baseline, you can only add,
Or only subtract, paths you once had,
You can’t do both, for nestedness,

Inside, the big one, the small one, is self-contained,
One has, extra paths, the other, does not maintain...

Look, shall we?
One model could have, paths “A” and “B,”
They would nest in, model “A/B/C,”
A/B’s contained, in A/B/C…

Maximum Likelihood
Lyrics by Alan Reifman
May be sung to the tune of “Pink Houses” (John Mellencamp)

The computer, runs your model, looking for a solution,
It seeks to maximize, or maybe minimize,
Some function, seen in, a distribution,

You have least squares, which tries to put, the best-fit line near the dots,
But ML, seeks equations, so your findings, will come out on top,

Oh, maximum likelihood, that’s what we use,
Maximum likelihood, it tends to confuse,
Maximum likelihood, underlying values, that make your results, most probable,
And that’s, big news!

Sir Ronald Fisher, statistician,
Developed the, ML perspective,
It will iterate, till it’s really great,
But it’s so, calculation intensive,

For a long time, ML sat there,
Its steps were, so hard to reckon,
But computers, came along, and sped things up,
And now ML, runs in mere seconds,

Oh, maximum likelihood, that’s what we use,
Maximum likelihood, it tends to confuse,
Maximum likelihood, underlying values, that make your results, most probable,
And that’s, big news!

Instrumental

Well there are data, and more data,
What do they show?
With its complex math, on a tricky path,
ML tells you, what you, need to know,

Oh yeah,

Well some data, might be missing,
But there’s no need, for frustration,
’Cause you can, estimate the means, and intercepts,
To get ML, with full, information,

Oh, maximum likelihood, that’s what we use,
Maximum likelihood, tends to confuse,
Maximum likelihood, underlying values, that make your results, most probable,
And that’s, big news!

2011-01-10T21:38:00.000-08:00

Welcome to the Spring 2011 semester. With all of my scattered postings on this blog over the past few years, the information is not that well organized. Below, I have attempted to organize the material in outline form, with links to the relevant postings from previous years.

Intro: The Pyramid of Success

Correlation and Regression (here, here, and here)

Path Analysis (here and here)

Exploratory Factor Analysis

Confirmatory Factor Analysis (here, here, and here)...

...and Associated Basic Concepts (model fit; reporting fit; degrees of freedom [here and here])

Full Structural Models (here, here, and here); also see the following article for discussion of what a "model" represents:

Rodgers, J. L. (2010). The epistemology of mathematical and statistical modelling. A quiet revolution. American Psychologist, 65, 1-12.

Beyond the Basics of SEM (contains all our topics for roughly the second half of the course)

Diagram for Assignment 2

SEM The Musical: 1, 2, 3, 4

Miscellaneous (here)

2010-04-20T10:17:00.000-07:00

Below is a sneak peek at our new songs for this year. We'll also be singing some "oldies" from SEM the Musical 1, 2, and 3 (just click directly on the numbers to access previous years' lyrics).

SEM Musical FOUR!
Lyrics by Alan Reifman
(May be sung to the tune of “Let’s Get it Started,” Will Adams et al. for the Black Eyed Peas)

(Softly) The models keep runnin-runnin, and runnin-runnin, and runnin-runnin, and runnin-runnin, and runnin-runnin, and runnin-runnin, and runnin-runnin, and runnin-runnin, and...

We’re back again, to have some fun,
We’re gonna bust some rhyme, have a good time,
We’re gonna sing some songs, about SEM technique,
Access your inner geek, let your voices speak,
SEM is different, your measurement model’s explicit,
The whole model, gets tested for fit,
Is it identified? We know how hard you’ve tried,
Knowns and unknowns, side by side,
It takes you on a ride, finally you’re satisfied,
Your output’s now just fine, you’ve arrived, you can take pride…

NFI, TLI, CFI,
Calculate estimates, let it run, have some fun, yeah…
SEM Musical (FOUR!), SEM Musical (HERE!),
SEM Musical (FOUR!), SEM Musical (HERE!),
SEM Musical (FOUR!), SEM Musical (HERE!),
SEM Musical (FOUR!), SEM Musical (HERE!),
Yeah,

Build your constructs, get this straight,
Make sure the indicators, correlate,
Draw your pathways, residuals too,
Don’t leave out, the fixed 1 value,
Take your time, think it through,
Don’t worry if you’re new, we’ll walk with you,
Step by step, right up the pyramid,
For SEM, we’re really groovin,’
Hope you get an acceptable solution,
Submit your model and get it movin,’

NFI, TLI, CFI,
Calculate estimates, let it run, have some fun, yeah…
SEM Musical (FOUR!), SEM Musical (HERE!),
SEM Musical (FOUR!), SEM Musical (HERE!),
SEM Musical (FOUR!), SEM Musical (HERE!),
SEM Musical (FOUR!), SEM Musical (HERE!),
Yeah…

Once You Work in AMOS
Lyrics by Alan Reifman
(May be sung to the tune of “Once in Love with Amy,” Frank Loesser)

Once you work, in AMOS,
Find every quirk, in AMOS,
Construct by construct, you can draw your picture,
Using all the gadgets, is fun,

Learn the rules, in AMOS,
Use all the tools, in AMOS,
Circles and boxes, and you can add arrows,
Soon your model’s, ready to run,

The moving truck, the sizer, and the bubble,
Your choices, are vast,
And even if, you find yourself in trouble,
You can fix things fast,

So, once you work, in AMOS,
Find every quirk, in AMOS,
Each time you use it, your skills are expanded,
And you’ll understand, what you see,
Cause, in the end, it’s fixed, or it's free…

Prof. Reifman
Lyrics by Kim Corson and Janis Henderson
(May be sung to the tune of "Fernando," Ulvaeus, Andersson, & Anderson, for ABBA)

Can you hear the songs, Prof. Reifman?
We remember long ago, in intro stats you sang like this,
At the front of class, Prof. Reifman,
You were humming to yourself, and softly strumming air guitar,
We could hear the distant drums,
And suddenly, the answers didn't seem so far,

We’re much closer now, Prof. Reifman,
Every box, every circle, seems to make more sense to us,
We are not afraid, Prof. Reifman,
We sit here so full of life; all of us are prepared to try,
And we're not ashamed to say,
The songs of SEM the Musical 4 helped us get by,

There was something in the air that day,
The fog went away, Prof. Reifman,
He was talking about SEM,
And our heads didn't swim, Prof. Reifman,

Though we never thought that we would grasp, degrees of freedom,
We can calculate them now, in fact, we just subtract, Prof. Reifman,
We can calculate them now, in fact, we just subtract, Prof. Reifman,

When we're old and grey, Prof. Reifman,
And for many years we haven't played in your "rock band,"
We'll still hear the strums, Prof. Reifman,
And we'll recall learning AMOS, like Emeril, can go "Bam!",
And we'll still call point-0-0-0 "Paula Abdul significance,"

There was something in the air that day,
The fog went away, Prof. Reifman,
He was talking about SEM,
And our heads didn't swim, Prof. Reifman,

Though we never thought that we would grasp, under-identification,
We now see it's when a model's flown, with too much unknown, Prof. Reifman,

There was something in the air that day,
The fog went away, Prof. Reifman,
He was talking about SEM,
And our heads didn't swim, Prof. Reifman,

Though we never thought that we would grasp, degrees of freedom,
We can calculate them now, in fact, we just subtract, Prof. Reifman,
We can calculate them now, in fact, we just subtract, Prof. Reifman,

2010-04-08T17:17:00.000-07:00

On the SEMNET discussion listserv around April 3-4, 2010, someone asked about graphics programs for drawing structural-equation-model diagrams, and other participants sent in suggestions. I, personally, find AMOS and PowerPoint to be good. However, if anyone wants to examine additional programs, here are some:

GraphViz

Concept Draw

Concept Map (perhaps more appropriate for illustrating theory construction than actual SEM drawing)

Easy Draw (seems like a very general graphic-arts program)

2010-03-07T17:41:00.000-08:00

Now that we've learned the basics of full structural models, we'll be taking up the following topics in the coming weeks:

Comparative model testing and nestedness

We'll use the following article to delve more deeply into comparative model testing:

Bryant, A. L., Schulenberg, J., Bachman, J. G., O'Malley, P. M., & Johnston, L. D. (2000). Understanding the links among school misbehavior, academic achievement, and cigarette use: A national panel study of adolescents. Prevention Science, 1, 71-87.

Maximum Likelihood Estimation

Phrasing of hypotheses

Equivalent models

Negative variances (Heywood Cases)

Equality constraints (these lecture notes also touch briefly on longitudinal models and multiple-group analyses)

Longitudinal (panel) models

Multiple-Group Modeling (see notes on equality constraints above; Kyle Gillett dissertation in links section to the right)

Dyadic analysis in SEM (Actor-Partner Interdependence Model)

Running an AMOS model off of a published correlation/covariance matrix from the literature

Advanced Applications

Latent Growth Modeling (here and here)

Example: Barnes, G. M., Reifman, A. S., Farrell, M. P., & Dintcheff, B. A. (2000). The effects of parenting on the development of adolescent alcohol misuse: A six-wave latent growth model. Journal of Marriage and the Family, 62, 175-186.

AMOS vs. Mplus

2010-03-04T09:27:00.000-08:00

The following is the model for the new assignment. You will run the model twice, once without the three red-dashed paths and once with them. We will learn about comparative model testing.

Here's a direct link to the figure we recently looked at regarding where variances are located in full structural models, as well as how degrees of freedom are determined in a full structural model.

The model also makes salient the issue of outliers, in particular that Harvard's endowment (and to a lesser extent those of a few other institutions) are so much larger than most others. Harvard and these other elite universities have endowments in the billions, whereas many other schools have endowments well under 1 billion. This document discusses approaches to handling outliers; in the past we've used winsorizing, but this year, we'll use a square-root transformation (already implemented in the data set).

There is some evidence that the depressed economy may be "winsorizing" Harvard's endowment, but this didn't occur early enough to be reflected in the data set.

2010-02-09T09:54:00.000-08:00

Today, we'll start covering exploratory factor analysis (EFA). We'll be drawing from three previous sets of lecture notes:

January 30, 2007 (these notes mainly)

January 22, 2008

February 2, 2007

UPDATE (February 10, 2010):

Here's a graphic display I created to illustrate the difference between using exact factor scores to make subscales vs. unit weighting. I also found a good online article by DiStefano et al. and a thorough PowerPoint show by Wuensch on the subject. You can click on the graphics below to enlarge them (note that there are TWO slides to click on, one on top of the other).

2010-01-28T09:44:00.001-08:00

Today's class is cancelled, per the university announcement of the weather-related shutdown.

2010-01-26T09:57:00.000-08:00

Today, we'll primarily be covering path analysis, after finishing up on the distinction between standardized and unstandardized regression solutions (for both topics, we will draw from my January 2007 blog postings).

By coincidence, Richard Florida, who will be speaking at Texas Tech on Friday night, February 5 (free for students), has used path analysis in his research (here).

2009-04-20T23:22:00.001-07:00

Here are some newly written songs (including one by a student) for SEM The Musical 3, which we'll perform on Wednesday. We'll also do some "oldies" from the first and second annual musicals.

The SEM Way
Lyrics by Alan Reifman
(May be sung to the tune of “Let’s Live for Today,” Mogol/Shapiro/Julien, popularized by the Grass Roots)

You’ve got your sets of measures, some constructs they could form,
Plus, indices of fitness, to compare to a norm,
You draw yourself a model, with circles, squares, and paths,
The AMOS program handles, the complicated math,
If you get too many errors, you can express your wrath,

1-2-3-4

Analyze your work, the SEM way,
Analyze your work, the SEM way,
Don’t forget to, check your RM-SEA,
Analyze your work, the SEM way…

AMOS is Ideal
Lyrics by Susan Murray
(May be sung to the tune of “Jesus, Take the Wheel,” James/Lindsey/Sampson, popularized by Carrie Underwood)

She was working last Friday on her laptop battery,
On homework to achieve,
Her constructs were getting muddy, with her model nowhere near complete,
Fifty specs to go and she was running low on patience and caffeine,

It was complex and unclear,
She had a constraint and the software caused the tension,
But they say SAS is unsurpassed,
Before she knew it she was closing down that darn software SAS,

She saw the latent variables flash before her eyes,
She didn’t even have time to imply,
She was so impaired,
She suddenly was aware,

AMOS is ideal,
Take causation from my hands,
‘Cause I can’t do this on my own,
I’m letting go,

No coding song and dance,
To learn about the population,
AMOS is ideal,

A cross-lagged panel model she pulled out of the folder,
And like Emeril she went BAM! nonstop,
She cried like a baby when she saw the RMSEA drop,
Her hypothesis and all the parameters,
She now could weigh,
She could model all day,
In a Paula Abdul light,
Software to exchange,
Arbuckle already did the fight,

AMOS is ideal,
Take causation from my hands,
‘Cause I can’t do this on my own,
I’m letting go,

No coding song and dance,
SAS I won’t depend upon,
Oh, AMOS is ideal,
SAS, I’m saying no,

No coding song and dance
SAS I won’t depend upon,
My loyalty is withdrawn,
AMOS is ideal,

Oh, don’t you take it from me,
Find my μ

Equal
Lyrics by Alan Reifman
(May be sung to the tune of “Mercy,” Duffy/Booker)

A-A-A, B-B-B, C-C-C, D-D-D

I’ve got two, paths that you can view,
I think their strength might be the same, and that’s the frame,
You’ve got, to see through,

Let’s run the model free, paths can be any sized,
Then run equalized,

We need a way to choose, a way to compare,
Test delta chi-square,

You’ve got me constrained to be equal,
Why won’t you release me?
You’ve got me constrained to be equal,
Why won’t you release me?
Can’t you rele-e-e-e-ase me?

Lower chi-square, will always be there,
When you let the paths go free, but you must see,
True sig-nif-i-cance,

When the chi-square’s non-sig, the one to retain,
Is where you constrain,

But if the change is big, p’s under oh-five,
Free paths shall survive,

You’ve got me constrained to be equal,
Why won’t you release me?
You’ve got me constrained to be equal,
Why won’t you release me?
Can’t you rele-e-e-e-ase me?

It’s Still SEM to Me
Lyrics by Alan Reifman
(May be sung to the tune of “It’s Still Rock and Roll to Me,” Billy Joel)

It’s a way to show, inter-relations,
In a set of, latent constructs,
It gives you, some global fit statistics,
Does your model, really stack up?
Squares and circles, now you’re off and you’re running,
Will your results, be routine or be stunning?

LISREL, M-PLUS, EQS, or AMOS,
It’s still SEM to me…..

2009-04-17T10:49:00.000-07:00

Yoona Chin, a newly minted Ph.D. recipient from our department, will be giving a guest presentation today on differences between AMOS, the program we've been using in class, and Mplus, an increasingly popular program. She's put together a very elaborate PowerPoint slide show, a few key graphics I wanted to put online (with Yoona's permission). You may click on the following images to enlarge them.

Professor Judy Fischer will also be with us in class today, presenting some recent multiple-group findings from her research.

2009-04-15T10:31:00.000-07:00

Today we will take up the topic of dyadic analysis in SEM, particularly something known as the Actor-Partner Interdependence Model (APIM). We will draw upon the following article, which is available via the TTU library.

Popp, D., Laursen, B., Burk, W. J., Kerr, M., & Stattin, H. (2008). Modeling homophily over time with an Actor-Partner Interdependence Model. Developmental Psychology, 44, 1028-1039.

The notes from my Methods class on unit of analysis may be helpful for this topic.

An important issue is whether the two partners in a dyad are distinguishable (i.e., non-exchangeable), as opposed to being indistinguishable (exchangeable). See David Kenny's webpage on dyadic analysis (particularly Topic 3). As Kenny, Kashy, and Cook (2006) state in their book Dyadic Data Analysis:

When dyad members are distinguishable, we estimate the path model or CFA model for each of the two members combined in a single model... However, when members are indistinguishable, it is less clear exactly how to do the analysis. The use of SEM with indistinguishable or exchangeable dyad members has generally been viewed pessimistically... (p. 111).

A suggested reference in this regard is:

Olsen, J. A., & Kenny, D. A. (2006). Structural equation modeling with interchangeable dyads. Psychological Methods, 11, 127-141.

As another example of an APIM-type model, see Hye-Sun Ro's dissertation in the online collection to the right.

UPDATE (7/28/09): I just found an online PowerPoint presentation that provides a nice introduction to dyadic analysis, including examples with SEM.

2009-03-18T19:03:00.000-07:00

One of our students this semester, Susan Murray, came up with a way to present model fit statistics that Kristina (the TA) and I both thought was very effective. With Susan's permission, here is the tabular format she came up with. The listed criteria for desirable values come from the Garson and Kenny documents in the links section to the right.

2009-03-04T09:38:00.001-08:00

As I alluded to recently, if one wanted to re-analyze a model from the published literature (or propose an entirely new specification of a model), one could directly type a correlation matrix (ideally with standard deviations) from an article into AMOS. In this way, SEM analyses can be done on a data set without having the actual raw data. We'll discuss this in class today. The necessary information can be found in the AMOS program, by going to the "Help" area and looking up: "To reformat a text file of sample moments."

[UPDATE (April 26, 2011): Make sure in your plain-text file with the necessary information to leave no blank lines beneath the last line of syntax. Thanks to "Hermione" for catching that!]

2009-02-13T11:04:00.000-08:00

The original post has been deleted. It pertained to a former course assignment and I want to avoid confusion with the updated form of the assignment.

2009-01-09T13:03:00.000-08:00

To summarize what we did today and provide easy access to the links, see the following:

1. We introduced the SEM Pyramid of Success (link).

2. We reviewed the Pearson correlation coefficient (link). We also looked at a graphical depiction of the least-squares criterion (link).

3. We transitioned from correlation to multiple-regression, with a focus on standardized and unstandardized relationships (link). We also looked at a picture (the second one down) of how an unstandardized regression equation would be depicted (link).

2009-01-07T10:14:00.000-08:00

Welcome to the Spring 2009 offering of QM IV. Over the past few years, I've created and posted a variety of write-ups, graphics, and links, intended to explain and expand on course concepts. I hope you'll find this blog a useful element of the course.

Our starting point will be what I call the SEM "Pyramid of Success."

2008-04-15T20:29:00.000-07:00

On Wednesday, April 23, we will present SEM The Musical 2. [Update: It's now been presented.] I've written some new songs, as has one of our students (shown below), plus we'll perform some "classics" from last year's musical.

It Do Run Run
Lyrics by Alan Reifman
(May be sung to the tune of “Da Doo Run Run,” Spector/Greenwich/Barry)

Got to draw a model that is error-free,
So it will run, run, so it will run,
Got to have constraints where they’re supposed to be,
So it will run, run, so it will run,

Oh, I got a Heywood Case,
Variance, I must replace,
Everything, is back on pace,
It do run, run, run; it do run, run,

Making sure my model is identified,
So it will run, run, so it will run,
Making sure conditions are all satisfied,
So it will run, run, so it will run,

Yeah, it runs so well,
The fit indices are swell,
No problems on which to dwell,
It do run, run, run; it do run, run...

And now, three songs about enumerating your degrees of freedom and the related issue of model identification.

Count ’Em Up
Lyrics by Alan Reifman
(May be sung to the tune of “Build Me Up Buttercup,” d'Abo/Macaulay, for The Foundations)

You’ve got to count ’em up (count ’em up), degrees of freedom,
So you’ll understand (understand), the model at hand,
And when you compare (you compare), two nested models,
When you add some paths (add some paths), to your arrow graphs,
You’ll know how (you’ll know how), to conduct the delta test,
And decide which model you’ll seize,
So count ’em up (count ’em up), all of your freedom’s degrees,

The measures in your trove, their variances and cov’s,
Are in a half-matrix, they’re your known elements,
From these you subtract, parameters you enact,
The model that you state, and freely estimate,

(Hey, hey, hey!) The df’s, are the difference,
(Hey, hey, hey!) Start out with known elements,
(Hey, hey, hey!) Then deduct,
The free parameters, and now it all makes sense,

You’ve got to count ’em up (count ’em up), degrees of freedom,
So you’ll understand (understand), the model at hand,
And when you compare (you compare), two nested models,
When you add some paths (add some paths), to your arrow graphs,
You’ll know how (you’ll know how), to conduct the delta test,
And decide which model you’ll seize,
So count ’em up (count ’em up), all of your freedom’s degrees…

D-of-F in SEM
Lyrics by Shera Jackson
(May be sung to the tune of "Flowers on the Wall," Lew DeWitt for the Statler Brothers)

I keep hearing about counting degrees of freedom for SEM,
Trying to keep it all straight is hard to do,
If I were a statistician, I wouldn’t worry none,
As I’m adding this up, I’m starting to have fun,

Counting degrees of freedom for SEM,
That don’t bother me at all,
Counting up the elements,
And now the parameters,
I’m adding all the knowns and subtracting the unknowns,
Now, don’t tell me I’ve nothing to do,

Last night I made a matrix, found the diagonal,
That’s my variances, and underneath are my co-v’s,
Please, don’t forget the means when using “means and intercepts,”
Square the elements, subtract them, divide by 2, and add them back,

Counting degrees of freedom for SEM,
That don’t bother me at all,
Counting up the elements,
And now the parameters,
I’m adding all the knowns and subtracting the unknowns,
Now, don’t tell me I’ve nothing to do,

Well, let’s count the unknowns, so many, free factor loadings,
Structural Paths, non-directional correlations,
Indicator residual variances, and
Construct residual variances ,and construct variances,

Counting degrees of freedom for SEM,
That don’t bother me at all,
Counting up the elements,
And now the parameters,
I’m adding all the knowns and subtracting the unknowns,
Now, don’t tell me I’ve nothing to do,

Now , counting degrees of freedom for SEM,
That don’t bother me at all,
Counting up the elements,
And now the parameters,
I’m adding all the knowns and subtracting the unknowns,
Now, don’t tell me I’ve nothing to do,

Don’t tell me I’ve nothing to do...

Over-identified
Lyrics by Alan Reifman
(May be sung to the tune of “Overjoyed,” Stevie Wonder)

To get your, structural diagram, to run fine,
Elements, and your parameters, must align, oh,
If you ask too much, your model will crash,
Plan it carefully, don’t let your choices be rash,

You cannot, have unknowns that number, more than knowns,
Negative, your degrees of freedom, cannot go, yeah,
You can use constraints, so free paths reduce,
Without more measures, that’s all you can do,

Under-iden-tified will not run,
What have you done?
You’ve posited,
More than known in-for-ma-tion,
To make sure that all is satisfied,
It has to be,
Overall, over-iii-dentified…

If you draw, all the curves and arrows, that you can,
You will have, mandated perfect fit, on your hands, oh,
If you saturate, you can’t judge the fit,
It’s always perfect, that’s automatic,

Just-i-den-ti-fied fit, will be one,
What have you done?
You’ve drawn all paths,
There could be under the sun,
To make sure that all is satisfied,
It has to be,
Overall, over-iii-dentified…

Chi-Square Rising
Lyrics by Alan Reifman
(May be sung to the tune of “Bad Moon Rising,” John Fogerty)

I see a chi-square rising,
I see the fit going astray,
You need to add more parameters,
What could be another pathway?

Parsimony’s nice, bad fit could be a price,
There’s a chi-square on the rise,

I see a chi-square rising,
I see there’s more that can be done,
Relations, you need to account for,
Then you can let the model run,

Parsimony’s nice, bad fit could be a price,
There’s a chi-square on the rise…

Curvy, Swervy, Dual-Connected, Correlation Bi-Directed
Lyrics by Alan Reifman
(May be sung to the tune of “Itsy-Bitsy, Teeny-Weeny, Yellow Polka-Dot Bikini,” Vance/Pockriss)

They were unsure whether A tends to precede B,
Or whether B occurs prior to A,
What could they do, to depict this in their model?
What symbolic notation could they portray?

It's not too late,
A and B could correlate,

They drew a curvy, swervy, dual-connected, correlation bi-directed,
That goes right in between A and B,
A curvy, swervy, dual-connected, correlation bi-directed,
So no one had to state causality...

Three Wave
Lyrics by Alan Reifman
(May be sung to the tune of “Heat Wave,” Holland/Dozier/Holland, popularized by Martha Reeves and the Vandellas)

You’re doing a survey,
Of how people change,
A trio of interviews,
With each person, you’ll arrange,

Autoregressive, and cross-lagged paths,
Equality constraints on the math,

You’ve got a three-wave,
Panel study design (three-wave!),
Not quite causation (three-wave!),
But, precedence of time…

Example of a three-wave panel model.

2008-03-31T12:16:00.000-07:00

This week, we'll learn about equality constraints. An equality constraint tells the SEM computer program that, in reaching its solution, it must provide the identical unstandardized coefficient for all parameters within a set that has been designated for equality (even when equality constraints have been imposed, standardized coefficients may not be exactly the same within the constrained set).

Designation of equality is done with the AMOS program via letters. For a given parameter, you can go to the "Object Properties" box and, for the parameter value, you can pre-specify a letter such as A, B, C, etc. Any two (or more) parameters to which you assign an A will all become constrained to take on the same unstandardized value; any two (or more) parameters to which you assign a B will be become constrained to take on identical unstandardized values, etc. Only parameters with the same letter will take on identical values. In other words, the uniform coefficient taken on by the set of A constraints will (almost certainly) be different from the uniform coefficient taken on by the B set.

Suppose that equality constraints are placed on the structural paths from two predictor constructs to an outcome construct. In the absence of constraints, one predictor might, for example, take on an unstandardized coefficient of .40 and the other predictor might take on a value of .30. Because of the constraints, however, the values must be identical, so both paths might take on a value of .35 (I don't know that the solution must always "split the difference," but it's probably a reasonable way to think about it).

Constraining the two (or more) values to equality, of necessity, harms model fit; the mathematically optimal MLE paths in the above example would have been .40 and .30. Giving the two paths the identical .35 is thus suboptimal mathematically, but it provides greater parsimony because we can say that a single value (.35) works reasonably well for both paths.

Equality constraints involve comparative model-testing and delta-chi square tests, as we've seen before. In this new context, what we do is run the model twice, once without constraints and once with (this is considered "nested"). If the delta-chi square test (with delta df) is significant, we say the constraints significantly harm model fit and we ditch them. If the rise in chi-square due to the constraints is not significant, we then retain the constraints in the name of parsimony.

Equality constraints have at least four purposes, as far as I can tell:

1. Theory/hypothesis testing. The following example, from one of my older articles, involves trying to test if three adolescent suicidal behaviors -- thoughts, communication, and attempts -- are three gradations on the same underlying dimension or are more qualitatively different. We reasoned that, if the behaviors were gradations along the same dimension, then each psychosocial predictor should relate equivalently to the three suicidal behaviors. If, on the other hand, the suicidal behaviors were qualitatively different, then a given predictor might relate significantly to one of them, but not all three.

2. Playing the "Devil's Advocate."

We will look at the example (near Figure 3) in the following article, which we can access online via the TTU Library.

Breckler, S.J. (1990). Applications of covariance structure modeling in psychology: Cause for concern? Psychological Bulletin, 107, 260-273.

3. Longitudinal/panel models.

We will go over several examples, including some from:

Farrell, A.D. (1994) Structural equation modeling with longitudinal data: Strategies for examining group differences and reciprocal relationships. Journal of Consulting and Clinical Psychology, 62, 477-487.

As the title notes, this article is also good for studying multiple-group modeling...

4. Multiple-group analyses. Here's a graphic I made to illustrate the use of equality constraints in this context.

2008-03-07T11:55:00.000-08:00

We'll next consider what is known as the "equivalent models" problem, and other cautions and limitations of SEM. In doing so, we will look at two articles that are available via the TTU library's website:

MacCallum, R.C., Wegener, D.T., Uchino, B.N., & Fabrigar, L.R. (1993). The problem of equivalent models in applications of covariance structure analysis. Psychological Bulletin, 114, 185-199.

(We'll also have a song about the equivalent models problem, entitled, "Your Model's Only One.")

Tomarken, A.J., & Waller, N.G. (2003). Potential problems with "well fitting" models. Journal of Abnormal Psychology, 112, 578-598.

2008-03-05T11:27:00.000-08:00

Today, I'd like to cover interpretive clarity in writing about your hypotheses and results. Many beginning writers on SEM simply restate the numerical information from their output, tables, and figures, without providing substantive interpretations.

Earl Babbie's textbook, The Practice of Social Research (2007, 11th ed.) contains a guest essay by Riley E. Dunlap, entitled "Hints for Stating Hypotheses" (p. 47). Here is an excerpt of what I believe is the key advice:

The key is to word the hypothesis carefully so that the prediction it makes is quite clear to you as well as others. If you use age, note that saying "Age is related to attitudes toward women's liberation" does not say precisely how you think the two are related... You have two options:"

1. "Age is related to attitudes toward women's liberation, with younger adults being more supportive than older adults"...

2. "Age is negatively related to support for women's liberation"...

As Dunlap demonstrates, these two statements of the hypothesis let the reader know with specificity which people are expected to hold which type of attitudes (I have added the color and bold emphases above).

Results should be described similarly -- not just that a standardized regression path between Construct A and Construct B was .46, but that (given the positively signed relationship) the more respondents do whatever is embodied in Construct A, the more they also do what is embodied in Construct B.

One of my SEM-based publications from several years ago, which is accessible on TTU computers via Google Scholar, can serve as a guide.

Thomas, G., Reifman, A., Barnes, G.M., & Farrell, M.P. (2000). Delayed onset of drunkenness as a protective factor for adolescent alcohol misuse and sexual risk-taking: A longitudinal study. Deviant Behavior, 21, 181-210.

2008-02-29T09:53:00.000-08:00

Today, let's take some time to talk about Maximum Likelihood Estimation (MLE), which is the default estimation procedure in AMOS and is considered the standard for the field. In my view, MLE is not as intuitively graspable as Ordinary Least Squares (OLS) estimation, which simply seeks to locate the best-fitting line in a scatter plot of data so that the line is as close to as many of the data points as possible. In other words, OLS minimizes the squared deviation scores between each actual data point and where an individual with a given score on the X-axis would fall on the best-fitting line, hence "least squares." However, Maximum Likelihood is considered to be statistically advantageous.

This site provides what I think is a very clear, straightforward introduction to MLE. In particular, we'll want to look at the second major heading on the page that comes up, Model-Fitting.

This other site lists some of the advantages of MLE, vis-a-vis OLS.

Lindsay Reed, our computer lab director, once loaned me a book on the history of statistics, the unusually titled, The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century (by David Salsburg, published in 2001).

This book discusses the many statistical contributions of Sir Ronald A. Fisher, among which is MLE. Writes Salsburg:

In spite of Fisher's ingenuity, the majority of situations presented intractable mathematics to the potential user of the MLE (p. 68).

Practically speaking, obtaining MLE solutions required repeated iterations, which was very difficult to achieve, until the computer revolution. Citing the ancient mathematician Robert Recorde, Salsburg writes:

...you first guess the answer and apply it to the problem. There will be a discrepancy between the result of using this guess and the result you want. You take that discrepancy and use it to produce a better guess... For Fisher's maximum likelihood, it might take thousands or even millions of iterations before you get a good answer... What are a mere million iterations to a patient computer? (p. 70).

2008-02-11T15:01:00.000-08:00

Below is the photograph Kristina took of the board, with the derivation of degrees of freedom for the Hendrick & Hendrick Love Styles model. You can click on the image to enlarge it, if you'd like.

One of the students in the class, noting the repeated references to "knowns" and "unknowns" in running the model, sent me this video link to provide some levity.

2008-01-30T18:14:00.001-08:00

During class earlier today, I encouraged everyone to think of a latent construct (such as the CONSERVATIVISM construct in this entry) as the shared variation (or correlatedness) between the manifest indicators. My reasoning, loosely stated, followed these three steps.

1. The standardized factor loadings are based upon the correlations between any two manifest indicators. For example, if one indicator has a standardized loading of .70 and another has a loading of .80, the Pearson correlation between the two indicators will be .56 or thereabouts (i.e., the product of the two loadings). High loadings go along with high correlations.

2. High loadings, which are considered desirable for having a strong factor, thus signify correlatedness (or shared variation) among the indicators.

3. Taking a little leap from step 2, one can think of the factor itself as representing shared variation among its indicators. The "tiny bubbles" pointing to each manifest indicator thus represent variation in a given indicator that is not due to the common factor.

2008-01-22T14:56:00.001-08:00

In tomorrow's class, we'll be covering exploratory factor analysis (EFA). There's a classic set of procedures for executing EFA, called "Little Jiffy," presumably because it works quickly. Many statisticians now advocate using procedures other than "Little Jiffy," however. In recognition of the controversy, I present you (below) with my first new SEM song of 2008 (plus some recent suggested readings, which we can access through the TTU Library site). The remaining content comes from last year's lecture notes on EFA.

Little Jiffy
Lyrics by Alan Reifman
(May be sung to the tune of “Desperado,” Frey/Henley)

“Little Jiffy,” you know your status is iffy,
Some top statisticians, think that you’re no good,
You are so simple, the users just take the defaults,
And thus halts the process, of finding structure,

(Bridge)
You are a three-stage, procedure,
For making, your data concise,
And upon some simple guidelines, you do rest,

But for each step, in the routine,
Little Jiffy’s not so precise,
And the experts say, other choices are best…

For extraction, you use Principal Components,
While all your opponents, advocate P-A-F,
You use Kaiser’s test, to tell the number of factors,
While all your detractors, support the Scree test,

On your behalf, some researchers claim,
Components and factors, yield almost the same,
But computers give, several more options today,
With “Little Jiffy,” you don’t have to stay,
You can experiment, with different ways…

Varimax is used, to implement your rotation,
There’s no correlation, among your axes,
If one goes oblique, like critics urge that you ought to,
The items you brought, ooh, will fall close to the lines…

---

Conway, J.M., & Huffcutt, A.I. (2003). A review and evaluation of exploratory factor analysis practices in organizational research. Organizational Research Methods, 6, 147-168.

Henson, R.K., & Roberts, J.K. (2006). Use of exploratory factor analysis in published research: Common errors and some comment on improved practice. Educational and Psychological Measurement, 66, 393-416.

---

UPDATE: During Wednesday's class, I said I'd make a PowerPoint graphic of my idea (which I had drawn on the board) that rotating axes in factor analysis was analogous to rotating the streets (or laying down new streets) to make them closer to people's houses. Here it is...

2008-01-09T11:23:00.001-08:00

Welcome back for the new semester and our Quantitative Methods IV (Structural Equation Modeling) course! This is the sixth time I've taught the course at Texas Tech and the second year with this blog. I wrote a lot of detailed entries last year, with my patented technicolor charts, so we'll be referring back to them a lot. I'll also write new entries as the need arises.

2007-04-30T10:07:00.000-07:00

Last Friday, April 27, we put on the debut performance of "SEM The Musical." We thank the former students of the course, faculty colleagues, and others who attended. It was like a Mitch Miller sing-along, but instead of following a bouncing ball over the lyrics, audience members could follow a laser pointer. Everyone seemed to enjoy it!

Several people videotaped the performance. I am adding links to the video clips (where available) by each song in the list of songs and their lyrics. You just need to scroll down a couple of entries, to the April 13 posting that contains the list of songs and lyrics. Enjoy!

2007-04-20T11:27:00.000-07:00

Some of the students wanted a review of determining the number of degrees of freedom in a model, so I made another of my "patented" color-coded diagrams below (you can click on the diagram to enlarge it). I aim to please!

Also, in your AMOS printouts, the number of freely estimated parameters can be observed by how many parameters have significance tests (i.e., estimates, critical ratios, and p levels).

2007-04-17T10:28:00.000-07:00

I'm sure I speak for the entire academic community in extending our thoughts and condolences to everyone at Virginia Tech and their families. The school has created a special website for information related to the tragedy.

In many fields of the Human Sciences (e.g., Human Development and Family Studies, Marriage and Family Therapy, and Personal/Family Financial Planning), Texas Tech and Virginia Tech are almost sibling organizations, as many Ph.D. recipients from one school are on the faculty at the other, and there are students who have gotten a degree from one school and are seeking another degree at the other.

2007-04-13T09:14:00.001-07:00

For roughly the last half-hour of the period on our last day of regular class (April 27, at 2:45), we will present the debut of "SEM The Musical."

Update 1: The musical was held, as scheduled, and we ended up with 19 songs (lyrics below). Some video clips are available below, shown by their respective songs and lyrics (thanks to Sothy Eng and Xiaozhi "Gigi" Zhou for their videography work).

Update 2: Derek Ross, an obviously talented video editor and husband of one of the SEM students, condensed our musical into a five-minute documentary, which is more like a spoof infomercial. Click here for the documentary/spoof infomercial. It's truly "must-see TV." Just to be clear: We are not selling videos! Full-length videos of many of our songs are available below.

SEM The Musical
By Dr. Alan Reifman and his Spring 2007 Quantitative Methods IV class
(Back-up vocals in parentheses)

Welcome to SEM The Musical
Lyrics by Alan Reifman
(May be sung to the tune of "Matchmaker," Bock/Harnick, from Fiddler on the Roof)

SEM, SEM, it can be sung,
You’ll be amazed, at what we’ve sprung,
We hope you’ll learn more ’bout this stats technique,
Through songs of which you’re among,

SEM, SEM, we like to run,
It takes awhile, but we get it done,
We hope you’ll learn of the steps that we take,
And take home from this, some fun…

I Am an Indicator
Lyrics by Alan Reifman
(May be sung to the tune of "The Entertainer," Billy Joel)

I am an indicator, a latent construct I represent,
I'm measurable, sometimes pleasurable,
A manifestation of what is meant,

I am an indicator, I usually come in a multiple set,
With other signs of the same construct, you may instruct,
I'm correlated with my co-indicators, you can bet,

I am an indicator, from my presence the construct is inferred,
I'm tap-able, the construct is not palpable,
The distinction should not be blurred

At Least Three
Lyrics by Alan Reifman
(May be sung to the tune of "Think of Me," Lloyd Webber/Hart/Stilgoe, from Phantom of the Opera)

(Cat Pause, lead vocals)

At least three, indicators are urged,
For each latent construct shown,
At least three, indicators should help,
Avoid output where you groan,

With less than three, your construct sure will be, locally unidentified,
Though the model might still run, you could have a rough ride

Gotta Fix It to 1
Lyrics by Alan Reifman
(May be sung to the tune of "Fortunate Son," John Fogerty)
You make a construct, with its loadings,
Can’t let them, all be free,
So that the model’s identified,
Fixing one is the key,

It ain’t free,
It ain’t free,
Gotta fix it to 1,

It ain’t free,
It ain’t free,
In AMOS, automatically done

The number of knowns in your model,
The unknowns can’t exceed,
Fixing a loading for each construct,
Will accomplish this need,

It ain’t free,
It ain’t free,
Gotta fix it to 1,

It ain’t free,
It ain’t free,
In AMOS, automatically done

Residual Variance
Lyrics by Alan Reifman
(May be sung to the tune of "I Say a Little Prayer," Bacharach/David)

Residual variance,
What variables do not share, hence,
I draw a little shape for you,

Residual variance,
What’s left after the R-square, hence,
I draw a little shape for you,

Small circles, to show the, unexplained variance,
...we will always use,
We’ll see what is left in the indicators,
And endogenous,
Constructs that we predict to...

Constrain, ’strain, ’strain...
Lyrics by Alan Reifman
(May be sung to the tune of "Chain of Fools," Don Covay, popularized by Aretha Franklin)

(Cat Pause, lead vocals)

Constrain, ’strain, ’strain (Constrain, ’strain, ’strain),
Constraints are tools (Constraints are tools),
Constrain, ’strain, ’strain (Constrain, ’strain, ’strain),
Constraints are tools (Constraints are tools),

You want to test, if two paths are equal,
You run the model once, then you run a sequel,

First, let the paths run free, they take on their own values,
A chi-square you will see, but what does it tell you?

You must...

Constrain, ’strain, ’strain (Constrain, ’strain, ’strain),
Constraints are tools (Constraints are tools),
Constrain, ’strain, ’strain (Constrain, ’strain, ’strain),
Constraints are tools (Constraints are tools),

You re-run your model, with paths fixed to be the same,
You get a new chi-square, higher than what before came,

You compare the two models, via the delta chi-square test,
If it’s significant, then the free version is best,

When you’ve...

Constrained, ’strained, ’strained (Constrained, ’strained, ’strained),
Constraints are tools (Constraints are tools),
Constrained, ’strained, ’strained (Constrained, ’strained, ’strained),
Constraints are tools (Constraints are tools)...

If it's Nested
Lyrics by Alan Reifman and Adam Munk
(May be sung to the tune of "Mandy," English/Kerr, popularized by Barry Manilow)

If you want to check and see,
If a path is necessary,
What you should do,
Is run a nested model,
Here's the steps to take,
You don't want to dawdle...

If it's nested,
You must only add paths without taking,
Or only take away paths without adding,

If it's nested,
You can compare chi-squares of the models,
And you'll see if the new path is worth adding...

Parsi-Mony
Lyrics by Alan Reifman (expanded for 2010, video)
(May be sung to the tune of "Mony Mony," Bloom/Gentry/James/Cordell)

Structural models need parsimony,
Don’t want to add paths that are phony,
Put the paths you need, now that’s all right, yeah,
You got to keep your model lean and tight, now,
...lean and tight now,
I said, yeah (audience joins), yeah, yeah, yeah, yeah,…

If you can account (PARSIMONY),
For (PARSIMONY),
The data (PARSIMONY),
With a (PARSIMONY),
Minimum of paths (PARSIMONY),
You’ve got (PARSIMONY)
Baby don't stop, seeking (PARSIMONY),
Hey, yeah, yeah, yeah, yeah, yeah, yeah,

Get up!
(brief break)

Few paths, sparse graphs, parsimony,
Above all, keep it small, parsimony,
You want to keep your model looking slim, yeah,
Don't stop now, seek out parsimony, seek parsimony!

Yeah, yeah, yeah…

If you can account (PARSIMONY),
For (PARSIMONY),
The data (PARSIMONY),
With a (PARSIMONY),
Minimum of paths (PARSIMONY),
You’ve got (PARSIMONY)
Baby don't stop, seeking (PARSIMONY),
Hey, yeah, yeah, yeah, yeah, yeah, yeah,

[interlude -- introduce our "band," thank-you's, etc.]

You want parsimony ...mo ...mo ...mony (audience repeats)
Parsimony ...mo ...mo ...mony (audience repeats)
Parsimony ...mo ...mo ...mony...(audience repeats)

Covariance
Lyrics by Alan Reifman (May be sung to the tune of "Aquarius," Rado/Ragni/MacDermot, from Hair, also popularized by the Fifth Dimension)
You draw paths to show relationships,
You hope align with the known r’s,
Your model will guide the tracings,
From constructs near to constructs far,

You will compare this with the data’s covariance,
The data’s covariance...
Covariance!
Covariance!

Similar to correlation,
With the variables unstandardized,
Does each known covariance match up with,
The one the model tracings will derive?

Covariance!
Covariance!

You’ve Got to Check Your R-M-S-E-A
Lyrics by Alan Reifman
(May be sung to the tune of "YMCA," Belolo/Morali/Willis, popularized by the Village People)
How well, does your model match up,
To the matrix of covariances? Yup,

I said, How well, can you reproduce the,
Structure... of the... variables... you see?

You’ve got to check your R-M-S-E-A,
You’ve got to check your R-M-S-E-A,
You want your value, to be very small,
Preferably below, .05 will it fall,

You’ve got to check your R-M-S-E-A,
You’ve got to check your R-M-S-E-A,
It’s one of the, best fit indices,
You can check it, with any others you please...

Check Your NFI
Lyrics by Alan Reifman
(May be sung to the tune of "Judy’s Turn to Cry," Lewis/Ross, popularized by Lesley Gore)
You’ve got to check your NFI,
...check your TLI,
...check your CFI,
’Cause none of them alone’s a hit...

You’ve just finished running your model,
And you want to know its goodness of fit,
But there’s no one single index,
That scholars consider a perfect hit,

You’ve got to check your NFI,
...check your TLI,
...check your CFI,
’Cause none of them alone’s a hit...

The standard advice is to look at,
A variety of measures of fit,
So you pick out a set of several,
Of indices, you form your own kit,

You’ve got to check your NFI,
...check your TLI,
...check your CFI,
’Cause none of them alone’s a hit...

(Instrumental)

Stand by Your Model
Lyrics by Alan Reifman
(May be sung to the tune of "Stand by Your Man," Wynette/Sherrill)
When there’s a path,
That comes out non-significant,
What should you do?
Should you eliminate this path?

Stand by your model,
It represents your best thinking,
Stand by your model,
Don’t want one that’s shrinking,

Just because you,
Didn’t find a certain result,
Keep the model intact,
A future study may support it,

Stand by your model,
It represents your best thinking,
Stand by your model,
Don’t want one that’s shrinking,

Ready to Run
Lyrics by Alan Reifman
(May be sung to the tune of "Ready to Run," Seidel/Hummon, popularized by the Dixie Chicks)

I’ve drawn my shapes and my arrows,
I’m gonna be ready this time (ready this time),
I’ve requested a standardized solution,
I’m gonna be ready this time (ready this time),

Ready, ready, ready, ready, ready, ready to run,
Error messages, I hope to see none...,
Will my assignment get done?

Gonna view my text output,
I hope that it’s correct this time (correct this time),
Got a chi-square and acceptable values,
It looks like it’s correct this time (correct this time),

Ready, ready, ready, ready, ready, ready to run,
Looking for error messages, I see none,
Running a model is fun...

If I Had Multiple Groups
Lyrics by Alan Reifman
(May be sung to the tune of “If I Had a Hammer,” Hays/Seeger)

If I had multiple groups,
I’d run them in the morning,
I’d run them in the evening,
All over this land,

I’d use cross-group constraints,
On the loadings and the structural paths,
I’d want to see if, the chi-square was so different,
From an unconstrained hand

You Had a Bad Fit
Lyrics by Lukas Dean
(May be sung to the tune of "Bad Day," Daniel Powter)
Where is the department statistician when needed the most?
Your research and theory kick up a model that's great,
You draw it in AMOS in a way that relates,
The little wand helps makes the structural paths straight,
You link it to data, now you're out of the gate,

You stand up in the lab to see how the results go,
You fake up a smile when you forgot to estimate means, oh no!

The output says your model is way off line,
Your theory's falling to pieces this time,
And I don't have no Heywood Case,

You had a bad fit,
You take some items down,
You correlate errors, just to turn it around,

You say you don't know, the numbers don't lie,
Your RMSEA was way too high,
You had a bad fit,
The numbers don't lie,
Kenny says it shouldn't be higher than .05,
You had a bad fit, you had a bad fit

An SEM Miracle
Lyrics by Alan Reifman (expanded for 2010)
(May be sung to the tune of "It’s a Miracle," Manilow/Panzer)

I ran this model hours on end,
And kept having problems with it,
There was negative variance,
Known as a Heywood Case,

Paths were unidentified,
Despite everything that I tried,
I did what, I thought would work,
The problem, I couldn't trace,

It’s a miracle (miracle),
All errors have gone away,
The model finally runs,

It was looking hazy, I was going crazy,
Till the output page came through,
It looks clear, and my fit will astound you,
So maybe, I no longer face defeat,

For the miracle (miracle),
I can start writing now,
My homework’s almost done,

I’m finally starting, now, to see,
Where I may have, really, gone astray,
I may have been missing,
A "1" for a fixed pathway,

You've got to use, the AMOS tool,
Or you're gonna look like a fool,
But now that I've done it right,
The errors just go away,

It’s a miracle (miracle),
All errors have gone away,
The model finally runs,

It was looking hazy, I was going crazy,
Till the output page came through,
It looks clear, and my fit will astound you,
And baby, I'll be dancing in the street!

Your Model’s Only One
Lyrics by Alan Reifman
(May be sung to the tune of "The Old Man Down the Road," John Fogerty)

You need a good conceptual model,
You need a nice, large sample size,
You need multiple indicators for,
Each latent construct you surmise,

Plus, you must realize,
That your model’s only one,
Of the many equal-fitting,
Models... that could be run,

Your model represents a best guess,
Causality you cannot show,
You may get some good ideas,
For an experimental way to go,

Thus, you must realize,
That your model’s only one,
You should probably look at,
The writings... of MacCallum

I Guess It Never Hurts to Winsorize
Lyrics by Kristina Keyton
(May be sung to the tune of "I Guess It Never Hurts to Hurt Sometimes," Randy VanWarmer, popularized by the Oak Ridge Boys)
Sometimes I feel the weight,
Of an outlier in my model,
It caused a Heywood case,
And it makes me want to cry,
Is there nothing we can do,
To fix this data problem,
But a memory,
Of Reifman's class saved me,

Outliers always hurt the mean,
And that's ruining my model,
But I won't give up on it,
Just because of one number,
Sometimes it makes me sad,
That we can't just say goodbye,
But I guess it never hurts to Winsorize,

We try and hold on to our moments,
But outliers can't stay,
But we can't just delete,
We lose information that way,
We can't look forward to our output,
And still hold onto bad data,
Oh I hope that you will hear me,
When I say...

Outliers always hurt the mean,
And that's ruining my model,
But I won't give up on it,
Just because of one number,
Sometimes it makes me sad,
That we can't just say goodbye,
But I guess it never hurts to Winsorize

SEM, Oh, SEM
Lyrics by Alan Reifman, dedicated to Peter Westfall (article of his)
(May be sung to the tune of "Galveston," Jimmy Webb, popularized by Glen Campbell)

Ultimately, SEM,
Your LV’s cannot be measured,
Which gives the critics some displeasure,
There’s nothing physical to grab on,
When you run SEM,

SEM, Oh, SEM,
You make many an assumption,
Is it recklessness or gumption?
Assume the e’s uncorrelated...
When you run SEM,

I can see the critics’ point of view, now,
They’re saying the models aren’t unique,

That, we must willingly acknowledge,
In response to the critique, if we want to keep on using...

SEM, Oh, SEM...

2007-04-10T15:12:00.000-07:00

Our next topic is longitudinal SEM, actually a particular type of longitudinal design called a panel study, where the same respondents are followed up over time. Within the panel design, we will also learn about autoregressive and cross-lagged paths. Equality constraints will play a major role here.

One of the major purposes of panel studies is to get a good approximation of causality. Short of actual experimentation, a panel study is probably as good a design as there is for inferring causation. A couple of lecture modules from my methods course (here and here) may be helpful.

The following article by Albert Farrell should also be helpful. We will go over sections of it in class.

Farrell, A.D. (1994). Structural equation modeling with longitudinal data: Strategies for examining group differences and reciprocal relationships. Journal of Consulting and Clinical Psychology, 62, 477-487.

The article actually covers both longitudinal/panel models and multiple-group models. The two are separate topics; a study can have one of these aspects and not the other. We'll also use the Farrell article to discuss multiple-group modeling, but later on.
I'm trying to find as many online documents on SEM panel analysis as I can. So far, I've found this.

Also, an article we've looked at before for other purposes, on which I'm a co-author (Thomas et al., 2000, in Deviant Behavior), illustrates some aspects of SEM panel analysis.

UPDATE (April 13, 2011): I've made some new graphics to illustrate two modeling conventions associated with panel SEM.

The correlated residuals are sometimes known as the "fountain effect" for their visual appearance. The fountain at Las Vegas's Bellagio Hotel nicely illustrates this, as seen below (from GoVegas.about.com).

UPDATE (March 18, 2010): Cameron McIntosh sent a list of references on longitudinal/panel analysis to the SEMNET listserv discussion group. The list, which I've lightly edited, may be helpful for students seeking to pursue the topic in greater detail.

Little, T.D., Preacher, K.J., Selig, J.P., & Card, N.A. (2007). New developments in latent variable panel analyses of longitudinal data. International Journal of Behavioral Development, 31, 357-365. [See Dr. Preacher's publications page.]

Collins, L.M. (2006). Analysis of longitudinal data: The integration of theoretical model, temporal design, and statistical model. Annual Review of Psychology, 57, 505-528.

Phillips, J.A., & Greenberg, D.F. (2007). A comparison of methods for analyzing criminological panel data. Journal of Quantitative Criminology, 24, 51-72.

Preacher, K.J., Wichman, A.L., MacCallum, R.C., & Briggs, N.E. (2008). Latent growth curve modeling (part of the series Quantitative Applications in the Social Sciences, vol. 157). Thousand Oaks, CA: Sage.

Bollen, K.A., & Brand, J.E. (2008). Fixed and random effects in panel data using structural equation models. Los Angeles, CA: California Center for Population Research, UCLA (online).

Wu, A.D., Liu, Y., Gadermann, A.M., & Zumbo, B.D. (2009). Multiple-indicator multilevel growth model: A solution to multiple methodological challenges in longitudinal studies. Social Indicators Research (published online).

More advanced:

Curran, P.J., & Bollen, K.A. (2001). The best of both worlds: Combining autoregressive and latent curve models. In L.M. Collins & A.G. Sayer (Eds.), New methods for the analysis of change (pp. 105-136). Washington, DC: American Psychological Association.

Bollen, K.A., & Curran, P.J. (2004). Autoregressive latent trajectory (ALT) models: A synthesis of two traditions. Sociological Methods and Research, 32, 336-383.

Delsing, M.J.M.H., & Oud, J.H.L. (2008). Analyzing reciprocal relationships by means of the continuous-time autoregressive latent trajectory model. Statistica Neerlandica, 62, 58-82.

Oud, J.H.L. (2002). Continuous time modeling of the cross-lagged panel design. Kwantitatieve Methoden 69, 1-26.

Hamaker, E.L. (2005). Conditions for the equivalence of the autoregressive latent trajectory model and a latent growth curve model with autoregressive disturbances. Sociological Methods and Research, 33, 404-416.

Voelkle, M. C. (2008). Reconsidering the use of autoregressive latent trajectory (ALT) models. Multivariate Behavioral Research, 43,564-591.

2007-03-03T19:29:00.000-08:00

As we've discussed, part of the latest assignment requires you to engage in comparative model testing. Specifically, you will run your model both with and without directed paths from three university properties (public/private status, years of existence, and endowment [square-root transformed]) to their Undergraduate Quality (UQ).

The more parsimonious model is, of course, the one without the additional paths. To override the preference for parsimony, therefore, you will have to show that the additional paths, as a set, significantly reduce the overall model chi-square, thus improving model fit. As you move along in your careers, you may wish to adopt additional criteria, such as whether the reduction in chi-square appears substantively large in addition to being statistically significant, but for now, we'll use statistically significant change as our criterion.

You can display your results in a table, as follows:

--------------------------------------------------------------

Model....................................X2.............df....

--------------------------------------------------------------

Model w/ fewer parameters.....----............---...

Model w/ added parameters.....----............---...

--------------------------------------------------------------
Delta (change)..........................----............---...
--------------------------------------------------------------

The chi-square change score can be treated like any other chi-square value and be referred to a chi-square table, with degrees of freedom equal to delta df.

UPDATE, March 5, 2008: Kristina photographed the decision-tree I drew on the board, to augment our discussion of comparative model testing. Here it is (you can click on the image to enlarge it).

And now, back to our regular programming...

An important condition for being able to conduct comparative model tests is that the two models being compared to each other must possess the property of nestedness. Two models are nested if they can be converted from one to the other either by only adding parameters to one to obtain the other, or only removing parameters from one to obtain the other. By parameters, we mean anything that is freely estimated in SEM (e.g., structural paths, non-directional correlations). If you start with one model and convert it to a new, second model by both adding and substracting parameters from the initial model, the two models will not fulfill the criteria for nestedness and thus cannot be compared via the delta chi-square test.

The following two diagrams provide examples of nested and non-nested models.

An analogous situation exists in multiple regression. You can do a delta R-square test to see, for example, if a model with predictor set A, B, C, D, and E accounts for significantly more variance in the dependent variable than does predictor set A, B, and C. ABC is contained -- that is nested -- within ABCDE, thus permitting the statistical comparison. You could not, however, test whether predictor set ABCDF accounts for more variance than set ABCDE, because the change in models would have required both dropping a predictor and adding one. If ABCDE was the starting point, we would have dropped E and added F.

[Edited March 4, 2010, to be consistent with the latest version of this assignment.]

2007-03-01T14:54:00.000-08:00

A problem specific to SEM (and factor-analytic models more generally), is that of negative residual variances. Variances, being squared entities (of a standard deviation), must be positive. Negative variances are known as "Heywood Cases." Garson's SEM page concisely defines what a Heywood Case is and suggests a simple remedy. Additional discussion of Heywood Cases is available here, here, and here.

[The original posting from this date has been edited, as it included information about an old assignment that is no longer relevant. The remaining information, on Heywood Cases, is still very much relevant.]

2007-02-19T19:13:00.000-08:00

At Wednesday's class, we'll make sure everyone can successfully run the model for your first assignment, and I'll answer any questions people have about the write-up.

Another thing I'd like to devote some time to is examining the meaning of several measures of overall model fit in greater depth than before. David Kenny's web-based summary of fit indices (links section on the right) is an excellent starting point (G.D. Garson's SEM page also has good information). As we discussed before, indices that have the word "fit" in their names should have high values (close to 1.0, which is the maximum for most of the fit indices) to signify a well-fitting model, whereas indices that have the words "error" or "residual" in their names should be small for a well-fitting model, the closer to 0.0 the better.

The AMOS output will report results for three models: the model you designed (also known as the default or proposed model); the independence (or null) model, which says that each measured variable is correlated exactly 0.0 with each other measured variable (with no latent constructs) and thus usually produces results indicative of poor fit with the data; and finally, the saturated model, which, as we've discussed, uses the maximum available parameters and thus is guaranteed to provide a perfect fit. As shown on Kenny's page, many of the fit indices involve comparisons of your model and the independence model.

Researchers typically will report three or four fit indices in an article or other scientific document. AMOS provides many more fit indices than that, so you'll need to form an opinion as to which ones you feel are the best.

St. Mary's University (Texas) professor Andrea Berndt performed a valuable service for the SEM community in her dissertation research at Old Dominion University. I received a copy of her study at the 1998 American Psychological Association convention and, to my knowledge, it has never been published (from the TTU library site, you can search in Dissertation Abstracts for BERNDT, ANDREA ELIZABETH, with her dissertation also available via Interlibrary Loan).

The background behind Berndt's research is that, in determining which are the optimal fit indices to use, we want ones that are not biased by study features, such as sample size. Fit indices should convey only the goodness of fit (or match) between the known, input correlation (covariance) matrix and the matrix derived from the SEM based on the tracing of paths. If a fit index provides a high or low value in good part just because of sample size or other study features, then it's probably misrepresenting the intrinsic fit of the model.

What Berndt did was search all issues of the journals Educational and Psychological Measurement, Journal of Applied Psychology, Journal of Personality and Social Psychology, and Structural Equation Modeling published between 1986-1996, extracting information on study design features and fit indices from all articles using SEM. Study features coded (as labeled by Berndt) were: sample size, number of indicators per latent variable, number of latent variables, number of estimated paths, and degrees of freedom. Fit indices recorded (if published in a given article or retrievable via other statistical information reported) were: Chi-Square, Comparative Fit Index, Critical Number, Goodness of Fit Index, Normed Fit Index, Non-Normed Fit Index (aka Tucker-Lewis Index), Root Mean Square Error of Approximation, and the Relative Noncentrality Index.

Berndt formed a dataset, with each line representing a study. From this set-up, she could conduct multiple-regression analyses, with each fit index serving as the dependent variable in a given analysis, and the study features serving as predictor variables. Again, because we want the fit indices to be free of any "contamination" from study features, a promising fit index will have the set of predictors produce a very small R-square (as close to 0.0 as possible). Here are the findings:

Chi-Square. R-square = .855 (degrees of freedom, Beta = .864, and sample size, Beta = .278, showed significant relations to Chi-Square size).

CFI. R-square = .084 (degrees of freedom, Beta = -.241, was significantly related to CFI).

CN. R-square = .038 (no study features significantly related to CN).

GFI. R-square = .253 (indicators per LV, Beta = -.283, and sample size, Beta = .174, significantly related to GFI).

NFI. R-square = .150 (df, Beta = -.288, and sample size, Beta = .176, significantly related to NFI).

NNFI. R-square = .027 (no study features significantly related to NNFI).

RMSEA. R-square = .038 (no study features significantly related to RMSEA).

RNI. R-square = .061 (df, Beta = -.238, significantly related to RNI).

I think it's clear which indicators have the smallest R-square values, and thus which ones would be recommended for you to use. If you would like to cite the Berndt paper in your writings to justify your choice of fit indices, the reference is:

Berndt, A.E. (1998, August). "Typical" model features and their effects on goodness-of-fit indices. Presented at the 106th Annual Convention of the American Psychological Association, San Francisco, CA.

2007-02-08T09:22:00.000-08:00

Now that we're beginning to learn how to draw models in AMOS (which, by the way, stands for Analysis of Moment Structures), I thought I'd list some of the technical aspects you'll see in the program. Most of the time, AMOS implements these technical aspects automatically, but it's important you know what is going on. [Update, Feb. 12: I've just added a diagram below to help illustrate the following principles; you can enlarge the diagram by clicking directly on it.]

1. Every manifest indicator (box) or latent construct (big circle) that has an incoming unidirectional "causal" arrow gets a residual (or error) term (small circle).

2. Every manifest indicator and latent construct (like any ordinary variable) gets a variance. If the indicator or construct has no incoming unidirectional arrow, its variance is located in the indicator or construct itself. However, if something has a residual, the variance is located only in the residual.

3. Non-directional, "curved" correlations can be inserted only between two entities that have variances. Thus, if two entities each have residual variances, it is the residual variances that get correlated, not the indicators or constructs themselves.

2007-02-03T11:42:00.000-08:00

When learning SEM, an important distinction to recognize from the start is that between a measurement model and a structural model.

A measurement model consists only of factor-loading paths from the latent constructs (factors) to their manifest indicators, non-directional correlations between constructs (like an oblique factor analysis), and in rare circumstances, correlations between the some of the indicators' residual (error) terms.

When the implementation of a measurement model involves only a single questionnaire instrument and the researcher is seeking to verify an a priori conceptualization of which constructs (subscales) go with which items, then that particular kind of model is a confirmatory factor analysis (CFA).

The differences between confirmatory (CFA) and exploratory (EFA) types of factor analysis should be apparent:

1. In EFA, the number of factors is empirically determined by consulting numerical values generated by the computer (i.e., Kaiser criterion, scree test, parallel analyses), whereas in CFA, the researcher decides the number of factors, based on conceptual/theoretical grounds or precedent in the literature.

2. In EFA, determination of which items go with which factors is, again, done empirically via the factor loadings generated by the computer (which can sometimes create problems if an item loads strongly on more than one factor). In CFA, the assignment of items (manifest indicators) to constructs is, again, done on conceptual grounds. Dual-loading items are avoided in CFA, as the researcher will have each manifest indicator receive an incoming factor-loading path from only one construct (factor).

Once the researcher settles on his or her measurement model (i.e., what the constructs are, and what the manifest indicators are of each), then he or she can develop the structural model. A structural model is simply the network of directional, "causal" paths between constructs. For example, a "life stress" construct (with, perhaps, manifest indicators for work stress, home stress, and miscellaneous stress) might have a directional arrow to a "physical symptoms" construct (with indicators for head and stomach ache, fatigue, and back and joint pain).

This study of family functioning and adolescent development by Cumsille et al. provides some nice diagrams of measurement and structural models (and will also be good to return to later in the semester when we cover multiple-group modeling).

2007-02-02T14:19:00.000-08:00

As we saw today, the Promax (oblique) factor-rotation technique in SPSS provides two different types of output, the Factor Structure matrix and Factor Pattern matrix. Russell (2002, Personality and Social Psychology Bulletin) provides a concise explanation of the difference (p. 1636):

The Factor Structure matrix provides the correlation between each of the measures and the factors that have been extracted and rotated; this is, of course, what we typically think of as factor loadings. However, given that the two factors [the number in Russell's example] are correlated with one another, there may be overlap in these loadings. Therefore the... Factor Pattern matrix, is designed to indicate the independent relationship between each measure and the factors. One can think of the values reported here as being equivalent to standardized regression coefficients, where the two factors are used as predictors of each measure.

2007-02-01T15:57:00.001-08:00

I'm anticipating that we'll use roughly the first half of Friday's class to learn implementation of exploratory factor analysis in SPSS, which will close out the unit on EFA.

We will then begin our coverage of SEM with a discussion of the conceptual basis of latent variables and their manifest (measured) indicators. The diagram below, which I developed several years ago, provides an "everyday" illustration.

There are some additional analogies that can be drawn upon:

In Freudian psychology, a distinction is made between the latent and manifest content of dreams.

In biology, a distinction is made between genotype and phenotype.

[ADDED January 16, 2008: The book Does Measurement Measure Up?, by John Henshaw, provides a concise summary of how observable, measurable manifestations are used to infer underlying, unobservable propensities: "Aristotle and others have contrasted the observed behavior of an individual with the underlying capacity on which that behavior depended. Intelligence, as one of those underlying capacities, is an ability that may or may not always be observed in everyday life. This underlying capacity must be deduced from observed behaviors" (p. 92).

2007-01-30T21:04:00.000-08:00

In Wednesday's class, we will work our way up the right-hand side of the "SEM Pyramid of Success," examining how the Pearson correlation gives rise to exploratory factor analysis (EFA).

Starting out with a fairly large set of variables (usually single items), EFA will arrange the variables into subsets, where the variables within each subset are strongly correlated with each other. These subsets are organized along axes (the plural of "axis," not the "axe" like a hatchet).

You could have a one-factor (one-dimensional) solution, in which case all the variables will be capable of being located along a single line (e.g., across from left to right, with low scores to the left and high scores to the right). Or there could be a two-factor (two-dimensional) solution, where the axes are across and up-and-down. Three-factor (three-dimensional) solutions are harder to describe verbally, so let's look at a picture. These examples hold only as long as the axes are orthogonal (at 90-degree angles) to each other (which denotes completely uncorrelated factors), an issue to which we'll return. Solutions can also exceed three factors, but we cannot visualize four spatial dimensions (at least I can't).

In conducting factor analyses with a program such as SPSS, there are three main steps, at each of which a decision has to be made:

(1) One must first decide what extraction method to use (i.e., how to "pull out" the dimensions). The two best-known approaches are Principal Axis Factoring (PAF; also known as common factor analysis) and Principal Components Analysis (PCA). There's only one difference, computationally, between PAF and PCA, as described in this document, yet some authors portray the two techniques as being very different (further, PCA is technically not a form of factor analysis, but many researchers treat it as such).

(2) Second, one must decide how many factors to retain. There is no absolute, definitive answer to this question. There are various tests, including the Kaiser Criterion (how many factors or components have eigenvalues greater than or equal to 1.00) and Scree Test (an "elbow curve," where one looks for drop-off in the sizes of the eigenvalues).

[ADDED January 16, 2008: The book Does Measurement Measure Up?, by John Henshaw, addresses the indeterminacy of factor analysis in the context of intelligence testing as follows: "Statistical analyses of intelligence test data... have been performed for a long time. Given the same set of data, one can make a convincing, statistically sound argument for a single, overriding intelligence (sometimes called the g factor) or an equally sound argument for multiple intelligences. In Frames of Mind, Howard Gardner argues that 'when it comes to the interpretation of intelligence testing, we are faced with an issue of taste or preference rather than one on which scientific closure is likely to be reached' " (p. 95).]

(3) The axes from the original solution will not necessarily come close to sets of data points (loosely speaking, it's sort of like the best-fitting line in a correlational plot). The axes can be rotated to put them into better alignment with the data points. The third decision, therefore, involves the choice of rotation method. Two classes of rotation methods are orthogonal (as described above) and oblique (in which the axes are free to intersect at other than 90-degree angles, which allows the factors to be correlated with each other). There's a web document on factor rotation in the links section (to the right) that has a nice color-coded depiction of orthogonal and oblique rotation, and we'll look at it in class.

[UPDATE: I created a graphic to illustrate factor rotation as an analogy to city planning of housing and roads.]

The particular combination of Principal Components Analysis for extraction, the Kaiser Criterion to determine the number of factors, and orthogonal rotation (specifically one called Varimax) is known as the "Little Jiffy" routine. I've always been a Little Jiffy guy myself, but in recent years, Little Jiffy has been criticized, both collectively and in terms of its individual steps.

An article by K.J. Preacher and R.C. MacCallum (2003) entitled "Repairing Tom Swift’s Electric FactorAnalysis Machine" (see link to papers by MacCallum) gives the following pieces of advice (shown in italics, with my comments inserted in between):

Three recommendations are made regarding the use of exploratory techniques like EFA and PCA. First, it is strongly recommended that PCA be avoided unless the researcher is specifically interested in data reduction... If the researcher wishes to identify factors that account for correlations among [measured variables], it is generally more appropriate to use EFA than PCA...

Another article we'll discuss (Russell, 2002, Personality and Social Psychology Bulletin) concurs that PAF is preferable to PCA, although it acknowledges that the solutions produced by the two extraction techniques are sometimes very similar. Also, data reduction (i.e., wanting to present results in terms of, say, three factor-based subscales instead of 30 original items) seems to be a respectable goal, for which PCA appears appropriate.

Second, it is recommended that a combination of criteria be used to determine the appropriate number of factors to retain... Use of the Kaiser criterion as the sole decision rule should be avoided altogether, although this criterion may be used as one piece of information in conjunction with other means of determining the number of factors to retain.

I concur with this, and Russell's recommendation seems consistent with this.

Third, it is recommended that the mechanical use of orthogonal varimax rotation be avoided... The use of orthogonal rotation methods, in general, is rarely defensible because factors are rarely if ever uncorrelated in empirical studies. Rather, researchers should use oblique rotation methods.

As we'll see, Russell has some interesting suggestions in this area.

One final area, discussed in the Russell article, concerns how to create subscales or indices based on your factor analysis. Knowing that certain items align well with a particular factor (i.e., having high factor loadings), we can either multiply each item by its factor loading before summing the items (hypothetically, e.g., [.35 X Item 1] + [.42 X Item 2] + [.50 X Item 3].......) or just add the items up with equal (or "unit") weighting (Item 1 + Item 2 + Item 3). Russell recommends the latter. It should be noted that, if one obtains a varimax-rotated solution, the newly created subscales will only have zero correlation (independence or orthogonality) with each other if the items are weighted by exact factor scores in creating the subscales.

UPDATE 1: I have created a diagram to explicate factor scoring in greater detail.

UPDATE 2: Here's another perspective on the issue of factor scores (see the heading "Factor Scores/Scale Scores" when the new page opens).

2007-01-29T13:50:00.000-08:00

Below are two photos from the recent lectures on path analysis (thanks again to Sothy).

The photo above is more conceptual, on how to identify the relevant sequences for multiplying path coefficients.

This other photo shows an actual example. Because the model was saturated (every possible linkage that could have been included, was included), the correlation between Age and Income implied by the tracings in the model is identical (within rounding) to the known, input correlation between Age and Income.

MAY 2009 UPDATE: A newly released Australian study indeed shows a positive relationship between height and earnings.

2007-01-19T10:40:00.000-08:00

NO CLASS TODAY!

According to this official university announcement, no classes past 1:00 pm will be held, due to weather conditions.

2007-01-17T09:46:00.000-08:00

Here is a photo from the previous class session, showing what I wrote on the board regarding the least-squares principle. The Wikipedia's entries on least-squares and residuals may also be helpful (here's another website, added on December 28, 2007, that provides a cute, interactive look at the least-squares criterion).

(Edited January 23, 2012.)

2007-01-12T11:10:00.000-08:00

Today, we’ll go over the left side of the SEM Pyramid of Success, from the correlation to multiple regression to path analysis, up to the brink of SEM. An important distinction applicable to all of these techniques is between standardized and unstandardized relationships.

The distinction is probably best illustrated, at this point, with multiple regression. Just to remind everyone, in multiple regression we test how well a number of predictor (independent) variables relate to an outcome (dependent) variable. For example, we could use (a) educational attainment, (b) experience on the job, and (c) performance evaluation as predictors of past-year earnings (outcome). The relationship between each predictor and earnings is computed holding constant the effect of the other predictors (e.g., assuming all respondents were equal in their educational attainment and experience on the job, are higher performance evaluations associated with higher earnings?).

[ADDED December 28, 2007: The following PowerPoint slide show provides an extensive review of multiple regression. I noticed an apparent error on the slide entitled, "The Overall Test...," occurring with slides numbered in the high teens to 20, so for discussion of null hypotheses, you should focus on the slide, with numbering in the 40's, that's titled "Test for Individual Terms."]

For each predictor variable in a multiple-regression analysis, the output will provide an unstandardized regression coefficient (usually depicted with the letter B) and a standardized coefficient (usually depicted with the Greek letter Beta, β). Unstandardized results are probably more straightforward to understand, so let’s discuss them first.

Unstandardized relationships are expressed in terms of the variables' original, raw units. Educational attainment would probably be measured in years of education, whereas earnings would probably be measured in dollars. Thus, the unstandardized (B) coefficient for educational attainment could be something like 2000. This would tell us that, for each increment of one raw unit (year) of education, projected earnings would increase by 2000 raw units of income (dollars).

Standardized results represent what happens after all of the variables (predictors and outcome) have initially been converted into z-scores (formula). As you'll recall from your earlier stat classes, z scores convey information in standard-deviation (SD) units; for example, someone who has a z score of +1 on a variable is one SD above the sample mean on that variable (to review SD's, see here and here). If we were measuring respondents' number of miles run per week in an athlete sample, the mean might be, say, 50 miles/week, with an SD of 10. Therefore, an athlete who ran 60 miles/week in training would be at z = +1, or 1 SD above the mean.

Another nice feature of z scores is that, if the data are distributed normally, you can relate them to a person's percentile ranking in the distribution. For example, someone with a z score of +1 on a given variable (84th percentile) is 34 percentile points ahead of someone who has a z score of 0 (50th percentile). This is illustrated in some of the above links.

Going back to our example of predicting people's earnings, years of experience may have a standardized regression coefficient (β) of .40. This finding would tell us that, for each increment of one SD of years experience, projected earnings would increase by .40 SD's of income.

To recap to this point:

Unstandardized relationships say that for a one-raw-unit increment on a predictor, the outcome variable increases (or if B is negative, decreases) by a number of its raw units corresponding to what the B coefficient is.

Standardized relationships say that for a one-standard deviation increment on a predictor, the outcome variable increasess (or decreases) by some number of SD's corresponding to what the β coefficient is.

When should you use the unstandardized solution and when should you use the standardized one? My own view is as follows: If the raw units are generally familiar (e.g., years, dollars, inches, miles, pounds), I'd go with the unstandardized solution. However, if the variables' raw units are not well-known in everyday usage (e.g., on a marital-satisfaction inventory with a maximum score of 50, what does one point really convey?), then I'd use the standardized solution.

This framework for unstandardized and standardardized solutions applies not only to multiple regression, but also to path analysis and SEM. What is not widely known is that the Pearson r, itself, is a statistic based on standardized variables. The correlation has an unstandardized "cousin," the covariance. The formula for converting between correlations and covariances, which is pretty simple, is shown in this document.

Update (1/19/07): Discussion during our previous class brought out an additional point that I didn't mention in my above write-up (thanks to Kristina).

Within the same regression equation, the different predictor variables' unstandardized B coefficients are not directly comparable to each other, because the raw units for each are (usually) different. In other words, the largest B coefficient will not necessarily be the most significant, as it must be judged in connection with its standard error (B/SE = t, which is used to test for statistical significance).

On the other hand, with standardized analyses, all variables have been converted to a common metric, namely standard-deviation (z-score) units, so the β coefficients can meaningfully be compared in magnitude. In this case, whichever predictor variable has the largest β (in absolute value) can be said to have the most potent relationship to the dependent variable, and this predictor will also have the greatest significance (smallest p value).

(Photo moved into this set of notes on January 23, 2012.)

2007-01-10T08:00:00.000-08:00

Welcome to Quantitative Methods IV, Structural Equation Modeling and Related Techniques. What we'll be covering in the first few class sessions is how SEM represents a culmination of earlier statistical techniques, building from the very basic Pearson's correlation coefficient (r) on up through more elaborate techniques, finally ending at SEM.

John Wooden, who coached men's basketball at UCLA from 1948-1975, winning 10 NCAA championships and garnering various honors as the greatest coach of the 20th century, developed a "Pyramid of Success," which is a guide not only for athletics, but for living a good all-around life.

I grew up a huge UCLA sports fan and went there for undergraduate college (1980-1984). Inspired as I am by Coach Wooden (who celebrated his 95th birthday in 2005 and is still going strong), I created what I call the "Structural Equation Modeling Pyramid of Success," which is shown below.

As we'll see later in the course, as complex as some of our structural equation models can get, the results can always be traced back to simple Pearson correlations.

UPDATE: Coach Wooden died in 2010, just a few months short of his 100th birthday (UCLA tribute).