Writing Up SEM Findings

The SEMNET e-mail discussion listserv had some useful references recently on how to write up SEM findings (thanks to Debbi Bandalos, Mbaye Fall Diallo, Jonathon Little, Amin Mousavi, Christian M. Ringle):

Bandalos, D.L., & Finney, S.J. (2010). Factor analysis: Exploratory and confirmatory. In G.R. Hancock & R.O. Mueller (Eds.), The reviewer’s guide to quantitative methods in the social sciences (pp. 125-155). New York: Routledge.

Boomsma, A. (2000). Reporting analyses of covariance structures (Teacher's Corner). Structural Equation Modeling, 7, 461-83.

Boomsma, A., Hoyle, R. H., & Panter, A. T. (2012).  The structural equation modeling research report.  In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp. 341-358).  New York: Guilford Press.

Gefen, D., Rigdon, E. E., & Straub, D.W. (2011). Editor's comment: An update and extension to SEM guidelines for administrative and social science research. MIS Quarterly, 35, iii-xiv.

Hoyle, R. H., & Panter, A. T. (1995). Writing about structural equation models. In R. H. Hoyle (Ed.), Structural equation modeling: Concepts, issues and applications (pp. 158-176). Thousand Oaks, CA: Sage.

Jackson, D. L., Gillaspy J. A., & Purc-Stephenson R. (2009). Reporting practices in confirmatory factor analysis: An overview and some recommendations. Psychological Methods, 14, 6–23.

Mueller, R. O., & Hancock, G. R. ( 2010). Structural equation modeling. In G. R.Hancock & R. O.Mueller ( Eds.), The reviewer's guide to quantitative methods in the social sciences (pp. 371– 383). New York: Routledge.

Raykov, T., Tomer, A., & Nesselroade, J. R. (1991). Reporting structural equation modeling results in Psychology and Aging: Some proposed guidelines. Psychology and Aging, 6, 499-503.

Schreiber, J. B., Nora, A., Stage, F. K., Barlow, E. A., & King, J. (2006). Reporting structural equation modeling and confirmatory factor analysis results: A review. Journal of Educational Research, 99,  323-337.

SEM The Musical 6

UPDATE May 10.  SEM The Musical 6 is now on the books. Shown below are the new songs written this year, along with some photos of the performances (thanks to Chris Bedard and Xiaohui Tang for the pictures).

We had a surprise "visit" via video feed from three friends of the musical who moved last August from Texas Tech to Virginia Tech: Professor Anisa Zvonkovic (below, center), now department chair at VT; Kyung-Hee Lee (left), who finished her Ph.D. at TTU and is now a post-doc at VT; and Andrea "Hermione" Swenson (right), who finished her Master's at TTU and is studying for her Ph.D. at VT. The song they wrote is shown further down the page.

We also sang several songs from the previous five musicals (links: 1, 2, 3, 4, 5). For the first time ever, we ran a poll (see right-hand column) to help determine which songs from previous years to sing. The song "Parsimony," from SEM The Musical 1, is always what we close the show with, so it was not included in the poll. That's enough background. Here are this year's new songs...

Let’s Sing About It

Lyrics by Alan Reifman

(May be sung to the tune of “Tell Her About It,” Billy Joel)

Listen all you, SEM users,

It’s our yearly, singing day,

To review the things we’ve learned, this semester,

In a most unusual, way,

From your model fit,

To max-i-mum likelihood,

You’ve learned practices that,

Expert modelers should,

Listen all you, SEM users,

You’ve really come, a long way,

From regression, correlation, and pathways,

Through factors and CFA,

From constraining paths,

And checking, delta chi-square,

To run multi-groups and,

Seeing what paths, they share,

Let’s sing about it,

Cover, all the things, we do,

So that, all the concepts,

Now seem old, although they’re new,

Let’s sing about it,

From the blueprint, to the fit,

How to judge, a model,

Knowing how, to interpret,

(brief interlude)

Listen all you, SEM users,

You can run a model, now,

You can draw the boxes, circles, and arrows,

Those are things, that you know how,

But to understand,

In depth, is what we seek,

To make each, of you,

A real, SEM geek,

Let’s sing about it,

Cover, all the things, we do,

So that, all the concepts,

Now seem old, although they’re new,

Let’s sing about it,

From the blueprint, to the fit,

How to judge, a model,

Knowing how, to interpret,

So, now and then, you get an error,

Cause you’ve drawn, something in AMOS, that is wrong,

Make sure to use, the correct tools,

So figuring out the right way, won’t take long,

Listen all you, SEM users,

You’ve, really, come, a long way,

From regression, correlation, and pathways,

Through factors and CFA,

From constraining paths,

And checking, delta chi-square,

To run multi-groups and,

Seeing what paths, they share,

Let’s sing about it,

Cover, all the things, we do,

So that, all the concepts,

Now seem old, although they’re new,

Let’s sing about it,

From the blueprint, to the fit,

How to judge, a model,

Knowing how, to interpret,

Let’s sing about it!

Sing about what, you’ve learned here,

We’ve got to, all sing about it,

How degrees-of-freedom work,

We’ve got to, all sing about it,

Finding out the, errors that lurk,

We’ve got to, all sing about it,

We’ve all got, models to run

We’ve got to, all sing about it

You know, we want to have fun,

We got to, all sing about it… 

You Have Not Shown It’s Causal

Lyrics (and performance) by Devin DuPree

(May be sung to the tune of “When You Say Nothing at All,” Overstreet/Schlitz, popularized by Alison Krauss)

Sometimes you may want, to show how, two constructs relate,

And you may want to show, that B’s caused by A.

Try as you may, it is hard to define,

If Y causes X, or if X causes Y.

Even with a, significant correlation,

There are two other things, you need to, show causation,

Time ordering and, ruling out third variables...

And without that,

You have not shown, it’s causal.

There's a link between, breast implants and suicide,

The risk for women, who do is, three times as high.

Is this because, they don’t like, what they’ve done,

Or are both, caused by, body dis-affection?

Even with a, significant correlation,

There are two other things, you need to, show causation,

Time ordering and, ruling out third variables...

And without that,

You have not shown, it’s causal...

Theory on Your Screen

Lyrics by Alan Reifman

(May be sung to the tune of “Music of the Night,” Lloyd Webber/Hart, from Phantom of the Opera)

(SLOWLY and SOFTLY)

Thinking over, every implication,
Direct pathways, maybe mediation,
Carefully consider, and overcome your jitter,

You have to convey, all that you mean,

For you record, the theory on your screen,

Slowly, gently, contemplate each linkage,
Parsimony, compels one, to shrinkage,
Draw the paths you say, underlie the works at play,
Think of every type of route, though serpent-teen,
And listen to, the theory on your screen,

Close your eyes and surrender to creative schemes!
Push notions beyond, what you’ve read before!
Close your eyes, let your thinking start to soar!
And draw something, that gets right to the core!

Constructs, measures, hypotheses surround you ...
Hear them, feel them, closing in around you ...
Open up your mind, new ideas let it find,
Try to make sense, of what all these things can mean,
The beauty of, the theory on your screen,

Try and model a journey, through a strange, new world!
Go beyond what, the world has known before!
Let arrows, fill in what you think you see!
Only then can you let the paths run free!

Theories, concepts, find your inspiration!
Later, check your, identification!
Let it calculate, the paths that you estimate,

Find the fruits, of the ideas, that you glean,

The power of, the theory on your screen,

ORCHESTRA CRESCENDO

You have to convey, all that you mean,

For you record, the theory on your screen...

S-E-M
Lyrics by Anisa Zvonkovic and Kyung-Hee Lee
(May be sung to the tune of "Edelweiss," Rodgers/Hammerstein, from The Sound of Music)

S-E-M, S-E-M,
Every model, we're building,

Small but right, fit and tight,
Keeps us happy, analyzing,

Helping our vitas, to bloom and grow,
Bloom and grow, for tenure,

Texas Tech, Virginia Tech,
Bless Dr. Reifman, forever

Ways to Treat Single-Indicator Variables

(Updated April 17, 2018)

Often, a researcher will have one or more single-indicator variables within his or her model. It could be a demographic variable such as gender or age, or a total scale score for some social/psychological questionnaire (e.g., Rosenberg Self-Esteem Scale).

With multiple manifest indicators for a latent construct, the construct is automatically rendered "error-free," with measurement error segregated out into each indicator's residual "tiny bubble." Relations between constructs will be stronger when they are error free. Single-indicator variables, when left to stand alone, usually have measurement error, but are assumed to be perfectly measured.

Here are five scenarios in which a researcher was interested in studying self-esteem (thanks to CRO for the photograph of the board).

As shown in the photo, reliability-corrected single-indicator constructs are a way to account for measurement error in single-indicator variables (lower-right). The following is a quote from Choi et al. (2011): “To account for imperfect reliability of the scale scores, we created latent variables to represent the … constructs with each latent variable being measured by its corresponding scale score and the residual variance of the scale score fixed to (1-scale reliability) * scale variance (Hayduk, 1987).” Cronbach's alpha (internal consistency) is often used as the reliability value. A made-up example of this procedure is shown in the photo.

I previously created the following graphic to illustrate further the difference between keeping single variables as they are and using reliability correction.

NEW! Video of Todd Little speaking at Texas Tech about parceling (February 2, 2018). Dr. Little discusses strategic parceling approaches, as opposed to random parceling. Follow this link to the video (limited to TTU); listed under Daniel Bontempo, organizer of IMMAP series.

References and Resources

Choi, K. H., Bowleg, L., & Neilands, T. B. (2011). The effects of sexism, psychological distress, and difficult sexual situations on U.S. women's sexual risk behaviors. AIDS Education and Prevention, 23(5), 397-411. (LINK)

Cole, D. A., & Preacher, K. J. (2014). Manifest variable path analysis: Potentially serious and misleading consequences due to uncorrected measurement error. Psychological Methods, 19, 300-315. (LINK)

Hayduk L. A. (1987). Structural equation modeling with LISREL: Essentials and advances. Baltimore, MD, USA: Johns Hopkins University Press. 

SEM The Musical 5

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!

SEM The Musical 4

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,

Graphics Programs for Drawing SEM Diagrams

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)

ADDED 6/10/2017: PowerPoint Tips for Displaying SEM Models

Further Issues for Full Structural Models

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

Maximum Likelihood Estimation

Equivalent models

Handling single-indicator variables

Negative variances (Heywood Cases)

Mediation

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; and this article, which we'll revisit from when we learned about measurement and structural models)

Dyadic analysis in SEM (Actor-Partner Interdependence Model)

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

Software comparison: AMOS vs. Mplus

Advanced Applications

Latent Growth Modeling (here and here)

Compared to the more piecemeal/incremental cross-lagged panel models for longitudinal analysis, latent growth models test for predictors and correlates of respondents' long-term growth trajectories (see cannon-ball analogy). Thanks to Tim and Xiaohui for photographing the board after class on May 1, 2012. Some illustrative references on latent growth modeling are:

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.

Wampler, R. S., Munsch, J., & Adams, M. (2002). Ethnic differences in grade trajectories during the transition to junior high. Journal of School Psychology, 40, 213-237.

Partial Least Squares (Alternative to Conventional SEM for Small Samples)