Welcome!

Here are links to our lecture notes on the different course topics...

Intro: The Pyramid of Success

Correlation, Least-Squares Principle, and Multiple Regression

Path Analysis

Exploratory Factor Analysis (here, here, and here)

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

...and Associated Basic Concepts (free/fixed parameters and model identification; degrees of freedom; model fit; reporting fit)

ONYX Program

Writing Up SEM/CFA Results

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. 

Video clip of legendary physicist Richard Feynman discussing conclusions one can draw from tests of theoretical models.

Comparative Model Testing and Nestedness

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

Diagram for Assignment 2

Refresher Diagram on SEM Terminology

SEM The Musical: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12.5

Graphic arts programs

SEM The Musical 12.5

We will be performing SEM The Musical 12.5 this upcoming Thursday, November 29. Why the designation "12.5"? For roughly the last dozen years, SEM had been the fourth course in our HDFS graduate statistics sequence (after Intro, ANOVA/Regression, and Multivariate) and always taught in the spring. However, we revamped the statistics sequence, knocking out the Multivariate course, moving SEM to third in the order, and adding Longitudinal in the fourth position. Starting with the current semester, SEM is now a fall course. Because only six months (rather than 12) have elapsed since the last SEM Musical, we are  therefore referring to the upcoming one as 12.5.

As always, we'll sing some new songs (shown below) and some classics of the previous 12 years. Just click on any of the following numbers to access a prior musical: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

Todd Little Parcels Indicators
Lyrics by Alan Reifman
(May be sung to the tune of “[My Baby Does the] Hanky Panky,” Greenwich/Barry, popularized by Tommy James & the Shondells)
[Video of performance; added 12/4/2018]

Todd Little parcels indicators,
Todd Little parcels indicators,
He’s one of modeling’s innovators,
He checks residuals’ “correlators*,”
Todd Little parcels indicators...

Todd Little parcels indicators,
Todd Little parcels indicators,
He’s one of modeling’s innovators,
He checks residuals’ “correlators,”
Todd Little parcels indicators...

You’ve got a bunch of indicators, you know,
You have to decide, how they’re gonna go,
Should you combine them, into smaller sets?
Do so at random or with other intent?
They’re still debating, yeah they’re still debating...

Todd Little parcels indicators,
Todd Little parcels indicators,
He’s one of modeling’s innovators,
He checks residuals’ “correlators,”
Todd Little parcels indicators...

(Guitar solo)

You’ve got a bunch of indicators, you know,
You have to decide, how they’re gonna go,
Should you combine them, into smaller sets?
Do so at random or with other intent?
They’re still debating, yeah they’re still debating...

Todd Little parcels indicators,
Todd Little parcels indicators,
He’s one of modeling’s innovators,
He checks residuals’ “correlators,”
Todd Little parcels indicators...

He’s one of modeling’s innovators,
He checks residuals’ “correlators,”
Todd Little parcels indicators ,
Todd Little parcels indicators... (fade out)

---
*There is, of course, no such term as “correlator.” I made it up to maintain the rhyme. What I’m referring to is how one may choose to combine into a parcel indicators that, while initially separate, show a residual correlation. Little et al. (2013, “Why the items versus parcels controversy needn’t be one,” Psychological Methods) note that: “...when a correlated residual is evident in an item-level solution, the most advantageous parcel solution may be one that aggregates those correlated items together” (p. 290).


Oh Mplus!
Lyrics by Alan Reifman
(May be sung to the tune of “Holy War,” Lukather/Vanston/Williams for Toto)
[Video of performance; added 12/18/2018]

(Guitar riff four times)

So, you need, some new, S-E-M software,
From lots of options, you can choose,
I’d say use AMOS, for the basics,
But Mplus, when things, are abstruse,

Ready, to start,
Your data, must,
Be in plain-text,
With no labels on top,

Run it, run it,
Then check, warnings,
So you can make sure,
That your run, didn’t stop,

For more, advanced stuff,
Check out all, the working papers,
Things should, be clear, eventually,

Oh Mplus!
Yes, you’re such a quirky program,
Adding covs, for which no one asked,
You’ve got quite, a learning curve,
To master, all the details,
It is a, substantial task!

(Guitar riffs twice)

Now, if you want latent classes,
Or, multilevel modeling,
Mplus keeps updating, its routines,
So it has got, the things you need,

You can get help,
In figuring, out, the details,
A book by Geiser’s, crystal clear,

There also is,
A website, to help you,
Where they do, Q & A,

Give it a try,
But don’t lose, your patience,
It takes some time, to find your way,

Oh Mplus!
Yes, you’re such a quirky program,
Adding covs, for which no one asked,
You’ve got quite, a learning curve,
To master, all the details,
It is a, substantial task!

(Guitar solos)

Oh Mplus!
Yes, you’re such a quirky program,
Adding covs, for which no one asked,
You’ve got quite, a learning curve,
To master, all the details,
It is a substantial task!

Oh Mplus!
Yes, you’re such a quirky program,
Adding covs, for which no one asked,
You’ve got quite, a learning curve,
To master, all the details,
It is a substantial task!

SEM The Musical 12

The twelfth annual SEM The Musical will be held on Thursday, May 3, during our class. The above logo was contributed by one of our students, Casey A. Smith. We'll sing new songs (to be listed below as they're written) and some favorites from the previous 11 musicals (links: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11).

Welcome to the SEM Parade
Lyrics by Jonathan Villarreal
(May be sung to the tune of “Welcome to the Black Parade”,Bryar, Iero, Way, Way, & Toro, for My Chemical Romance) 

When we were, a new class,
Our professor, took us through the basics,
To see a working model,

He said, class, when you’re finished,
Would you make, a structural equation model,
To find RMSEA,

He said, will you, connect them,
The pathways, and get degrees of freedom,
And compare your delta chi-squares?

Because one day, you’ll leave here,
As scholars, to do your own research,
And join the SEM Parade,

When we were, a new class,
Our professor, took us through the basics,
To see a working model,

He said, class, when you’re finished,
Would you make, a structural equation model,
To find RMSEA,

(Drum-led speed-up)

Sometimes we get the feeling,
These are good factor loadings,
And other times, it feels like it’s all wrong,

When through it all,
The models we draw,
In Onyx and AMOS,
And when it’s done, we want you all to know,

We’ll run Mplus, we’ll run Mplus,
And though this class is done, believe us,
We’ll continue to run Mplus,

We’ll run Mplus,
And three people cannot run it,
Remote desktop will not allow it,

An error that sends you reeling,
Iterations exceeded,
This model will not run at all,

So do the math,
And change a path,
Let’s make our syntax clear,

Triumphant in the end,
We heed the call,

To run Mplus,
We’ll run Mplus,
And though this class is done, believe us,
We’ll continue to run Mplus,

We’ll run Mplus,
And through maximum likelihood estimation,
We’ll accept the adjusted,

Model, and write up our results (oh, oh, oh)
Make sure it’s in STDYX (oh, oh, oh)

Take a look at analyses,
Cause it does not fit at all,

The CFI, Is below point 90,
Didn’t calculate, Degrees of freedom,
It’s too low, The Tucker-Lewis,

We compared it all,
We want to cite our source,
For best fit,
David Kenny,
Don’t forget,
To discuss correlations,
“Causal” paths,
For all our factors,

List them all,
Or at least if significant,
We’re just a class,
We’re not statisticians,
Just a class, who had to run these tests,
We’re just a class,
We’re not Todd Little,
WE – DID – IT,

We’ll run Mplus,
We’ll run Mplus,
And though this class is done, believe us,

We’ll continue to run Mplus,
We’ll run Mplus,
And through maximum likelihood estimation,
We’ll accept the adjusted model,

The CFI,
Is below point 90,
Didn’t calculate,
Degrees of freedom,
It’s too low,
The Tucker-Lewis,
We compared it all,
We want to cite our source,

The CFI (we’ll run Mplus),
Is below point 90 (we’ll run Mplus),
Didn’t calculate (we’ll run Mplus),
Degrees of freedom,
It’s too low,
The Tucker-Lewis,
We compared it all,
We want to cite our source (we’ll run Mplus)

The CFI
Lyrics by Alan Reifman
May be sung to the tune of “English Eyes” (Kimball/Paich/J. Porcaro /S. Porcaro for Toto)

What you’ve run, you want to see how well it fits,
Do the known, and the implied r’s, match bit-by-bit?
The NFI, is one way, but it rises just by adding paths,
Can parsimony, be embedded, right there in the maths?

It takes account, com-plex-i-ty, CFI,
You want to get, values above point-9-5,

(Instrumentals)

It compares, your model to the null version, which has no links,
To ensure, your model fits better, than one you know that stinks,
In the formula, the df track, how many paths you use,
In this way, the more you saturate, the more you lose,

It takes account, com-plex-i-ty, CFI,
You want to get, values above point-9-5,
CFI!
CFI!

(Keyboard/guitar back-and-forth)

[Slow and quiet:
How's your fit?
What indices should you be using now?]

Of sample-size bias, the CFI is relatively free,
As a fit index, it enjoys great popularity,
It’s in programs, such as AMOS, Onyx, and Mplus,
So you can find it, without going through, any fuss,

It takes account, com-plex-i-ty, CFI,
You want to get, values above point-9-5,
It takes account, com-plex-i-ty, CFI,
You want to get, values above point-9-5,
CFI!
CFI!

(More instrumentals)

CFI!

(Guitar solo)

CFI!

CFI!

Dr. Cong (pronounced “Tsong” like tsunami)
Lyrics by Alan Reifman
May be sung to the tune of “Miss Sun” (David Paich, popularized by Boz Scaggs)

Been teaching stats, a long time,
Since you, came from U-S-C,
You’ve taught, lots of students,
In QM 1, and 2, and 3,

Dr. Cong, what can we say?
We wish you, all the best, out at U-T-A,
We hope it isn’t long,
Before our paths, will cross again, in some way,

You’ve served, on our committees,
Methods quals, won’t be the same,
Who’s going to, ask the students,
With a sample, what’s your aim?

Dr. Cong, what can we say?
We wish you, all the best, out at U-T-A,
We hope it isn’t long,
Before our paths, will cross again, in some way (Cross again in some way)

(Guitar solo)

Dr. Cong, what can we say?
You’ve been a, friend of ours, for 10 years, every day,
We hope it isn’t long,
Before our paths, will cross again, in some way,
...In some way...

(Brief interlude)

Dr. Cong, what can we say?
We wish you, all the best, out at U-T-A,
We hope it isn’t long,
Before our paths, will cross again, in some way...

(Instrumentals)

SEM The Musical 11

(Updated May 5, 2017)

The eleventh annual SEM The Musical was held Thursday, May 4, during our class. We had two new songs this year, one by Dr. Reifman and one by student Derrick Holland. Derrick's song keeps our streak alive of having at least one student-written song every year. We also, of course, sang a bunch of favorites from the previous ten musicals (links: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10). See below for this year's new songs...

Why Won’t It Run? 
Lyrics by Alan Reifman
May be sung to the tune of “On the Run” (Lukather/Paich/Waybill for Toto)

SEM involves, some complex math,
Before you go, you need to check, you’ve added each path,
Lots of little details, for you to keep in sight,
Formatting the data, and making sure, your syntax is right,

Have you verified, what you’ll fix to one?
Otherwise, your troubles, have just begun,

Why won’t it run, why all these error signs?
Why won’t it run? Only hints, of what could’ve, gone wrong,
Why won’t it run? Check your steps, line-by-line,
You can put, all your angst, into song!

(Instrumental)

You never know, what a new model, can bring,
Punctuation, constraints, it could be anything,
Maybe what you have, is a Heywood Case?
You’ll need a sharp eye, to keep things in place,

Maximum likelihood, seeks a minimum to achieve,
Gonna take a miracle, for you to receive,

Oh, oh, oh, why won’t it run, why won’t the steps converge?
Why won’t it run, are the magnitude scales far apart?
Why won’t it run, when will I be, on the verge?
Doing this, can tax your heart!

(Instrumental and Guitar Solo)

Hungry Like a Low Chi-Square
Lyrics by Derrick Holland
May be sung to the tune of "Hungry Like a Wolf" (Duran Duran)

Open the connection, get ready to run,
Make sure variables, are in a dot-dat file,
Do do do do do do do dodo dododo dodo,

List, all your variables, that you will use,
Make sure all missing variables, are -99*,
Do do do do do do do dodo dododo dodo,

All pathways are free,
Unless you fix a path to 1,
The very end goal,
It’s important you know,
And I'm hungry, like a low chi-square,

Straddle the line,
With comparative models,
I'm on the hunt, for a good CFI,
Check your TLI, and RMSEA,
And I'm hungry, like a low chi-square,

You get an error, so you start to freak out,
Mplus tells you, that variables are not defined,
Do do do do do do do dodo dododo dodo,

You get a low CFI, important paths are behind,
You search in theory, for paths that are not benign,
Do do do do do do do dodo dododo dodo,

All pathways are free,
Unless you fix one path to 1,
The very end goal,
It’s important you know,
And I'm hungry, like a low chi-square,

Straddle the line,
With comparative models,
I'm on the hunt, for a good CFI,
Check your TLI, and RMSEA,
And I'm hungry, like a low chi-square,

Searching for paths,
I break from theory,
I'm on the hunt,
But I won’t get pubbed,

Latent constructs, made up of manifest
And I'm hungry like low chi-squares,

Draw many lines,
If you use ONYX,
I'm on the hunt, for a good CFI,
Check your TLI, and RMSEA,
And I'm hungry, like a low chi-square,

---

*This is a specification in the Mplus program

Partial Least Squares (Small-Sample Alternative to Conventional SEM)

Partial Least Squares (PLS) is a variation on Structural Equation Modeling (SEM). Riou, Guyon, and Falissard (2016) state that, relative to conventional SEM, PLS “is more suitable to … work with smaller sample sizes.” PLS is recommended for exploratory purposes, and is often used with single-indicator constructs. The technique seems to be used predominantly within the field of Management Information Systems (MIS).

Significance testing is done through bootstrapping, with 100 random variations of the original data set being generated and the model rerun in each random data set. An actual path coefficient from one’s model can then be evaluated for extremity, relative to the distribution of the same coefficient estimated 100 times from the bootstrap.

Though PLS may have reputation for making it easier to obtain significant results, this view appears overstated; a simulation study found that “for N = 40, PLS had 3% and 1% higher power than regression for strong and medium effect sizes [and…] the same power as regression at weak effect size” (Goodhue, Lewis, & Thompson, 2006).

Fit indices, such as NFI, CFI, RMSEA, are not available.

WarpPLS (Kock, 2015) is a program I've found useful and that has a three-month free trial version. Note that the probabilities given in WarpPLS output are one-tailed, so that if you want to report two-tailed p-values, you must double the printed value (e.g., p = .02 one-tailed represents p = .04 two-tailed).

Discussion of the pros and cons of PLS, and of the circumstances for which it may -- or may not -- be appropriate, is available in Goodhue, Thompson, and Lewis (2013); Marcoulides, Chin, and Saunders (2009); McIntosh, Edwards, and Antonakis (2014); and other sources. See also this discussion piece by Kock.

References

Goodhue, D., Lewis, W., & Thompson, R. 2006. “PLS, small sample size and statistical power in MIS research,” in Proceedings of the 39th Hawaii International Conference on System Sciences, R. Sprague Jr. (ed.), Los Alamitos, CA: IEEE Computer Society Press. (link)

Goodhue D. L., Thompson R. L., & Lewis W. (2013). Why you shouldn’t use PLS: Four reasons to be uneasy about using PLS in analyzing path models. In 46th Hawaii International Conference on System Sciences (pp. 4739–4748). Wailea, HI: HICSS.

Kock, N. (2015). WarpPLS 5.0 User Manual. Laredo, TX: ScriptWarp Systems. (link)

Marcoulides, G. A., Chin, W. W., & Saunders, C. (2009). A critical look at partial least squares modeling. MIS Quarterly, 33(1), 171-175. (link)

McIntosh, C. N., Edwards, J. R., & Antonakis, J. (2014). Reflections on partial least squares path modeling. Organizational Research Methods, 17, 210-251. (abstract)

Riou, J., Guyon, H., & Falissard, B. (2016). An introduction to the partial least squares approach to structural equation modelling: A method for exploratory psychiatric research. International Journal of Methods in Psychiatric Research, 25, 220-231. Published online first at doi: 10.1002/mpr.1497.

Reminder of Terminology in an SEM

To the beginning SEM practitioner, terms such as "parameter," "factor loading," and "directional path" may be confusing. Here's a drawing on the whiteboard (with some touch-ups in PowerPoint) to help clarify proper usage. Thanks to the students who photographed the board!

Introduction to ONYX

ONYX is a free SEM package developed in Germany. We will use it for Assignment 1, a CFA on the Hendrick and Hendrick love styles. ONYX is a graphic-arts-based program (like the commercial product AMOS), so your first experience designing a structural equation model will involve what I hope is an intuitive approach of drawing a picture (before we switch to the more technical, but more broadly applicable, Mplus for later assignments).  Here are some tips I have come up with for using ONYX, given its differences from other SEM programs:

1. Everything is done through right-clicking to bring up menus.

2. You can use an SPSS data file or a plain-text (tab-delimited) ".dat" file saved from an SPSS data file. The ONYX user manual lists available options for designating missing data. Once you've drawn your model, you can use "Load Data" to connect to the .dat file, yielding what's called a "Data Panel."

3. Use the "Create Variable" option to generate either latent or observed variables.

4. You should name latent variables (in ALL CAPITALS) via the right-clicking. However, but you’ll have to drag in the measured variables from the "Data Panel" to the variables' respective boxes in the model. By hovering over the measured-variable boxes, you can verify that the data have been linked.

5. By right-clicking on top of a variable, you can use the "Add Path" tool (the default is to draw unidirectional "causal" paths, whereas holding down the Shift key while using "Add Path" yields dual-headed correlational arrows).

6. All unstandardized factor loadings start out fixed at 1; you should free all of them (i.e., letting them take on freely estimated values). To identify the model (i.e., make sure you're not estimating more quantities than you have information for), construct variances should be fixed to 1.*

7. The default settings yield an unstandardized solution, whereas usually we're interested in a standardized one. You can obtain a standardized solution by right-clicking on each indicator’s box and selecting “z-score Transform.”

8. Covariances (correlations between factors) are also fixed and should be freed.

9. Unlike other programs, which have you submit a "job" or a "run," ONYX is constantly running in the background and responds to changes you make in model specifications. Right-clicking and selecting “Show Estimate Summary” will show current results.

10. The following article provides a concise summary of ONYX, including the claim that its method of seeking a solution is superior to that of other programs (see Figure 6 and the text beginning on the prior page at "Multiple Optima").

von Oertzen, T., Brandmaier, A. M., & Tsang, S. (2015). Structural equation modeling with Ωnyx, Structural Equation Modeling, 22, 148-161. 

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*Fixing (or constraining) variables and (under)identification are discussed here.