SEM The Musical 10

The tenth annual SEM The Musical was held on Thursday, May 5. We performed a few new songs this year, as shown below. We also performed songs from previous SEM Musicals. Three older songs we performed this year are available on YouTube (thanks to SH for filming). These songs are "Common Model Mistakes" (originally from SEM The Musical 9), "Saturated Your Model" and "If You Wanna Join My Construct (You've Gotta Load with My Friends)," the latter two from SEM The Musical 8. To see the lyrics from these (and other) older songs, just click on the year number of the musical: 1, 2, 3, 4, 5, 6, 7, 8, 9.


SEM Musical TEN!
Lyrics by Alan Reifman (retread from previous years)
(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 (TEN!), SEM Musical (HERE!),
SEM Musical (TEN!), SEM Musical (HERE!),
SEM Musical (TEN!), SEM Musical (HERE!),
SEM Musical (TEN!), 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 (TEN!), SEM Musical (HERE!),
SEM Musical (TEN!), SEM Musical (HERE!),
SEM Musical (TEN!), SEM Musical (HERE!),
SEM Musical (TEN!), SEM Musical (HERE!),
Yeah...


The Part That’s Error-Free 
Lyrics by Alan Reifman
(May be sung to the tune of “Biggest Part of Me,” David Pack for Ambrosia)

Boxes, they hold the manifestations,
Bubbles, are error locations,
Constructs, house the shared variation,
They're the part, that’s error-free,

Loadings, show measures, are correlated,
That makes, indicators validated,
Errors, in the bubbles, they are gated,
So constructs, are error-free,

Well...
You remove error,
And the paths, become more true*,
This is such, a key thing,
Latent constructs, do for you,

So draw it now,
Tell measurement error, to shoo.
You can estimate, the paths,
Without error, troubling you,

Sometimes, you have just, total-scale measures,
Of those, certain constructs, that you treasure,
Alpha, gives a way to block displeasure,
Controls, unreliability,

Parcels, a technique, that can’t be plainer,
Items, placed into, random containers,
These sets, can then serve, as indicators,
Constructs now, are error-free,

Well-l-l-l-l...
You remove error,
And the paths, become more true,
This is such, a key thing,
Latent constructs, do for you,

So draw it now,
Tell the measurement error, to shoo,
You can estimate, the paths,
Without error, troubling you,

(Instrumentals)

It’s an SEM hallmark,
Going back to CFA,
It’s a major advantage, of using LV’s,

Not all techniques, give you this,
Measurement error, doesn’t go away,
So use latent constructs, to be error-free,
Be error-free,
Be error-free...

*Stephenson, M. T., & Holbert, R. L. (2003). A Monte Carlo simulation of observable versus latent variable structural equation modeling techniques. Communication Research, 30, 332-354.

See also previous lecture modules here and here.


Those Kinds of Paths (Are Autoregressive)
Lyrics by Alan Reifman
(May be sung to the tune of “Because the Night,” Springsteen/Smith)

Panel models, longitudinally,
Follow the same people, over time,
Each major construct, we include repeatedly,
It gets us the time-ordering, of causality,

So, come on now, no hand-calculated math,
In cross-lagged models, we run paths,
From Construct A at one time, to B at the next,
We also have paths, from the same construct,
Time 1 to Time 2, and Time 2 to Time 3,

Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,
Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,

Autoregressive paths, play a crucial role,
They control for earlier levels, of a later DV,
So when a cross-lagged path, is significant,
It shows association, beyond stability,

So, come on now, no hand-calculated math,
In cross-lagged models, we run paths,
From Construct A at one time, to B at the next,
We also have paths, from the same construct,
Time 1 to Time 2, and Time 2 to Time 3,

Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,
Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,

(Guitar solo)

These kinds, of paths,
Predict, to later versions, of themselves,
Without them, analyses would lack rigor,
So include them...

Time 1 to Time 2, Time 2 to Time 3,
Time 1 to Time 2, Time 2 to Time 3,
Time 1 to Time 2, Time 2 to Time 3,

Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,
Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,

Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,
Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,

Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,
Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,

Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability,
Those kinds of paths, are auto-regressive,
Those kinds of paths, test stability...


Constructs (Don’t be Afraid of Changing!)
Lyrics by Diane Wittie
(May be sung to the tune of “Landslide,” Stevie Nicks)

Gathered the data, and they abound,
I cleaned them up, then I went to town,
And I saw some variables, that looked interesting,
And now my sleep, would be sound,

Oh, yes I can begin, naming latent constructs,
But will those, constructs make any sense?
Will they adequately represent,
What I envision?
Can I implement, my central concepts?

Uh-hum, I do think so,

Well, don’t be, afraid of changing,
’Cause your constructs, need to make sense,
Think through, your decisions,
You may need, revisions,
Don't do anything, you'll rue,

(Brief guitar)

So, don’t be, afraid of changing,
’Cause your constructs, need to make sense,
Think through, your decisions,
You may need, revisions,
Don't do anything, you'll rue,
To your theory, be true,

So, analyze your data, see what you've found,
Your model, may earn great renown!
If you see factor loadings, at plus or minus .4,
Well maybe, high points you will score,
If you see structural paths, that are significant,
Yes, high points, you will score!

Mediational Models

(Updated April 2, 2024)

Many SEM-based studies examine mediation between variables. To mediate is to go in the middle, like a negotiation mediator comes between the labor union and management.

In statistical analysis, we often start out with a relationship between two variables. Using an example from one of my grad-school mentors, Patricia Gurin, cigarette smoking and lung cancer are positively associated.

Cigarette Smoking ==> Lung Cancer

Why does this relationship exist? A more fine-grained understanding would be that smoking leads to lung tissue damage, and tissue damage leads to cancer. Tissue damage would thus be considered the mediator or mechanism.

Cigarette Smoking ==> Tissue Damage ==> Lung Cancer

Reuben Baron and David Kenny published an article in 1986 on mediation that has been cited over 126,000 times (as of April 2024)! Kenny summarizes the process in a nutshell here. In the following figure, I apply Baron and Kenny's "old school" method to Gurin's example. Note that one would run the model twice.

(Illustration of Baron and Kenny's [1986] logic. Example from Patricia Gurin, University of Michigan, circa 2002-2003, link)

This worksheet from Jason Newsom contains a nice four-step chart for implementing Baron and Kenny's framework for mediation.

The above diagram presents the scenario of full mediation (i.e., the initially significant direct path from antecedent to outcome becomes nonsignificant). One can then say that the mediator accounts fully for the antecedent-outcome relationship. If the initial direct path from antecedent to outcome remains significant after addition of the two mediational paths, but the initial direct path is reduced in magnitude, one can claim partial mediation (see Huselid and Cooper, 1994, "Gender roles as mediators of sex differences in expressions of pathology").

(As noted above, my perspective is that the antecedent and outcome should first be shown to relate significantly, before one pursues the further steps to test mediation. For an opposing view, that a significant antecedent-outcome path should not be a "gatekeeper" for mediation analyses, see here.)

As Kenny writes on his website, "More contemporary analyses focus on the indirect effect." The leading names associated with contemporary mediational analysis are Andrew Hayes and Kristopher Preacher, who indeed emphasize indirect effects. The indirect effect can be calculated by multiplying the standardized paths from antecedent to mediator, and from mediator to outcome (think back to our unit on path-analysis tracing rules).

The indirect effect is .15 in the above example. If each of the two segments of the indirect effect (A to M, and M to O) is each statisically significant (i.e., different from zero), we would be confident that the indirect effect also is significant. As Hayes (2009, "Beyond Baron and Kenny: Statistical mediation analysis in the new millennium") notes, however, "it is possible for an indirect effect to be detectably different from zero even though one of its constituent paths is not." What is called for is a significance test of the indirect effect of .15 (or whatever value one has).

The problem is that there is no existing theoretical distribution such as the z, t, F, or chi-square distribution to judge the statistical significance of an indirect effect (i.e., whether or not one's obtained indirect effect falls in the upper or lower 2.5% of the distribution for a two-tailed p < .05 significance level). Therefore, researchers use a "synthetic" statistical distribution for testing the significance of indirect effects, known as a "bootstrap" distribution. Kenny discusses this on his website and it is also illustrated in slide 6 of this slideshow. In 2022, my colleague Sylvia Niehuis and I published an encyclopedia entry on bootstrapping, which can be obtained via ResearchGate.

So where do things stand in the field regarding the "classic" Baron and Kenny approach (with its emphasis on the significance of different paths) vs. the "modern" Hayes and Preacher approach (with its focus on the significance of the indirect pathway from antecedent to mediator to outcome)? As the above quote from Kenny suggests, the Hayes/Preacher method predominates in the field, with some commentators seemingly thinking the Baron/Kenny framework is as outmoded as viewing movies through VHS tapes or using a rotary phone.

The Hayes/Preacher method is not without its skeptics, however. Vincent Yzerbyt and colleagues point out that, under certain circumstances, the overall indirect effect can be significant even when only one of the components (antecedent-to-mediator or mediator-to-outcome) is large. Further, they recommend that, "Claims of mediation should properly be guided by the component approach and be based on joint-significance tests [i.e., showing that the antecedent-to-mediator and mediator-to-outcome paths are both significant] to avoid spurious mediation claims" (p. 941).

Yzerbyt, V., Muller, D., Batailler, C., & Judd, C. M. (2018). New recommendations for testing indirect effects in mediational models: The need to report and test component paths. Journal of Personality and Social Psychology, 115, 929–943.

For an earlier illustration of how one might draw on both the classic and modern approaches to mediation, please see the following:

Niehuis, S., Reifman, A., Fischer, J. L., & Lee, K.-H. (2016). Do episodic self- and partner-uncertainty mediate the association between attachment orientations and emotional responses to relationship-threatening events in dating couples? Cognition and Emotion, 30, 1232–1245.

An additional source for studying mediation in SEM is:

Li, S. (2011). Testing mediation using multiple regression and structural modeling analyses in secondary data. Evaluation Review, 35, 240-268.

One sometimes posits multiple mediators in a model. For an illustration of what are known as parallel mediation (i.e., the mediators have no causal relations amongst themselves) and serial mediation (i.e., one mediator causes another mediator and so on, in a sequence), see:

Guthrie Yarwood, M. F. (2013).The relationship between love styles and digital dating outcomes: A multiple mediation test of the Perceived Importance of Dating Features Scale. Dissertation, Texas Tech University (LINK). See especially pp. 56-62.