Dyadic Analysis

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) and the slides from a talk he gave. 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.

In honor of David Kenny's contribution to dyadic analysis, I've written a song.

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

UPDATE 1 (4/24/12): As we discussed in class, data from both members of a couple (or from a repeated-measures/panel design of individual participants) can be organized two ways. I have created the following graphic to illustrate (you may click on it to enlarge). The "elbow" arrows are meant to convey that, in moving from the "long" to the "wide" format, the second line of couple data (in this case, the wife's) is raised to join the husband's data on the first line. In the longitudinal/panel example, each participant's Time 2 and Time 3 data are moved up to the first line, to join the Time 1 data.

Here's a YouTube video I discovered on SPSS's restructuring technique, which can be used to convert between the above two formats.

UPDATE 2 (4/28/16): A potentially confusing situation can arise when one wants to compare the paths of men and women. Scenarios can exist for using multiple-group modeling or dyadic/APIM analysis. As I wrote on the board (shown below), multiple-group modeling is done when the groups (such as men and women) are independent of each other. Dyadic analysis, on the other hand, is done when there is a connection or interdependence between paired individuals, such as a man and woman in a heterosexual marriage (thanks to OR for taking the picture).

Presenting Model Fit Statistics

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.

Running AMOS Off of a Published Correlation Matrix

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!]

SEM The Musical 2

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.

Equality Constraints

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).

Implementation of equality constraints in Mplus is illustrated at this site (see section 3.0).*

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."

At first glance, one path may look like it is significantly larger than another. However, this proposition must be tested by constraining the paths in question to equality and conducting a delta chi-square test. The following articles illustrate the process.

Foster, T. D., Froyen, L. C., Skibbe, L. E., Bowles, R. P., & Decker, K. B. (2016). Fathers’ and mothers’ home learning environments and children’s early academic outcomes. Reading and Writing, 29, 1845–1863.

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. (AMOS uses letters to denote paths constrained to equality, whereas Mplus uses numbers.)



---

*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.

"Equivalent Models" Problem

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.

In addition, regarding one of MacCallum et al.'s suggestions, here's an article that incorporates participation in a randomized experimental program (yes/no) into an SEM model.

Florsheim, P., Burrow-Sánchez, J., Minami, T., McArthur, L. & Heavin, S. (2012). The Young Parenthood Program: Supporting positive paternal engagement through co-parenting counseling. American Journal of Public Health, 102, 1886-1892.

Stating Hypotheses

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.

(Added April 29, 2015): This webpage explains the distinction between a hypothesis (when you have a prediction) and a research question (when you don't).