Formative (causal) measurement model... PreviousNext
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 Christian Klode posted on Friday, December 01, 2006 - 6:11 am
hi linda and
hi bengt!

i have a question concerning fixed parameters within formative (or causal) measurement models (FMM) like SES. my aim is to perform a SEM with exogenous FMM and endogen ordinal reflective measurement model (RMM). the model fit of WRMR = .907 looks appropriate, and comparable parameter estimates structure were obtained by a PLS-model.
my problem for now is the following: apparantly the FMM regression weights are very sensitive due to the fixing value. the following model provide a value of .01 as a nice approximation.

is there a rule-of-thumb for the fixing value within FMMs?
here is the corresponding model:

categorical: pride1 pride2 pride3;
model: repu by;
repu on repu1@0.01 repu2 repu3;
pride by pride1 pride 2 pride3;
pride on repu;

thanks in advance!
 Linda K. Muthen posted on Friday, December 01, 2006 - 9:53 am
If you set the metric using .01 instead of 1, then the FMM regression weights will be changed. Their relative size should remain constant.
 Christian Klode posted on Friday, December 08, 2006 - 6:44 am
linda, thank you for the immediate reply.
i have an additional question concerning the 'statistical fit' of formative indicators with a potential regard to scale purification. let's assume one has no access to PLS (within a theory 'building'-approach), then there would be no guidance for setting the metric via a discrete indicator-weight (in order to obtain trustful t-values).

is there an alternative to the PLS-pre-'testing' respectively would you denote this procedure as appropriate?

 Bengt O. Muthen posted on Friday, December 08, 2006 - 6:23 pm
The choice of constant or variable for setting the metric does not affect the model fit - the same restrictions on the covariances between the FMM indicators and the DVs are obtained.
 Richard E. Zinbarg posted on Thursday, March 15, 2012 - 8:15 am
Hi Linda and/or Bengt,
A student and I are trying to fit our first causal indicator measurement model. From the syntax on slide 246 of the Mplus Short Courses Topic 1 Handout, it looks to us as if there is no disturbance term for f - your formative construct. Are we interpreting that correctly? If so, this would seem to us to be more akin to what Bollen would call a composite indicator model than a causal indicator model and we would be interested in guidance regarding syntax we should use to give the construct a disturbance term?
Thanks very much!
 Linda K. Muthen posted on Thursday, March 15, 2012 - 9:41 am
I am not sure what Bollen means by a causal indicator model. Can you send the paper to
 Bengt O. Muthen posted on Saturday, March 17, 2012 - 1:28 pm
Yes, the formative model we give the specification for is what Bollen-Bauldry (2011) Psych Meth call composite indicators. What they refer to as causal indicators can simply be specified as a MIMIC-type model; no special syntax needed. See their Figure 4, where the causal indicators behave like regular covariates - to me, it is more of a conceptual distinction.
 Randall MacIntosh posted on Thursday, December 27, 2012 - 10:02 am
What alterations would I have to make to the model in slide 246 (topic 1) to obtain the model shown in Fig. 4a of the Bollen-Baudary paper?

Is it possible?

 Bengt O. Muthen posted on Thursday, December 27, 2012 - 2:05 pm
What's the reference? You don't mean the Bollen-Bauldry (2010) SM&R paper, right?
 Randall MacIntosh posted on Thursday, December 27, 2012 - 3:51 pm
Bollen & Bauldry (2011) Three Cs in Measurement Models:Causal Indicators, Composite Indicators, and Covariates. Psychological Methods 2011, Vol. 16, No. 3, 265–284

Figure 4a is on page 276.
 Bengt O. Muthen posted on Friday, December 28, 2012 - 5:37 pm
So I assume you want the parameterization of eq. (13) on page 11. So say:

eta1 BY y1@1; [y1@0];
eta1 ON x1-x3; [eta1];
 Randall MacIntosh posted on Saturday, January 05, 2013 - 12:27 pm
Bollen and Bauldry suggest on p.279 (or p.14) looking at the indicator's unique validity variance, which they define as the difference between the r-square for eta with all causal indicators and r-square for eta, less causal variable x_i.

Mplus provides r-square for the latent variable. How can I get (or compute) the second value to obtain the unique validity variance?

Thank you.
 Bengt O. Muthen posted on Saturday, January 05, 2013 - 4:17 pm
Are you referring to eqn (21)?
 Randall MacIntosh posted on Sunday, January 06, 2013 - 9:48 am
Eqn. 24
 Bengt O. Muthen posted on Sunday, January 06, 2013 - 5:23 pm
I think you would have to do 2 different runs and get the 2 R-square values.
 Lois Downey posted on Thursday, June 30, 2016 - 1:20 pm
I used a suggestion from Bollen and Bauldry's article to build the following syntax for estimating a model with causal indicators:

USEVARIABLES = y1 y2 x1-x4;
Factor by;
Factor on x1@1 x2-x4;
y1 y2 on Factor;
y1 with y2@0;

Although that is a different method for achieving model identification than the one you suggest in your course handout, it avoids having to use a composite variable between the indicators and the latent variable of interest, and the model ran as expected.

I then wanted to test the model for between-group invariance of the factor indicators. To do that, I added a grouping variable and made changes to the MODEL statement as follows:

GROUPING = country (0=US 1=Canada);

Factor by;
Factor on x1@1
x2 (1)
x3 (2)
x4 (3);
y1 y2 on Factor;
y1 with y2@0;

MODEL Canada:

However, this model resulted in an error message, indicating that the model may not be identified – and pointing to the factor intercept in the Canada group as the problematic component.

What additional constraint(s) does the model need for statistical identification?

 Bengt O. Muthen posted on Friday, July 01, 2016 - 11:39 am
Your single-group model seems to have a free residual variance for the factor - is that really identified?

Your two-group model is not identified because the two intercepts of y1, y2 in their regressions on the factor cannot be identified together with the factor intercepts. It would be identified if you hold those y intercepts equal across the groups.
 Lois Downey posted on Saturday, July 02, 2016 - 9:18 am

I get a solution for the single-group model, so it is presumably identified. It includes estimates for 14 free parameters:
the factor on the 3 causal indicators
the 2 outcomes on the factor
the 8 thresholds for the 2 outcomes
the residual variance for the factor

With regard to the additional constraints needed for the two-group model: Since the two y-variables are polytomous, do I constrain the thresholds, rather than the intercepts, to equality between groups? (I did that earlier, and that model was identified. I just wasn't sure whether that constraint was "reasonable.")

Thank you again!
 Bengt O. Muthen posted on Saturday, July 02, 2016 - 6:08 pm
I see now that you have 2 y's that the factor points to and that are uncorrelated conditioned on the factor so they are like 2 factor indicators which of course makes the factor residual variance identified - this is just a MIMIC model if you fix the y1 on factor coefficient instead of fixing the x1 coefficient.

Right, hold the thresholds equal across groups.
 Lois Downey posted on Tuesday, July 05, 2016 - 10:29 am
Thanks. One more question. I think that in testing for between-group measurement invariance with reflective indicators, one is supposed to fix the delta scale factors at 1.0 in the first group, and estimate them in the remaining groups. Should this also be done when the indicators are causal?
 Bengt O. Muthen posted on Tuesday, July 05, 2016 - 5:56 pm
No, they act as covariates. Deltas are only for DVs.
 Lois Downey posted on Wednesday, July 06, 2016 - 12:39 pm
Yes -- I meant delta scale factors for the outcome variables. But I think one shouldn't constrain those in any way. Correct?

In terms of parameterization, is there any reason not to use theta parameterization for these models with causal indicators (and for testing them for between-group invariance)? I suspect that there is a reason why delta parameterization is the Mplus default, and there may be some advantage to using it. However, residual variance is much easier to understand than scale factors, so is appealing to some of us who aren't statistically sophisticated.
 Bengt O. Muthen posted on Wednesday, July 06, 2016 - 5:37 pm
Q1. Like you say, fix the delta scale factors at 1.0 in the first group, and estimate them in the remaining groups.

Q2. Theta param'n is fine when it works - sometimes convergence has problems (discussed in Web Note 4).
 Ronald L. Hess Jr. posted on Thursday, July 07, 2016 - 7:29 am
I am following a procedure discussed by Diamantopoulos (2011) MISQ for handling formative indicators using CB-SEM. The model contains 6 IVs, 1 mediator, and 1 outcome variable. All variables are measured with formative items. I use the @1 command to fix the error variance of each of the variables.

I hope this (@1) is the correct command to handle these types of items.

My problem involves the fit of the model.

CFI = 0.88 (fine)
TLI = 0.87 (fine)
RMSEA = .073 (fine)

I would also be happier if I had a couple more significant paths in the model.

Why do I have such an inflated SRMR? Is there a way to reduce this. It seems odd to me that the other fit indices are acceptable but SRMR is way off.
 Bengt O. Muthen posted on Thursday, July 07, 2016 - 9:40 am
We recommend fixing the formative factor residual variance at zero.

You may want to ask your fit question on SEMNET.
 Alexander Tokarev posted on Friday, October 07, 2016 - 10:28 am
Dear Drs Muthen,
I have a model with 3 latent variables: P->B->D (all latent ‘effect’ variables, 4 ordinal indicators each) and I also have 1 observed continuous moderator variable (FI) that moderates the P->B path. As a first step I am trying to fit a measurement model(MM). My understanding is that I need to first
A) Fit the MM with the latent constructs (i.e. P, B, D), and then
B) I somehow need to incorporate an observed moderator variable (FI) into the MM. Given that I plan to use FI as a moderator variable, I am not sure how to incorporate it into the MM. I read from earlier posts that I need to include it as a covariate using an ‘ON’ statement. If my measurement model is:
P BY P1-P4;
B BY B1-B4;
D BY D1-D4;

Do I incorporate FI into the MM by 1) regressing all latent variables on FI or by only regressing B on FI, since FI will soon moderate the P->B path? 2) do I only regress the latent variable(s) on FI, or do I need to regress all indicators of latent variable(s) onto FI, or is it both? 3) Do I need to constrain variance of any factor to 0 or set a metric of an observed FI to 1? 4) Does the interaction variable between a latent and observed variable – result in a latent or observed interaction term? I am sorry for many questions, I read the Mplus handouts but none of 4 formative models perfectly resembled this case.

Thank you
 Alexander Tokarev posted on Friday, October 07, 2016 - 10:45 am
sorry for the second consecutive post.

Addition to the last one: since the moderator is an observed continuous variable, or would that rather be correct to simply correlate it with all latent variables within the measurement model by using a 'WITH' command?
 Bengt O. Muthen posted on Friday, October 07, 2016 - 4:37 pm
UG Chapter 14, p 697-698 mentions that an interaction between an observed variable and a latent variables is carried out using XWITH.

Yes, it is good to evaluate fit first without the XWITH addition because when added there is no longer a global fit statistic available.
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