reflective first order, formative second order model

[Sylvia López Davis]

Dear all:

I am trying to estimate a reflective first order, and formative second order model and I am having some problems with it. The syntax used to test the model has been the following:

# Estimation of the structural model

PSR2 <- ' 
# measurement model 
ECO =~ ECO_01 + ECO_02 + ECO_03
LEG =~ LEG_01 + LEG_02 + LEG_03
ETI =~ ETI_01 + ETI_02 + ETI_03 + ETI_04
FIL =~ FIL_01 + FIL_02 + FIL_03 + FIL_04 + FIL_05  
MED =~ MED_01 + MED_02 + MED_03 + MED_04
SAT =~ SAT_01 + SAT_02 + SAT_03 + SAT_04 + SAT_05
SEST =~ SEST_01 + SEST_02 + SEST_03 + SEST_04
PSR <~ ECO + LEG + ETI + FIL + MED
# regressions 
PSR ~ SAT + SEST
'
fitS1 <- sem(PSR2, data = dataM)
summary(fitS1, fit.measures = TRUE)
fitMeasures(fitS1)
inspect (fitS1, "cor.lv")
standardizedSolution(fitS1)
MI <- modificationIndices(fitS1)
subset(MI, mi > 10)

But then I got the next message:

Warning messages:
1: In lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats,  :
 lavaan WARNING: could not compute standard errors!
 lavaan NOTE: this may be a symptom that the model is not identified.

2: In lav_model_test(lavmodel = lavmodel, lavpartable = lavpartable,  :
 lavaan WARNING: could not compute scaled test statistic

Does anyone have an idea of the problem? We do not have the formative latent variable (PSR) measured, but therefore we are including two reflectively-measured constructs (SAT, SEST). 

I would appreciate if you could help me. 

Thank you very much in advance. 

Sylvia.

[Mikko Rönkkö]

Hi

Your model is not identified because PSR does not have any observed outcomes. That is what the error message says “lavaan NOTE: this may be a symptom that the model is not identified.” You basically have a regression without dependent variable. 

But more fundamentally, I do not think that the model can be interpreted like you want to interpret it. You seem to want to interpret it so that you have a latent variable PSR, which derives it’s meaning from the formative “dimensions", and then regress that on two other LVs. However, the regression equation that you define is:

PSR = ECO + LEG + ETI + FIL + MED + SAT + SEST (note that there is no error term.)

So how to get the result  that you want is the next question? On this line 

PSR <~ ECO + LEG + ETI + FIL + MED

you define PSR as a weighted sum of five latent variables. The weights are not identified, so you must choose some values for them. If you have no a priori reason to not do so, you should use equal weights. 

Because PSR is an exact sum of other LVs, then all effects that go toward that LV must be completely mediated by the LVs that form PSR. The simplest way to model this is to omit the PSR LV from the model altogether and instead specify a model where each of the ECO-MED LVs are regressed on SAT and SESTs. The total effects of SAT and SEST on PSR are then simply sums of their effects on the ECO-MED LVs. 

For a fairly accessible explanation of the math of this types of models, see

Edwards, J. R. (2001). Multidimensional constructs in organizational behavior research: An integrative analytical framework. Organizational Research Methods, 4(2), 144–192.

Mikko

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[Edward Rigdon]

If you mean for the two factors SAT and SEST to assist in identification of the parameters related to PSR, then they have to be dependent on PSR, as in

SAT + SEST ~ PSR

not 

PSR ~ SAT + SEST

--Ed Rigdon

--

[Sylvia López Davis]

Thank you very much, Mikko, for your clarifying and prompt response. 

Kind regards,

Sylvia.

[Sylvia López Davis]

Thank you Edward for the clarification of the mistake. 

Just another question:

Does it mean that the reflectively measured variables that I would use to identify the model MUST be outcomes of the construct (consequences)?

What happens when the literature identifies mainly antecedents of related constructs, but not consequences or outcomes? could I ever use antecedents in this first step to identify the model?

Thanks a lot in advance, again. 

Sylvia.

[Edward Rigdon]

Sylvia—

     Yes, you must have outcomes to identify a factor model, for two reasons.  Effectively, we are representing the covariance matrix of the variables as functions of model parameters and then solving for the model parameters as functions of the elements of the covariance matrix.  With the focal variable itself not observed, we must find the parameters related to it in the variances and covariances of other variables.

     So the first reason is that you will not find all of the needed parameters represented in this way.  In particular, by the way regression works, you will not find the residual variance of the focal variable associated with the variable’s antecedents, because we generally assume that the residual is orthogonal to the predictors.

     The second reason is that an element of the covariance matrix is consumed or expended each time we identify a parameter.  The elements must be described by few enough parameters that we can borrow a covariance from over *here* to estimate a parameter from over *there*.  With antecedents, their variances and covariances are just enough to cover their own parameters.  But with multiple dependent variables and a constrained factor model, we often have covariances left over, among the outcomes, that we can use to identify parameters related to unobserved variables.  Without outcomes, however, these opportunities do not exist.

[Mikko Rönkkö]

Hi,

Yes, a latent variable MUST always have some observed consequences to be identified.

If you use antecedents, then you must fix all paths toward the latent variable to some values. In this case, it would be easier not to have the latent variable at all in the model just calculate the estimates related to the omitted latent variable by hand after model estimation.

Mikko