# Identification problems with a structural equation model of experimental data

I have performed an experiment in which I manipulated three factors and I would like to model latent variables that those factors affect and then estimate the effects of the latents on response variables I measured (two gases). I am new to SEM and am working on implementing this model in R with lavaan. I have spent a few weeks working on this and got a basic mediation model without latents working but cannot get this second model to work. I believe the problem is that the three exogenous variables (ie, manipulated factors) do not have adequate variance so the model is under-identified. See the model structure below.

So there are 3 exogenous variables (all are experimentally manipulated with multiple levels), 2 latent variables (technically I think they are composite latent variables), and 2 measured endogenous variables (Gas1 and Gas2). I believe my model is setup to estimate 8 parameters. By the rule that t<=p(p+1)/2, where t=# params and p=# variables, I should be able to estimate up to 15 parameters with my 5 variables. However, I keep getting errors that indicate that my model is likely not identified and I also get negative degrees of freedom (DF=-3).

Ultimately my interest is to estimate the effects that OD and OC have on Gas2. However, my questions at this point are:

1. Is under-identification due to low variance of the manipulated variables likely the cause this problem?
2. Is there a better way to go about structuring such a SEM based on experimental (ie, manipulated not observational) data? Either specific to this model or more generally.

Sorry if this is a basic question- Thanks for any help!

Code of the model:

SEM3<-'
OC<~Ltype+H2Opercent+MD
OD<~H2Opercent+MD

Gas1~OC
Gas2~OC+OD
'
SEM3_full<-lavaan::sem(SEM3,data=nc_full_DF)


Errors and output:

SEM3_full<-lavaan::sem(SEM3,data=nc_full_DF_SO)

Warning message:

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.

> summary(SEM3_full)
lavaan (0.5-17) converged normally after  13 iterations

Used       Total
Number of observations                           139         157

Estimator                                         ML
Minimum Function Test Statistic                   NA
Degrees of freedom                                -3
Minimum Function Value               0.7803616416535

Parameter estimates:

Information                                 Expected
Standard Errors                             Standard

Estimate  Std.err  Z-value  P(>|z|)
Composites:
OC <~
Ltype             0.000
H2Opercent        0.000
MD                0.000
OD <~
H2Opercent        0.000
MD                0.000

Regressions:
Gas1 ~
OC                0.000
Gas2 ~
OC                0.000
OD                0.000

Covariances:
OC ~~
OD                0.000
Gas1 ~~
Gas2              0.125

Variances:
Gas1              0.042
Gas2              3.626
OC                0.000
OD                0.000


Do you have any missing data?

I don't think this part is right:

Gas1~O2cons
Gas2~OC+OD


Models with formative factors are very tricky to identify, and you're not even close. You can't have a formative latent with a single indicator. I think your model would be identified if you constrain the residual variance of the two latents to zero, and constrain the paths coming out to 1.00, but then they won't do anything, your model would be equivalent to a model where they weren't there. (Just regress the two gases on the three variables on the left.

Formative indicators are usually used in MIMIC models. A latent with only formative indicators is kind of weird.

• Thanks, Jeremy. You're right about the mistyped formula. (The error was not in my code so I've fixed it here). What you're calling formative factors I knew as composite variables and now I'm reading up on them in Treiblmaier's article "Formative Constructs Implemented via Common Factors" – DirtStats Mar 26 '15 at 20:46
• When you state that "A latent with only formative indicators is kind of weird" did you mean the gases as latents? Gas1 and Gas2 were measured. I'm going to look into MIMIC models to better understand what you're saying about them. If you have any suggested reading please let me know! – DirtStats Mar 26 '15 at 20:55
• I mean OC and OD. It's my understanding that they are latent variables. But they're not really serving any function. Think about it this way: The saturated model would have 6 regression parameters (gas1 and gas2 regressed on the 3 predictors). Your model has 8 parameters. You are trying to get out more than you are putting in. You can add some restrictions to get those df back, but then you're just estimating a model without those latents. – Jeremy Miles Mar 26 '15 at 23:07
• I see. Now I understand why it's not identified. I guess the "t rule" doesn't work with formative factors. Conceptually, OC and OD represent distinct (but unmeasured) mechanisms through which the experimental factors affect Gas1 and Gas2. Their inclusion allows me to compare the influence of each on Gas2 and test related hypotheses. – DirtStats Mar 27 '15 at 19:16
• I have more data (ie, other measured responses) that I can add to the model as predictors of the gases or as responses from the latent variables. I was holding off on that to keep the model simple but since the additions will move the model towards saturation by giving me more df I'll build it up. Thanks again! – DirtStats Mar 27 '15 at 19:17