Best Practices for 3lvl measurement SEM

I'm currently building a 3lvl MSEM including measurement models. The Pritikin et al. (2017) paper is relevant and helpful but whereas they have observed variables on every level, I mainly want to use levels to control for the between-part of the variance of items I measured only on the 1st level. Hence, I want to make sure I don't misclassify my model (see below). Assuming the model is right, I'm also grateful on tips how to deal with a non-convex Hessian when trying to fit it.

**The overall goal**: My goal is to build a model with two variables measured on the 1st "individual" level (X, Y), two variables measured on the 2nd "team" level (V, W) and a 3rd "district" level to control for shared variance between teams. All X, Y, V, W will be related through regressions on the 2nd level. I compare my models with lavaan as far as feasible (lavaan isn't able to do 3lvl MSEMs).

**In this step, I start with a simple measurement model of one latent variable, X, measured through n items. I want to build a 3-lvl measurement model to do justice to the team and district structure underlying the data.** After building these models also for Y, V, and W, I want to build the full regression model by adding paths between the variables.

**2 Levels**: I get the single-level regressions and 2lvl-measurement models to agree between OpenMx and lavaan. This means lavaan seems to make the following implicit assumptions:
- fixing first factor loading to 1 while freeing the variance of latent variables (ok)
- fixing the means of variables to zero on the within level (why is this a good choice?)
- adding variables as latent variables to the between level (sure, this is what "levels" is all about)
- adding a path from between-level (latent) variables to the corresponding manifest variables on the within-level, fixed to 1 (ok)

**3 Levels**: I now added a third level ("district") and followed these basic steps. **Please see the attached file.**
There are two modeling choices I'm not 100% sure about - if you know about best practices I'd be very happy as I wasn't able to find any. (E.g., Pritikin et al. link latent variables, as manifest variables differ on each level; I'm not an expert in MPLUS but it seems many of their assumptions regarding these choices are not made explicit when defining the model.)
- should I fix means of the items to zero on the between level? I am assuming different team means so opted not do to so (but I also thought there would be different means on the individual level)
- should I link the latent variables corresponding to the items between the district-lvl and the between-lvl (lvls 3 and 2)? Teams are clustered in districts, so this makes sense. (Or do I link lvls 3 and 1 directly as this is the actual relationship I'm interested in?)

With these decisions out of the way, here is the code I used.
**Running the code** I get the following warning and standard errors are through the roofs: In model 'within' Optimizer returned a non-zero status code 5. The Hessian at the solution does not appear to be convex. See ?mxCheckIdentification for possible diagnosis (Mx status RED).

Please note that I have used the fitted model to eye-ball starting values of the right order of magnitude. Overall district-level variances are very very small, several of them turn negative in the estimates. Might this be a reason why mxRun() struggles with the model?

I'm grateful for your support. I hope I've just made a simple mistake somewhere and hope it's not too stupid.


data$team <- as.factor(as.character(data$team))
data$district <- as.factor(as.character(data$district))
dataL1 <- data
dataL2 <- data[!duplicated(data$team),]
dataL3 <- data[!duplicated(data$district),]
mxDataL1 <- mxData(observed=dataL1, type="raw")
mxDataL2 <- mxData(observed=dataL2, type="raw", primaryKey = "team")
mxDataL3 <- mxData(observed=dataL3, type="raw", primaryKey = "district")

nx <- 14
xItems <- paste0("x",1:nx)

# Define Paths
# Paths X lvl 1
pathXItemsVar <- mxPath(from=xItems, arrows=2, free=TRUE, values = diag(var(data[xItems], na.rm=TRUE)), labels=paste0("e_x",1:nx)) # residual variances
pathXItemsMeans <- mxPath(from = "one", to = xItems, free=TRUE, values=1, labels=paste0("m_x",1:nx)) # means items
pathXItemsMeans0 <- mxPath(from = "one", to = xItems, free = FALSE, values=0, labels=paste0("m_x",1:nx)) # manifest means, constrained to 0
pathXLoadings <- mxPath(from="x", to=xItems, free=c(FALSE,rep(TRUE,nx-1)), values=1, labels=paste0("fl_x_w",1:nx)) # factor loadings, fixing first factor to 1
pathXLatentVar <- mxPath(from="x", arrows=2, free=TRUE, values=0.2, labels="var_x") # latent variance

# Paths X lvl 2
pathXItemsVarL2 <- mxPath(from=xItems, arrows=2, free=TRUE, values = 0.1, labels=paste0("e_l2x",1:nx)) # (latent) items variance
pathXItemsMeansL2 <- mxPath(from = "one", to = xItems, free=TRUE, values=1, labels=paste0("m_l2x",1:nx)) # (latent) items means
pathXItemsMeansL20 <- mxPath(from = "one", to = xItems, free=FALSE, values=0, labels=paste0("m_l2x",1:nx)) # (latent) items means, constrained to 0 (not used here)
pathXLoadingsL2 <- mxPath(from = "xTeam", to = xItems,free = c(FALSE,rep(TRUE,nx-1)), values = 1, labels = paste0("fl_x_b",1:nx)) # latent Team factor, loadings constrained to same as on within level, first fixed to 1
pathXLatentVarL2 <- mxPath(from = "xTeam", arrows = 2, values = 0.1, free = TRUE, labels = "xTeam_var") # latent team factor, variance

# Paths X lvl 3
pathXItemsVarL3 <- mxPath(from=xItems, arrows=2, free=TRUE, values = 0.1, labels=paste0("e_l3x",1:nx)) # (latent) items variance
pathXItemsMeansL3 <- mxPath(from = "one", to = xItems, free=TRUE, values=1, labels=paste0("m_l3x",1:nx)) # (latent) items means
pathXLoadingsL3 <- mxPath(from = "xDistrict", to = xItems, values = 1, free = c(FALSE,rep(TRUE,nx-1)), labels = paste0("fl_x_d",1:nx)) # latent Team factor, loadings constrained to same as on within level, first fixed to 1
pathXLatentVarL3 <- mxPath(from = "xDistrict", arrows = 2, values = 1, free = TRUE, labels = "xDistrict_var") # latent team factor, variance

# OpenMx - 3 lvl
modelXLatentL3 <- mxModel(
model = "district",
type = "RAM",
manifestVars = c(),
latentVars = c("xDistrict",xItems),
mxData(observed = dataL3, type="raw", primaryKey = "district"),
pathXItemsVarL3,
pathXItemsMeansL3,
pathXLoadingsL3,
pathXLatentVarL3
)

modelXLatentL2 <- mxModel(
model = "between",
type = "RAM",
modelXLatentL3,
manifestVars = c(),
latentVars = c("xTeam",xItems),
mxDataL2,
mxPath(from = paste0("district.",xItems), to = xItems, values=1, free=FALSE, joinKey="district"), #linking (latent) district lvl items
pathXItemsMeansL2, pathXItemsVarL2,
pathXLatentVarL2, pathXLoadingsL2
)

modelXLatentL123 <- mxModel(
model = "within",
type = "RAM",
modelXLatentL2,
manifestVars = c(xItems),
latentVars = c("x"),
mxDataL1,
mxPath(from = paste0("between.",xItems), to = xItems, values=1, free=FALSE, joinKey="team"), #linking (latent) between lvl Items to manifest vars
pathXItemsMeans0, pathXItemsVar,
pathXLatentVar, pathXLoadings
)

fitXLatentL123 <- mxRun(modelXLatentL123, intervals = TRUE)
summary(fitXLatentL123)