My understanding is that one should not be able to use both a regression and covariance formula between two variable in lavaan. Nevertheless, I am able to estimate the below model. Is this an issue with syntax, identification, or a bug?
library(lavaan)
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Specify model
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
mod1 <- '
# Measurement model
# - 2 factors {rs, wr} at 2 time points {1,2}, 4 indicators apiece
# - fix first loading to 1 for identification
# - constrain factor loadings to be equal across time
r1 =~ 1*x1_1 + x1_2 + x1_3 + x1_4
r2 =~ 1*x2_1 + x2_2 + x2_3 + x2_4
w1 =~ 1*x1_5 + x1_6 + x1_7 + x1_8
w2 =~ 1*x2_5 + x2_6 + x2_7 + x2_8
# Estimate latent factor variances
# - only bc model is identified via fixed 1st loading
r1 ~~ NA*r1
r2 ~~ NA*r2
w1 ~~ NA*w1
w2 ~~ NA*w2
# Estimate covariance between contemporaneously-measured latent factors
r1 ~~ w1
r2 ~~ w2
# Regressions
r2 ~ r1
w2 ~ w1
# Estimate covariance between factors across time
# NOTE: unclear why I can estimate these parameters given regression formulas
r2 ~~ r1
w2 ~~ w1
'
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Create sample covariance matrix
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Variable names
vars <- c("x1_1","x1_2","x1_3","x1_4","x1_5","x1_6","x1_7","x1_8",
"x2_1","x2_2","x2_3","x2_4","x2_5","x2_6","x2_7","x2_8")
# Create matrix
sample_cov <- matrix(data = c(0.999833293078379,0.487956444582057,0.483664056467545,0.400998426513142,0.491202933952403,
0.353931449316117,0.285944906976342,0.259668866614031,0.597588804705785,0.389983101571606,
0.389986282460954,0.352688728884064,0.413556920977731,0.310728635134553,0.265334745911224,
0.275968834910899,0.487956444582057,0.987898167268851,0.597282474952359,0.624490928441204,
0.247125607027116,0.329128073457,0.292935925099769,0.248056475192883,0.410618949682518,
0.491198808670857,0.428587638131653,0.415870887874728,0.263648376842619,0.335405511090691,
0.299434930929212,0.303516673953949,0.483664056467545,0.597282474952359,0.987591489156997,
0.70697049346826,0.273060714265672,0.32252711049228,0.476538022724238,0.405092823248217,
0.383809997408897,0.420133217722261,0.532069928685633,0.472834561550818,0.281130854462545,
0.320879117346738,0.394828887309731,0.367727058341432,0.400998426513142,0.624490928441204,
0.70697049346826,1.01074290449809,0.196256330906052,0.263382784920277,0.375529025588328,
0.416310437347982,0.323177010703894,0.396708104792945,0.448962585030441,0.485905309303198,
0.214011581764438,0.286993920549238,0.324133600586384,0.376046637537441,0.491202933952403,
0.247125607027116,0.273060714265672,0.196256330906052,0.998610835001528,0.573990161485373,
0.442329722010083,0.40108886269627,0.344987740402206,0.22482827808808,0.216597438677272,
0.155839106657515,0.497912057832072,0.350927415574806,0.287668780643262,0.256277533580356,
0.353931449316117,0.329128073457,0.32252711049228,0.263382784920277,0.573990161485373,
1.00444245494004,0.666026932163401,0.607357662014435,0.248469574577396,0.288661477004284,
0.288770354292082,0.236124802603109,0.391592159472882,0.530091538181447,0.416582877505298,
0.407030707616969,0.285944906976342,0.292935925099769,0.476538022724238,0.375529025588328,
0.442329722010083,0.666026932163401,1.00046601549333,0.782931683531597,0.213627568752322,
0.283215107581028,0.366472357153765,0.288307993754945,0.326451332805336,0.432225371555803,
0.537107909732078,0.507474265680886,0.259668866614031,0.248056475192883,0.405092823248217,
0.416310437347982,0.40108886269627,0.607357662014435,0.782931683531597,1.00617979684262,
0.223617186711885,0.264438350331771,0.339792125173085,0.355808519681323,0.284308372166869,
0.401670251542325,0.475918180445249,0.591423836570424,0.597588804705785,0.410618949682518,
0.383809997408897,0.323177010703894,0.344987740402206,0.248469574577396,0.213627568752322,
0.223617186711885,0.989779924090633,0.653160349325124,0.63806316639147,0.594152250976385,
0.467091528287757,0.395182555441065,0.368953579075386,0.37879303032014,0.389983101571606,
0.491198808670857,0.420133217722261,0.396708104792945,0.22482827808808,0.288661477004284,
0.283215107581028,0.264438350331771,0.653160349325124,0.991757099571506,0.762963568718599,
0.762480884806328,0.384610284210552,0.50670524190018,0.487150198379972,0.477553716549035,
0.389986282460954,0.428587638131653,0.532069928685633,0.448962585030441,0.216597438677272,
0.288770354292082,0.366472357153765,0.339792125173085,0.63806316639147,0.762963568718599,
1.00053469496961,0.822834292300233,0.374705998031281,0.501332849270603,0.573598299835578,
0.543045239225744,0.352688728884064,0.415870887874728,0.472834561550818,0.485905309303198,
0.155839106657515,0.236124802603109,0.288307993754945,0.355808519681323,0.594152250976385,
0.762480884806328,0.822834292300233,1.00022307320538,0.335192622088254,0.462799373720387,
0.543255621639804,0.601852486639232,0.413556920977731,0.263648376842619,0.281130854462545,
0.214011581764438,0.497912057832072,0.391592159472882,0.326451332805336,0.284308372166869,
0.467091528287757,0.384610284210552,0.374705998031281,0.335192622088254,0.990336396260981,
0.649930931016302,0.548528520259829,0.48836742442375,0.310728635134553,0.335405511090691,
0.320879117346738,0.286993920549238,0.350927415574806,0.530091538181447,0.432225371555803,
0.401670251542325,0.395182555441065,0.50670524190018,0.501332849270603,0.462799373720387,
0.649930931016302,0.997339136613221,0.733410325563141,0.685756835860867,0.265334745911224,
0.299434930929212,0.394828887309731,0.324133600586384,0.287668780643262,0.416582877505298,
0.537107909732078,0.475918180445249,0.368953579075386,0.487150198379972,0.573598299835578,
0.543255621639804,0.548528520259829,0.733410325563141,0.994147669703169,0.811982851534474,
0.275968834910899,0.303516673953949,0.367727058341432,0.376046637537441,0.256277533580356,
0.407030707616969,0.507474265680886,0.591423836570424,0.37879303032014,0.477553716549035,
0.543045239225744,0.601852486639232,0.48836742442375,0.685756835860867,0.811982851534474,
0.996970290203717),
nrow = length(vars),
ncol = length(vars),
dimnames = list(vars, vars))
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Fit model with sample covariance matrix
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
set.seed(123)
mod_fit <- sem(sample.cov = sample_cov,
sample.nobs = 969,
model = mod1)
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Summarize parameters
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
summary(mod_fit)
After some probing, I found that this is due to false convergence. While different starting values, numbers of iterations, optimizers, etc. replicate the issue, setting the threshold for convergence even slightly lower appropriately tells us the model failed to converge.