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lavaan multi-level SEM in OpenMX

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twoertwein's picture
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Joined: 12/11/2019 - 23:04
lavaan multi-level SEM in OpenMX

I'm trying to replicate this example two-level SEM from lavaan in OpenMx http://lavaan.ugent.be/tutorial/multilevel.html

Lavaan's log-likelihood is -23309.87 but with the following OpenMx code I get only -26495.56. I have also tried to use the estimated parameters from lavaan as fixed parameters in the OpenMx model - the log-likelihood gets even worse then. I assume that the model structure in OpenMx is not the same as the structure model in lavaan.

A few notes on the code: lavaan doesn't allow a connection from an observed variable to a latent variable (I introduced 'pseudo-latent' variables (I prefixed them with '_'): they have have one loading (with factor 1) to the observed variable, have its mean, its (co)variance, and the observed variable has no residual - lavaan does the same internally, see inspect(lavaan_fit, what='est')); lavaan seems to use the biased (co)variance (I resale the covariance matrix with scale/cluster_scale).

 
library('lavaan')
library('OpenMx')
 
data <- Demo.twolevel
data$cluster <- sapply(data$cluster, as.factor)
 
y <- c('y1', 'y2', 'y3')
x <- c('x1', 'x2', 'x3')
x_ <- c('_x1', '_x2', '_x3')
w <- c('w1', 'w2')
w_ <- c('_w1', '_w2')
manifest_l1 <- c(y, x)
latent_l1 <- c('fw', x_)
manifest_l2 <- c(y, w)
latent_l2 <- c('fb', w_)
level2 <- data[!duplicated(data$cluster),]
 
scale <- (dim(data)[1] - 1.0) / dim(data)[1]
cluster_scale <- (dim(level2)[1] - 1.0) / dim(level2)[1]
 
between <- mxModel(
    "between",
    type="RAM",
    manifestVars=manifest_l2,
    latentVars=latent_l2,
    mxData(observed=level2, type="raw", primaryKey="cluster"),
    mxPath(from=y, arrows=2, free=TRUE, values=diag(var(data[y], na.rm=TRUE)) / 2),
    mxPath(from=w, arrows=2, free=FALSE, values=0),
    mxPath(from='fb', arrows=2, free=FALSE, values=1),
 
    mxPath(from=w_, to=w, arrows=1, free=FALSE, values=1),
    mxPath(from=w_, arrows=2, free=FALSE, values=diag(var(level2[w], na.rm=TRUE)) * cluster_scale),
    mxPath(from='_w1', to='_w2', arrows=2, free=FALSE, values=var(level2[w], na.rm=TRUE)['w1', 'w2'] * cluster_scale),
 
    mxPath(from='fb', to=y, arrows=1, free=TRUE, values=1, labels=c('fb_y1', 'fb_y2', 'fb_y3')),
    mxPath(from=w_, to='fb', arrows=1, free=TRUE, values=1, labels=c('w1_fb', 'w2_fb')),
 
    mxPath(from='one', to=y, free=TRUE, values=colMeans(data[y], na.rm=TRUE)),
    mxPath(from='one', to=w, free=FALSE, values=0),
    mxPath(from='one', to=w_, free=FALSE, values=colMeans(level2[w], na.rm=TRUE)))
 
within <- mxModel(
    "Demo_twolevel",
    type="RAM",
    between,
    manifestVars=manifest_l1,
    latentVars=latent_l1,
    mxData(observed=data, type="raw"),
    mxPath(from=y, arrows=2, free=TRUE, values=diag(var(data[y], na.rm=TRUE)) / 2),
    mxPath(from=x, arrows=2, free=FALSE, values=0),
    mxPath(from='fw', arrows=2, free=FALSE, values=1),
 
    mxPath(from=x_, to=x, arrows=1, free=FALSE, values=1),
    mxPath(from=x_, arrows=2, free=FALSE, values=diag(var(data[x], na.rm=TRUE)) * scale),
    mxPath(from='_x1', to='_x2', arrows=2, free=FALSE, values=var(data[x], na.rm=TRUE)['x1', 'x2'] * scale),
    mxPath(from='_x1', to='_x3', arrows=2, free=FALSE, values=var(data[x], na.rm=TRUE)['x1', 'x3'] * scale),
    mxPath(from='_x2', to='_x3', arrows=2, free=FALSE, values=var(data[x], na.rm=TRUE)['x2', 'x3'] * scale),
 
    mxPath(from='fw', to=y, arrows=1, free=TRUE, values=1, labels=c('fw_y1', 'fw_y2', 'fw_y3')),
    mxPath(from=x_, to='fw', arrows=1, free=TRUE, values=1, labels=c('x1_fw', 'x2_fw', 'x3_fw')),
 
    mxPath('between.y1', 'y1', values=1, free=FALSE, joinKey="cluster"),
    mxPath('between.y2', 'y2', values=1, free=FALSE, joinKey="cluster"),
    mxPath('between.y3', 'y3', values=1, free=FALSE, joinKey="cluster"),
 
    mxPath(from='one', to=manifest_l1, free=FALSE, values=0),
    mxPath(from='one', to=x_, free=FALSE, values=colMeans(data[x], na.rm=TRUE)))
 
fit <- mxRun(within)
 
summary(fit)
 
 
AdminRobK's picture
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Joined: 01/24/2014 - 12:15
mxVersion() ?

I'm guessing you're using an out-of-date version of OpenMx. When I try to run your MxModel, I get this:

Error in runHelper(model, frontendStart, intervals, silent, suppressWarnings,  : 
  between.data: primary key must be an integer or factor column in raw observed data

The variable named "cluster" in the dataset is of type "numeric", but it needs to be of type "integer" or "factor". My mxVersion() :

OpenMx version: 2.15.5.2 [GIT v2.15.5-2-g471224d3-dirty]
R version: R version 3.6.1 (2019-07-05)
Platform: x86_64-pc-linux-gnu
Default optimizer: CSOLNP
NPSOL-enabled?: Yes
OpenMP-enabled?: Yes

I'm running a build of OpenMx that's a few commits newer than the most recent release.

twoertwein's picture
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Joined: 12/11/2019 - 23:04
Thank you for quick reply!

Thank you for quick reply!
I'm sorry, I forgot the line to convert the cluster column in the above example. I get the same error message without it.

data <- Demo.twolevel
data$cluster <- sapply(data$cluster, as.factor)

I updated my original post.

mxVersion()

OpenMx version: 2.15.4 [GIT v2.15.4]
R version: R version 3.5.2 (2018-12-20)
Platform: x86_64-pc-linux-gnu
Default optimizer: CSOLNP
NPSOL-enabled?: No
OpenMP-enabled?: Yes
AdminRobK's picture
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Joined: 01/24/2014 - 12:15
reproduced

Well, once I convert "cluster" to datatype "factor", I reproduce what you report. I don't know enough about lavaan or OpenMx's multilevel feature to be able to guess what's going on, though.

jpritikin's picture
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Joined: 05/24/2012 - 00:35
confused

Hm, you specified the regressions incorrectly. You need to use definition variables. After I fixed that, I noticed that the data for the indicators (y1,y2,y3) varies by row. You use data[!duplicated(data$cluster),] to get the level2 data, but is this what lavaan is doing? I don't think so. I think the between level is entirely latent. If make the between level latent and link the y1,y2,y3 indicators across levels then I get the same parameter estimates.

library('lavaan')
library('OpenMx')
 
data <- Demo.twolevel
 
model <- '
        level: 1
            fw =~ y1 + y2 + y3
            fw ~ x1 + x2 + x3
        level: 2
            fb =~ y1 + y2 + y3
            fb ~ w1 + w2
    '
lfit <- sem(model = model, data = Demo.twolevel, cluster = "cluster")
 
data$cluster <- sapply(data$cluster, as.factor)
 
y <- c('y1', 'y2', 'y3')
x <- c('x1', 'x2', 'x3')
w <- c('w1', 'w2')
latent_l1 <- c('fw', x)
manifest_l2 <- c()
latent_l2 <- c('fb', y, w)
level2 <- data[!duplicated(data$cluster),]
 
between <- mxModel(
  "between",
  type="RAM",
  manifestVars=manifest_l2,
  latentVars=latent_l2,
  mxData(observed=level2, type="raw", primaryKey="cluster"),
  mxPath(from=y, arrows=2, free=TRUE, values=diag(var(data[y], na.rm=TRUE)),
         labels=paste0("y",1:3,"_var")),
  mxPath(from='fb', arrows=2, free=TRUE, values=1, labels="fb_var"),
  mxPath(from=y, arrows=2, free=TRUE, values=1,
         labels=paste0("y",1:3,"_var_b")),
 
  mxPath(from='fb', to=y, arrows=1, free=c(FALSE, rep(TRUE,2)),
         values=1, labels=c('fb_y1', 'fb_y2', 'fb_y3')),
 
  mxPath(from=w, to='fb', free=TRUE, labels=paste0('w',1:2,"_fb")),
  mxPath(from='one', to=w, free=FALSE, labels=paste0("data.w",1:2)),
  mxPath('one', y))
 
within <- mxModel(
  "Demo_twolevel",
  type="RAM",
  between,
  manifestVars=manifest_l1,
  latentVars=latent_l1,
  mxData(observed=data, type="raw"),
  mxPath(from=y, arrows=2, free=TRUE, values=diag(var(data[y], na.rm=TRUE)),
         labels=paste0("y",1:3,"_var_w")),
  mxPath(from='fw', arrows=2, free=TRUE, values=1, labels="fw_var"),
 
  mxPath(from='fw', to=y, arrows=1, free=c(FALSE,rep(TRUE,2)),
         values=1, labels=c('fw_y1', 'fw_y2', 'fw_y3')),
  mxPath(from=x, to='fw', arrows=1, free=TRUE, values=1,
         labels=c('x1_fw', 'x2_fw', 'x3_fw')),
 
  mxPath(paste0('between.y',1:3), paste0('y',1:3), values=1, free=FALSE, joinKey="cluster"),
 
  mxPath(from='one', to=manifest_l1, free=FALSE),
  mxPath(from='one', to=x, free=FALSE, labels=paste0("data.x",1:3)))
 
fit <- mxRun(within)
 
summary(fit)

The logLik doesn't match at all. It looks like a lavaan bug or maybe there is some other way to compute the likelihood? I'm not sure.

AdminRobK's picture
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Joined: 01/24/2014 - 12:15
normalized?

Maybe lavaan drops normalizing constants from its likelihood?

AdminNeale's picture
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Joined: 03/01/2013 - 14:09
Might differ by a constant?

If so, the LRT of model vs submodel should be equivalent across the two programs. That the estimates are the same suggests that both have found the MLEs.

twoertwein's picture
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Joined: 12/11/2019 - 23:04
Thank you!

I didn't know that you can specify definition-variables like that, I assumed you need to do that manually the way I had tried to do it :)

I'm sorry, I should have posted how I use lavaan (yes, it doesn't distinguish between data for level1 and level2 in its interface; and I set the variance of the latent variables to 1 instead of one fixed factor loading).

library('lavaan')
 
model <- '
        level: 1
            fw =~ y1 + y2 + y3
            fw ~ x1 + x2 + x3
        level: 2
            fb =~ y1 + y2 + y3
            fb ~ w1 + w2
'
 
fit <- sem(model = model, data = Demo.twolevel, cluster = "cluster", auto.fix.first = FALSE, std.lv = TRUE)

Yes, the estimated parameters match but the logL is different (OpenMx -12076.57, lavaan -23309.87). The constant -0.5nlog(2*pi) is only -2297.35.

AdminRobK's picture
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Joined: 01/24/2014 - 12:15
constant

The loglikelihoods differ by some constant. If I fix one of the factor loadings to zero like this,

model <- '
        level: 1
            fw =~ y1 + 0*y2 + y3
            fw ~ x1 + x2 + x3
        level: 2
            fb =~ y1 + y2 + y3
            fb ~ w1 + w2
    '
lfit <- sem(model = model, data = Demo.twolevel, cluster = "cluster")

, I get an lfit with a loglikelihood about 350 units lower than the full model,

> logLik(lfit) - -23309.87
'log Lik.' -350.4593 (df=19)

. If I likewise fix that loading in the MxModel,

foo <- omxSetParameters(model=within,labels="fw_y2",free=F,values=0)
foo <- mxRun(foo)

, I get a foo with a loglikelihood again about 350 units lower than the full model,

> logLik(foo) - -12076.57
'log Lik.' -350.455 (df=19)

. In other words, the log of the likelihood ratio is the same out to five significant figures in both cases, indicating that the two packages use proportional likelihood functions.

esei's picture
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Joined: 08/05/2016 - 02:03
Different model

Hi,

Could it be that lavaan is fitting the joint distribution of the endogenous and exogenous variables?

if i change the OpenMx script i get similar results.

library('lavaan')
library('OpenMx')
 
data <- Demo.twolevel
 
model <- '
level: 1
fw =~ y1 + y2 + y3
level: 2
fb =~ y1 + y2 + y3
fb ~ w1 + w2
'
 
model <- '
level: 1
fw =~ y1 + y2 + y3
fw ~ x1 + x2 + x3
level: 2
fb =~ y1 + y2 + y3
fb ~ w1 + w2
'
 
lfit <- sem(model = model, data = Demo.twolevel, cluster = "cluster", fixed.x = T)
 
data$cluster <- sapply(data$cluster, as.factor)
y <- c('y1', 'y2', 'y3')
x <- c('x1', 'x2', 'x3')
w <- c('w1', 'w2')
manifest_l1 <- c(y, x)
latent_l1 <- c('fw')
manifest_l2 <- c(w)
latent_l2 <- c('fb', y)
level2 <- data[!duplicated(data$cluster),]
 
vc_w = cov(level2[,w])
vc_x = cov(data[,x])
 
between <- mxModel(
  "between",
  type="RAM",
  manifestVars=manifest_l2,
  latentVars=latent_l2,
  mxData(observed=level2, type="raw", primaryKey="cluster"),
  mxPath(from=y, arrows=2, free=TRUE, values=diag(var(data[y], na.rm=TRUE)),
         labels=paste0("y",1:3,"_var")),
  mxPath(from='fb', arrows=2, free=TRUE, values=1, labels="fb_var"),
  mxPath(from=y, arrows=2, free=TRUE, values=1,
         labels=paste0("y",1:3,"_var_b")),
  mxPath(from="w1", to = "w1", arrows=2, free=F, values = vc_w[1,1],
         labels="w11"),
  mxPath(from="w1", to = "w2", arrows=2, free=F, values = vc_w[2,1],
         labels="w12"),
  mxPath(from="w2", to = "w2", arrows=2, free=F, values = vc_w[2,2],
         labels="w22"),
  mxPath(from='fb', to=y, arrows=1, free=c(FALSE, rep(TRUE,2)),
         values=1, labels=c('fb_y1', 'fb_y2', 'fb_y3')),
  mxPath(from=w, to='fb', free=TRUE, values = 1, labels=paste0('w',1:2,"_fb")),
  mxPath(from='one', to=w, free=FALSE, values = colMeans(level2[,w])),
  mxPath('one', y))
 
within <- mxModel(
  "Demo_twolevel",
  type="RAM",
  between,
  manifestVars=manifest_l1,
  latentVars=latent_l1,
  mxData(observed=data, type="raw"),
  mxPath(from=y, arrows=2, free=TRUE, values=diag(var(data[y], na.rm=TRUE)),
         labels=paste0("y",1:3,"_var_w")),
  mxPath(from='fw', arrows=2, free=TRUE, values=1, labels="fw_var"),
  mxPath(from='fw', to=y, arrows=1, free=c(FALSE,rep(TRUE,2)),
         values=1, labels=c('fw_y1', 'fw_y2', 'fw_y3')),
  mxPath(from="x1", to = "x1", arrows=2, free=F, values = vc_x[1,1],
         labels="x11"),
  mxPath(from="x1", to = "x2", arrows=2, free=F, values = vc_x[1,2],
         labels="x12"),
  mxPath(from="x1", to = "x3", arrows=2, free=F, values = vc_x[1,3],
         labels="x13"),
  mxPath(from="x2", to = "x2", arrows=2, free=F, values = vc_x[2,2],
         labels="x22"),
  mxPath(from="x2", to = "x3", arrows=2, free=F, values = vc_x[2,3],
         labels="x23"),
  mxPath(from="x3", to = "x3", arrows=2, free=F, values = vc_x[3,3],
         labels="x33"),
  mxPath(from=x, to='fw', arrows=1, free=TRUE, values=1,
         labels=c('x1_fw', 'x2_fw', 'x3_fw')),
  mxPath(paste0('between.y',1:3), paste0('y',1:3), values=1, free=FALSE, joinKey="cluster"),
  mxPath(from='one', to=manifest_l1, values = colMeans(data[,x]), free=FALSE) 
  )
fit <- mxTryHard(within)
 
logLik(fit)
logLik(lfit)
AdminRobK's picture
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Joined: 01/24/2014 - 12:15
I think you're right.

I think you're right.

bwiernik's picture
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Joined: 01/30/2014 - 19:39
Unbiased variance in lavaan
lavaan seems to use the biased (co)variance (I resale the covariance matrix with scale/cluster_scale).

To use the unbiased covariance estimator, add likelihood = "wishart" to your lavaan call. http://lavaan.ugent.be/tutorial/est.html