Baseline model is the user model in 1-factor CFA?

[Robert Wilcom]

I fit a simple 1-factor model with three observed variables and 1,000 observations. The bivariate correlations between the three variables range from 0.65 to 0.86. I used cfa() with std.lv = TRUE, which should identify the model by fixing the latent factor's variance.

What I'm not sure about is why the model fit indices all show perfect fit or produce missing values. Any ideas?  It seems to assume the baseline model is the same as the model I'm fitting, but I'm not sure why.

Below, I provide (1) excerpts of the summary output from my actual modeling and (2) code that reproduces the issue using the example data provided in (3).

1. Actual Output

2. Example code to reproduce the issue

# Model syntax
example <- "fac1 =~ x1 + x2 + x3"

# Get fit statistics
fit <- cfa(model = example, df, std.lv = TRUE) 
summary(fit, fit.measures = T, standardized = T)

# These bivariate correlations are roughly representative
cor(df$x1, df$x2, use = "complete.obs")
cor(df$x1, df$x3, use = "complete.obs")
cor(df$x2, df$x3, use = "complete.obs")

3. Example data to reproduce data.frame used in above model

(run the code below to produce a data.frame)

df <- structure(list(x1 = c(0.560264465798536, 0.933211877385453, 0.979830303833817,

                            -0.139011930926932, -0.651814621858942, 0.513646039350172, -0.372104063168755,
                            0.793356598040359, 0.513646039350172, -1.35109101858441, 0.746738171591994,
                            0.0940802013148908, -0.232248783823662, -0.372104063168755, -1.67742000372296,
                            -0.232248783823662, 0.793356598040359, 0.280553907108349, 0.839975024488723,
                            -0.185630357375297, 0.140698627763255, 0.70011974514363, 0.373790760005078,
                            -0.0457750780302031, 0.933211877385453, -0.791669901204036, 0.513646039350172,
                            -0.41872248961712, -1.21123573923932, 0.653501318695266, 0.839975024488723,
                            0.467027612901807, -0.884906754100765, 0.606882892246901, 0.886593450937088,
                            0.280553907108349, 0.606882892246901, 0.746738171591994, 0.513646039350172,
                            0.327172333556713, -3.21582807651899, 0.560264465798536, -0.0923935044785677,
                            0.606882892246901, -0.278867210272026, 0.606882892246901, -0.325485636720391,
                            0.0474617748665261, -0.511959342513849, -0.278867210272026, 0.746738171591994,
                            0.979830303833817, -3.26244650296736, -2.60978853269025, 0.327172333556713,
                            0.839975024488723, -0.0923935044785677, 0.606882892246901, 0.0940802013148908,
                            0.233935480659984, -0.325485636720391, 0.420409186453442, 0.513646039350172,
                            -0.139011930926932, -0.185630357375297, -0.41872248961712, 0.187317054211619,
                            -0.325485636720391, -2.51655167979353, -2.79626223848371, -0.139011930926932,
                            0.933211877385453, -0.558577768962213, -1.53756472437787, -1.30447259213605,
                            -0.372104063168755, 0.233935480659984, 0.420409186453442, -2.2834595475517,
                            0.280553907108349, 0.979830303833817, 0.979830303833817, 0.886593450937088,
                            0.327172333556713, 0.839975024488723, -1.67742000372296, -0.139011930926932,
                            0.140698627763255, -1.44432787148114, 0.979830303833817, -0.0457750780302031,
                            0.000843348418161536, 0.560264465798536, 0.606882892246901, 0.0940802013148908,
                            0.979830303833817, -1.16461731279095, 0.979830303833817, 0.140698627763255,
                            -2.93611751782881, 0.0474617748665261, 0.793356598040359, 0.187317054211619,
                            0.513646039350172, -0.605196195410578, 0.0940802013148908, -0.698433048307307,
                            0.187317054211619, -0.41872248961712, 0.746738171591994, 0.979830303833817,
                            0.793356598040359, 0.746738171591994, 0.793356598040359, 0.513646039350172,
                            -0.838288327652401, 0.513646039350172, 0.933211877385453, 0.70011974514363,
                            0.886593450937088, -2.74964381203535, -3.21582807651899, 0.467027612901807,
                            0.70011974514363, 0.327172333556713, 0.560264465798536, -0.511959342513849,
                            -3.21582807651899, 0.70011974514363, 0.979830303833817, 0.70011974514363,
                            0.839975024488723, 0.560264465798536, 0.467027612901807, 0.653501318695266,
                            0.933211877385453, 0.886593450937088, 0.560264465798536, -0.185630357375297,
                            -1.44432787148114, 0.513646039350172, 0.886593450937088, 0.513646039350172,
                            -0.185630357375297, 0.979830303833817, -0.185630357375297, 0.653501318695266,
                            0.979830303833817, 0.327172333556713, -1.21123573923932, 0.979830303833817,
                            0.886593450937088, 0.140698627763255, -3.02935437072554, 0.373790760005078,
                            0.793356598040359, 0.513646039350172, 0.839975024488723, -1.11799888634259,
                            -2.4233148268968, 0.0940802013148908, 0.280553907108349, -0.838288327652401,
                            0.0940802013148908, -1.77065685661969, 0.140698627763255, -0.185630357375297,
                            0.513646039350172, 0.979830303833817, 0.513646039350172, 0.187317054211619,
                            0.839975024488723, 0.746738171591994, 0.933211877385453, 0.0474617748665261,
                            -3.21582807651899, -0.185630357375297, 0.0940802013148908, 0.513646039350172,
                            -0.978143606997495, -0.698433048307307, 0.746738171591994, 0.0474617748665261,
                            -2.23684112110334, -0.278867210272026, 0.979830303833817, 0.886593450937088,
                            0.513646039350172, -1.4909462979295, 0.467027612901807, 0.327172333556713,
                            0.839975024488723, 0.187317054211619, -1.16461731279095, 0.653501318695266,
                            0.513646039350172, 0.839975024488723, -0.185630357375297, 0.280553907108349
), x2 = c(0.136902564711936, 0.717047589790198, 0.910429264816285,
          0.407636909748458, 0.794400259800633, 0.523665914764111, 0.136902564711936,
          0.83307659480585, 0.523665914764111, -0.0951554453193692, -2.76382256067938,
          0.0208735596962833, 0.0595498947015008, 0.368960574743241, -0.0564791103141517,
          -0.0951554453193692, 0.717047589790198, 0.407636909748458, 0.83307659480585,
          -0.0951554453193692, 0.601018584774546, 0.639694919779763, -0.636624135392414,
          0.0595498947015008, 0.871752929811068, -1.4875035055072, 0.407636909748458,
          0.175578899717153, -1.02338748544459, 0.0595498947015008, 0.871752929811068,
          0.562342249769328, -0.0564791103141517, 0.639694919779763, 0.755723924795415,
          0.523665914764111, 0.562342249769328, 0.368960574743241, 0.717047589790198,
          0.407636909748458, -2.57044088565329, 0.523665914764111, -0.0564791103141517,
          0.755723924795415, -0.0178027753089342, 0.717047589790198, -0.133831780324587,
          0.136902564711936, -0.0564791103141517, -1.99029586057503, 0.67837125478498,
          0.910429264816285, -2.53176455064807, -2.10632486559068, 0.368960574743241,
          0.794400259800633, 0.330284239738023, -0.288537120345457, 0.523665914764111,
          0.368960574743241, -0.172508115329804, 0.523665914764111, 0.523665914764111,
          0.910429264816285, -0.0564791103141517, 0.446313244753676, 0.523665914764111,
          0.0982262297067184, -2.18367753560111, -2.33838287562198, -0.0564791103141517,
          0.83307659480585, -0.327213455350674, -1.79691418554894, -2.91852790070025,
          -1.99029586057503, 0.252931569727588, 0.484989579758893, -1.02338748544459,
          0.175578899717153, 0.910429264816285, 0.910429264816285, 0.910429264816285,
          0.330284239738023, -2.87985156569503, -1.25544549547589, 0.523665914764111,
          0.523665914764111, -1.06206382044981, 0.910429264816285, 0.562342249769328,
          -0.0951554453193692, -2.87985156569503, 0.717047589790198, 0.0208735596962833,
          0.83307659480585, -0.365889790355891, 0.910429264816285, 0.910429264816285,
          0.562342249769328, 0.562342249769328, 0.717047589790198, 0.136902564711936,
          0.368960574743241, -0.443242460366326, 0.601018584774546, -0.752653140408067,
          -1.02338748544459, 0.446313244753676, 0.562342249769328, 0.910429264816285,
          -1.99029586057503, 0.717047589790198, 0.871752929811068, 0.523665914764111,
          -0.713976805402849, 0.562342249769328, -0.0951554453193692, 0.871752929811068,
          0.639694919779763, -2.3770592106272, -2.57044088565329, 0.562342249769328,
          0.562342249769328, 0.523665914764111, 0.562342249769328, -1.64220884552807,
          -2.53176455064807, 0.601018584774546, 0.910429264816285, 0.601018584774546,
          0.717047589790198, 0.484989579758893, 0.523665914764111, 0.755723924795415,
          -2.68646989066894, -0.946034815434154, 0.601018584774546, -0.0564791103141517,
          -1.02338748544459, 0.523665914764111, -1.06206382044981, 0.67837125478498,
          -0.0178027753089342, 0.910429264816285, 0.407636909748458, 0.794400259800633,
          0.910429264816285, 0.407636909748458, 0.0208735596962833, -2.57044088565329,
          0.871752929811068, 0.446313244753676, -2.33838287562198, 0.446313244753676,
          0.794400259800633, 0.717047589790198, 0.871752929811068, 0.639694919779763,
          -0.984711150439371, 0.368960574743241, 0.21425523472237, -0.559271465381979,
          0.523665914764111, -1.41015083549676, 0.562342249769328, 0.0208735596962833,
          0.562342249769328, -2.95720423570546, 0.562342249769328, 0.601018584774546,
          0.755723924795415, 0.639694919779763, 0.83307659480585, 0.523665914764111,
          -2.57044088565329, 0.523665914764111, 0.67837125478498, 0.562342249769328,
          -0.481918795371544, -0.211184450335022, 0.717047589790198, 0.910429264816285,
          -1.56485617551763, -0.0178027753089342, 0.910429264816285, 0.794400259800633,
          0.523665914764111, -1.13941649046024, 0.717047589790198, 0.252931569727588,
          0.871752929811068, 0.330284239738023, -0.791329475413284, 0.717047589790198,
          0.523665914764111, 0.83307659480585, -1.41015083549676, 0.0595498947015008
), x3 = c(0.376284465894342, 0.74833264743926, 0.93435673821172,
          -0.367811897195496, 0.794838670132375, 0.469296511280572, 0.0507423070425382,
          0.701826624746145, 0.469296511280572, -1.34443837375091, 0.701826624746145,
          0.329778443201227, 0.283272420508112, -0.50732996527484, -2.08853473684074,
          -0.367811897195496, 0.84134469282549, 0.469296511280572, 0.887850715518605,
          -0.0887757610368065, 0.469296511280572, 0.376284465894342, -1.99552269145451,
          -0.0422697383436916, 0.887850715518605, -0.693354056047299, 0.283272420508112,
          -0.274799851809266, -1.15841428297845, -0.925884169512874, 0.701826624746145,
          0.329778443201227, -0.228293829116151, 0.608814579359916, 0.84134469282549,
          -0.228293829116151, 0.701826624746145, 0.701826624746145, 0.74833264743926,
          0.283272420508112, -3.25118530416862, 0.469296511280572, -0.0422697383436916,
          0.84134469282549, -0.181787806423036, 0.562308556666801, -0.181787806423036,
          0.00423628434942329, 0.236766397814997, -0.321305874502381, 0.655320602053031,
          0.93435673821172, -3.2046792814755, -2.8791371226237, 0.236766397814997,
          0.701826624746145, -0.460823942581726, 0.236766397814997, -0.135281783729921,
          0.236766397814997, -0.135281783729921, 0.608814579359916, 0.469296511280572,
          -1.34443837375091, -0.181787806423036, -0.135281783729921, 0.329778443201227,
          0.0972483297356525, -3.01865519070304, -2.8791371226237, 0.469296511280572,
          0.84134469282549, -1.06540223759222, -2.22805280492009, -0.832872124126644,
          0.143754352428767, -0.135281783729921, 0.422790488587457, -0.925884169512874,
          0.562308556666801, 0.887850715518605, 0.93435673821172, 0.93435673821172,
          0.283272420508112, 0.794838670132375, -1.39094439644402, -0.181787806423036,
          -1.6234745099096, -1.25142632836468, 0.93435673821172, 0.74833264743926,
          0.701826624746145, 0.422790488587457, 0.701826624746145, -0.181787806423036,
          0.887850715518605, -1.29793235105779, 0.93435673821172, 0.562308556666801,
          -2.64660700915812, 0.701826624746145, 0.701826624746145, -0.0422697383436916,
          0.422790488587457, -1.25142632836468, 0.283272420508112, -0.879378146819759,
          -3.15817325878239, 0.469296511280572, 0.469296511280572, 0.93435673821172,
          0.794838670132375, 0.93435673821172, 0.74833264743926, 0.469296511280572,
          -1.06540223759222, 0.562308556666801, 0.84134469282549, 0.794838670132375,
          0.655320602053031, -2.32106485030632, -3.25118530416862, 0.701826624746145,
          0.701826624746145, 0.469296511280572, 0.469296511280572, -2.60010098646501,
          -3.25118530416862, 0.608814579359916, 0.93435673821172, 0.0972483297356525,
          -0.135281783729921, 0.515802533973686, 0.469296511280572, 0.562308556666801,
          0.93435673821172, 0.93435673821172, 0.283272420508112, 0.236766397814997,
          -0.460823942581726, 0.469296511280572, 0.887850715518605, 0.562308556666801,
          -0.321305874502381, 0.93435673821172, -0.181787806423036, 0.794838670132375,
          0.93435673821172, 0.422790488587457, -0.553835987967955, 0.701826624746145,
          0.84134469282549, -0.414317919888611, -2.92564314531681, 0.236766397814997,
          0.701826624746145, 0.701826624746145, 0.422790488587457, 0.794838670132375,
          -2.64660700915812, 0.515802533973686, 0.329778443201227, -2.46058291838566,
          0.329778443201227, -1.34443837375091, 0.283272420508112, -0.460823942581726,
          0.469296511280572, 0.93435673821172, 0.469296511280572, 0.0972483297356525,
          0.701826624746145, 0.84134469282549, 0.84134469282549, 0.469296511280572,
          -3.25118530416862, 0.469296511280572, 0.376284465894342, 0.236766397814997,
          0.143754352428767, 0.794838670132375, 0.655320602053031, 0.469296511280572,
          -2.55359496377189, -0.228293829116151, 0.93435673821172, 0.655320602053031,
          0.515802533973686, -1.48395644183025, 0.701826624746145, 0.283272420508112,
          0.701826624746145, -0.181787806423036, -1.20492030567156, 0.376284465894342,
          0.469296511280572, 0.84134469282549, -0.181787806423036, 0.562308556666801
)), class = c("tbl_df", "tbl", "data.frame"), row.names = c(NA,
                                                            -199L))

[Shu Fai Cheung]

A CFA model with three indicators, identified by setting the factor variance to 1, has 6 free parameters: 3 loadings and 3 error variances. The number of unique elements in the covariance matrix is 6 (3 variances + 3 covariances). The model is a saturated model with df is zero, regardless of the data. The fit is "perfect," in this sense (but it's also "perfect" for any 0-df model in this sense). You will find the same results in other SEM program.

P.S.: This is also true if you identify the factor by setting a loading to 1 because it still has 6 free parameters: 2 (free) loadings, 3 error variances, and 1 factor variance.

-- Shu Fai

[Shu Fai Cheung]

Sorry. I meant "A 1-factor CFA model with three indicators ..."

-- Shu Fai

[Edward Rigdon]

To clarify, the perfect values for CFI and TLI do not indicate that "the baseline model is the same as..." The perfect values indicate no evidence of misspecification in the estimated model. There is no evidence of misspecification because, as Shu Fai noted, the estimated model is saturated with 0 DF.

--
You received this message because you are subscribed to the Google Groups "lavaan" group.
To unsubscribe from this group and stop receiving emails from it, send an email to lavaan+un...@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/lavaan/9f017eab-1909-4875-9033-ddf7dcce9a51n%40googlegroups.com.

[Robert Wilcom]

Yes, I completely forgot that the model is just-identified in the typical 3-item, 1 fact case where residual variances are uncorrelated and only one parameter is fixed!

Does anyone have any guidance on reporting the results of fitting in the just-identified case? Currently reporting (a) the parameter estimates and (b) measures of scale reliability (alpha, omega, etc.). Since standard relative and absolute fit measures aren't insightful, currently omitting. Maybe some sort of cross-validation comparison can avoid this issue since we're passing "new data" to the model?

[Shu Fai Cheung]

How about searching SEMNET?

https://listserv.ua.edu/cgi-bin/wa?REPORT=SEMNET&z=4&L=SEMNET&1=SEMNET&X=&Y=

Not that this topic, an interesting and practical one, cannot be discussed here. I recall there are several interesting threads on this topic on SEMNET in the past. I did a quick search using "just-identified" (at the risk of missing posts with "just identified" and other variants) and still found over 4000 posts. You may find some useful comments and advice there.

-- Shu Fai