Accounting for measurement error diminish fit

[Shajar]

Hi, in my research I started with a simple model without latent variables. I have put up a model that has a good fit (P-value = 0.696). But since I have knowledge of the measurement error, I wanted to put them in. I have defined each variable as a single- indicator that controls for measurement error – see code below. But the result is - no fit. I was expecting similar results, just distinguishing the measurement error from the structural error. Did I define the model correctly with regards to measurement error? If so, why is there such significant difference in the fit?

The code below contains connection to my data in google drive so you can replicate the issue.

### models with and without measurement error

# Load the data from a csv file I have put in google drive

id<-"1BQ0Q0QUYQdYRXNVxmfs92CIPWqCKlQ4S"# google file ID

dat<-read.csv(sprintf("https://docs.google.com/uc?id=%s&export=download",id),stringsAsFactors=FALSE)

 

library(lavaan)

 

### model a: simple model, no measurement error

mod.a <- '

NH4toTN ~ Crane_Count + Temp + pH_Outlet

Chl ~  season +  NH4toTN + pH_Outlet + Temp

TDP2TP ~  Crane_Count + Chl + Temp + Rain8d

'

# Fit model

mod.a.fit <- sem(mod.a, dat)

summary(mod.a.fit, rsq=TRUE, standardized=TRUE)

# Result: good fit. P-value = 0.696

 

### model b: same model but add measurement error

mod.b <- '

#latent variables

NH4toTN_L     =~ NH4toTN

Crane_Count_L =~ Crane_Count

Temp_L        =~ Temp

Chl_L         =~ Chl

pH_Outlet_L   =~ pH_Outlet

TDP2TP_L      =~ TDP2TP

 

#Regrression

NH4toTN_L ~ Crane_Count_L + Temp_L  + pH_Outlet_L

Chl_L ~  season +  NH4toTN_L + pH_Outlet_L + Temp_L

TDP2TP_L ~  Crane_Count_L + Chl_L + Temp_L + Rain8d

 

#fix error variances

NH4toTN     ~~ 0.507*NH4toTN

Crane_Count ~~ 0.146*Crane_Count

Temp        ~~ 0.034*Temp

Chl         ~~ 0.224*Chl

pH_Outlet   ~~ 0.031*pH_Outlet

TDP2TP      ~~ 0.399*TDP2TP

'

# Fit model

mod.b.fit <- sem(mod.b, dat)

summary(mod.b.fit, rsq=TRUE, standardized=TRUE)

# Result: no fit.

 

[Terrence Jorgensen]

You can check the model-implied covariances

lavInspect(fit, "cov.ov")

and residuals

resid(fit)

to see how the models differ.

Terrence D. Jorgensen

Assistant Professor, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

http://www.uva.nl/profile/t.d.jorgensen

[Shajar]

Thanks Terrence!

your suggestion helped me figure out the problem: I didn't apply the measurement error for 2 variables (season and Rain8d) because it is close to zero. Once I added the single indicator latent variable and defined the measurement error to be zero - everything falls into place... and the model has P-value=0.478.

[Shajar]

I continue with my model but now I encounter a similar problem but different.

The model has composite variables. The model has good fit and df=6.

Now I take the exact same model and add the measurement error estimation as I did in the original question. But now the df= (-10).

I can't figure out why the addition of single factor latent variable and setting the error variance would change the df in the model.

It did not change the df on a model without composite variables. 

Where do I get it wrong?...

The model is a bit complex:

### model 1f: no measurement error

mod.1f <- '

# Latent variable  - composite

EndoP <~ 1*pH_Outlet  + pH_td_in + Temp + SunRad8d

ExoP  <~ 1*PP_td_in + Crane_Count

#Regrression

TDP2TP ~  EndoP + ExoP + Chl + Sh_TDP2TP

Chl ~  Sh_Chl + TDP2TP + EndoP 

EndoP ~~ 0*ExoP

EndoP ~~  2.36*EndoP

'

# Fit model

mod.1f.fit <- sem(mod.1f, dat1, estimator = "MLR")

# Examine model fit

fitMeasures(mod.1f.fit, c("chisq", "df", "pvalue","rmsea","cfi","srmr"))

summary(mod.1f.fit, rsq=TRUE, standardized=TRUE)

# good fit. df=6

# model 2f - add measurement error

mod.2f <- '

#latent variables for measurement error

L_TDP2TP      =~ TDP2TP

L_Chl              =~ Chl

L_pH_Outlet   =~ pH_Outlet

L_pH_td_in     =~ pH_td_in

L_Temp           =~ Temp

L_SunRad8d   =~ SunRad8d

L_PP_td_in      =~ PP_td_in

L_Crane_Count =~ Crane_Count

L_Sh_TDP2TP   =~ Sh_TDP2TP

L_Sh_Chl          =~ Sh_Chl

# Latent variable  - composite

EndoP <~ 1*L_pH_Outlet  + L_pH_td_in + L_Temp + L_SunRad8d

ExoP  <~ 1*L_PP_td_in + L_Crane_Count

#Regrression

L_TDP2TP ~  EndoP + ExoP + L_Chl + L_Sh_TDP2TP

L_Chl ~  L_Sh_Chl + L_TDP2TP + EndoP 

#Covariances

EndoP ~~ 0*ExoP

EndoP ~~  2.36*EndoP

#fix error variances

TDP2TP      ~~ 0.399*TDP2TP

Chl         ~~ 0.224*Chl

pH_Outlet   ~~ 0.031*pH_Outlet

pH_td_in    ~~ 0.602*pH_td_in

Temp        ~~ 0.034*Temp

SunRad8d    ~~ 0.013*SunRad8d

PP_td_in    ~~ 0.012*PP_td_in

Crane_Count ~~ 0.146*Crane_Count

Sh_TDP2TP   ~~ 0.345*Sh_TDP2TP

Sh_Chl      ~~ 0.212*Sh_Chl

'

# Fit model

mod.2f.fit <- sem(mod.2f, dat1, estimator = "MLR")

# Examine model fit and parameters

fitMeasures(mod.2f.fit, c("chisq", "df", "pvalue","rmsea","cfi","srmr"))

summary(mod.2f.fit, rsq=TRUE, standardized=TRUE)

# df= -10...

[Terrence Jorgensen]

A composite is not a latent variable, and a formative construct is not a common factor (reflective construct).  The rules of identification are different.

https://scholar.google.com/scholar?q=related:Li5XXqmcqF8J:scholar.google.com/&hl=en&as_sdt=0,5

[Shajar]

Thanks Terrence,

I read some of the links you attached. Although there are some arguments against using composites I believe that it is right for my biogeochemical system. But even A.Diamantopoulos in "The error term in formative measurement models: interpretation and modeling implications" does not suggest how to model measurement error of each formative measure and relate to a model such as mine as "Spurious model" which I fail to understand why.

( Since this is not a lavaan issue but rather a general SEM - I will try SEMNET. )

Thanks again!