fixed error variance for single indicator in CFA model with multiple and single indicators for constructs

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Павел Валединский

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Feb 4, 2020, 9:37:59 AM2/4/20
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Hi everyone,
Now I have a model with 9 variables and 3 constructs. 1st and 2nd factors have 4 indicators each, but the 3rd factor (Fac3) has only 1 variable as indicator (v9). I found in Kline's book that for the model (where there are constructs with multiple indicators and single indicators in one model) to be identified, single indicator shoud have a fixed error variance. How can I fix the error variance of 20% to this single indicator in syntax? 
Model <- '#measurement model
Fac1 =~ v1 + v2 + v3 + v4
Fac2 =~ v5 + v6 + v7 + v8
Fac3 =~ v9
#regressions
v10 ~ Fac1 +Fac2 +Fac3'

Thank you 

Terrence Jorgensen

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Feb 4, 2020, 3:14:41 PM2/4/20
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How can I fix the error variance of 20% to this single indicator in syntax? 

Are you saying the reliability of v9 is 0.8?  Then the ratio of the factor variance (psi3) to error variance (theta9) should be 0.8 / 0.2, and you can exploit this to specify a model constraint in your syntax:

Fac3 =~ 1*v9
Fac3 ~~ psi3*Fac3
v9
~~ theta9*v9
## model constraint
theta9
== psi3 * 0.2 / 0.8 # implies theta9 / psi3 == 0.2 / 0.8


Terrence D. Jorgensen
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam

Павел Валединский

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Feb 5, 2020, 12:56:49 PM2/5/20
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Thank you very much! The model now works differently giving appropriet estimates, different from the previous ones. 

вторник, 4 февраля 2020 г., 23:14:41 UTC+3 пользователь Terrence Jorgensen написал:

Valeria Ivaniushina

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Feb 10, 2020, 10:12:30 AM2/10/20
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Hi Terrence,

This formula 

Fac3 =~ 1*v9
Fac3 ~~ psi3*Fac3 
v9 ~~ theta9*v9
## model constraint
theta9 == psi3 * 0.2 / 0.8 # implies theta9 / psi3 == 0.2 / 0.8

is different from the script that was discussed previously in many other threads on this forum:
F =~ x
x ~~ a*x,  where a = (1-Reliability)*variance of x

I am confused that I don't see the variance of the indicator in your formula.
Maybe I am missing something obvious?

Regards,
Valeria Ivaniushina

Terrence Jorgensen

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Feb 11, 2020, 7:24:31 AM2/11/20
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I am confused that I don't see the variance of the indicator in your formula.
Maybe I am missing something obvious?

Not necessarily obvious, but they are equivalent.  Specifying the observed variance of the indicator allows you to explicitly set the constraint on the residual variance as a numeric value.  The formula above implicitly makes the same constraint, by constraining the ratio of psi / theta to be the ratio of reliability / unreliability.  A little algebra shows that this is equivalent to saying the total variance is equal to the sum of those 2 variance components.  

The limitation of the implicit approach in this thread is that it relies on the model-implied rather than the observed indicator variance.  But with a single-indicator construct, I don't see a lot of room for such misspecification that would make the observed and model-implied indicator variance differ.

Valeria Ivaniushina

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Feb 11, 2020, 10:08:33 AM2/11/20
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Thank you Terrence, this point is clear now. Two follow-up questions:

1) Is it possible to express your formula in the Mplus language?

2) On the Statmodel forum Linda said: "With categorical outcomes, residual variances are not parameters in the model. I would not recommend trying to correct these variables using reliability".  
But including a single indicator as a manifest variable (or as a latent with error fixed to 0, as lavaan does by default) would mean that we assume this indicator to be perfectly measured, right? Is it a valid assumption?  Kenneth Bollen, Les Hayduk and some other are clearly against this practice.

Valeria

Terrence Jorgensen

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Feb 12, 2020, 4:48:55 AM2/12/20
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1) Is it possible to express your formula in the Mplus language?

Yes, you can use the NEW command to name the new variable, then define it below (I think in a MODEL CONSTRAINT command).  You can search for examples in the user guide or forum, especially in the mediation models, for defining new parameters.
 
2) On the Statmodel forum Linda said: "With categorical outcomes, residual variances are not parameters in the model.

Well, they are if you use parameterization = "theta", although they are nonetheless fixed (to 1) by default for identification.  But the issue with identification is that with a single continuous indicator, you only have 1 piece of observed information (the variance), so you can only estimate 1 piece of information (either the factor variance or loading, fixing the other to 1).  With a categorical indicator, you don't have an observed variance.  You are fitting the model to a polychoric correlation matrix, in which the (total) variances are fixed to 1.  So you cannot estimate anything in a single-indicator construct.

Using the default parameterization = "delta", the residual variances are fixed to 1 minus the common-factor variance for identification, such that the model-implied total variances == 1.  So the trick is simply to fix the factor loading to the square-root of the reliability, in which case the residual variance will be set to 1 minus the reliability.  For example, suppose the reliability of "u4" is 0.64 (however that is supposed to have been estimated...), the square-root of which is 0.8:

myData <- read.table("http://www.statmodel.com/usersguide/chap5/ex5.16.dat")
names
(myData) <- c("u1","u2","u3","u4","u5","u6","x1","x2","x3","g")
model
<- '
f1 =~ u1 + u2 + u3
f2 =~ 0.8*u4 # reliability == 0.64
'

summary
(cfa(model, data = myData, ordered = paste0("u", 1:4), std.lv = TRUE))

Notice the the residual variance in the summary is fixed to 1 - 0.64 = 0.36

But including a single indicator as a manifest variable (or as a latent with error fixed to 0, as lavaan does by default) would mean that we assume this indicator to be perfectly measured, right? Is it a valid assumption?  Kenneth Bollen, Les Hayduk and some other are clearly against this practice.

It's up to you.  I agree it is highly dubious to assume it is error-free, but it is also suspicious to use a reliability estimate for a single categorical item.  When we calculate scale reliability, it is the reliability of the composite (e.g., a scale sum or scale mean), not the reliability of each individual scale item.  If you are using a single categorical indicator, I doubt it is a composite of many items.  There is nothing wrong with choosing a few different values (including perfect reliability) and comparing results as a sensitivity analysis to different reliability assumptions.

Valeria Ivaniushina

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Feb 12, 2020, 10:34:23 AM2/12/20
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Thank you very much, Terrence!
it's very helpful
Valeria

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