Binary Single Indicator Latent Variables
Message/Author
 Tom Booth posted on Monday, September 24, 2012 - 6:57 am
Is it possible to create a single indicator latent variable based on a binary indicator?

I am trying to develop a measurement model where my primary interest is in a second-order construct. However, my indicator set is sub-optimal for estimating a second order construct as I only have two well measured latent factors(>3 indicators with reasonable loadings >0.40). In addition I have a single observed binary variable which could act as a third indicator.

I want to discuss and demonstrate/estimate the alternatives whilst acknowledging the problems. I am comfortable describing the issues in the 2 first order factor model (non-identification at the 2nd order without additional constraints), and can present a bi-factor model as an alternative way to get at the general construct (with its different assumptions about the nature of the construct).

But it is the estimation and discussion of a model with two first order latent factors and, potentially, a single indicator latent variable from a binary observed indicator with which I am struggling.

Thank you in advance for any suggestions.
 Linda K. Muthen posted on Tuesday, September 25, 2012 - 8:49 am
You can simply use the binary indicator as a factor indicator for the second-order factor. It is not necessary to put a factor behind it. If you did, you would obtain the exact same results.
 Tom Booth posted on Tuesday, September 25, 2012 - 8:55 am
Thanks Linda. Sometimes the answer is the simplest one!

When I run this model, I don't get a p-value for the loading - is this expected?
 Linda K. Muthen posted on Tuesday, September 25, 2012 - 1:48 pm
If is is the first factor indicator, it is fixed at one to set the metric of the factor so it is not estimated and you would not get a p-value.
 Tom Booth posted on Tuesday, September 25, 2012 - 11:15 pm
Hi Linda,

The model is set up as follows;

usevariable = x1-x8 u1 ;
categorical = u1 ;
model:
f1 by x1* x2-x3;
f1@1 ;

f2 by x4* x5-x8 ;
f2@1 ;

f3 by f1* f2 u1;
f3@1 ;

I do not believe in this coding I have set u1 to be fixed at 1, but in the model output, it is showing as fixed on F3.

I am sure I am missing something obvious.

Thank you for your help and swift responding.
 Linda K. Muthen posted on Wednesday, September 26, 2012 - 6:10 am
 Tom Booth posted on Thursday, September 27, 2012 - 4:32 am
Thanks Linda, will do. I am in the process of upgrading to version 7 and getting a new help license so will have to send once this is sorted.
 jiesi G posted on Wednesday, December 18, 2013 - 3:36 am
Hello,
I am running a 4-factor CFA using continuous and only one binary indicator. Three of the latent factors have continuous indicators, while the binary indicator was correlated with these 3 factors.
The CFA model is set up as follows;
usevariable = x1-x9 u1 ;
categorical = u1 ;
ANALYSIS: estimator=WLSMV;
model:
f1 by x1* x2 x3;
f1@1 ;
f2 by x4* x5 x6 ;
f2@1 ;
f3 by x7* x8 x9;
f3@1 ;
u1 with f1 f2 f3;

in the following up SEM analysis,
f1 and f2 -> f3 -> u1,
I want to treat the binary indicator as the outcome variable, and use MLR and link=logit to test indirect effect from f1 to u1.

my question is whether I can report the correlation matrix in my paper based on this set-up CFA input.
 Bengt O. Muthen posted on Wednesday, December 18, 2013 - 10:39 am
I don't know why you want to report a correlation matrix when your analysis estimates parameters of a path model. The model is sufficient. If you ask for TECH4, Mplus will give you correlations among your 3 factors. I don't think you want a correlation with the binary outcome.