Binary Single Indicator Latent Variables PreviousNext
Mplus Discussion > Confirmatory Factor Analysis >
 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 ;
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
Please send the output and your license number to
 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
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;
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.
 Eric M. posted on Friday, February 16, 2018 - 1:40 pm
Hello. Quick question on this topic. If I am attempting to use a binary variables (sex) as a single indicator latent variable that is an exogenous --nothing predicts it. Is this acceptable within Mplus given that the latent variable is exogenous. When I run it like that it gives a message that SE's of the model my not be trustworthy. And says that This is most likely due to the binary variable being dichotomous but declared as continuous.

Does this imply that these results can not be trusted?

Thank you so much.
 Bengt O. Muthen posted on Friday, February 16, 2018 - 6:05 pm
I don't know why you would want to put a factor behind the binary sex variable. The message about dich's vbles being treated as cont's is ok and ignorable when the variable is binary.
 Eric M. posted on Friday, February 16, 2018 - 6:34 pm
Thank you. I'm not quite sure if I need to. But, in attempt to construct a measurement model prior to evaluating my structural model- as I was advised to follow a two-step approach. I couldn't figure out how to represent the single indicator variables in a measurement model (including the binary one), without creating a single indicator latent variable. Regarding sex, I read in Kline (2017), that some people recommend even controlling for possible error in demographics (Hayduk & Littvay, 2012) that might result from erroneous or accidental responses.

Is there are way to construct a measurement model with latent factors and observed variables?
 Bengt O. Muthen posted on Saturday, February 17, 2018 - 12:01 pm
With a single factor indicator you can't take into account measurement error - unless you know the indicator's reliability.

I don't know what you mean by your question; perhaps you can give an example. Also, see the SEM videos and handouts for our Short Course Topics 1 and 2 on our website.
 Eric M. posted on Saturday, February 17, 2018 - 2:05 pm
Thank you so much. You have know idea how much you are helping me. My phrasing was a bit off. Let me provide example:
Theoretical Model, were Obs1 (continuous) and sex (binary) are observed.
X1 BY A1 A2 A3
X2 BY B1 B2
X3 BY C1 C2 C3
Y BY D1 D2 D3
X1 ON Obs1 sex
X2 ON X1
X3 ON Obs1 sex X1
Y ON X2 X3
I was told that need to follow a two-step approach (anderson & gerbing 1998) where the measurement model (CFA) is evaluated before evaluating structural model. Obs1 and sex can not be included in a measurement model unless single latent indicators are used. Which would be done by fixing the error variance of Obs1 to some number derived from (1-r)var and sex to 0. Correct? Only then could these observed variables be included measurement model that is equivalent to a saturated theoretical model.

FObs1 ON Obs1@1

Fsex ON sex@1

Is there any other way in Mplus where Obs1 and sex could be included in the CFA? I understand this is mainly an MPlus help board. But, I also value your opinion, is it even necessary to include the observed variables in the CFA? If there any reference that says you shouldn't. Does Mplus hav any features that eliminates this need?
 Bengt O. Muthen posted on Sunday, February 18, 2018 - 2:58 pm
The idea of testing the measurement model first is reasonable. But it doesn't mean that you have to create factors behind single indicators. What it means is that you want to not have any restrictions on the structural part of the model so that any model misfit is due to the measurement part. In a CFA context this means that the factor covariance matrix is unrestricted, that is, all factor covariances are freely estimated. in your context is means that this part of the model should have no left-out arrows:

X1 ON Obs1 sex
X2 ON X1
X3 ON Obs1 sex X1
Y ON X2 X3

So for instance, X2 and Y should be regressed on Obs1 and Sex (there are more).
 Eric M. posted on Sunday, February 18, 2018 - 3:53 pm
Thank you Bengt. So unrestricting all paths of a structural model is equivalent to perforfoming a cfa? Does that mean that a cfa where factors are formed and intercorrelated (leaving off the structural part)) is not really necessary bc an unrestricted (all paths estimated) structural model tells me the exact same thing? It seems that in sem papers people often report a cfa first (without structural part). Is this because people use different stratefies? Different software? Iíve been all through semnet and looked at examples in attempt to satisfy advisors request?
 Bengt O. Muthen posted on Sunday, February 18, 2018 - 4:33 pm
Q2 (second ?): Right. The underlying principle is really about an unrestricted structural model part. Often all factors have multiple indicators where that's the same as a CFA with unrestricted factor covariance matrix and that's probably how the CFA strategy started. But it would be silly to create single-indicator factors just to fall into that mold.

I think it is good to do a CFA for each part of the model that has factors with multiple indicators.
 Eric M. posted on Sunday, February 18, 2018 - 6:00 pm
Thank you Bengt. This clarification helps a lot.
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