DELTA vs. THETA parameterization
Message/Author
 Dan Putka posted on Wednesday, September 01, 2004 - 3:01 pm
I'm trying to fit a path model that has two dichotomous variables as outcomes. When I run the syntax, I get the following:

ERROR in Model command
The model is not supported by DELTA parameterization. Use THETA parameterization.

How do I implement the THETA parameterization in MPlus?
 bmuthen posted on Wednesday, September 01, 2004 - 3:05 pm
Say

parameterization = theta;

 Sanjoy posted on Friday, June 10, 2005 - 6:39 pm
Dear Professor/s

Q1. Under "Theta parameterization" theta the residual variance (associated with measurement section of SEM) is our model parameter, right Professor! ... however, the corresponding "Tech 1" output showing NO parameter assignment ... why is it so, I can't get it ... I thought because my analysis is Single group, I tried to make it Multiple group, there also it's NOT assigning any parameters in Tech1 (I could have written code wrong)

(following your Mplus web note 4) ... We get Scale factor as remainder, which is = (factor loading^2)*factor variance + theta

we have factor loading values from Lambda Matrix (tech 1 showing the parameter assignment), we have factor variance from Psi matrix (tech 1 showing parameter asignment here as well)

Q2. Mplus output report "scale factors" ... how does it calculate scale factor ...it's written in the web note that "Theta approach standardizes to theta=1" ... in that case scale factor should be greater than one (since factor loading^2 as well as factor variance are positive number) ... however my results are different ... confused once more

R-SQUARE

Observed Scale
Variable Factors R-Square

D31 0.769 0.432
D32 0.778 0.418
D33 0.883 0.238
D43 0.827 0.331
D53 0.846 0.299
D63 0.986 0.030
BID1A 0.014 1.000

Q3. What does the R-square associated with scale factor stand for? The reason I am asking Q3 in particular ... our dependent variables are categorical (ordinal) .... can we interpret "r-square" values for the categorical outcomes as the proportion of variance explained as we do in the case with continuous outcomes

thanks and regards
 bmuthen posted on Saturday, June 11, 2005 - 6:05 am
Theta is only identifiable in groups > 1 in multiple-group analysis and time point > 1 in growth modeling.

The scale factor delta is the inverted y* SD, not the y* variance.

R2 is the regular R2 for y*.

See also Appendix 1 of the technical appendices on the Mplus web site.
 Sanjoy posted on Saturday, June 11, 2005 - 10:36 pm
Thank you very much Professor
 jenny yu posted on Saturday, September 09, 2006 - 12:34 pm
Dear Drs. Muthen,

When we build a MIMIC model with ordinal indicators and categorical background variables, how should we choose the parameterization? Dr. Bengt mentioned that either delta or theta can be used. But I am wondering whether there are some criteria we need to consider to choose between delta and theta? How can I justify my selection over the other?

Thank you very much.
 Bengt O. Muthen posted on Saturday, September 09, 2006 - 2:08 pm
The Delta parameterization has been chosen as the default in Mplus because there are some cases where it works better than Theta - see Mplus Web Note#4.
 jenny yu posted on Sunday, September 10, 2006 - 1:59 pm
Thank you, Professor Muthen.
I have another question on adding direct effect in MIMIC model--

Can the model fit obtained after adding the direct effect be used as one of the criteria to decide the existence of DIF effect? Say, if the model fit gets worse, then the variable is not appropriate to add for examining direct effect.

Thank you very much.
 Bengt O. Muthen posted on Sunday, September 10, 2006 - 2:14 pm
I would recommend looking at the z score (Est/SE) of the direct effect instead.
 Jungeun Lee posted on Monday, August 20, 2007 - 5:42 pm
Hi,

I am working on a SEM model in which a categorical (ordinal) dependent variable is influenced by and influences another latent variable. I looked at the Mplus user's guide(v.3) parameterization section and thought that THETA would be the way to go. I ran the model with THETA. Out of curiousity, I re-ran the model with DELTA. Both models ran fine but gave me somewhat different results. Usually, I go with a default setting in statistical packages, unless I have a clear idea about why the default setting wouldn't work... Since models with either DELTA and THETA worked fine and I am not super clear about when to use which parameterization method, my usual self tells me to go with results from the model with DELTA parameterization... do you think it is a good way to go? Thanks in advance!!
 Linda K. Muthen posted on Thursday, August 23, 2007 - 11:18 am
We recommend the Delta parameterization unless the model cannot be run without using the Theta parameterization. I would use Delta.
 Sanjoy Bhattacharjee posted on Saturday, July 26, 2008 - 1:00 pm
Professors Muthen,

I have a simple model to estimate:

Y1* = f (Y2*, X1);
Y2* = g(X3);

Y1* is related to a binary variable, Y1 and Y2* is related to an ordinal variable Y2 (3 categories). X's are strictly exogenous.

I am struggling to understand the following.

The R-square value of Y1 increases significantly (.32 to .50) when we use theta instead of delta parameterization.

Q1. I am wondering why?

Q2. Coefficient values change from one parameterization to the other one. However, their relative strength remains same. I think that's how it should be; because parameterization affects the limit of integration through the scaling factor. Is not it?

Q3. As an extension of the model, I add X4 in the Y2 equation. As expected, the R-square value of Y2 goes up but R-square value of Y1 goes down. Given Y2 is significant in explaining Y1, why does it happen? I think only "adjusted R-square" could go down with addition of new variable but I am not sure what is Mplus doing here.

I fit WLSMV estimator and I am using Mplus 4.1

Thanks and regards
 Linda K. Muthen posted on Sunday, July 27, 2008 - 8:54 am
You should run this in Version 5.1. There have been changes between Version 4.1 and 5.1 related to categorical data modeling.
 Sanjoy Bhattacharjee posted on Wednesday, July 30, 2008 - 4:58 pm
Hi Madam, I do understand there have been changes between version 4.1 and 5.1, but the results derived using 4.1 should not be different. Is not it? After all, the WLSMV estimator remains same.

Thanks and regards
 Linda K. Muthen posted on Wednesday, July 30, 2008 - 6:09 pm
There have been changes that could change the results. The only way to know is to use the latest version and to send the files and your license number to support@statmodel.com. The information you give is not sufficient to answer your question.
 Dorothee Durpoix posted on Wednesday, March 18, 2009 - 9:29 pm
Hello,

I was wondering what one should do when one or several estimates are significant with Delta parameterization but not significant with Theta parameterization (in multigroup CFA analysis)?

thanks in advance for any input.
 Dorothee Durpoix posted on Thursday, March 19, 2009 - 10:15 pm
Sorry, i put this thread here by mistake. I tried again in the topic CFA.

Cheers,
Do.
 Paul A.Tiffin posted on Saturday, November 13, 2010 - 2:59 am
Dear Team,

I just wanted to quickly check something with you:
I am running a MC simulation of a CFA with categorical factor indicators (5 response choices; i.e. 4 thresholds) using your two step process. In step one I specify the parameterization as theta and set the metric of the factor by fixing factor variance to 1. I specify the same with the model and model population in step 2. The results from the MC seem plausible but I just wanted to check I wasn't missing something important. I note in example 4.2 you set the values to make delta and theta parameterizations equivalent but presumably this is because the original CFA used the delta parameterization. Is this correct? Many thanks for your help.
best wishes
Paul
 Bengt O. Muthen posted on Saturday, November 13, 2010 - 3:55 pm
Ex4.2 uses parameter values so that the u* variances are 1, which is what the Delta parameterization uses. Delta is used in Ex4.2. The Theta parameterization instead has theta residual variances 1.
 Alexei Tretiakov posted on Thursday, February 03, 2011 - 7:56 pm
I am using Mplus version 6.

I'd like to fit a model to real data and save parameter estimates (as in the example 12.7 Step 1 of the User's Guide), and then read the parameters to use in a Monte Carlo simulation (as in the example 12.7 Step 2).

This works fine for as long as I am treating the indicators as continuous.

However, when I declare indicators as ordered categorical, I can no longer save estimates, and I am getting an error message "PARAMETERIZATION=THETA is required when estimates are to be saved".

Once I switch to THETA, I can save parameter estimates, but I cannot read them from the Monte Carlo simulation.

I wonder if there is somewhere an example similar to the example 12.7, but using ordered categorical indicators?
 Linda K. Muthen posted on Friday, February 04, 2011 - 9:57 am
This feature does not work for WLSMV. You need to use the results as population parameter values in MODEL POPULATION.
 Alexei Tretiakov posted on Sunday, February 06, 2011 - 9:22 pm
I created a Monte Carlo model by adding SVALUES in the model I used to analyse real data (the model declares all indicators as ordered categorical). I copy the SVALUES output both under MODEL POPULATION and under MODEL in the Monte Carlo model (for the latter, minus the part where threshold values are set).

I also needed to set residual variances for the indicators under MODEL POPULATION, which I found under R-SQUARE section of the output generated by the model I used to analyse real data (hopefully, these are the right ones - it is not very clear from the model output).

The resulting Monte Carlo model does run. However, I found that on most iterations, it fails with a "CATEGORICAL VARIABLE" "HAS ZERO OBSERVATIONS IN CATEGORY" message. Indeed, some of the categories on some of the items occur very rarely in the original data, so one would expect that in a Monte Carlo simulation fairly often they would not occur at all.

The Monte Carlo simulation does generate some reasonably looking result based on the iterations that do not fail. (Coverage values, though, are not as good as I expected - often rather far from 0.95).

What would be the best way to resolve the "ZERO OBSERVATIONS IN CATEGORY" problem?
 Alexei Tretiakov posted on Sunday, February 06, 2011 - 9:24 pm
An appendix to my previous message:

I tried saving Monte Carlo data and analysing it from a separate Mplus program, but that did not work - I'm just getting a different error message - something about categories being inconsistent.
 Linda K. Muthen posted on Monday, February 07, 2011 - 9:28 am
In your Monte Carlo study, you can either increase the sample size or change the thresholds.

 Sara posted on Tuesday, February 15, 2011 - 4:49 pm
When I conduct a CFA with binary (0,1) indicators using WLSMV estimator and specify "parameterization is Theta", how does one intepret the first set of parameter estimates Mplus produces (under "Model Results")?

I realize the Standardized parameters(which come after the first set of parameters) indicate the standardized relationship between y* for the indicator and the latent factor. The standardized thresholds can be intepreted as z-scores. Can I call these "standardized probit regression coefficients"?

Can the values above the standardized coefficients (first set produced) be interpreted and labeled as the "unstandardized probit coefficients"? That is, can they be intepreted as the unit change in the probit of the indicator for every unit change in factor? Can the unstandardized threshold be interpreted as the expected unstandardized probit that the observed response equals lower category (zero in this case) when the factor is zero?

I didn't know if there was a naming convention that was used (or should be used) when the Theta parameterization is employed and the 2 sets of parameters are reported (and intepreted).

Thanks
 Bengt O. Muthen posted on Tuesday, February 15, 2011 - 6:03 pm
I think you can characterize the unstandardized Theta estimates as unstandardized probit estimates in the sense that both probit regression and Theta parameterization fix the residual variance to 1. So your interpretations are on target. The Delta param. instead fixes the u* variance to 1 (which means that the residual variance is not 1). That also gives probit estimates, but not in the usual probit regression metric. The Delta advantage is that the loading estimates are in usual standardized factor analysis metric.
 Sara posted on Wednesday, February 16, 2011 - 4:51 am
Thanks so much!
Then can you clarify the interpretation of the standardized vs unstandardized probit coefficients in this context.
Can I interpret the standardized threshold of a binary item (0, 1) as the z-score that indicates the proportion of respondents answering 0 (thus the probability of answering a 0 for that item)?
Then how is the unstandardized probit threshold interpreted? It does NOT correspond to the cumulative area under the normal curve for that response (like the standardized probit thresholds), correct? If not, how does one interpret these values? This is the main parameter I am having a difficult time interpreting.
As you said, I can interpret the standardized probit regressions (loadings) as standardized factor loadings (if factor is standardized, the loading indicated the SD change in y* for the item for every SD change in factor).
Then how should one interpret the unstandardized probit regression/loading. If factor is standardized, do I say the value represents the change in the probit of the item for every 1 SD in the factor? Is saying "change in probit Item 1" correct, because the item isn't standardized (that is, total variance of y* doesn't equal 1) can't say change in standard deviation of Item 1?

I just want to be sure I accurately represent the parameters Mplus reports with the Theta parameterization.

Thank you!
 Bengt O. Muthen posted on Wednesday, February 16, 2011 - 10:19 am

Yes, the standardized threshold refers to a z-score at the factor value(s) of zero.

The unstandardized threshold refers to a normal score where the variance is not 1 as for the z score - so the interpretation is more involved. That's why the default Delta parameterization is easier to interpret.

With Theta param., I would focus on the standardized loadings. The unstandardized loadings refer to DV's (namely the u*'s) that don't have variance 1 and therefore are harder to interpret.
 Sara posted on Thursday, February 17, 2011 - 2:12 pm
Thanks for the clarification.

Given the difficulty in interpretation, when would one want to employ and report the parameters from the Theta scaling?
 Bengt O. Muthen posted on Thursday, February 17, 2011 - 4:31 pm
In Mplus, Theta is used as a backup to Delta. There are certain mediation models where only Theta can be used (see also UG). The Theta vs Delta parameterization is further discussed in Mplus Web Note #4.
 Fernando H Andrade posted on Monday, April 04, 2011 - 8:15 am
Dear Dr. Muthen
i am fitting a multi group cross lagged model between negative affect (latent) and smoking (observed 1 item). When I use the delta parametrization and i got the error:
“ The model is not supported by DELTA parameterization. Use THETA parameterization.”

Then when I use the theta parametrization and i got the error:

“Scale factors for categorical outcomes can only be specified using
PARAMETERIZATION=DELTA with estimators WLS, WLSM, or WLSMV.”

Is there something i could do.
thank you
Fernando
 Linda K. Muthen posted on Monday, April 04, 2011 - 9:07 am
In the Delta parametrization, scale factors are parameters in the model. In the Theta parametrization, residual variances are parameters in the model. You should refer to residual variances instead of scale factors.
 Fernando H Andrade posted on Tuesday, April 05, 2011 - 7:57 am
Dear Linda
Thank you very much, i will refer to the residual variances
Fernando
 Jinsong Chen posted on Sunday, August 05, 2012 - 1:28 pm
In pg. 485-486 of the version 5 UG, it said:" In addition, there are certain models that can be estimated using only the THETA parameterization because they have been found to impose improper parameter constraints with the DELTA parameterization. These are models where a categorical dependent variable is both influenced by and influences either another observed dependent variable or a latent variable."

Could you please elaborate more what improper parameter constraints it might be with the delta para.? I understand that when we are interested in the indirect or mediated effect with categorical data, we have to fix the scaling factor (theta para.) b/c the indirect effect (i.e., product of regression coefficients) are not scale free. However, in your scenario that a categorical dependent variable is both influenced by and influences other variables, AND if I am only interested at the regression coefficients, there should not be any scaling problem. So why I can't use the delta para.?
 Linda K. Muthen posted on Monday, August 06, 2012 - 10:00 am
If you need the Theta parametrization, you will receive a message if you are using Delta. You do not need Theta for mediation, for example,

x -> m -> y

You need it for

x -> y2 -> u -> y1
 Jinsong Chen posted on Wednesday, August 08, 2012 - 6:24 pm
Here's the setting:
M=a*X+e1
Y=b*M+c*X+e2
All M, X, Y are latent variables with ordinal observed responses (e.g., Likert scale). I generate data using both the Delta and Theta par.

In both cases:
when I use the Delta par. to estimate, all a, b, c, var(e1), and var(e2) tend to be underestimated;
when I use the Theta par., they all seem to be correct.

I double check but cannot find any problem in the way the data is generated. Any possible explanation?
 Linda K. Muthen posted on Thursday, August 09, 2012 - 10:32 am
 Theda rose posted on Sunday, March 31, 2013 - 8:15 am
With an SEM analysis that has a mediation and categorical indicators of latent factors, is it recommended that the theta parameterization be used instead of the default delta parametrization?
 Linda K. Muthen posted on Sunday, March 31, 2013 - 9:04 am
We recommend Delta unless Theta is required. Run you model using Delta. If Theta is required, you will receive a message telling you this.
 Tavinder Ark posted on Friday, June 07, 2013 - 1:40 pm
Yes, but why use theta over delta? Other programs such as Lavaan can estimate the same models without a theta parameterization. I guess I want to know exactly what the theta paramertization is doing, what type of transformation is happening etc.,
 Tavinder Ark posted on Friday, June 07, 2013 - 1:48 pm
That is, why is this necessary for categorical situations as I understand why theta is needed for the multiple group case and latent growth model. Thanks.
 Bengt O. Muthen posted on Friday, June 07, 2013 - 2:02 pm
There are certain models where the default Delta parameterization is not suitable and Theta is needed. These include multiple-group models where you have across-group hypotheses about the residual variances for the factor indicators and path models where a categorical dependent variable is both influenced by and influences another variable.

The Theta parameterization lets you access the residual variances of the factor indicators as parameters, whereas in the Delta parameterization, the delta parameters are functions of factor variances, factor loadings, and residual variances. So if for instance you are interested in testing group invariance of residual variances that is not achieved by holding Delta equal across groups because you may have group differences in factor variances and/or loadings.
 Tyler Mason posted on Thursday, October 03, 2013 - 5:04 pm
I ran a mediation model with two IVs, two mediators, and a categorical DV (2 categories). The analysis failed when I ran it with the normal delta parameter. The error said I needed to do Monte Carlo. However, I tried the theta parameter and it worked. Is it okay for me to use the theta parametrized results? Thanks!
 Bengt O. Muthen posted on Thursday, October 03, 2013 - 5:14 pm
Please send the 2 outputs to Support.
 Tait Medina posted on Monday, May 05, 2014 - 9:31 am
My question is regarding different results obtained when using different parameterizations of Multiple-Group Factor Analysis when outcome variables are dichotomous. Two groups are considered.

I have 4 models. Models 1 and 2 use the Theta parameterization and Models 3 and 4 the Delta parameterization. The metric of the factor is set by fixing one loading at one in each group in Models 1 and 3, and the metric of the factor is set by fixing the factor variance at one in the first group in Models 2 and 4. In all models the factor mean is fixed at zero in the first group. In all models factor loadings and item thresholds are constrained to be invariant over group.

I was expecting to find the Wald test of differences in the factor variance between groups to be similar (asymptotically equivalent) regardless of the parameterization. But this doesn't appear to be the case.

Wald test of factor variance:
Model 1: 15.212 (1) p=0.0001
Model 2: 46.423 (1) p=0.0000
Model 3: 31.584 (1) p=0.0000
Model 4: 46.434 (1) p=0.0000

I understand from Webnote 4 that the Delta parameterization may perform better. However, my intention is to, in a next step, fix residual variances at one in both groups and compare the results I obtain using the "traditional" multiple group testing approach to the the results from the new Alignment approach.
 Tihomir Asparouhov posted on Monday, May 05, 2014 - 8:23 pm
Testing
z1-1=0 (model 2 and 4)
or testing
L1*L1*(z1-1)=0 (model 1)
are two different tests and will have different power to detect significance

If you want to avoid that use the DIFFTEST command.

Var(f1)-Var(f2)=0
you can test
Var(f1)/Var(f2)=1
or
Var(f2)/Var(f1)=1
that will make the test scale independent, and you will get the same value, i.e., the formulation of the test will be independent of the parameterization.
 Tait Medina posted on Wednesday, May 14, 2014 - 1:18 pm
Is Var(f1)/Var(f2)=1 possible useng the MODEL TEST command? I have tried labeling the parameters, say p1 and p2, and using the syntax:

MODEL TEST:
p1/p2=1

However, I recieve this warning:
*** ERROR in MODEL TEST command
A parameter label or the constant 0 must appear on the left-hand side of a MODEL TEST statement. Problem with the following:
p1/p2 = 1
 Linda K. Muthen posted on Wednesday, May 14, 2014 - 3:36 pm
You should do this is MODEL CONSTRAINT.
 Bengt O. Muthen posted on Thursday, May 15, 2014 - 5:54 am
Or, do what the ERROR suggests:

MODEL TEST:
0 = 1-p1/p2;
 Keri Wong posted on Monday, August 18, 2014 - 4:26 am
Dear Dr Muthen,
I ran a CFA 3-factor model with intercorrelated latent factors in two separate samples. I get the R-squares for individual items but not estimates or R-squares for individual factors. They all appear to be 1.00 in the output. Is there a way to freely estimate that?

Also, I would like to know whether the R-squares explained by each item is significantly different across my samples. Is there a way to do this? Or would this be the equivalent of assessing measurement invariance at the item level?

Many thanks,
 Linda K. Muthen posted on Monday, August 18, 2014 - 10:46 am
In a CFA, the factors are independent variables. R-squares are given for dependent variables.

After you establish measurement invariance of intercepts, factors loadings, and residual variances, you can use MODEL CONSTRAINTS to create the R-squares and test if they are equal.
 Naomi Friedman posted on Wednesday, November 26, 2014 - 10:08 am
In prior posts you mention that Mplus will tell me if I need to use the theta parameterization, and that theta is necessary to access the residual variances of the categorical dependent variables and in "path models where a categorical dependent variable is both influenced by and influences another variable." I have the latter case (single group), specifically a panel design longitudinal model, in which I allow the residuals for the categorical indicators to correlate with residuals for other latent variables within time points. I thought I would need to use theta for this, but Mplus does allow me to use delta. Should it actually be possible to use delta in this case (or have I perhaps specified the model incorrectly?) and is it appropriate?
 Bengt O. Muthen posted on Wednesday, November 26, 2014 - 6:13 pm
Sounds like you have WITH statements and not the ON statements that would trigger the disallowance.
 james unnever posted on Thursday, May 28, 2015 - 5:10 pm
a path model with continuous and categorical dependent variables does not require theta (pg 35).I run the model it says that it is required.

This is my model....

NAMES ARE
WISCSCAL AGE3 SJ8 SJ11 SJ12 EASI_1 EXT_Y3 EXT_C male
disc2 church religsal highsch faminc

Missing = WISCSCAL AGE3 SJ8 SJ11 SJ12 EASI_1 EXT_Y3 EXT_C male
disc2 church religsal highsch faminc

USEVARIABLES = WISCSCAL AGE3 SJ11 sj8 sj12 EASI_1 EXT_Y3 EXT_C male
disc2 church religsal highsch faminc

CATEGORICAL IS SJ11 ;

ANALYSIS: PARAMETERIZATION = theta;

MODEL:
EXT_Y3 ON sj11 sj8 sj12 disc2 mom dad
age3 male married highsch faminc
prison EASI_1 wiscscal religsal church ext_C ;

age3 male married highsch faminc
prison EASI_1 wiscscal religsal church EXT_C;

OUTPUT: STDYX;

It terminates normally but I get this error message???

WARNING: VARIABLE MALE MAY BE DICHOTOMOUS BUT DECLARED AS CONTINUOUS.

WARNING: VARIABLE HIGHSCH MAY BE DICHOTOMOUS BUT DECLARED AS CONTINUOUS.

WARNING: VARIABLE MARRIED MAY BE DICHOTOMOUS BUT DECLARED AS CONTINUOUS.

NO CONVERGENCE. NUMBER OF ITERATIONS EXCEEDED.

Any suggestions?...thanks...
 Bengt O. Muthen posted on Thursday, May 28, 2015 - 6:06 pm
Bottom of page 605 and top pf 606 apply to your SJ11 variables being "in the middle" of a chain, thereby requiring Theta.

To be able to diagnose the rest, please send output and license number to support.
 Denise Haynie posted on Wednesday, March 08, 2017 - 7:49 am
I am running a cross-lagged model examining the bidirectional effects between a binary and a continuous variable. The analysis type = complex to account for the survey design, and the parameterization is theta. Looking at the standardized coefficients, are they all probit regression coefficients? Is it okay interpret differences in magnitude as differences in the strength of the association across the different paths? Thanks much.
 Bengt O. Muthen posted on Wednesday, March 08, 2017 - 5:53 pm
Q1: The cont's DV regression is regular linear. The binary DV regression is probit for the binary DV and regular linear for the continuous latent response variable underlying the binary observed DV.

Q2: Yes if you are considering the continuous latent response variable.