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 Nicholas Turner posted on Wednesday, June 26, 2013 - 5:54 am
I have run an IRT model using binary indicator variables using the WLSMV estimator and theta parameterization. The output gives me the difficulty and discrimination parameters. I'm now running a graded response IRT model using 3 level categorical indicator variables using the MLR estimator and I wanted to now if similarly the difficulty and discrimination parameters can be obtained?
Many thanks
 Bengt O. Muthen posted on Wednesday, June 26, 2013 - 1:05 pm
That is not automatically given, but you can do it in Model constraint by comparing (1) and (2) of the paper:
 Nicholas Turner posted on Friday, June 28, 2013 - 6:53 am
Thanks for the reply. I've had a look at the paper and I'm not sure I follow how I can use Model constraint to get the difficulty and discrimination parameters?
If it helps here is some of the syntax for my graded response IRT model:

estimator = MLR;

Dep by q18_a*1 q18_c q18_d q18_e q18_f q18_g q18_i q18_j q18_l q18_m q18_n
q18_o q18_p ;

Dep@1 ;

And here it is for the model when each indicator variable was recategorised as binary:

estimator = WLSMV;
parameterization = theta;
Dep by q18_a*1 q18_c q18_d q18_e q18_f q18_g q18_i q18_j q18_l q18_m q18_n
q18_o q18_p ;

Dep@1 ;
 Bengt O. Muthen posted on Friday, June 28, 2013 - 4:06 pm
I actually don't think you need to translate results but stay with the Mplus parameterization for the graded response model. It seems to be the parameterization used in the IRT literature. That is, with ordinal response as opposed to binary response, they seem to switch to the Mplus factor analytic parameterization. See for example section in the book

Reckase (2011). Multidim. IRT. Springer.

as well as eqn (6) of the Psych Method article

Cai et al (2011). Generalized full-info....

Reckase gives a discussion of interpretations. In my view, the fact that IRT makes a parameterization switch when going from binary to ordinal speaks to using the factor analytic parameterization all the time.
 Bengt O. Muthen posted on Friday, June 28, 2013 - 4:08 pm
To comment on your inputs, you can use ML or WLSMV irrespective of binary or ordinal response. With ML you can use the default logit link or you can use probit link.
 Emily posted on Monday, March 03, 2014 - 11:26 pm
I have run the following model using MLR estimation:

f1 BY U1@1 U2-U20*;
f2 BY T1-T20@-1;
f1 on f2;

This portion of the model (f1 BY U1@1 U2-U20*;) is the 2PL IRT model.

I am wondering what the appropriate conversion formula would be to get from the MPlus parameters to the IRT parameters. Is it simply the factor loading*sqrt(f1 var) or is it more complicated because of the second factor?

 Bengt O. Muthen posted on Tuesday, March 04, 2014 - 7:22 pm
You use the regular formula you point to when an item loads on only one factor. It doesn't matter that you have other items loading on other factors.

For multiple factors, see a new FAQ to be posted tomorrow.
 Alvin  posted on Thursday, August 14, 2014 - 10:20 pm
Hi Dr Muthen,

I ran a two-parameter IRT model - with four factors. Given the number of integration points, it seems to take a long time for estimation ... My question is, in addition to using monte carlo integration (is this recommended?), are there alternative estimator methods (e.g. Bayesian) that may reduce computational time? Or does it make sense to run separate models for each component of the measure? but I am interested in the item response properties as well as the structure of this psychiatric construct. Also, the other question, in IRT models, is factor variance always fixed to 1, as demonstrated in mplus users guide?
 Bengt O. Muthen posted on Friday, August 15, 2014 - 7:51 am
With four factors you can use integration = montecarlo(5000). You can also use Bayes.

Factor variances are often fixed at 1 in IRT, but you can set the metric differently (e.g. fixing the first loading) in Mplus.
 Alvin  posted on Wednesday, August 20, 2014 - 1:30 am
Thanks Dr Muthen - I ran a 2-parameter LRT with a four-factor model with binary variables using Monte Carlo (5000) - mplus says I need to increase miterations which I did up to 1000 but still couldn't get the parameter estimates?
 Linda K. Muthen posted on Wednesday, August 20, 2014 - 5:58 am
Please send the output and your license number to
 Alvin  posted on Saturday, August 30, 2014 - 7:22 pm
Hi Dr muthen can I clarify - in my 2-parameter binary IRT output, the discrimination parameters for some of my items are not significant with a high SE, is it correct to interprete this as evidence of the items not being able to statistically discriminate positive from negative cases? The threshold parameters however are significant. The other question is re multidimensional IRT, parameterization switches around automatically when a model of this kind is estimated? Many thanks
 Bengt O. Muthen posted on Sunday, August 31, 2014 - 11:03 am
Q1. That's right.

Q2. I think you are saying that with a single factor Mplus offers additional output with a re-parameterization into IRT parameters, but with multiple factors Mplus does not offer re-parameterized output. The reason is given in the FAQ:

IRT parameterization using Mplus thresholds

I recommend always using the default Mplus parameterization of thresholds and factor loadings.
 Alvin  posted on Sunday, August 31, 2014 - 11:02 pm
Thanks Dr Muthen - When running a multidimensional version of the 2PL model (with two factors), factor loadings can be interpreted as discrimination parameters? is the key distinction between multi- and unidimensional 2-PL that in a multidimensional model, the probability of endorsing a positive response depends on M latent variables and that item responses can be plotted against different latent variables (in this case severity based two sets of symptoms or factors)? when plotting CCIs one has to look at item charateristics as a function factor 1 and factor 2?
 Bengt O. Muthen posted on Monday, September 01, 2014 - 3:18 pm
Q1. The factor loadings are simply slopes in a logit or probit regression, so the interpretation is just like for such regression (see our Topic 2 handout or standard textbooks).

Q2. You want to plot against one factor holding other factors constant at for example their means (typically zero).
 J.D. Haltigan posted on Thursday, June 23, 2016 - 10:42 pm
Quick question. I am working with very sparse data estimating a 2PLM IRT model (binary indicators). Some ML estimates are very poorly-behaved with extreme values and even further extreme SEs; Bootstrapping has been one way I have dealt with this (cf. Albanese & Knott, 1994) but I have also tried using the Bayes estimator.

When I estimate the model with the Bayes estimator the estimates seem to behave much better...however, does one get the IRT parameterization in the usual way when working with Bayes estimates of factor loadings and thresholds (i.e., to item discrimination and difficulty).

Thank you!
 Linda K. Muthen posted on Friday, June 24, 2016 - 10:43 am
If you are using Bayes with non-informative priors, you should not get very different results. You may be seeing the difference between probit and logistic regression. Bayes and ML handle missing data in the same way.

Because Bayes is probit, the translation is somewhat different. It is described on our IRT page.
 Owis Eilayyan posted on Sunday, August 07, 2016 - 2:56 pm
Hello Dr. Muthen,

I am running 1-parameter IRT model using the following input:

Interference BY Personal* Lifting Walking sitting
Standing Sleeping Sex Social Travelling


but I got different discrimination values! I think in 1-parameter model I should get same discrimination value for all items.

Thank you,
 Bengt O. Muthen posted on Monday, August 08, 2016 - 9:44 am
It sounds like you don't get the loading equality by putting (1) on only one of the lines. Instead, write your BY statement as

Interference BY Personal-BPI_9G (1);
 Owis Eilayyan posted on Monday, August 08, 2016 - 11:01 am
Ok, Thank you
 Jo Cotton posted on Thursday, August 03, 2017 - 2:22 am

I am working on an item response model that meets the assumptions for 2PL; binary data, unidimensional, monotonic, local independence, no guessing likely. The underlying latent trait is non-normally distributed as it is a psychiatric condition that is rare in the population. I have read literature on simulations that suggests failing to adjust for non-normal leads to increased estimation error:
Sass, D. A., Schmitt, T. A., & Walker, C. M. (2008). Estimating non-normal latent trait distributions within Item Response Theory using true and estimated item parameters.
And other lit suggests an empirical histogram (EH), a Ramsay curve (RC-IRT) or Davidian curve (DC-IRT) should be used to adjust. See:
Woods, C. M. (2015). Estimating the Latent Density in Unidimensional IRT to Permit Non-Normality.

I have searched through your online documentation and forum, and cannot find reference to any of these methods. Can you advise if any these are accommodated by Mplus?

Best wishes,
 Bengt O. Muthen posted on Thursday, August 03, 2017 - 5:18 pm
We have an article on our website which shows a mixture approach to non-normal trait modeling:

Wall, M. M., Park, J. Y., & Moustaki, I. (2015). IRT modeling in the presence of zero-inflation with application to psychiatric disorder severity. Applied Psychological Measurement. DOI: 10.1177/0146621615588184
view abstract
 Jo Cotton posted on Friday, August 04, 2017 - 6:30 am
This is slightly different to what I was looking for, but still very helpful, thank you. I missed it in my search.
 Madison Silverstein posted on Monday, September 04, 2017 - 9:08 am

I am attempting to run a multidimensional IRT model using the Plot: type is plot 2; command to examine the associated item response function, information function, and test information function. The model runs, but the plot button on the top of the mplus window does not appear. I also do not see any errors or warnings about the plots. In the past, I have used a pc when examining plots. I am now using a mac. Do you have to do something different to get the plots on a mac, or do you think this is a function of something else? Any advice would be much appreciated.

Thank you!

 Bengt O. Muthen posted on Monday, September 04, 2017 - 4:03 pm
As we state on our website, the Mac version does not come with plotting. Instead, see our webpage for R plotting:
 Channing Mathews posted on Sunday, November 26, 2017 - 3:44 pm
Hello Dr. Muthen,

Similar to Nicolas Turner's question, I am having trouble getting MPLUS to provide difficulty and discrimination parameters for the graded response IRT model that I am trying to run. I also looked at the paper that was cited and I am also still unclear on how to use Model Constraint to obtain these two parameters. Could you guide me as to how I should proceed to get these parameters in the output?

Thank you!
 Bengt O. Muthen posted on Sunday, November 26, 2017 - 5:19 pm
The graded response model for a certain item observed in category k is

P(y=k | f) = F(k) - F(k-1),

where f represents the factor and where

F(k) = 1/[1+exp(tau_k - lambda*f)].

As in equations (21) and (22) of our 2016 IRT document as well as in the Topic 2 handout, slide 94, the translation to IRT parameters with theta having mean zero and variance one is analogous to the translation for the binary logistic response case,

a = lambda*sqrt(psi),

b_k = (tau_k - lambda*alpha)/lambda*sqrt(psi),

where tau_k is a threshold, lambda is a factor loading, and alpha and psi are the mean and variance of the factor f, respectively.

The a and b_k parameters can be expressed in the Model Constraint command using parameter labels in the Model command for tau_k, lambda, alpha, and psi (the latter two may already be fixed to zero and one).
 Harun Kızılay posted on Saturday, July 07, 2018 - 9:22 am
Dear Muthen

Where is the problem

Here is the syntax

Model Constraint: New(a b);
a = lambda*sqrt(psi);
b_k = (tau_k-lambda*alpha)/lambda*sqrt(psi);

Unknown parameter label in MODEL CONSTRAINT: LAMBDA
 Harun Kızılay posted on Saturday, July 07, 2018 - 10:55 am
Dear Muthen

I have solved,

Thank you
 Madison Silverstein posted on Thursday, November 01, 2018 - 2:43 pm

I am running a unidimensional IRT model using WLSMV. Is there a way to calculate the discrimination parameter at certain levels of theta?

Thank you!

 Bengt O. Muthen posted on Friday, November 02, 2018 - 2:56 pm
Yes. See our Short Course Topic 2 on IRT and also section 3 of our IRT paper:
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