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 Richard E. Zinbarg posted on Tuesday, August 15, 2006 - 7:16 pm
Hi Bengt,
On p. 259 of his 1999 book, Rod McDonald gives an equation relating the factor loading, lambda, to the IRT slope parameter b. In 12.17a, he states that lamba = b/(sqrt(1 + b^2)). Would the b in this equation correspond to the probit model slope that Mplus provides as factor loadings? Or should I think of the probit model slope that Mplus provides as the lambda in this equation?
Thanks very much!
Rick Zinbarg
 Bengt O. Muthen posted on Tuesday, August 15, 2006 - 8:37 pm
Here is what we say in our teachings:

2-parameter normal ogive IRT model uses
P (u = 1 | theta) = [a (theta b)]
a discrimination
b difficulty

2-parameter logistic IRT model uses
P (u = 1 |theta ) =
1/(1 + exp(-D a (theta - b)))

with D = 1.7 to make a, b close to those of probit.
 Richard E. Zinbarg posted on Tuesday, August 15, 2006 - 9:56 pm
thanks for the very speedy reply! And in Mplus Web Notes #4, I know you give a different equation relating the IRT discrimination parameter to lambda from factor analysis. If I am understanding that Web Note correctly, the lambda in your equation 19 is the factor loading from an analysis of the tetrachoric correlations using a probit link rather than of the phi correlations (or observed covariances) among the observed variables using a linear regression model. Is that correct? If so, are you aware of any work that relates a factor loading from an analysis of tetrachorics using a probit link to a loading from an analysis of the phi correlations (or observed covariances) among the observed variables using a linear regression model? It is clear to me that for the purposes of model testing and comparison, the analysis of tetrachorics using a probit link is the most appropriate but in terms of estimating factor-analytically derived indices of reliability of composite scores, what quantities one should use in these reliability formulas is less clear. Rod McDonald's advice seems to be, that for the purpose of estimating factor-analytically derived reliability indices such as omega, to just fit
the linear model to the sample item covariance matrix. I am trying to figure out if this is a strategy that is both reasonable and the only one feasible and that should (presumably) satisfy reviewers.
 Bengt O. Muthen posted on Thursday, August 17, 2006 - 11:14 am
For some reason, my earlier post included only half of what I intended, but as you say Web Note #4 has the formula in (19). There is not a simple relationship between the tetrachoric/probit-based loadings and loadings from linear modeling using phi's. I think Rod has written about such relations. I think one has to define what reliability should mean - if it refers to how well a factor in a probit/logit IRT model is captured by a sum of binary items, then I think one has to use a non-linear model, but that would not necessarily be the case if one has another definition.
 Salma Ayis posted on Tuesday, September 05, 2006 - 5:17 am
Hi, I am a new user of IRT and still have few questions for which I very much appreciate answers/advice/references!
1- for a set of binary items, I would like to interpret my results in term of logits for each item, is it possible to get these logits as an output without needing to compute them seperately?. If so please let me know!; if not I can see in the output, in Model Results, that there is a formula stated as: IRT PARAMETERIZATION IN TWO-PARAMETER LOGISTIC METRIC
WHERE THE LOGIT IS 1.7*DISCRIMINATION*(THETA - DIFFICULTY), what is theta exactly, I can see theta in my output but I am unable to link this with other parametrs-please advice!.
2- If I use more than two categories would I still have estimated difficulty and discrimination parameters for each item? your advice is most appreciated!
 Linda K. Muthen posted on Tuesday, September 05, 2006 - 10:26 am
The regression coefficients obtained using the CATEGORICAL option with the maximum likelihood estimator are logits. Theta is a factor score. The Theta in your output refers to a parameter in the model. If you use more than two categories, you will obtain difficulty and discrimiation.
 Salma Ayis posted on Friday, January 05, 2007 - 5:35 am
Dear Linda, Further to your response on Tuesday, September 05, 2006, I am afraid, still unsure where to find the logits?, I am using example 5.5, and have specified the CATEGORICAL option for my set of binary indicators, when you say the regression coefficients, what are these called in the output? are they the estimates? or another command is needed to do these calculations? many thanks for your anticipated response!
 Linda K. Muthen posted on Sunday, January 07, 2007 - 9:06 am
The parameter estimates are shown under the column labeled Estimates. The output is described in the beginning of Chapter 17.
 Lisa M. Yarnell posted on Wednesday, March 12, 2014 - 12:47 pm
Hello, I employed the D(1.7) output option for my IRT model with dichotomous indicators. We are using WLSMV estimation and hence a probit link. The output for the factor model and IRT parameterization sections do not show the same values for the loadings/discrimination levels and thresholds/difficulties.

I had thought that in employing the D(1.7) option, the factor model section of output would be translated to the IRT parameterization, and that the two sections of output would hence contain the same numbers.

Or is it true that the output continues to show the two different sets of values even when employing the D(1.7) output option?

Thank you and regards.
 Lisa M. Yarnell posted on Wednesday, March 12, 2014 - 12:58 pm
Also, I have just found that with theta parameterization, the factor model and IRT parameterization output sections contain the same values, whether employing the D(1.7) option or not.

But when delta parameterization is used, I do not receive the same values in the factor model and IRT parameterization sections of output, whether I employ D(1.7) or not.

So is receiving the same solution across the two parameterizations more an issue of theta vs. delta, or using the translation constant?

Thank you.
 Bengt O. Muthen posted on Friday, March 14, 2014 - 12:46 pm
The regular output and the IRT translation are not expected to show the same results since they are different parameterizations. The D=1.7 is another matter - it has to do with making probit and logit close.
 Lisa M. Yarnell posted on Monday, March 17, 2014 - 7:41 pm
I was under the impression that when probit estimation is being done, employing D(1.7) makes the factor model and IRT parameterization outputs equal (not that it makes the probit and logit solutions close).

The default is probit for both the factor model and IRT output sections when WLSMV estimation is used, which happens with dichtomous items.

Can you comment on my statements here?
 Bengt O. Muthen posted on Tuesday, March 18, 2014 - 2:02 pm
Your first paragraph is incorrect: 1.7 is to make the IRT logit close to the IRT probit. It has nothing to do with the factor model parameterization.

Your second paragraph is correct.
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