Easy way to fit a Rasch model in Stata

[Rafal Raciborski (StataCorp)]

Thanks to Kit Baum, raschify is now available on SSC.

raschify transforms 1PL model parameters into Rasch model parameters. All the postestimation tools that are available after irt 1pl can be used on a Rasch-ified model.

Here is an example:

ssc install raschify

webuse masc1, clear
irt 1pl q*

// convert to Rasch parameterization
raschify
// display the transformed results
irt

[ericmelse]

Dear Rafal,

Thank you for your contribution.

I am exploring the functionality of raschify and to this purpose studied the Standard dataset 3 from the Rasch org website:

input item1 item2 item3
1 0 0
0 1 1
0 1 1
end

Next, I ran the IRT 1PL model, raschify and created their test information function plot, which clearly illustrate the purpose and benefit of raschify (i.e. '...in that it represents the structure which data should exhibit in order to obtain measurements from the data...' Quotation from Wikipedia).

* Fit a 1PL model to binary items
irt 1pl item1-item3
irtgraph tif, xtitle("Theta 1PL Standard Dataset 3", margin(l-0 r+0 b-2 t+2))
* Transform the 1PL model coefficients to the Rasch metric
raschify
* Display parameters of the Rasch model
irt , cformat(%9.5f)
irtgraph tif, xtitle("Theta Rasch Standard Dataset 3", margin(l-0 r+0 b-2 t+2))

Now, I have two questions.
When I compare the parameters of the Rasch model, calculated by raschify, with the reference published on the Rasch org website, i.e. Pairwise Maximum Likelihood Estimation (PMLE), the column estimates are:
.00000, -0.69315, -0.69315
whereas from Stata/raschify these are:
.69315, -0.69315, -0.69315
I assume the difference is probably not substantive, but I am interested to learn why this difference occurs (e.g. is the first item 'anchored' to be 0 in the reference models?).

My second question is if it is possible, somehow, to calculate row estimates, like those published on the Rasch org website.

[Rafal Raciborski (StataCorp)]

Yes, in the example on the Rasch website, the first item is anchored at 0. You can replicate their estimates in Stata using gsem, however you will not be able to use irtgraph afterwards.

clear
input item1 item2 item3
1 0 0
0 1 1
0 1 1
end

irt 1pl *
raschify
irt

Note that the underlying model is fit by gsem:

di e(cmdline2)
gsem (Theta@a1 -> item1 item2 item3, logit)   , variance(Theta@1) latent(Theta) constraints( )

See how the coefficients are labeled:

gsem, coeflegend
------------------------------------------------------------------------------
             |      Coef.  Legend
-------------+----------------------------------------------------------------
item1        |
       Theta |          1  _b[item1:Theta]
       _cons |  -.6931471  _b[item1:_cons]
-------------+----------------------------------------------------------------
item2        |
       Theta |          1  _b[item2:Theta]
       _cons |   .6931471  _b[item2:_cons]
-------------+----------------------------------------------------------------
item3        |
       Theta |          1  _b[item3:Theta]
       _cons |   .6931471  _b[item3:_cons]
-------------+----------------------------------------------------------------
   var(Theta)|   3.67e-15  _b[/var(Theta)]
------------------------------------------------------------------------------

Constrain the difficulty of item1 to 0 and run the model using gsem massaging the syntax to fit a Rasch model:

constraint 1 _b[item1:_cons] = 0
gsem (Theta@1 -> item1 item2 item3, logit), latent(Theta)  constraints(1)
------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
item1        |
       Theta |          1  (constrained)
       _cons |          0  (omitted)
-------------+----------------------------------------------------------------
item2        |
       Theta |          1  (constrained)
       _cons |   .6931472   1.224745     0.57   0.571    -1.707309    3.093603
-------------+----------------------------------------------------------------
item3        |
       Theta |          1  (constrained)
       _cons |   .6931472   1.224745     0.57   0.571    -1.707309    3.093603
-------------+----------------------------------------------------------------
   var(Theta)|   2.63e-35   1.20e-17                             .           .
------------------------------------------------------------------------------

Note that the model is fit using the slope-intercept form, a'*Theta + b'. If we were able to use irt to display these gsem estimates, we would see a = a' and b = -b'/a, which would match the coefficients on the Rasch website.

Alternatively, reverse the item coding to match the estimates without the need of a transformation:

recode item* (1=0) (0=1)
constraint 1 _b[item1:_cons] = 0
gsem (Theta@1 -> item1 item2 item3, logit), latent(Theta)  constraints(1)
------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
item1        |
       Theta |          1  (constrained)
       _cons |          0  (omitted)
-------------+----------------------------------------------------------------
item2        |
       Theta |          1  (constrained)
       _cons |  -.6928628   1.224687    -0.57   0.572    -3.093205    1.707479
-------------+----------------------------------------------------------------
item3        |
       Theta |          1  (constrained)
       _cons |  -.6928628   1.224687    -0.57   0.572    -3.093205    1.707479
-------------+----------------------------------------------------------------
   var(Theta)|   3.57e-34   5.39e-17                             .           .
------------------------------------------------------------------------------

I suspect it is possible to calculate row estimates in Stata using AMLE or WMLE, however that would require further programming.