1. Presented at Measured Progress
IRT Fixed Parameter Calibration and Other Approaches to Maintaining Item Parameters on a Common Ability Scale
Seonghoon Kim, PhD
Keimyung University
Presented at Measured Progress
on July 10, 2008

2. Overview I. Nature of IRT Ability Scale
II. Three Approaches to Maintaining Item Parameters on a Common Scale
III. Principle of Fixed Parameter Calibration (FPC)
IV. Use of Computer Programs for FPC
V. Applications of FPC for Scaling and Equating

3. Reference Guide This presentation was prepared based on my articles,
Kim, S. (2006a). A comparative study of IRT fixed parameter calibration methods. Journal of Educational Measurement, 43 (4),
Kim, S. (2006b). A study on IRT fixed parameter calibration methods using BILOG-MG. Journal of Educational Evaluation, 19 (1),
Kim, S., & Kolen, M. J. (2006). Robustness to format effects of IRT linking methods for mixed-format tests. Applied Measurement in Education, 19 (4),
Kim, S., & Lee, W. (2006). An extension of four IRT linking methods for mixed-format tests. Journal of Educational Measurement, 43 (1),
Kim, S., & Kolen, M. J. (2007). Effects on scale linking of different definitions of criterion functions for the IRT characteristic curve methods. Journal of Educational and Behavioral Statistics, 32 (4),
and my recent thoughts and works on FPC

4. I. Nature of IRT Ability Scale Indeterminacy in IRT modeling
Item response function (IRF) and metrics
Two-parameter logistic (2PL) model
IRF = P(θ | a, b) = 1/[1+exp(-Da(θ-b))]
Suppose that θO = A θN + B
If aO = aN /A and bO = A bN + B, P(θO | aO, bO) = P(θN | aN, bN) Therefore, IRF and item parameters are invariant conditional on linear transformation
Thus, in practice, either θO or θN can be used, which means scale indeterminacy.

5. I. Nature of IRT Ability Scale “0, 1” Scaling vs. Rasch Scaling
Scaling by arbitrarily assuming that the mean (M) and standard deviation (SD) of the ability distribution are equal to 0 (origin) and 1 (unit).
Such arbitrary but “standardized” fixing is unavoidable when the M and SD are unknown.
Rasch scaling
Setting the origin (0) of the scale at the average difficulty of all items involved, while fixing the unit at 1.
The fixed unit is guaranteed by the Rasch modeling.

6. I. Nature of IRT Ability Scale Need for a Fixed Common Ability Scale
A fixed common scale should be used across test administrations for several reasons
To check the invariance property of item parameters
To achieve comparability between item parameters from different administrations
To develop an item pool
To conduct IRT equating

7. I. Nature of IRT Ability Scale Need for a Fixed Common Ability Scale
To develop a common ability scale requires all new scales to be linked to the fixed old scale θO.
θN1
θN2
θN3 θO

8. I. Nature of IRT Ability Scale Factors for Development of a Common Scale
Development of a fixed common scale is subject to
Data collection design for IRT scaling and equating test forms
Random groups design vs. Common-item nonequivalent groups design
Scaling convention
“0,1” scaling vs. Rasch scaling
Item parameter estimation method
Marginal maximum likelihood (MML) estimation vs. Joint maximum likelihood (JML) estimation

9. The Context Assumed in This Presentation
Data collection design for IRT scaling and equating test forms
Common-item nonequivalent groups (CING) design
Anchor items (i.e., common items) link two test forms
Scaling convention
“0, 1” scaling
Group dependent
In a random groups design, two “0, 1” scales from alternative forms may be considered equivalent.
Marginal Maximum Likelihood (MML) Estimation
Estimation of Item parameters
Estimation of Underlying Ability Distribution
Quadrature weights are estimated at quadrature points.

10. Data Structure Illustration for the CING Design
Old Form
(Group 1)
New Form
(Group 2)
New Form Unique Items to New Group (2)
Old Form Unique Items to Old Group (1)
Items
Common Items (Anchor) to Old and New Groups

11. II. Three Approaches to Maintaining a Common Scale
Separate calibration by form and linking
Estimate transformation coefficients A and B using two sets of item parameter estimates for the anchor items
Use A and B to transform new form item parameter estimates into those on the old scale
Fixed parameter calibration (FPC)
Holding the old form anchor item parameters fixed and estimating the new form non-anchor items
Concurrent calibration (aka multiple-group estimation)
Combining new and old form data and estimating both all item parameters and underlying ability distributions, with the old group being designated as the reference-scale group
Will not be addressed in details in this presentation

12. II. Maintaining the Old Scale Separate Calibration by Form and Linking
“0, 1” scales from two test forms
Old form scale: θO (reference)
New form scale: θN (arbitrary)
Scheme of linking two “0, 1” scales
θO = A θN + B
θN (arbitrary origin & unit) -1 1 B A
θO (fixed origin & unit)

13. II. Maintaining the Old Scale Separate Calibration by Form and Linking
Linking ability scales is completed by placing all item parameters from separate calibrations onto the fixed old scale.
In the case of the 2PL model, given A and B, aN and bN parameters from a new scale are transformed into a* = aN /A and b* = A bN + B
In practice, A and B are estimated with item parameter estimates from the old and new scales.
Mean-Sigma Method (Marco, 1977)
Mean-Mean Method (Loyd & Hoover, 1980)
Haebara Method (Haebara, 1980)
Stocking-Lord Method (Stocking & Lord, 1983)

14. II. Maintaining the Old Scale Comparative Performance
Suppose that the characteristic curve (Haebara or Stocking-Lord) method is employed as a linking method for the “separate calibration and linking” approach.
The performance of the three alternative approaches to maintaining the old scale is differential depending on whether the new form items are common or not (Hanson & Béguin, 2002; Kim, 2006b; Kim & Kolen, in process).
For the common items, concurrent calibration would perform best, due mainly to larger sample size (new group + old group), compared to the non-common items.
For the non-common items, the three approaches would perform almost equally.

15. II. Maintaining the Old Scale Comparative Performance

16. II. Maintaining the Old Scale When is FPC most appropriate?
When using the “stable” old form anchor item parameters to obtain or diagnose the parameters of new form non-anchor items on the fixed old scale
Note
Placing the parameters of new form non-anchor items on the old scale is the focus.
Updating of the old form item parameters is not concerned at all.
The old form anchor items are assumed to have stable parameter estimates because a large sample was used for obtaining them.

17. III. Principle of FPC Basics
Why
To place the parameters of new form non-anchor items onto the fixed old scale
How
Holding the old form anchor item parameters fixed and estimating the new form non-anchor items
Critical Process
Estimating the underlying distribution of ability for the new form on the fixed old scale so that the new item parameters may be properly expressed on the old scale.
By the IRT modeling, the underlying distribution can be estimated using both the new form data and the fixed anchor item parameters.

18. Estimated New Item Parameters on the θO Scale
III. Principle of FPC Schematic Illustration of Updating Priors and Underlying Distributions of Ability
1st Est. Ability Dist. = 2nd Initial Prior
a1N b1N … bJN θO
EM Iterations
2nd Est. Ability Dist. = 3rd Initial Prior
1st Initial Prior
Fixing a1O, b1O, a2O, b2O, …
Final Est. Ability Dist.
Estimated New Item Parameters on the θO Scale

19. Refer to Kim (2006a) for numerical details.
III. Principle of FPC Numerical Expression: Multiple Prior Weights Updating and Multiple EM Cycles (MWU-MEM)
Likelihood Function for Estimating New Form Non-Anchor Item Parameters (Iteration s, quadrature point k, person i, data y, parameters Δ)
Closed-Form Formula for Estimating Quadrature Weights of the Underlying Ability Distribution from the New Form Data
Refer to Kim (2006a) for numerical details.

20. III. Principle of FPC Summary of Key Points
The values of the fixed anchor item parameters are expressed on the fixed old scale, so the origin and unit of the ability scale for the new form data have been already set. That is, we do not need to use “0, 1” scaling for the new form data.
New form non-anchor item parameters should be estimated using the new form underlying distribution that is properly recovered on the fixed old scale.
As with ability estimates, the underlying distribution can be estimated using the new form data and the fixed anchor item parameters.
Fixing the anchor item parameters pulls the underlying distribution onto the old scale gradually. Accordingly, the new form item parameters are also pulled onto the old scale.

21. III. Principle of FPC Concerns about the Unstable Estimates of Anchor Item Parameters
Unstable estimates of the fixed item parameters might adversely affect the performance of FPC.
However, Kim (2006a) showed that FPC is robust to sampling errors of the fixed item parameter estimates in calibrating non-anchor items.
This seems to be because the new form data collaborate with the fixed item parameters in “revealing” the old scale.
In other words, as long as the sample size of the new group is large enough, unstable estimates of the fixed item parameters would not much affect the proper estimation of both the underlying distribution for the new group and the non-anchor item parameters.

22. III. Principle of FPC Two Alternatives to the MWU-MEM Method
Some computer programs, such as BILOG-MG, do not update the prior quadrature weights during EM cycles when conducting FPC.
The resulting posterior (quadrature) weights would not properly represent the underlying ability distribution for the new form data.
Two ad-hoc methods can be used to obtain good estimates of the quadrature weights for the underlying distribution.
Simple Transformation Prior Update (STPU) Method
Iterative-Run Prior Update (IRPU) Method

23. III. Principle of FPC Two Alternatives to the MWU-MEM Method
Simple Transformation Prior Update (STPU) Method
Uses A and B from a linking method to simply update the prior ability distribution by transforming the posterior distribution from the regular, separate calibration with the new form. Then, conduct FPC with the updated prior ability distribution.
Iterative-Run Prior Update (IRPU) Method
Uses iteratively updated prior ability distributions through multiple FPC runs of BILOG-MG. An estimated posterior distribution in a calibration run is used as a prior distribution in the next calibration until the sequential procedure minimizes the difference between the two distributions.

24. III. Principle of FPC Two Alternatives to the MWU-MEM Method
Kim (2006b) shows that the two ad hoc methods for updating the prior ability distribution work very well.
In recovering the parameters of non-anchor items, the two methods perform almost equally to the Stocking-Lord linking method and concurrent calibration.
In practice, the STPU method may be preferred due to simplicity.
The IRPU method has the same feature as the MWU-MEM method, except for multiple runs of FPC. Thus, theoretically, the IRPU method may be more acceptable than the STPU method.

25. III. Principle of FPC Caveats against Using “Constrained” Estimation for FPC
Someone might think that imposing strong Bayesian priors on the fixed item parameters and freeing the non-anchor item parameters would function similarly to FPC.
A rationale for such constrained estimation can be found in, for example, the BILOG (Mislevy & Bock, 1990) manual.
In theory, it sounds reasonable.
But, my experiences suggest that using strong priors to fix the anchor item parameters tends to distort the non-fixed item parameters.

26. III. Principle of FPC Caveats against Using “Constrained” Estimation for FPC
Note that in constrained estimation the anchor item parameters are to be estimated (although almost fixed), while in FPC they are excluded from the parameter list to be estimated.
Without a facility to update ability prior weights, both the underlying distribution and non-anchor item parameters would be distorted.

27. IV. Use of Computer Programs for FPC
BILOG-MG 7.0 (Zimowski et al., 2003)
The “FIX” option does not function properly because the prior weights are not updated during EM cycles (Kim, 2006a).
The STPU or IRPU method can be used.
PARSCALE 4.1 (Muraki & Bock, 2003)
For FPC to work properly, the “POSTERIROR” option should be used (Kim, 2006a).
Without the “POSTERIOR” option, the STPU or IRPU method can be used.

28. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
Data
3,000 examinees for the new form data
The data were obtained by simulating examinees from Normal (1, 1) distribution, against the old group of N(0, 1) distribution.
25-item multiple-choice (MC) test
FPC
First 20 items fixed (item parameters are ready for use)
Last 5 items freed
The three-parameter logistic (3PL) model is used for item analyses.
Comparison of Default, STPU, and IRPU FPC methods

29. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
Command File (to Use the Default FPC Facility)
Default FPC with BILOG-MG
The examinee group (2) was sampled from N(1,1)
>COMMENT
Fixed-parameter calibration
>GLOBAL DFNAME=‘New.txt', PRNAME='Sample.PRM',
NPARM=3, SAVE;
>SAVE PAR='itempar';
>LENGTH NITEMS=25;
>INPUT NTOT=25, SAMPLE=3000, NALT=5, NID=4;
>ITEMS INUM=(1(1)25), INAMES=(O01(1)O20, P01, P02, P03, P04, P05);
>TEST TNAME=G2_FIX, INUM=(1(1)25), FIX=(1(0)20, 0(0)5);
(4A1, T1, 25A1)
>CALIB NQPT=31, CYCLE=100, CRIT=0.001, NEWTON=1,NOADJUST;
>SCORE NOPRINT;

30. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
Data File (New.txt)
Item Responses for Anchor Items

31. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
No. of Fixed Items
Fixed Parameter File (Sample.PRM) a b c 20

32. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
Command File for the STPU Method (Before Transformation)
Single Group “0, 1” Scaling,
Although the examinee group was sampled from N(1,1).
>COMMENT
STPU FPC before Transformation of Ability Points
>GLOBAL DFNAME='New.txt',
NPARM=3, SAVE;
>SAVE PAR='sampleSim01.PAR';
>LENGTH NITEMS=25;
>INPUT NTOT=25, SAMPLE=3000, NALT=5, NID=4;
>ITEMS INUM=(1(1)25),
INAMES=(O01(1)O20, P01, P02, P03, P04, P05);
>TEST TNAME=NO_FIX;
(4A1, T1, 25A1)
>CALIB NQPT=31, CYCLE=100, CRIT=0.001, NEWTON=0, IDIST=0;
>SCORE NOPRINT;

33. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
Posterior Distribution from “0, 1” Scaling for the STPU Method
QUADRATURE POINTS, POSTERIOR WEIGHTS, MEAN AND S.D.:
POINT E E E E E+01
POSTERIOR E E E E E-02
POINT E E E E E+01
POSTERIOR E E E E E-01
POINT E E E E E+00
POSTERIOR E E E E E+00
POINT E E E E E+01
POSTERIOR E E E E E-01
POINT E E E E E+01
POSTERIOR E E E E E-02
POINT E E E E E+01
POSTERIOR E E E E E-04 31
POINT E+01
POSTERIOR E-04
MEAN
S.D

34. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
Command File for the STPU Method (After Transformation)
STPU FPC with Transformed Prior Points
The examinee group was sampled from N(1,1).
Omitted (The same as the commands for before-transformation “0, 1” calibration)
>TEST TNAME=G2_FIX, INUM=(1(1)25), FIX=(1(0)20, 0(0)5);
(4A1, T1, 25A1)
>CALIB NQPT=31, CYCLE=100, CRIT=0.001, NEWTON=0, IDIST=1,
NOADJUST;
>QUAD POINTS=(
E E E E E+000
E E E E E-001
E E E E E-001
1.0330E E E E E+000
2.4328E E E E E+000
3.8323E E E E E+000
5.2329E+000),
WEIGHTS=(
2.1630E E E E E-003
3.0300E E E E E-002
4.2810E E E E E-001
1.0740E E E E E-002
4.3430E E E E E-003
2.9570E E E E E-005
3.2120E-005);
>SCORE NOPRINT;
Rescaled points by θ* = Aθ+B,
A =
B =
From “0, 1” Scaling
(Not Transformed)

35. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
2nd Command File for the IRPU Method
IRPU FPC with Updated Prior Weights
The examinee group was sampled from N(1,1).
Omitted (The same as the commands for the default FPC run)
>TEST TNAME=G2_FIX, INUM=(1(1)25), FIX=(1(0)20, 0(0)5);
(4A1, T1, 25A1)
>CALIB NQPT=31, CYCLE=100, CRIT=0.001, NEWTON=0, IDIST=1,
NOADJUST;
>QUAD POINTS=(
E E E E E+000
E E E E E+000
E E E E E-001
E E E E E+000
1.3330E E E E E+000
2.6670E E E E E+000
4.0000E+000),
WEIGHTS=(
8.8370E E E E E-005
2.5560E E E E E-003
1.1060E E E E E-002
8.4190E E E E E-001
9.2970E E E E E-002
1.1400E E E E E-004
1.5790E-004);
>SCORE NOPRINT;
Fixed Points
(-4.0 to 4.0)
Updated Weights
(= Posterior weights from the 1st run of IRPU FPC)

36. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
History of Updated Posterior Distributions by the IRPU Method
Iter# Mean Std. Dev.
From Default FPC
Iterations stopped because the M and SD were not changed beyond the limit

37. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
FPC Estimates of Non-Anchor Item Parameters on the Fixed Old Scale
Mean/Sigma Default FPC
Item a b c a b c
STPU FPC IRPU FPC
Item a b c a b c

38. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
FPC Estimates of Mean and SD of the Underlying Distribution on the Fixed Old Scale
Under-estimation
Method Mean Std. Dev.
Default FPC
STPU FPC
IRPU FPC
Mean-Sigma B = A = 1.041
Note. The new group examinees were from a N(1,1) distribution that was expressed on the fixed old scale.

39. IV. Use of Computer Programs for FPC Illustration of FPC with PARSCALE
Data
3,000 examinees for the new form data
The data were obtained by simulating examinees from Normal (0.5, 1.22) distribution, against the old group of N(0, 1) distribution.
A mixed-format test of 15 MC items and 2 five-category constructed-response (CR) items
FPC
First 10 MC items fixed (item parameters are ready for use)
Last 5 MC and 2 CR items freed
The 3PL model for MC items and the generalized partial credit (GPC) model for CR items
Comparison of STPU and MWU-MEM methods

40. IV. Use of Computer Programs for FPC Illustration of FPC with PARSCALE
Command File (MWU-MEM FPC)
MWU-MEM FPC with PARSCALE
The examinee group was sampled from N(0.5, 1.2^2)
>COMMENT
10 common items fixed and 2 CR items calibration
>FILE DFNAME='new.txt',
IFNAME='MC10FIX.IFN',
SAVE;
>SAVE PARM='MC10FIX';
>INPUT NTOT=17, TAKE=3000, NID=5, NTEST=1, LENGTH=17;
(5A1, T1, 17A1)
>TEST TNAME=I10FIX, ITEMS=(1(1)45), NBLOCK=17;
>BLOCK BNAME=FIXED, NITEMS=1, NCAT=2,
ORI=(0, 1), MOD=(1, 2), REP=10, SKIP;
>BLOCK BNAME=FREEMC, NITEMS=1, NCAT=2,
ORI=(0, 1), MOD=(1, 2), GPARM=0.2, GUESS=(2, EST), REP=5;
>BLOCK BNAME=FREED, NITEMS=1, NCAT=5,
ORI=(0,1,2,3,4), MOD=(1,2,3,4,5), REP=2;
>CALIB NQPT=41, PAR, LOG, SCALE=1.7, CYCLE=200, NEWTON=0,
FREE=(NOADJUST, NOADJUST), ESTORDER, SPRI, GPRI, POSTERIOR;
>SCORE NOSCORE;

41. IV. Use of Computer Programs for FPC Illustration of FPC with PARSCALE
Data File (New.txt)
Item Responses for CR Items
Item Responses for Anchor Items

42. IV. Use of Computer Programs for FPC Illustration of FPC with PARSCALE
Command File to Prepare IFNAME File (MC10FIX.IFN)
MWU-MEM FPC with PARSCALE
No Fix, “0, 1” Scaling
>COMMENT
10 common items fixed and 2 CR items calibration
>FILE DFNAME='new.txt',
SAVE;
>SAVE PARM='MC10FIX';
>INPUT NTOT=17, TAKE=3000, NID=5, NTEST=1, LENGTH=17;
(5A1, T1, 17A1)
>TEST TNAME=I10FIX, ITEMS=(1(1)45), NBLOCK=17;
>BLOCK BNAME=FIXED, NITEMS=1, NCAT=2,
ORI=(0, 1), MOD=(1, 2), GPARM=0.2, GUESS=(2, EST), REP=10;
>BLOCK BNAME=FREEMC, NITEMS=1, NCAT=2,
ORI=(0, 1), MOD=(1, 2), GPARM=0.2, GUESS=(2, EST), REP=5;
>BLOCK BNAME=FREED, NITEMS=1, NCAT=5,
ORI=(0,1,2,3,4), MOD=(1,2,3,4,5), REP=2;
>CALIB NQPT=41, PAR, LOG, SCALE=1.7, CYCLE=200, NEWTON=0,
FREE=(NOADJUST, NOADJUST), ESTORDER, SPRI, GPRI, POSTERIOR;
>SCORE NOSCORE;
No IFNAME
No SKIP

43. IV. Use of Computer Programs for FPC Illustration of FPC with PARSCALE
Item Parameter Output File from “0, 1” Scaling
MWU-MEM FPC with PARSCALE
No Fix, “0, 1” Scaling
I10FIX
GROUP 01
FIXED
BLOCK
BLOCK
(Omitted)
FREED
BLOCK

44. IV. Use of Computer Programs for FPC Illustration of FPC with PARSCALE
Modified Item Parameter File (MC10FIX.IFN)
Replaced with fixed a
Replaced with fixed b
Replaced with fixed c
MWU-MEM FPC with PARSCALE
No Fix, “0, 1” Scaling
I10FIX
GROUP 01
FIXED
BLOCK
BLOCK
(Omitted)
FREED
BLOCK
Replacing for the 10 fixed items

45. IV. Use of Computer Programs for FPC Illustration of FPC with PARSCALE
Command File for the STPU Method (After Transformation)
STPU FPC with Transformed Prior Points
The examinee group was sampled from N(1,1).
Omitted (The same as the commands for MWU-MEM
>CALIB NQPT=31, PAR, LOG, SCALE=1.7, CYCLE=200, NEWTON=0,
FREE=(NOADJUST, NOADJUST), ESTORDER, SPRI, GPRI,
DIST=4, QPREAD;
>QUADP POINTS=(
E E E E E+000
E E E E E+000
E E E E E-001
2.3929E E E E E+000
2.0844E E E E E+000
3.9310E E E E E+000
5.7761E+000),
WEIGHTS=(
1.2430E E E E E-004
9.1150E E E E E-002
4.2780E E E E E-002
8.6880E E E E E-002
6.4860E E E E E-003
3.3710E E E E E-004
4.1380E-005);
>SCORE NOSCORE;
Rescaled points by θ* = Aθ+B,
A = 1.38
B = 0.24
From “0, 1” Scaling
(Not Transformed)

46. IV. Use of Computer Programs for FPC Illustration of FPC with PARSCALE
FPC Estimates of Non-Anchor Item Parameters on the Fixed Old Scale
STPU Method
a b c
Item c c c c5
MWU-MEM Method
Item c c c c5

47. IV. Use of Computer Programs for FPC Illustration of FPC with BILOG-MG
FPC Estimates of Mean and SD of the Underlying Distribution on the Fixed Old Scale
Method Mean Std. Dev.
STPU FPC
MWU-MEM FPC
Mean-Sigma B = A = 1.384
Note. The new group examinees were from a N(0.5,1.22) distribution that was expressed on the fixed old scale.
Over-estimation
Under-estimation

48. V. Applications of FPC for Scaling and Equating
Online Calibration in Computerized Adaptive Testing (CAT)
Calibration of Pretest Items on the Fixed Operational Scale in Regular, Non-CAT Administration
In a Mixed-Format Test, Separate Calibration of CR Items from MC Items To Minimize Effects of Bad CR Items on MC Item Calibration
Equating Test Forms in the CING Design

49. V. Applications of FPC Online Calibration in CAT
In CAT, different sets of operational items are adaptively administered to examinees, with pretest items “seeded” in a certain common block of examinee groups.
Because the operational items were already calibrated, their parameters are known in CAT
Thus, FPC may be the best way to calibrate and diagnose the pretest items on the scale of the operational items, without affecting the operational item parameters.

50. V. Applications of FPC Calibration of Pretest Items on the Fixed Operational Scale
To develop test forms, pretest items are often administered together with operational items to examinees.
However, it would be wise to calibrate operational items separately from pretest items, because the operational item parameters could be contaminated by bad pretest items.
In this case, the ability distribution that is estimated using only the operational items can be reasonably used as the prior ability distribution for FPC with the pretest items, while the operational item parameters are used to fix the operational items in the FPC.

51. V. Applications of FPC FPC with Different Formats of Items
A mixed-format test contains different types of items; for instance, some are MC items and others are CR items.
Simultaneous calibration with both types of items can be conducted, assuming that a dominant factor underlies examinees’ responses to items.
However, practitioners may want to calibrate MC items separately from CR items, because calibration with bad CR items might adversely affect the estimation of MC item parameters.
In this case, MC items are first calibrated and then CR items are calibrated while fixing the MC item parameters.

52. V. Applications of FPC Equating Test Forms in the CING Design
Test equating using IRT requires all item parameters to be placed on a common scale (which is usually the old form scale).
Once all item and ability parameters are placed on a common scale, IRT true score or observed score equating is conducted.
Thus, FPC can be effectively used for placing all item parameters on the fixed old scale. Surely, the anchor is the common items between the new and old forms.

53. EXPLORE FPC
END
Thank You