Stocking–Lord Equating Simulator

From the anchor set to parameter transformation, equated forms and score equivalents.

12 anchor candidates

The selected items make up the initial anchor set. Turn candidates on or off to examine how the linking solution changes.

Transformation constants

Constant A

0.701.001.40

Constant B

−1.500.001.50

Current result

Active anchor items

Objective F(A,B)

TCC of the anchor items

2016 reference 2017 original 2017 transformed

Difference after transformation

TCC_Ref(θ) − TCC_Trans(θ)

Parameter transformation

a* = aA b* = Ab + B c* = c

Anchor-set characteristic curve

TCC(θ) = ∑ P_i(θ)

Stocking–Lord criterion

F(A,B) = ∑ φ(θ)[TCC_Ref(θ) − TCC_Trans(θ)]²

Criteria: θ from −4 to 4 in increments of 0.1, weighted by the standard normal density φ(θ).

Compare an anchor candidate

Item

2016 reference

2017 original

2017 transformed

A and B are the current values from the Stocking–Lord tab. The transformation is applied to the 2017 item parameters, not to student responses.

ICC of the selected item

2016 reference 2017 original 2017 transformed

Parameters for the 12 candidates

ItemID	Active anchor	a Ref	b Ref	c Ref	a 2017	b 2017	c 2017	a*	b*	c*

From anchors to the full forms

Constants A and B are estimated using the active anchor items and then applied to the parameters of all 35 items in the 2017 form. The transformed TCC can then be compared directly with the 2016 reference form.

Items in 2016 form

Items in 2017 form

Mean absolute difference

Maximum absolute difference

TCCs of the full forms

2016 reference 2017 original 2017 transformed

Difference between equated forms

TCC₂₀₁₆(θ) − TCC_2017*(θ)

Applying the link

a*_i = a_i/A b*_i = Ab_i + B

Form TCC

TCC(θ) = ∑_i=1³⁵P_i(θ)

The sum now uses all items in each form, not only the anchors.

Interpretation

If the linking works, the forms can be expressed on a common metric even when they do not contain exactly the same items.

Explore a score equivalent

2017 raw score

035

Procedure: locate θ on the transformed 2017 TCC using the 0.1 grid, then obtain the expected 2016 score at that θ.

Equating chain

2017 raw

θ on the common metric

Expected 2016

Rounded equivalent

Scale score

Score-equivalence function

Expected 2016 equivalent Identity line

Raw score and scale score by θ

Minimum scale score

Maximum scale score

Transformed 2017 TCC · left axis Scale score · right axis

Equivalents for 2017 raw scores

2017 raw	θ	Expected 2016	Rounded 2016	Scale S