The main goal of this section is to define the problem of linking IRT parameters from different calibration studies and derive the specific linking functions necessary for the 3PL model. More specifically, we address the problem of post hoc linking; that is, mapping the parameters from one study onto the values they would have had if they had been included in another after both calibrations have been conducted. This type of linking is common in the testing industry. As both the item and ability parameters are to be linked simultaneously, the linking automatically is for the fixed-effects specification of the 3PL model. In principle, it is possible to avoid the problem by concurrent recalibration of the response data collected in different studies, capitalizing on the presence of common items or test takers in them and imposing constraints on the parameters. For this approach, it has even been proposed to tentatively impose constraints, e.g., linear constraints as in (3)–(5) and check on their appropriateness using Lagrange multiplier tests (von Davier & von Davier, 2011). But such strategies are not always practical for testing programs that have to link their parameters continuously across multiple test administrations.

Our current treatment of the linking problem only deals with its mathematical aspects at the level of the model parameters, without any bothering about the fact that these parameters are unknown. In order to actually use them in practice, linking functions need to be estimated from response data. But we are only able to find defendable estimators and evaluate their statistical quality once we have an explicit definition of their estimand.

We begin with considering the more general case of a parametric response model used to calibrate the responses for a set of P test takers on I items with the vector of success probabilities \({\varvec{\pi }}=(\pi _{pi})\) in (20). Let \(f(\cdot )\) be the response function specified by the model, \({\varvec{\xi }}_{pi}\) its vector of parameters for the combination of test taker p and item i, and \({\varvec{\xi }} =({\varvec{\xi }}_{pi})\) the vector of parameters for all test takers and items. The choice of model amounts to the adoption of a system of \(P\times I\) equations \(\pi _{pi}=f({\varvec{\xi }}_{pi})\). As the probabilities \(\pi _{pi}\) are identified and thus have fixed values for all combinations of p and i, each of the equations introduces a level surface (contour) in the domain of f, which is the subset of all values of \({\varvec{\xi }}\) for which \(f({\varvec{\xi }}_{pi})=\pi _{pi}\) is true. The solution set for the system of equations is the intersection of all \(P\times I\) surfaces. As the system lacks identifiability of the model parameters, the set consists of more than one point. Identifiability restrictions are extra equations added to the system. The intersection of their solution sets with the set for the system reduces the latter to a unique point, whose coordinates are the true values of the item and test taker parameters for the calibration.

Now, suppose we have conducted two separate calibration studies that had both unique and common test takers and/or items in them. Both studies are assumed to have used appropriate sets of identifiability restrictions. Obviously, the use of different restrictions implies different intersections of their solution sets with those determined by the two systems of model equations, and hence different true values for the common parameters in the two calibrations. But different true values can also arise if formally identical sets of identifiability restrictions have been imposed on the two calibrations. The presence of unique test takers and/or items in the calibrations implies different vectors of success probabilities \({\varvec{\pi }}^{*}

e \) \({\varvec{\pi }}\) for them and thus different solutions sets for their model equations. Consequently, their intersections with the solution set for the identifiability restrictions generally differ, and the same items or test takers assigned to the two calibrations can therefore have different true parameter values. The critical factor is the scope of the restrictions. For example, if they fix the values of some of the common parameters to the same known constants in the two calibrations, obviously their impact on them is identical. But if they include unique parameters and leave the common parameters free, they yield different true values for the latter—an observation confirmed by the educational testing industry, where invariably different parameter values are found for common parameters in separate calibrations with large samples of test takers for the restrictions in (13) and (14), as well as by our example later in this paper. In fact, if this differential effect did not exist, we would not have to link any parameters.

Consider a hypothetical combination of a test taker and item assigned to two of these calibration studies with identified parameters. Let \({\varvec{\xi }} ^{*}\) and \({\varvec{\xi }}\) denote the vectors with the unique true values for the combination in the calibrations (where the indices have been omitted for notational convenience, as well as to emphasize the hypothetical nature of the combination). For example, for the 3PL model, \({\varvec{\xi }} ^{*}=(\theta ^{*},a^{*},b^{*},c^{*})\) and \({\varvec{\xi }} =(\theta ,a,b,c)\). The question of how to map \({\varvec{\xi }}\) and \({\varvec{\xi }}^{*}\) onto one another is the topic of this section. Observe that, although different, both \({\varvec{\xi }}^{*}\) and \({\varvec{\xi }}\) are associated with the same success probability \(\pi \) for the combination of test taker and item. This fact is key in our derivation of the mapping below.

Our first theorem is for a general response model that specifies success probability \(\pi \) for each combination of a test taker and item only as a monotone continuous function of their parameters, where the monotonicity is taken to mean that \(\pi \) is strictly increasing or decreasing in each of the components of \({\varvec{\xi }}\) with all other components fixed at any of their admissible values. We then present our results for the 3PL model. The version of this model for the regular parameterization in (1)–(2) is both continuous and monotone in each of its parameters, provided we exclude the case of \(\theta =b\) for the a parameter; the version with the slope-intercept parameterization is monotone in each of its parameters without any further restriction.

Again, except for an illustrative example below, the current paper only deals with the mathematical aspects of linking functions; the problem of how to actually estimate them and evaluate their estimation error deserves separate treatment.

Theorem 3

Assume a response model with a fixed parameter structure that (i) specifies \(\pi \) as a monotone continuous function of its parameters and (ii) has been used in two separate calibration studies with identified parameters \({\varvec{\xi }}\) and \({\varvec{ \xi }}^{{*}}\). Then \({\varvec{\xi }}^{*}\) is linked to \({\varvec{ \xi }}\) by a vector function

$$\begin{aligned} {\varvec{\xi }}^{*}\mathbf {=}\varphi ({\varvec{\xi }})\mathbf {=(}\varphi _{1}(\xi _{1}),\ldots ,\varphi _{d}(\xi _{d})) \end{aligned}$$ (31)

with components \(\varphi _{1},\ldots ,\varphi _{d}\) that are both monotone and continuous.

Proof

Let \(\xi ^{*}\) and \(\xi \) be an arbitrary pair of corresponding components of \({\varvec{\xi }}^{*}\) and \( {\varvec{\xi }}\) . Fixing all other components, the monotonicity of the model implies the existence of monotone functions \(\pi \mathbf {=}f(\xi ^{*})\) and \(\pi \mathbf {=}g(\xi ).\) Hence, there also exists a function \( \xi ^{*}=f^{-1}(g(\xi ))=\varphi (\xi )\). Being a composite of continuous functions, \(\varphi \) is continuous. Further, as both f and g are bijective, \(\varphi \) is bijective as well. Suppose that \(\varphi \) is not monotone. It then has an interior point \(\xi _{0}\) in its domain with a local optimum. But this implies the existence of points \(\xi ^{\prime }<\xi _{0}<\) \(\xi ^{\prime \prime }\) with \(\varphi (\xi ^{\prime })=\varphi (\xi ^{\prime \prime })\), which contradicts the fact that \(\varphi \) is bijective. Thus, \(\varphi \) is monotone. \(\square \)

The feature of monotonicity should not come as a surprise. If it did not hold, the two sets of the identifiability restrictions would imply a different order of some of the parameters, for instance, a reversal of the difficulties of two items, which is impossible without violating the requirement of observational equivalence. Observe, however, that \(\varphi ({\varvec{\xi }})\) is only required to be componentwise monotone; it does not need to hold that all components be increasing or all of them be decreasing.

It is thus possible to view the impact of the use of different sets of identifiability restrictions as a componentwise reparameterization of the response model, which leaves the structure of the model intact. However, unlike the earlier case of a known function in (16) applied to unknown parameter values, we now have to address the reverse problem: This time the two sets of parameter values are given, and we have to find the componentwise bijective function that maps them onto one another. Observe again that the specific identifiability restrictions used in the calibrations need not be known at all; neither do we need to assume anything about the statistical estimation method through which the restrictions might have been imposed. Only their impact on the item and test taker parameters counts.

For the 3PL model, the linking function \({\varvec{\varphi }}\mathbf {=(}\varphi _{\theta }(\theta ),\varphi _{a}(a),\varphi _{b}(b),\varphi _{c}(c))\) has to be derived from (1)–(2) for two arbitrary sets of values (\(\theta ^{*},a^{*},b^{*},c^{*}\)) and (\(\theta ,a,b,c\)). However, it is simpler to use the first equation in (24), and find \( \varphi _{\theta }\), \(\varphi _{a}\), \(\varphi _{b}\), and \(\varphi _{\gamma }\) as the solution of

$$\begin{aligned} \frac{\varphi _{\gamma }(\gamma )}{1+\exp [\varphi _{a}(a)(\varphi _{\theta }(\theta )-\varphi _{b}(b))]}=\frac{\gamma }{1+\exp [a(\theta -b)]}, \end{aligned}$$ (32)

with additional back transformation of \(\varphi _{\gamma }\) to \(\varphi _{c}\) . The required linking function is thus the solution of a functional equation in four unknowns (for relevant theory of functional equations, see, for instance, Sahoo & Kannappan, 2011, or Small, 2007). The next theorem shows the solution:

Theorem 4

Given the conditions in Theorem 3), the linking function for the 3PL model is

$$\begin{aligned} \varphi _{a}(a)= & {} u^{-1}a, \end{aligned}$$ (33)

$$\begin{aligned} \varphi _{b}(b)= & {} ub+v, \end{aligned}$$ (34)

$$\begin{aligned} \varphi _{c}(c)= & {} c, \end{aligned}$$ (35)

and

$$\begin{aligned} \varphi _{\theta }(\theta )=u\theta +v, \end{aligned}$$ (36)

with

$$\begin{aligned} u\equiv \frac{\varphi _{\theta }(\theta )-\varphi _{b}(b)}{\theta -b},\text { }\theta

e b, \end{aligned}$$ (37)

and

$$\begin{aligned} v=\varphi _{b}(b)-ub=\varphi _{\theta }(\theta )-u\theta . \end{aligned}$$ (38)

Proof

From (32),

$$\begin{aligned} \varphi _{\gamma }(\gamma )=\frac{1+\exp [\varphi _{a}(a)(\varphi _{\theta }(\theta )-\varphi _{b}(b))]}{1+\exp [a(\theta -b)]}\gamma . \end{aligned}$$ (39)

As this function is monotone in \(\gamma \), \(\varphi _{\gamma }\) is a monotone component of \({\varvec{\varphi }}\), and therefore

$$\begin{aligned} \frac{1+\exp [\varphi _{a}(a)(\varphi _{\theta }(\theta )-\varphi _{b}(b))]}{ 1+\exp [a(\theta -b)]}=\kappa >0, \end{aligned}$$ (40)

is a constant independent of \(\gamma \). Thus, \(\varphi _{\gamma }(\gamma )=\kappa \gamma .\) However, since \(\varphi _{\gamma }\) is a monotone mapping from [0, 1] onto itself, \(\kappa =1\) and (35) follows. We now have to find \(\varphi _{\theta }\), \(\varphi _{a}\), and \(\varphi _{b}\) as the solution of (40) for \(\kappa =1\); that is,

$$\begin{aligned} \varphi _{a}(a)[\varphi _{\theta }(\theta )-\varphi _{b}(b)]=a(\theta -b) \text {.} \end{aligned}$$ (41)

Rewriting the equation,

$$\begin{aligned} \varphi _{a}(a)=\frac{\theta -b}{\varphi _{\theta }(\theta )-\varphi _{b}(b)} a, \end{aligned}$$ (42)

with \(\varphi _{\theta }(\theta )

e \varphi _{b}(b)\). But, as \(\varphi _{a} \) is a @@@monotone@@@ component of \({\varvec{\varphi }}\) as well ,

$$\begin{aligned} \frac{\varphi _{\theta }(\theta )-\varphi _{b}(b)}{\theta -b}=\text {const,} \end{aligned}$$ (43)

which is our key equation. First, (33) follows directly from (43) along with the definition of its constant in (37). Further, (43) shows that \(\varphi _{\theta }(x)-\varphi _{b}(x)\) is equal to a constant times \(\theta -b\). Substituting \(x=\theta =b\) yields \(\varphi _{\theta }(x)-\varphi _{b}(x)=0,\) and it thus holds that \(\varphi _{\theta }=\varphi _{b}=\varphi \). Observe also that (43) implies a constant difference quotient for \(\varphi \). Hence, \(\varphi \) is linear, and (34) and (36) hold. Finally, (38) follows from (34)–(36). \(\square \)

Although the proof did not make any assumptions as to the general shape of the functions that map the values of the \(a_{i}, b_{i},\) and \(\theta _{p}\) parameters in a new calibration onto those in an earlier calibration, they appear to be linear, just as currently assumed in the literature; see our review in (3)–(5). However, a new result is the definition of linking parameter u, and consequently of v. Unlike (6)–(8), (37) defines it as the ratio of the differences between the test taker’s ability and the difficulty of the item in the two calibrations. The reason for the difference between these new and old definitions may be the failure in the current literature to distinguish between the formal definitions of u and v and their solutions from the system of linking equations implied by the choice of linking design. As demonstrated in the next section, separating the two does give us large flexibility to derive alternative solutions for u and v from alternative designs. Another new result is the derivation of the identity function for the \(c_{i}\) parameters @@@. We will further reflect on its practical implications in the last section of this paper.

In addition, it is important to note the different nature of these functions for the four different types of parameters. The one for the \(c_{i}\) parameters is an identity function, which does not involve either of the linking parameters u and v. On the other hand, the function for the \( a_{i}\) parameters involves one linking parameter, u, whereas those for the \(b_{i}\) and \(\theta _{p}\) parameters depend both on u and v. Thus, unlike the \(c_{i}\) parameters, the latter can be linked only when the numerical values of the linking parameters are known. This obvious point takes us to another identifiability requirement, namely for the system of linking equations to be derived from (33)–(38) for the specific design adopted in the linking study.