Appendix 1: Partial Derivatives Necessary to Obtain the Asymptotic Standard Errors
(a) MM-GM Method
The derivatives in Equation (10), necessary to compute the covariance matrix of the equating coefficients and the synthetic item parameters with the MM-GM method, are given by
$$\begin{aligned}&\frac{\partial \hat{\varvec{\beta }}_1}{\partial \hat{\varvec{a}}^\top }=\mathrm {diag}(\hat{\varvec{\beta }}_1) ({\varvec{X}}_1^\top {\varvec{X}}_1)^{-1} {\varvec{X}}_1^\top \left( \mathrm {diag}(\hat{\varvec{a}}) \right) ^{-1}, \end{aligned}$$
(37)
$$\begin{aligned}&\frac{\partial \hat{\varvec{\beta }}_1}{\partial \hat{\varvec{b}}^\top }={{\varvec{0}}},\end{aligned}$$
(38)
$$\begin{aligned}&\frac{\partial \hat{\varvec{\beta }}_2}{\partial \hat{\varvec{a}}^\top }= \frac{\partial \hat{\varvec{\beta }}_2}{\partial \hat{\varvec{A}}^\top } \frac{\partial \hat{\varvec{A}}}{\partial \hat{\varvec{a}}^\top }= ({\varvec{X}}_2^\top {\varvec{X}}_2)^{-1} {\varvec{X}}_2^\top \mathrm {diag}(\hat{\varvec{b}}) {\varvec{T}}^\top \frac{\partial \hat{\varvec{A}}}{\partial \hat{\varvec{a}}^\top }, \end{aligned}$$
(39)
$$\begin{aligned}&\frac{\partial \hat{\varvec{\beta }}_2}{\partial \hat{\varvec{b}}^\top }=({\varvec{X}}_2^\top {\varvec{X}}_2)^{-1} {\varvec{X}}_2^\top \mathrm {diag}(\hat{\varvec{A}}_n), \end{aligned}$$
(40)
where \(\frac{\partial \hat{\varvec{A}}}{\partial \hat{\varvec{a}}^\top }\) is given in the first \(T-1\) rows of \(\frac{\partial \hat{\varvec{\beta }}_1}{\partial \hat{\varvec{a}}^\top }\).
(b) MM-M Method
The derivatives of the equating coefficients \(A_t\), \(t=2,\dots ,T\), with respect to the estimated item discrimination parameters \(\hat{a}_{js}\), \(j=1,\dots ,v\), \(s=1,\dots ,T\), namely \(\frac{\partial \hat{A}_t}{\partial \hat{a}_{js}}\), cannot be found in closed form but can instead be determined numerically. These derivatives are included in the matrix \(\frac{\partial \hat{\varvec{A}}}{\partial \hat{\varvec{a}}^\top }\) that corresponds to the first \(T-1\) rows of \(\frac{\partial \hat{\varvec{\beta }}_1}{\partial \hat{\varvec{a}}^\top }\). The derivatives of the synthetic discrimination parameters \(\hat{a}_j^*\) with respect to the discrimination parameter estimates obtained from each calibration can be then found as follows:
$$\begin{aligned} \frac{\partial \hat{a}^*_j}{\partial \hat{a}_{jt}}= & {} \frac{1}{ \sum _{s\in U_j} \hat{A}_s } -\frac{\sum _{s\in U_j} \hat{a}_{js}}{\left( \sum _{s\in U_j} \hat{A}_s \right) ^2} \sum _{s\in U_j}\frac{\partial \hat{A}_s}{\partial \hat{a}_{jt}} , \end{aligned}$$
(41)
$$\begin{aligned} \frac{\partial \hat{a}^*_j}{\partial \hat{a}_{it}}= & {} -\frac{\sum _{s\in U_j} \hat{a}_{js}}{\left( \sum _{s\in U_j} \hat{A}_s \right) ^2} \sum _{s\in U_j}\frac{\partial \hat{A}_s}{\partial \hat{a}_{it}} , \quad \forall i\ne j. \end{aligned}$$
(42)
These derivatives form the matrix \(\frac{\partial \hat{\varvec{a}}^*}{\partial \hat{\varvec{a}}^\top }\) that corresponds to the last v rows of \(\frac{\partial \hat{\varvec{\beta }}_1}{\partial \hat{\varvec{a}}^\top }\). The derivatives \(\frac{\partial \hat{\varvec{\beta }}_1}{\partial \hat{\varvec{b}}^\top }\), \(\frac{\partial \hat{\varvec{\beta }}_2}{\partial \hat{\varvec{a}}^\top }\) and \(\frac{\partial \hat{\varvec{\beta }}_2}{\partial \hat{\varvec{b}}^\top }\) can then be determined as explained in Appendix 1(a) for the MM-GM method, using the appropriate matrices \(\frac{\partial \hat{\varvec{A}}}{\partial \hat{\varvec{a}}^\top }\) and \(\hat{\varvec{A}}_n\).
(c) MIRF and MTRF Methods
In order to obtain the partial derivatives necessary to compute the asymptotic standard errors of the equating coefficients, \(P^*_{mjt}\) will be written as follows:
$$\begin{aligned} P^*_{mjt}=\hat{c}_{jt}+(1-\hat{c}_{jt})\frac{\exp ( LP_{mjt} ) }{1+\exp ( LP_{mjt} )}, \end{aligned}$$
(43)
where
$$\begin{aligned} LP_{mjt}=D\frac{1}{u_j} \left( \sum _{\begin{array}{c} s\in U_j \\ s\ne t \end{array}}\frac{\hat{a}_{js}}{\hat{A}_s} \hat{A}_t + \hat{a}_{jt}\right) y_m -D\frac{1}{u_j} \sum _{s\in U_j}\frac{\hat{a}_{js}}{\hat{A}_s} \left( \frac{1}{u_j}\sum _{s\in U_j}(\hat{b}_{js} \hat{A}_s + \hat{B}_s)-\hat{B}_t \right) .\nonumber \\ \end{aligned}$$
(44)
In the following, all the derivatives entering in Equations (26), (27) (28), (32), (33), and (34) will be given as
$$\begin{aligned} \frac{\partial P^*_{mjt}}{\partial \hat{A}_t} = \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \cdot \frac{\partial LP_{mjt}}{\partial \hat{A}_t}, \end{aligned}$$
(45)
where
$$\begin{aligned} \frac{\partial P^*_{mjt}}{\partial LP_{mjt}}= & {} (P^*_{mjt}-\hat{c}_{jt})\left( 1-\frac{P^*_{mjt}-\hat{c}_{jt}}{1-\hat{c}_{jt}}\right) , \end{aligned}$$
(46)
$$\begin{aligned} \frac{\partial LP_{mjt}}{\partial \hat{A}_t}= & {} \frac{D}{u_j} \left[ \sum _{\begin{array}{c} s\in U_j \\ s\ne t \end{array}}\frac{\hat{a}_{js}}{\hat{A}_s} y_m + \frac{\hat{a}_{jt}}{\hat{A}_t^2}(\hat{b}_j^*-\hat{B}_t) I_{U_j}(t) -\hat{a}_j^*\hat{b}_{jt} I_{U_j}(t) \right] , \end{aligned}$$
(47)
and \(I_{U_j}(t)\) is an indicator function, which is 1 if \(t\in U_j\).
$$\begin{aligned} \frac{\partial P^*_{mjt}}{\partial \hat{A}_k} = \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \cdot \frac{\partial LP_{mjt}}{\partial \hat{A}_k}, \quad \forall k\ne t, \end{aligned}$$
(48)
where
$$\begin{aligned} \frac{\partial LP_{mjt}}{\partial \hat{A}_k}= & {} \frac{D}{u_j} \left[ -\frac{\hat{a}_{jk}}{\hat{A}_k^2} \hat{A}_t y_m + \frac{\hat{a}_{jk}}{\hat{A}_k^2}(\hat{b}_j^*-\hat{B}_t)-\hat{a}_j^*\hat{b}_{jk} \right] I_{U_j}(k), \end{aligned}$$
(49)
$$\begin{aligned} \frac{\partial P^*_{mjt}}{\partial \hat{B}_t}= & {} \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \cdot \frac{\partial LP_{mjt}}{\partial \hat{B}_t}, \end{aligned}$$
(50)
where
$$\begin{aligned} \frac{\partial LP_{mjt}}{\partial \hat{B}_t}= & {} D \hat{a}_j^* \left( 1-\frac{1}{u_j} I_{U_j}(t) \right) , \end{aligned}$$
(51)
$$\begin{aligned} \frac{\partial P^*_{mjt}}{\partial \hat{B}_k}= & {} \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \cdot \frac{\partial LP_{mjt}}{\partial \hat{B}_k}, \quad \forall k\ne t, \end{aligned}$$
(52)
where
$$\begin{aligned} \frac{\partial LP_{mjt}}{\partial \hat{B}_k}= & {} - D \hat{a}_j^* \frac{1}{u_j} I_{U_j}(k), \end{aligned}$$
(53)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{B}_t \partial \hat{B}_k}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{B}_k} \frac{\partial LP_{mjt}}{\partial \hat{B}_t} , \quad \forall k, \end{aligned}$$
(54)
where
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{B}_k} = \frac{\partial P^*_{mjt}}{\partial \hat{B}_k} -2 \frac{P^*_{mjt}-\hat{c}_{jt}}{1-\hat{c}_{jt}} \frac{\partial P^*_{mjt}}{\partial \hat{B}_k}. \end{aligned}$$
(55)
All other second derivatives of \(P^*_{mjt}\) with respect to \(LP_{mjt}\) and one of these variables \(\hat{B}_h, \hat{B}_t, \hat{A}_k, \hat{A}_h, \hat{A}_t, \hat{a}_{jk}, \hat{a}_{jh}, \hat{a}_{jt}, \hat{b}_{jk}, \hat{b}_{jh}, \hat{b}_{jt}\) are analogous and can be obtained by substituting \(\hat{B}_k\) with the appropriate variable in (55). The other derivatives entering in Equations (27), (28), (33) and (34) are
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{B}_t \partial \hat{A}_k}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{A}_k} \frac{\partial LP_{mjt}}{\partial \hat{B}_t} - \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \frac{\hat{a}_{jk}}{\hat{A}_k^2} \left( 1-\frac{1}{u_j} I_{U_j}(t) \right) I_{U_j}(k), \quad \forall k, \end{aligned}$$
(56)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{B}_k \partial \hat{B}_h}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{B}_h} \frac{\partial LP_{mjt}}{\partial \hat{B}_k} , \quad \forall k \ne t, \; \forall h, \end{aligned}$$
(57)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{B}_k \partial \hat{A}_h}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{A}_h} \frac{\partial LP_{mjt}}{\partial \hat{B}_k} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j^2} \frac{\hat{a}_{jh}}{\hat{A}_h^2} I_{U_j}(k) I_{U_j}(h) , \quad \forall k\ne t, \; \forall h, \end{aligned}$$
(58)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_t^2}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{A}_t} \frac{\partial LP_{mjt}}{\partial \hat{A}_t}+ \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \left[ -2 \frac{\hat{a}_{jt}}{\hat{A}_t^3} (\hat{b}_j^*-\hat{B}_t) +2\frac{\hat{a}_{jt}\hat{b}_{jt}}{u_j \hat{A}_t^2} \right] I_{U_j}(t), \end{aligned}$$
(59)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_t \partial \hat{A}_k}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{A}_k} \frac{\partial LP_{mjt}}{\partial \hat{A}_t}+ \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \nonumber \\&\quad \left[ - \frac{\hat{a}_{jk}}{\hat{A}_k^2} y_m + \frac{\hat{a}_{jt}\hat{b}_{jk}}{u_j \hat{A}_t^2} I_{U_j}(t) + \frac{\hat{a}_{jk}\hat{b}_{jt}}{u_j \hat{A}_k^2} I_{U_j}(t) \right] I_{U_j}(k) , \quad \forall k\ne t, \end{aligned}$$
(60)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_t \partial \hat{B}_t}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{B}_t} \frac{\partial LP_{mjt}}{\partial \hat{A}_t}+ \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \frac{\hat{a}_{jt}}{\hat{A}_t^2} \left( \frac{1}{u_j}-1 \right) I_{U_j}(t), \end{aligned}$$
(61)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_t \partial \hat{B}_k}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{B}_k} \frac{\partial LP_{mjt}}{\partial \hat{A}_t} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j^2} \frac{\hat{a}_{jt}}{\hat{A}_t^2} I_{U_j}(t) I_{U_j}(k) , \quad \forall k\ne t, \end{aligned}$$
(62)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_k^2}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{A}_k} \frac{\partial LP_{mjt}}{\partial \hat{A}_k} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \nonumber \\&\quad \left[ 2 \frac{\hat{a}_{jk}}{\hat{A}_k^3}\hat{A}_t y_m -2 \frac{\hat{a}_{jk}}{\hat{A}_k^3} (\hat{b}_j^*-\hat{B}_t) + 2 \frac{\hat{a}_{jk}\hat{b}_{jk}}{u_j \hat{A}_k^2} \right] I_{U_j}(k) , \quad \forall k\ne t, \end{aligned}$$
(63)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_k \partial \hat{A}_t}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{A}_t} \frac{\partial LP_{mjt}}{\partial \hat{A}_k} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \nonumber \\&\quad \left[ - \frac{\hat{a}_{jk}}{\hat{A}_k^2} y_m + \frac{\hat{a}_{jk}\hat{b}_{jt}}{u_j \hat{A}_k^2} I_{U_j}(t) + \frac{\hat{a}_{jt}\hat{b}_{jk}}{u_j \hat{A}_t^2} I_{U_j}(t) \right] I_{U_j}(k) , \quad \forall k\ne t, \end{aligned}$$
(64)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_k \partial \hat{B}_h}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{B}_h} \frac{\partial LP_{mjt}}{\partial \hat{A}_k} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j^2} \frac{\hat{a}_{jk}}{\hat{A}_k^2} I_{U_j}(k) I_{U_j}(h) , \quad \forall k, \; h\ne t, \end{aligned}$$
(65)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_k \partial \hat{B}_t}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{B}_t} \frac{\partial LP_{mjt}}{\partial \hat{A}_k} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \frac{\hat{a}_{jk}}{\hat{A}_k^2} \left( \frac{1}{u_j} I_{U_j}(t) - 1 \right) I_{U_j}(k), \quad \forall k\ne t, \qquad \end{aligned}$$
(66)
$$\begin{aligned} \frac{\partial P_{mjt}}{\partial \hat{c}_{jt}}= & {} 1- \frac{P_{mjt}-\hat{c}_{jt}}{1-\hat{c}_{jt}}, \end{aligned}$$
(68)
$$\begin{aligned} \frac{\partial P_{mjt}}{\partial \hat{a}_{jt}}= & {} (P_{mjt} - \hat{c}_{jt}) \left( 1- \frac{P_{mjt}-\hat{c}_{jt}}{1-\hat{c}_{jt}} \right) D (y_m -\hat{b}_{jt}), \end{aligned}$$
(69)
$$\begin{aligned} \frac{\partial P_{mjt}}{\partial \hat{b}_{jt}}= & {} -(P_{mjt} - \hat{c}_{jt}) \left( 1- \frac{P_{mjt}-\hat{c}_{jt}}{1-\hat{c}_{jt}} \right) D \hat{a}_{jt}, \end{aligned}$$
(70)
$$\begin{aligned} \frac{\partial P^*_{mjt}}{\partial \hat{c}_{jt}}= & {} 1- \frac{P^*_{mjt}-\hat{c}_{jt}}{1-\hat{c}_{jt}}, \end{aligned}$$
(71)
$$\begin{aligned} \frac{\partial P^*_{mjt}}{\partial \hat{a}_{jt}}= & {} \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \left[ y_m - \frac{1}{\hat{A}_t} (\hat{b}^*_j - \hat{B}_t) \right] I_{U_j}(t), \end{aligned}$$
(72)
$$\begin{aligned} \frac{\partial P^*_{mjt}}{\partial \hat{a}_{jk}}= & {} \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \left[ \frac{\hat{A}_t}{\hat{A}_k} y_m - \frac{1}{\hat{A}_k} (\hat{b}^*_j - \hat{B}_t) \right] I_{U_j}(k), \quad \forall k\ne t, \end{aligned}$$
(73)
$$\begin{aligned} \frac{\partial P^*_{mjt}}{\partial \hat{b}_{jk}}= & {} - \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \hat{a}^*_j \hat{A}_k I_{U_j}(k), \quad \forall k, \end{aligned}$$
(74)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{B}_t \partial \hat{c}_{jt}}= & {} -\frac{1}{1-\hat{c}_{jt}} \frac{\partial P^*_{mjt}}{\partial \hat{B}_t}, \end{aligned}$$
(75)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{B}_t \partial \hat{a}_{jk}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{a}_{jk}} \frac{\partial LP_{mjt}}{\partial \hat{B}_t} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \frac{1}{\hat{A}_k} \left( 1-\frac{1}{u_j} I_{U_j}(t) \right) I_{U_j}(k) , \quad \forall k, \end{aligned}$$
(76)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{B}_t \partial \hat{b}_{jk}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{b}_{jk}} \frac{\partial LP_{mjt}}{\partial \hat{B}_t}, \quad \forall k, \end{aligned}$$
(77)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{B}_k \partial \hat{c}_{jt}}= & {} -\frac{1}{1-\hat{c}_{jt}} \frac{\partial P^*_{mjt}}{\partial \hat{B}_k}, \end{aligned}$$
(78)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{B}_k \partial \hat{a}_{jh}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{a}_{jh}} \frac{\partial LP_{mjt}}{\partial \hat{B}_k} - \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} D \frac{1}{u_j^2} \frac{1}{\hat{A}_h} I_{U_j}(k) I_{U_j}(h), \quad \forall k\ne t , \; \forall h, \end{aligned}$$
(79)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{B}_k \partial \hat{b}_{jh}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{b}_{jh}} \frac{\partial LP_{mjt}}{\partial \hat{B}_k} , \quad \forall k\ne t , \; \forall h, \end{aligned}$$
(80)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_t \partial \hat{c}_{jt}}= & {} -\frac{1}{1-\hat{c}_{jt}} \frac{\partial P^*_{mjt}}{\partial \hat{A}_t}, \end{aligned}$$
(81)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_t \partial \hat{a}_{jt}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{a}_{jt}} \frac{\partial LP_{mjt}}{\partial \hat{A}_t} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \left[ \frac{1}{\hat{A}_t^2} (\hat{b}_j^*-\hat{B}_t) -\frac{\hat{b}_{jt}}{u_j \hat{A}_t} \right] I_{U_j}(t), \end{aligned}$$
(82)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_t \partial \hat{a}_{jk}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{a}_{jk}} \frac{\partial LP_{mjt}}{\partial \hat{A}_t} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \left[ \frac{1}{\hat{A}_k}y_m -\frac{\hat{b}_{jt}}{u_j \hat{A}_k} I_{U_j}(t) \right] I_{U_j}(k) , \quad \forall k\ne t , \nonumber \\\end{aligned}$$
(83)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_t \partial \hat{b}_{jt}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{b}_{jt}} \frac{\partial LP_{mjt}}{\partial \hat{A}_t} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \left[ \frac{\hat{a}_{jt}}{u_j \hat{A}_t} - \hat{a}_j^* \right] I_{U_j}(t), \end{aligned}$$
(84)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_t \partial \hat{b}_{jk}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{b}_{jk}} \frac{\partial LP_{mjt}}{\partial \hat{A}_t} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \left[ \frac{\hat{a}_{jt} \hat{A}_k}{u_j \hat{A}_t^2} \right] I_{U_j}(t) I_{U_j}(k) , \quad \forall k\ne t , \end{aligned}$$
(85)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_k \partial \hat{c}_{jt}}= & {} -\frac{1}{1-\hat{c}_{jt}} \frac{\partial P^*_{mjt}}{\partial \hat{A}_k}, \quad \forall k\ne t, \end{aligned}$$
(86)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_k \partial \hat{a}_{jk}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{a}_{jk}} \frac{\partial LP_{mjt}}{\partial \hat{A}_k} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \left[ -\frac{\hat{A}_t}{\hat{A}_k^2}y_m + \frac{1}{\hat{A}_k^2} (\hat{b}_j^*-\hat{B}_t) - \frac{\hat{b}_{jk}}{u_j \hat{A}_k} \right] I_{U_j}(k) , \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\quad \forall k\ne t , \nonumber \\\end{aligned}$$
(87)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_k \partial \hat{a}_{jh}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{a}_{jh}} \frac{\partial LP_{mjt}}{\partial \hat{A}_k} - \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j^2} \frac{\hat{b}_{jk}}{\hat{A}_h} I_{U_j}(k) I_{U_j}(h) , \quad \forall k \ne t, \; \forall h \ne k , \end{aligned}$$
(88)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_k \partial \hat{b}_{jk}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{b}_{jk}} \frac{\partial LP_{mjt}}{\partial \hat{A}_k} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j} \left( \frac{\hat{a}_{jk}}{u_j \hat{A}_k} - \hat{a}^*_j \right) I_{U_j}(k) , \quad \forall k\ne t , \end{aligned}$$
(89)
$$\begin{aligned} \frac{\partial ^2 P^*_{mjt}}{\partial \hat{A}_k \partial \hat{b}_{jh}}= & {} \frac{\partial ^2 P^*_{mjt}}{\partial LP_{mjt} \partial \hat{b}_{jh}} \frac{\partial LP_{mjt}}{\partial \hat{A}_k} + \frac{\partial P^*_{mjt}}{\partial LP_{mjt}} \frac{D}{u_j^2} \frac{\hat{a}_{jk} \hat{A}_h}{\hat{A}_k^2} I_{U_j}(k) I_{U_j}(h), \quad \forall k\ne t, \; \forall h\ne k , \end{aligned}$$
(90)
The derivatives of the synthetic discrimination parameters with respect to the item parameter obtained from separate calibration can be found as follows:
$$\begin{aligned} \frac{\partial \hat{a}^*_j}{\partial \hat{a}_{jt}}= & {} \frac{1}{u_j} \left( \frac{1}{\hat{A}_t} I_{U_j}(t) - \sum _{s\in U_j} \frac{\hat{a}_{js}}{\hat{A}_s^2} \frac{\partial \hat{A}_s}{\partial \hat{a}_{jt}} \right) , \end{aligned}$$
(91)
$$\begin{aligned} \frac{\partial \hat{a}^*_j}{\partial \hat{a}_{it}}= & {} \frac{1}{u_j} \left( - \sum _{s\in U_j} \frac{\hat{a}_{js}}{\hat{A}_s^2} \frac{\partial \hat{A}_s}{\partial \hat{a}_{it}} \right) , \quad \forall i\ne j, \end{aligned}$$
(92)
$$\begin{aligned} \frac{\partial \hat{a}^*_j}{\partial \hat{b}_{it}}= & {} \frac{1}{u_j} \left( - \sum _{s\in U_j} \frac{\hat{a}_{js}}{\hat{A}_s^2} \frac{\partial \hat{A}_s}{\partial \hat{b}_{it}} \right) , \quad \forall i, \end{aligned}$$
(93)
$$\begin{aligned} \frac{\partial \hat{b}^*_j}{\partial \hat{a}_{it}}= & {} \frac{1}{u_j} \sum _{s\in U_j} \left( \hat{b}_{jt}\frac{\partial \hat{A}_s}{\partial \hat{a}_{it}} + \frac{\partial \hat{B}_s}{\partial \hat{a}_{it}} \right) , \quad \forall i, \end{aligned}$$
(94)
$$\begin{aligned} \frac{\partial \hat{b}^*_j}{\partial \hat{b}_{jt}}= & {} \frac{1}{u_j} \left[ \hat{A}_t I_{U_j}(t) + \sum _{s\in U_j} \left( \hat{b}_{js}\frac{\partial \hat{A}_s}{\partial \hat{b}_{jt}} + \frac{\partial \hat{B}_s}{\partial \hat{b}_{jt}} \right) \right] , \end{aligned}$$
(95)
$$\begin{aligned} \frac{\partial \hat{b}^*_j}{\partial \hat{b}_{it}}= & {} \frac{1}{u_j} \sum _{s\in U_j} \left( \hat{b}_{js}\frac{\partial \hat{A}_s}{\partial \hat{b}_{it}} + \frac{\partial \hat{B}_s}{\partial \hat{b}_{it}} \right) , \quad \forall i\ne j. \end{aligned}$$
(96)
Appendix 2: Proof of the Correspondence Between the MM-M Method for Two Forms and the Mean-Mean Method
When \(T=2\) the estimator of the equating coefficient \(A_2\) with the MM-M method is given by
$$\begin{aligned} \hat{A}_2=\frac{\sum _{j\in J_2}\hat{a}_{j2}}{\sum _{j\in J_2} \hat{a}^*_j}= \frac{\sum _{j\in J_2}\hat{a}_{j2}}{\sum _{j\in J_1\cap J_2} \frac{\hat{a}_{j1}+\hat{a}_{j2}}{1+\hat{A}_2}+ \sum _{j\in J_2\setminus J_1} \frac{\hat{a}_{j2}}{\hat{A}_2}}, \end{aligned}$$
(97)
from which we obtain
$$\begin{aligned} \hat{A}_2 \sum _{j\in J_1\cap J_2} \frac{\hat{a}_{j1}+\hat{a}_{j2}}{1+\hat{A}_2}+ \hat{A}_2 \sum _{j\in J_2\setminus J_1} \frac{\hat{a}_{j2}}{\hat{A}_2} = \sum _{j\in J_1\cap J_2}\hat{a}_{j2}+\sum _{j\in J_2\setminus J_1}\hat{a}_{j2}, \end{aligned}$$
(98)
and
$$\begin{aligned} \frac{\hat{A}_2 }{1+\hat{A}_2} \sum _{j\in J_1\cap J_2} \hat{a}_{j1}+\hat{a}_{j2}= \sum _{j\in J_1\cap J_2}\hat{a}_{j2}. \end{aligned}$$
(99)
We then obtain
$$\begin{aligned} \hat{A}_2 \sum _{j\in J_1\cap J_2} \hat{a}_{j1}= \sum _{j\in J_1\cap J_2}\hat{a}_{j2}. \end{aligned}$$
(100)
The estimator of the equating coefficient \(A_2\) is then equal to
$$\begin{aligned} \hat{A}_2 = \frac{\sum _{j\in J_1\cap J_2}\hat{a}_{j2}}{\sum _{j\in J_1\cap J_2} \hat{a}_{j1}}, \end{aligned}$$
(101)
which corresponds to the mean-mean estimator of the equating coefficient A for two forms.
Appendix 3: Proof of the Symmetry Property of MIRF and MTRF Methods
In order to convert item parameters on the scale of Form r, the equating coefficients are transformed as follows:
$$\begin{aligned} \hat{A}_t' = \frac{\hat{A}_t}{\hat{A}_r} \quad \text {and} \quad \hat{B}_t' = \frac{\hat{B}_t - \hat{B}_r}{\hat{A}_r},\quad \text {for } t=1,\dots ,T, \end{aligned}$$
so that \(\hat{A}_r'=1\) and \(\hat{B}_r'=0\). If \(\hat{A}_t\) is replaced with \(\hat{A}_t' \) and \(\hat{B}_t\) is replaced with \(\hat{B}_t'\) in Equation (22), it is simple to verify that \(\hat{a}_{jt}^*\) and \(\hat{b}_{jt}^*\) do not vary after this substitution. Consequently, Equations (17) and (29) are invariant with respect to changes of the base form, thus proving the symmetry property.
Appendix 4: Variability of Estimated Abilities
The following equation gives the conversion of estimated abilities from the scale of Form t to the scale of the base form
$$\begin{aligned} \theta ^*=\theta _t A_t + B_t. \end{aligned}$$
The estimated ability \(\hat{\theta }_t\) can be transformed using the estimated equating coefficients
$$\begin{aligned} \hat{\theta }^*=\hat{\theta }_t \hat{A}_t + \hat{B}_t. \end{aligned}$$
The variance of \(\hat{\theta }^*\) given \(\hat{\theta }_t\) is
$$\begin{aligned} \mathrm {var}(\hat{\theta }^*|\hat{\theta }_t)=\hat{\theta }_t^2 \mathrm {var}(\hat{A}_t)+\mathrm {var}(\hat{B}_t)+ 2\hat{\theta }_t \mathrm {cov}(\hat{A}_t,\hat{B}_t), \end{aligned}$$
while the conditional expected value is
$$\begin{aligned} \mathrm {E}(\hat{\theta }^*|\hat{\theta }_t)=\hat{\theta }_t \mathrm {E}(\hat{A}_t)+\mathrm {E}(\hat{B}_t)= \hat{\theta }_t (A_t+o(1))+ B_t+o(1), \end{aligned}$$
provided that the estimators \(\hat{A}_t\) and \(\hat{B}_t\) are consistent. So, the variance of \(\hat{\theta }^*\) is
$$\begin{aligned} \mathrm {var}(\hat{\theta }^*)&=\mathrm {E}\{\mathrm {var}(\hat{\theta }^*|\hat{\theta }_t)\}+ \mathrm {var}\{\mathrm {E}(\hat{\theta }^*|\hat{\theta }_t)\} \\&= \mathrm {var}(\hat{A}_t)+\mathrm {var}(\hat{B}_t)+\mathrm {var}(\hat{\theta }_t) A_t^2+ o(1), \end{aligned}$$
where \(\mathrm {E}(\theta _t)\) and \(\mathrm {var}(\theta _t)\) are assumed to be 0 and 1, respectively, as usual with the marginal maximum likelihood estimation method. Hence, if the reliability of \(\hat{\theta }_t\) is
$$\begin{aligned} \rho (\hat{\theta }_t)=\frac{\mathrm {var}(\theta _t)}{\mathrm {var}(\hat{\theta }_t)}= \frac{1}{\mathrm {var}(\hat{\theta }_t)}, \end{aligned}$$
(102)
the reliability of \(\hat{\theta }^*\) is
$$\begin{aligned} \rho (\hat{\theta }^*)=\frac{\mathrm {var}(\theta ^*)}{\mathrm {var}(\hat{\theta }^*)}\simeq \frac{\mathrm {var}(\theta _t) A_t^2}{\mathrm {var}(\hat{A}_t)+\mathrm {var}(\hat{B}_t)+\mathrm {var}(\hat{\theta }_t) A_t^2}= \frac{ A_t^2}{\mathrm {var}(\hat{A}_t)+\mathrm {var}(\hat{B}_t)+\mathrm {var}(\hat{\theta }_t) A_t^2}.\qquad \quad \end{aligned}$$
(103)
The reliability of \(\hat{\theta }^*\) is then always greater than the reliability of \(\hat{\theta }_t\), due to variability of the estimated equating coefficients. These reliabilities can be estimated by substituting the true values with their estimates in (102) and (103). An estimate of \(\mathrm {var}(\hat{\theta }_t)\) is \(1+\hat{se}^2(\hat{\theta }_t)\), where \(\hat{se}(\hat{\theta }_t)\) is the estimated standard error of \(\hat{\theta }_t\).
Another quantity of interest is the standard error of \(\hat{\theta }^*\), which can be obtained as follows:
$$\begin{aligned} se(\hat{\theta }^*)=\{ \mathrm {var}(\hat{\theta }^*)-\mathrm {var}(\theta ^*) \} ^{1/2} \simeq \{ \mathrm {var}(\hat{A}_t)+\mathrm {var}(\hat{B}_t)+se^2(\hat{\theta }_t) A_t^2 \} ^{1/2}. \end{aligned}$$