2 Longitudinal Rasch Poisson Gamma counts model

This section outlines the longitudinal RPGCM and the estimation of its parameters. A more detailed technical description can be found in Parsamaram et al. (Reference Parsamaram, Holling and Schwabe2026).

2.1 Model specification

Let $Y_{ijk}$ denote the response of test person $i$ to item $k$ administered at the $j$ th measurement occasion taking place at time $t_j$ . The response is assumed to follow a Poisson distribution with mean $\theta _{ij}\sigma _{jk}$ , where $\theta _{ij}$ denotes the ability of person $i$ at time $t_j$ , $i = 1, \ldots , N$ , $j = 1, \ldots , J$ , and $\sigma _{jk}$ denotes the easiness of item $k$ administered at time $t_j$ , $k = 1, \ldots , n_j$ .

The subjects are assumed to receive the same items with easiness $\sigma _{jk}$ and, in particular, the same total number $n_j$ at occasion $j$ , $\sum _{j=1}^{J} n_j = n$ , each. Furthermore, the item easiness parameters $\sigma _{jk}$ are assumed to be known, which does not constitute a restriction when calibrated tests are employed, and the mean item easiness $\bar {\sigma } = \frac {1}{n}\sum _{j=1}^{J}\sum _{k=1}^{n_j}\sigma _{jk}$ is fixed. This condition is trivially satisfied if the same item parameters are used across time points. When automatically generated, rule-based items are administered, this assumption typically does not pose practical problems.

The primary objective of the analysis is the estimation of the development of ability over time, represented by a common mean learning curve $\theta (t)$ evaluated at $t_1, \ldots , t_J$ . Optimal designs allow $\theta (t)$ to be estimated with maximal precision as a function of the allocation of items across measurement occasions, that is, the proportions $n_j / n$ and the items $\sigma _{jk}$ , $k = 1, \ldots , n_j$ . In empirical applications, the same number $n_j = n / J$ of items is often presented with common mean item easiness $\bar {\sigma }_j = \frac {1}{n_j}\sum _{k=1}^{n_j}\sigma _{jk}$ at each measurement occasion, that is, $n_j = n / J$ and $\bar {\sigma }_j = \bar {\sigma }$ . This practice raises the question of whether such a design is efficient or whether alternative allocations can reduce the total number of items required without compromising statistical accuracy.

Abilities $\theta _{ij}$ are modeled as random effects. Specifically, $\theta _{ij}$ is expressed as a realization of

$$\begin{align*}\Theta_{ij} = \Lambda_{ij} \, \theta(t_j), \end{align*}$$

where $\Lambda _{ij}$ is a gamma-distributed latent variable capturing individual deviations from the mean trajectory $\theta (t)$ .

To specify a longitudinal RPGCM based on compound symmetry, analogous to linear mixed models with multivariate normal random effects, a permanent person effect and time-specific person effects are introduced. Accordingly, the latent variable $\Lambda _{ij}$ is decomposed as

$$\begin{align*}\Lambda_{ij} = \Gamma_{i0} + \Gamma_{ij}, \qquad j = 1, \ldots, J, \end{align*}$$

where the time-specific effects $\Gamma _{i1}, \ldots , \Gamma _{iJ}$ are assumed to be independently and identically distributed as $\Gamma (\alpha , \tau )$ , the permanent person effect $\Gamma _{i0} \sim \Gamma (\alpha _0, \tau )$ shares the same scale parameter $\tau $ , and all components are mutually independent.

Under these assumptions, $\Lambda _{ij} \sim \Gamma (\alpha _0 + \alpha , \tau )$ , and the vector $\boldsymbol {\Lambda }_i = (\Lambda _{i1}, \ldots , \Lambda _{iJ})^\top $ follows a multivariate gamma distribution with correlated components induced by the permanent person effect.

By standardizing the latent effects by $\tau = 1 / (\alpha _0 + \alpha )$ such that $\mathrm {E}(\Lambda _{ij}) = 1$ , the marginal mean of the responses is given by

$$\begin{align*}\mathrm{E}(Y_{ijk}) = \theta(t_j)\sigma_{jk}. \end{align*}$$

Thus, the growth curve $\theta (t)$ represents the marginal mean ability, satisfying $\mathrm {E}(\Theta _{ij}) = \theta (t_j)$ .

Furthermore, $\mathrm {Var}(\Lambda _{ij}) = \tau $ and $\mathrm {Cov}(\Lambda _{ij}, \Lambda _{ij'}) = \rho \tau $ , where $\rho = \alpha _0 \tau $ denotes the intraclass correlation coefficient. Consequently,

$$\begin{align*}\mathrm{Cov}(\boldsymbol{\Lambda}_i) = \tau \left[(1 - \rho)\mathbf{I}_J + \rho \mathbf{1}_J \mathbf{1}_J^\top \right]. \end{align*}$$

To model the mean structure, we adopt an unstructured one-way ANOVA-type specification $\theta (t_j) = \theta _j$ embedded in a generalized linear mixed model with a log link,

$$\begin{align*}\theta_j = \exp(\beta_j), \qquad j = 1, \ldots, J, \end{align*}$$

with vector $\boldsymbol {\theta } = (\theta _1, \ldots , \theta _J)$ of mean abilities over time.

In the model outlined above, the latent variables $\Gamma _{ij}$ are assumed to be identically distributed across measurement occasions. Consequently, the variances of the ability parameters $\Theta _{ij}$ remain constant over time. For many studies investigating retest effects in cognitive abilities, this assumption appears plausible. We inspected the studies included in the meta-analysis by Scharfen et al. (Reference Scharfen, Peters and Holling2018) and found that the variances of the observed test scores were generally stable across repeated measurement occasions. However, gains in ability may not be strictly additive. For example, individuals with higher initial abilities may exhibit different growth trajectories than individuals with lower initial abilities. In such situations, the assumption of constant variance may be violated. To accommodate this possibility, we extend the model by allowing the latent variables to follow occasion-specific gamma distributions, $ \Gamma _{ij} \sim \Gamma (\alpha _j,\tau ), j = 1,\ldots ,J$ , thereby permitting the variances of the ability parameters to vary over time. The extended model, together with the corresponding optimal designs, is presented in the Appendix. It should be noted, however, that the resulting A-optimal designs depend on the variance parameters of the random effects, whereas the designs obtained under the assumption of identically distributed $\Gamma _{ij}$  do not. Thus, they require substantially more prior information, which may be difficult to obtain in practical applications.

2.2 Parameter estimation

In the generalized linear mixed model setup, the focus is on estimating the parameter vector $\boldsymbol {\beta } = (\beta _1, \ldots , \beta _J)^\top $ . Even if the parameters $\tau $ and $\rho $ are assumed to be known, the likelihood involves integrals which are analytically difficult to handle. Consequently, we employ maximum quasi-likelihood estimation (Wedderburn, Reference Wedderburn1974), which relies only on the first- and second-order marginal moments of the response vector.

The maximum quasi-likelihood estimator $\hat {\boldsymbol {\beta }}_{\mathrm {MQL}}$ is defined as the solution to

$$\begin{align*}\sum_{i=1}^{N} \mathbf{D}_i^\top \mathbf{V}_i^{-1} (\mathbf{Y}_i - \boldsymbol{\mu}_i) = \mathbf{0}, \end{align*}$$

where $\boldsymbol {\mu }_i = \mathrm {E}(\mathbf {Y}_i)$ , $\mathbf {V}_i = \mathrm {Cov}(\mathbf {Y}_i)$ , and $\mathbf {D}_i = \partial \boldsymbol {\mu }_i / \partial \boldsymbol {\beta }^\top $ . The explicit expressions for these terms are provided in the Appendix.

The quasi-information matrix is given by

$$\begin{align*}\mathbf{M}_{\mathrm{Q}} = \sum_{i=1}^{N} \mathbf{D}_i^\top \mathbf{V}_i^{-1} \mathbf{D}_i. \end{align*}$$

Similar to the situation in maximum likelihood estimation, the asymptotic covariance matrix of the maximum quasi-likelihood estimator $\hat {\boldsymbol {\beta }}_{\mathrm {MQL}}$ is proportional to the inverse $\mathbf {M}_{\mathrm {Q}}^{-1}$ of the quasi-information matrix.

As shown in Theorem 4.2 of Parsamaram et al. (Reference Parsamaram, Holling and Schwabe2026), the quasi-information simplifies, under a saturated mean model and identical test conditions across individuals, to

$$\begin{align*}\mathbf{M}_{\mathrm{Q}} = N \left[ \mathrm{diag}\!\left(\frac{1}{\theta_j \sum_{k=1}^{n_j}\sigma_{jk}}\right) + \tau \left((1 - \rho)\mathbf{I}_J + \rho \mathbf{1}_J \mathbf{1}_J^\top \right) \right]^{-1}. \end{align*}$$

In the fixed-effects RPCM, the maximum quasi-likelihood estimator coincides with the common maximum likelihood estimator, and the likelihood-based Fisher information for $\boldsymbol {\beta }$ is obtained by formally setting $\tau = 0$ in the quasi-information.

3 A-optimal design

Parsamaram et al. (Reference Parsamaram, Holling and Schwabe2026) developed locally D-optimal designs for estimating mean abilities in this model. According to their results, the D-optimal designs, which minimize the volume of the confidence ellipsoid of the parameters, require not only the specification of the parameter vector of interest, $\boldsymbol {\theta }$ , but also prior knowledge of the variance $\tau $ and the correlation $\rho $ . Such information, however, is rarely available in practice. In contrast, A-optimal designs, which minimize the sum of the variances of the parameter estimators or specified functions thereof, rely on substantially weaker assumptions and are therefore better suited to practical applications. We will discuss this topic in more detail in the fifth section.

The standardized (per response) quasi-information matrix is

$$\begin{align*}\mathbf{M}_{\mathrm{Q}}(\xi) = \left[ \mathrm{diag}\!\left(\frac{1}{\theta_j w_j \bar{\sigma}}\right) + n \tau \left((1 - \rho)\mathbf{I}_J + \rho \mathbf{1}_J \mathbf{1}_J^\top \right) \right]^{-1}, \end{align*}$$

where $\xi = (w_1, \ldots , w_J)$ denotes the vector of proportions $w_j = \sum _{k=1}^{n_j}\sigma _{jk} / \sum _{j=1}^{J}\sum _{k=1}^{n_j}\sigma _{jk}$ of item easiness allocated to the measurement occasions $j = 1, \ldots , J$ , $\sum _{j=1}^{J} w_j = 1$ .

The vector $\xi $ of proportions of easiness constitutes the test design and can be chosen to optimize certain characteristics of the estimation. These proportions of easiness can be interpreted as follows: When all items have the same easiness parameters $\sigma _{jk} = \bar {\sigma }= \frac {1}{n} \sum _{j=1}^J \sum _{k=1}^{n_j} \sigma _{jk}$ , then $w_j = n_j / n$ is the proportion of items presented at occasion $j$ . Alternatively, when the same number $n_j = n / J$ of items is presented at each occasion, then $w_j = \frac {1}{J \bar {\sigma }}\bar {\sigma }_j$ is proportional to the mean easiness $\bar {\sigma }_j = \frac {1}{n_j}\sum _{k=1}^{n_j}\sigma _{jk}$ of items to be administered at occasion $j$ .

Note that, in the unstructured mean model, the proportions $w_j$ are only involved in the first term in the representation of the quasi-information. In particular, the asymptotic covariance matrix of $\hat {\boldsymbol {\beta }}$ equals $c_1 \mathrm {diag}(1/(\theta _j w_j)) + \mathbf {C}_2$ for some generic constant terms $c_1> 0$ and $\mathbf {C}_2$ . This explains why the A-optimal weights neither depend on the variance parameters $\tau $ and $\rho $ nor on $\bar {\sigma }$ and $n$ .

We derive A-optimal designs for the most relevant parameters of the unstructured longitudinal RPGCM. We begin with the basic parameter vector $\boldsymbol {\beta }$ of the unstructured mean model and subsequently develop A-optimal designs for the mean ability vector $\boldsymbol {\theta } = (\theta _1, \ldots , \theta _J)$ as well as for the square-root-transformed mean abilities $\boldsymbol {d} = (d_1, \ldots , d_J)$ , $d_j = \sqrt {\theta _j}$ .

In many applications, hypotheses are formulated in terms of contrasts rather than absolute parameter values. Accordingly, we also derive A-optimal designs for commonly used contrasts, including differences relative to a baseline (e.g., $\theta _j - \theta _1$ , $j = 2, \ldots , J$ ) and successive differences (e.g., $\theta _j - \theta _{j-1}$ , $j = 2, \ldots , J$ ).

All A-optimality criteria considered in the following can be reduced to the minimization of an objective function of the form

$$\begin{align*}C(\xi) = \sum_{j=1}^{J} \frac{h_j}{w_j}, \qquad \text{for some constants}\ h_j> 0, \end{align*}$$

in the proportions of easiness $w_j$ allocated to measurement occasion $j$ . Minimization of this function yields the optimal weights

$$\begin{align*}w_j^\ast = \frac{\sqrt{h_j}}{\sum_{k=1}^{J} \sqrt{h_k}}, \qquad j = 1, \ldots, J. \end{align*}$$

This general result considerably simplifies the derivation of the optimal designs presented below.

As an illustration, consider A-optimal designs for the parameters $\beta _j = \log (\theta _j)$ , $j = 1, \ldots , J$ . In this case, the trace of the asymptotic covariance matrix has to be minimized. This is equivalent to minimizing

$$\begin{align*}C_{\beta_1, \ldots, \beta_J}(\xi) = \sum_{j=1}^{J} \frac{1}{\theta_j w_j}. \end{align*}$$

Setting $h_j = 1/\theta _j$ immediately yields the A-optimal weights

$$\begin{align*}w^\ast_{j,\beta_j} = \dfrac{\sqrt{1/\theta_j}}{\sum_{k=1}^{J} \sqrt{1/\theta_k}}, \quad j = 1, \ldots, J \end{align*}$$

for estimating the parameters $\boldsymbol {\beta } = (\beta _1, \ldots , \beta _J)$ .

3.1 A-optimal designs for $\boldsymbol{\beta }$

For contrasts relative to the baseline, $\beta _j - \beta _1$ , $j = 2, \ldots , J$ , the objective function

$$\begin{align*}C_{\beta_2 - \beta_1, \ldots, \beta_J - \beta_1}(\xi) = \frac{J-1}{\theta_1 w_1} + \sum_{j=2}^{J} \frac{1}{\theta_j w_j} \end{align*}$$

has to be minimized, and for successive differences, $\beta _2 - \beta _1, \ldots , \beta _J - \beta _{J-1}$ , the objective function

$$\begin{align*}C_{\beta_2 - \beta_1, \ldots, \beta_J - \beta_{J-1}}(\xi) = \frac{1}{\theta_1 w_1} + \sum_{j=2}^{J-1} \frac{2}{\theta_j w_j} + \frac{1}{\theta_J w_J} \end{align*}$$

has to be minimized. The resulting A-optimal weights for parameters and contrasts of $\boldsymbol {\beta }$ are listed in lines 1–3 of Table 2.

Table 2

A-optimal weights for parameters and contrasts

Table 2 Long description

The table contains 13 numbered rows.

Row 1: Parameter beta sub j. Weight for all columns is the square root of 1 over theta sub j all over the summation from k equals 1 to J of the square root of 1 over theta sub k.

Row 2: Contrast beta sub j minus beta sub 1. Weight for w sub 1 superstar is the square root of J minus 1 over theta sub 1 all over the square root of J minus 1 over theta sub 1 plus the summation from k equals 2 to J of the square root of 1 over theta sub k. Weight for other columns is the square root of 1 over theta sub j all over the same denominator.

Row 3: Contrast beta sub j over beta sub j minus 1. Weight for w sub 1 super star is the square root of 1 over theta sub 1 all over a denominator D. Weight for w sub j super star is the square root of 2 over theta sub j all over D. Weight for w sub J super star is the square root of 1 over theta sub J all over D. D equals the square root of 1 over theta sub 1 plus the summation from k equals 2 to J minus 1 of the square root of 2 over theta sub k plus the square root of 1 over theta sub J.

Row 4: Parameter theta sub j. Weight for all columns is the square root of theta sub j all over the summation from k equals 1 to J of the square root of theta sub k.

Row 5: Contrast theta sub j minus theta sub 1. Weight for w sub 1 super star is the square root of J minus 1 times theta sub 1 all over the square root of J minus 1 times theta sub 1 plus the summation from k equals 2 to J of the square root of theta sub k. Weight for other columns is the square root of theta sub j all over the same denominator.

Row 6: Contrast theta sub j minus theta sub j minus 1. Weight for w sub 1 super star is the square root of theta sub 1 all over a denominator D sub 2. Weight for w sub j super star is the square root of 2 times theta sub j all over D sub 2. Weight for w sub J super star is the square root of theta sub J all over D sub 2. D sub 2 equals the square root of theta sub 1 plus the summation from k equals 2 to J minus 1 of the square root of 2 times theta sub k plus the square root of theta sub J.

Row 7: Contrast theta sub j over theta sub 1. Weight for w sub 1 super star is the square root of the summation from k equals 2 to J of theta sub k squared over theta sub 1 all over a denominator D sub 3. Weight for other columns is the square root of theta sub j all over D sub 3. D sub 3 equals the square root of the summation from k equals 2 to J of theta sub k squared over theta sub 1 plus the summation from k equals 2 to J of the square root of theta sub k.

Row 8: Contrast theta sub j over theta sub j minus 1. Weights involve complex square roots of theta ratios. Weight for w sub 1 super star is the square root of theta sub 2 squared over theta sub 1 cubed all over a summation denominator.

Row 9: Parameter d sub j. Weight for all columns is 1 over J.

Row 10: Contrast d sub j minus d sub 1. Weight for w sub 1 super star is the square root of J minus 1 all over the square root of J minus 1 plus J minus 1. Weight for other columns is 1 all over the square root of J minus 1 plus J minus 1.

Row 11: Contrast d sub j minus d sub j minus 1. Weights for w sub 1 super star and w sub J super star are 1 over 2 plus J minus 2 times the square root of 2. Weight for w sub j super star is the square root of 2 over 2 plus J minus 2 times the square root of 2.

Row 12: Contrast d sub j over d sub 1. Weight for w sub 1 super star is the square root of the summation from k equals 2 to J of theta sub k over theta sub 1 all over the square root of the same summation plus J minus 1. Weight for other columns is 1 over the same denominator.

Row 13: Contrast d sub j over d sub j minus 1. Weights involve square roots of theta sub j plus 1 over theta sub j squared plus 1 over theta sub j minus 1.

3.2 A-optimal designs for $\boldsymbol{\theta }$

In applications of longitudinal Poisson models, the primary parameter of interest is often the trajectory $\theta $ . In repeated testing, this may correspond, for example, to the expected number of correctly processed stimuli, in epidemiology to the expected number of rare disease cases, and in criminology to the expected number of aggressive acts.

For estimation of $\boldsymbol {\theta }$ , the plug-in estimators $\hat {\theta }_j = \exp (\hat {\beta }_j)$ may be used. By the $\delta $ -method, the asymptotic covariance matrix of $\hat {\boldsymbol {\theta }}$ equals $c_1 \mathrm {diag}(\theta _j / w_j) + \mathbf {C}_2$ . Again, applying the general objective function, the following A-optimal designs for $\boldsymbol {\theta }$ can be derived.

For mean abilities $\theta _j$ , $j = 1, \ldots , J$ , the objective function is

$$\begin{align*}C_{\theta_1, \ldots, \theta_J}(\xi) = \sum_{j=1}^{J} \frac{\theta_j}{w_j}. \end{align*}$$

For baseline differences $\theta _j - \theta _1$ , $j = 2, \ldots , J$ , the objective function is

$$\begin{align*}C_{\theta_2 - \theta_1, \ldots, \theta_J - \theta_1}(\xi) = \frac{(J-1) \theta_1}{w_1} + \sum_{j=2}^{J} \frac{\theta_j}{w_j}. \end{align*}$$

For successive differences $\theta _2 - \theta _1, \ldots , \theta _J - \theta _{J-1}$ , the objective function is

$$\begin{align*}C_{\theta_2 - \theta_1, \ldots, \theta_J - \theta_{J-1}}(\xi) = \frac{\theta_1}{w_1} + \sum_{j=2}^{J-1} \frac{2 \theta_j}{w_j} + \frac{\theta_J}{w_J}. \end{align*}$$

The corresponding A-optimal weights are given in lines 4–6 of Table 2.

Ratios of expected values are often more stable and interpretable than differences. Again, plug-in estimators can be used and the asymptotic covariance is obtained by the $\delta $ -method. For ratios relative to the baseline, $\theta _j/\theta _1$ , $j = 2, \ldots , J$ , the objective function is given by

$$\begin{align*}C_{\theta_2 / \theta_1, \ldots, \theta_J / \theta_1}(\xi) = \frac{1}{\theta_1^2} \left(\frac{\sum_{j=2}^{J} \theta_j^2 / \theta_1}{w_1} + \sum_{j=2}^{J} \frac{\theta_j}{w_j}\right) \, , \end{align*}$$

and for successive ratios, it is

$$\begin{align*}C_{\theta_2 / \theta_1, \ldots, \theta_J / \theta_{J-1}}(\xi) = \frac{\theta_2^2/\theta_1^3}{w_1} + \sum_{j=2}^{J-1} \frac{\theta_{j+1}^2/\theta_j^3 + \theta_j/\theta_{j-1}^2}{w_j} + \frac{\theta_J/\theta_{J-1}^2}{w_J}. \end{align*}$$

The A-optimal weights are in lines 7 and 8 of Table 2.

3.4 Numerical results

All A-optimal designs for $\boldsymbol {\beta }$ , $\boldsymbol {\theta }$ , and their contrasts are locally optimal designs. They depend on the relative change in the mean trajectory between measurement occasions, but not on the overall mean. The latter is in accordance with the property that the combination of ability and easiness may be scaled at will. For $\boldsymbol {d}$ and the corresponding difference contrasts, however, the optimal designs are independent of the expected values and, hence, globally optimal for all parameter values. They are functions of the number $J$ of measurement occasions only, which substantially simplifies experimental planning.

Table 3 illustrates the derived A-optimal designs for four example growth patterns, each with $J = 4$ repeated measurements and a baseline with a mean of 4. The first example corresponds to a constant value for the growth curve, which may be considered as a null hypothesis setup of no retest effects. The second example exhibits growth increments motivated by the meta-analysis of retest effects reported by Scharfen et al. (Reference Scharfen, Peters and Holling2018). The third example represents a linear increase in expected values, whereas the fourth example follows an exponential growth curve, as commonly observed in other application domains such as epidemiology.

The results demonstrate once again that A-optimal designs for square-root-transformed parameters and their differences are independent of the actual expected values. In general, a uniform allocation of items across measurement occasions is suboptimal and can lead to a loss of efficiency, in particular, when a baseline measurement serves as a reference. This topic will be analyzed in more detail in the next section.

It should be noted that alternative expected values, growth rates, or numbers of measurement occasions may yield different A-optimal designs. However, the corresponding optimal weights can always be obtained directly from the closed-form expressions derived above.

Table 3

A-optimal weights for four sets of growth patterns with four measurement occasions each

Table 3 Long description

The table is organized into 15 columns. The first column identifies the Set (1 through 4). The subsequent columns represent mathematical parameters and weights: theta, w super star beta sub j, w super star beta sub j minus beta sub 1, w super star beta sub j minus beta sub j minus 1, w super star theta sub j, w super star theta sub j minus theta sub 1, w super star theta sub j minus theta sub j minus 1, w super star theta sub j over theta sub 1, w super star theta sub j over theta sub j minus 1, w super star d sub j, w super star d sub j minus d sub 1, w super star d sub j minus d sub j minus 1, w super star d sub j over d sub 1, and w super star d sub j over d sub j minus 1.

* Set 1: All theta values are 4. Weights for w super star beta sub j and w super star theta sub j are constant at 0.25. Other weights alternate between 0.37, 0.21, and 0.29.

* Set 2: Theta values increase from 4 to 6.06. Weights for w super star beta sub j decrease from 0.29 to 0.23.

* Set 3: Theta values increase linearly (4, 8, 12, 16). Weights for w super star beta sub j decrease from 0.36 to 0.18. Weights for w super star theta sub j increase from 0.16 to 0.33.

* Set 4: Theta values increase exponentially (4, 8, 16, 32). Weights for w super star beta sub j decrease from 0.39 to 0.14. Weights for w super star theta sub j increase from 0.14 to 0.39. Weights for w super star d sub j remain constant at 0.25 across all sets.

4 A-efficiency

In practice, experimenters frequently employ a uniform allocation of observations across measurement occasions, either for convenience or due to limited prior knowledge about expected values. The previously derived A-optimal designs for the parameter vectors $\boldsymbol {\beta }$ , $\boldsymbol {\theta }$ , and $\boldsymbol {d}=\sqrt {\boldsymbol {\theta }}$ provide a minimal estimator variance. In this section, we analyze the gain of these optimal designs compared to uniform designs $\xi _{\mathrm {uniform}} = (1/J, \ldots , 1/J)$ . Commonly used uniform designs allocate an equal proportion of observations with equal mean easiness to each measurement occasion. Specifically, we consider the A-efficiency of the uniform design compared to the previously obtained A-optimal designs for the parameters $\boldsymbol {\beta }$ , $\boldsymbol {\theta }$ , and $\boldsymbol {d}$ as well as the contrasts thereof presented in Table 2.

The efficiency, which will be considered here, is defined by

$$\begin{align*}\mathrm{Eff}_A(\xi_{\mathrm{uniform}};\boldsymbol{\delta}) = \frac{C_{\boldsymbol{\delta}}(\xi_{\boldsymbol{\delta}}^\ast)} {C_{\boldsymbol{\delta}}(\xi_{\mathrm{uniform}})}, \end{align*}$$

for the uniform design $\xi _{\mathrm {uniform}}$ , where $\boldsymbol {\delta }$ stands for the parameter vector of interest and $\xi _{\boldsymbol {\delta }}^\ast$ is the corresponding A-optimal design.

The A-efficiency ranges between 0 and 1 and quantifies the amount of within-test-person information of the uniform design compared to the A-optimal design. Based on the efficiency, the corresponding A-deficiency can be computed as $\mathrm {Def}_A(\xi _{\mathrm {uniform}};\boldsymbol {\delta }) = \mathrm {Eff}_A(\xi _{\mathrm {uniform}};\boldsymbol {\delta })^{-1} - 1$ , which represents the proportional increase in the total number $n$ of items required when using design $\xi _{\mathrm {uniform}}$ instead of the A-optimal design $\xi _{\boldsymbol {\delta }}^\ast$ to achieve the same estimation precision while keeping the mean easiness $\bar {\sigma }$ fixed. In the case of the fixed-effects RPCM, the above A-deficiency may be interpreted likewise as the proportional increase in the number $N$ of test persons required when the design $\xi _{\mathrm {uniform}}$ is used instead of the A-optimal design $\xi _{\boldsymbol {\delta }}^\ast$ to obtain the same value for the trace of the covariance matrix while the total easiness $n \bar {\sigma }$ per test person is kept fixed.

For the four data examples introduced in Table 3, we compute the A-efficiency of the uniform design relative to the A-optimal design for the parameters $\boldsymbol {\theta }$ and $d$ , as well as for the contrasts considered above. The results are summarized in Table 4 (left). Furthermore, we compute the A-efficiency for another example of four sets of analogous growth patterns, but each with seven measurement occasions. The first set of $\boldsymbol {\theta }$ values is constant $4$ , the second consists of (4, 8, 11, 13, 14, 14.5, 14.75), the third of (4, 8, 12, 16, 20, 24, 28), and the fourth of (4, 8, 16, 32, 64, 128, 256); see Table 4 (right).

Table 4

A-efficiency of the uniform design relative to the A-optimal design for the four sets of growth patterns with four measurement occasions as outlined in Table 3 (left) and with seven measurement occasions as outlined in the text (right)

Table 4 Long description

The table is organized by Parameter in the first column, followed by two main sections: J equals 4 (Sets 1 to 4) and J equals 7 (Sets 1 to 4).

Row 1: beta sub j. J equals 4: 1.00, 0.99, 0.93, 0.87. J equals 7: 1.00, 0.94, 0.89, 0.70.

Row 2: beta sub j minus beta sub 1. J equals 4: 0.93, 0.88, 0.76, 0.70. J equals 7: 0.85, 0.62, 0.56, 0.43.

Row 3: beta sub j minus beta sub j minus 1. J equals 4: 0.97, 0.98, 0.94, 0.89. J equals 7: 0.98, 0.97, 0.92, 0.75.

Row 4: theta sub j. J equals 4: 1.00, 0.99, 0.94, 0.87. J equals 7: 1.00, 0.97, 0.93, 0.70.

Row 5: theta sub j minus theta sub 1. J equals 4: 0.93, 0.97, 0.99, 0.94. J equals 7: 0.85, 0.98, 0.96, 0.75.

Row 6: theta sub j minus theta sub j minus 1. J equals 4: 0.97, 0.96, 0.92, 0.89. J equals 7: 0.98, 0.94, 0.92, 0.75.

Row 7: theta sub j over theta sub 1. J equals 4: 0.93, 0.88, 0.73, 0.61. J equals 7: 0.85, 0.60, 0.51, 0.24.

Row 8: theta sub j over theta sub j minus 1. J equals 4: 0.97, 0.96, 0.88, 0.89. J equals 7: 0.98, 0.83, 0.81, 0.75.

Row 9: d sub j. J equals 4: 1.00, 1.00, 1.00, 1.00. J equals 7: 1.00, 1.00, 1.00, 1.00.

Row 10: d sub j minus d sub 1. J equals 4: 0.93, 0.93, 0.93, 0.93. J equals 7: 0.85, 0.85, 0.85, 0.85.

Row 11: d sub j minus d sub j minus 1. J equals 4: 0.97, 0.97, 0.97, 0.97. J equals 7: 0.98, 0.98, 0.98, 0.98.

Row 12: d sub j over d sub 1. J equals 4: 0.93, 0.88, 0.75, 0.67. J equals 7: 0.85, 0.62, 0.54, 0.32.

Row 13: d sub j over d sub j minus 1. J equals 4: 0.97, 0.97, 0.91, 0.89. J equals 7: 0.98, 0.91, 0.87, 0.75.

The results indicate that a uniform allocation of observations is generally suboptimal. A-optimal designs lead to reductions in variance, although these gains are small in several of the examples considered. In such situations, practitioners may rely on uniform designs with only minor loss in efficiency. However, substantially larger reductions are observed for parameters representing contrasts with respect to a reference category. For such contrasts, which are frequently of practical interest, A-optimal designs should therefore be preferred. For any specific situation arising in an empirical study, the formulas presented above allow the derivation of A-optimal designs and the corresponding A-efficiency.

Overall, these findings highlight the practical relevance of A-optimal design strategies for longitudinal and experimental studies when contrasts relative to a reference category are of primary interest.

5 D-optimal versus A-optimal designs for the longitudinal RPGCM

Recently, Parsamaram et al. (Reference Parsamaram, Holling and Schwabe2026) developed locally D-optimal designs for estimating mean abilities in the RPGCM under compound symmetry of the covariance matrix of the random effects. This work motivated us to derive A-optimal rather than D-optimal designs here, although the latter constitute the most prominent class of optimal designs. While a D-optimal design maximizes the determinant of the Fisher information matrix and thereby minimizes the volume of the asymptotic confidence ellipsoid for the parameter vector, an A-optimal design minimizes the trace of the inverse information matrix, this means that it minimizes the average asymptotic variance of the parameter estimates. A major reason for the predominance of D-optimal designs is the invariance of the D-criterion under nonsingular reparameterizations of the model; in particular, the designs do not depend on the scaling of the parameters. Hence, Parsamaram et al. (Reference Parsamaram, Holling and Schwabe2026) needed to derive D-optimal designs only for $\boldsymbol {\beta } = \log (\boldsymbol {\theta })$ ; the resulting designs are also D-optimal for the parameters $\boldsymbol {\theta }$ and $\boldsymbol {d}$ , which, as shown above, does not hold for A-optimal designs. However, the derived D-optimal designs require knowledge of the unknown parameter vector $\boldsymbol {\beta }$ and, in addition, of $\tau $ and $\rho $ . The A-optimal designs derived above, in contrast, only require the specification of the parameter of interest, $\boldsymbol {\beta }$ or $\boldsymbol {\theta }$ , while the A-optimal designs for $\boldsymbol {d}$ do not presuppose any a priori knowledge at all. Moreover, the A-criterion is easier to interpret, as it is directly aligned with the substantive research objective, namely, the precise estimation of the parameters themselves, without taking the covariances of the estimators into account. Altogether, both types of designs have their merits; for practical purposes, however, A-optimal designs may be considered more advantageous for the longitudinal RPGCM and are only feasible because of their parsimonious requirements regarding prior knowledge.

In addition to the unstructured model considered in the present study, Parsamaram et al. (Reference Parsamaram, Holling and Schwabe2026) derived general conditions for D-optimal designs for models with structured functional relationships describing the evolution of the mean ability $\theta (t)$ over time, such as generalized linear and nonlinear models. Extending the present A-optimality results to such models would be substantially more challenging because the invariance property of the D-criterion no longer applies and the A-optimal designs additionally depend on the variance parameters. The development of A-optimal designs for these settings therefore may remain a topic for future research. It should be noted that an unstructured model may often be adequate, in particular, when only a few measurement occasions are analyzed. As the meta-analysis of Scharfen et al. (Reference Scharfen, Peters and Holling2018) shows, increases in the abilities are to be expected in most studies only to the third or fourth measurement occasion. Furthermore, of the studies included in this meta-analysis, most studies (75%) analyzed retest effects for two occasions, while 19% and 4% of the studies included three and four repetitions, respectively. Only 2% of the studies included in this meta-analysis had more than four measurement points. Nevertheless, the analysis of ability changes may be interesting for a larger set of repetitions, in particular, in clinical settings.

A direct comparison between the D-optimal designs derived by Parsamaram et al. (Reference Parsamaram, Holling and Schwabe2026) and the A-optimal designs developed in the present study is difficult. As noted above, nearly all D-optimal designs for the RPGCM require knowledge not only of the parameters of interest, such as $\boldsymbol {\beta }$ , but also of additional quantities, including $\tau $ and $\rho $ . In contrast, the derivation of the A-optimal designs considered here does not depend on such prior information.

One exception is the unstructured model with equal means considered in Example 5.1 by Parsamaram et al. (Reference Parsamaram, Holling and Schwabe2026). In this case, the D-optimal design yields uniform weights, $w_j = 1/J$ , for the parameters $\beta _j$ . Owing to the invariance property of the criterion, the same weights are obtained for $\theta _j$ and $d_j$ . These weights coincide with those obtained from the corresponding A-optimal designs for these parameters (see Table 2).

6 Summary

In this article, we developed A-optimal designs for longitudinal RPGCMs. The proposed designs also apply to more general longitudinal Poisson Gamma models and are therefore relevant across disciplines in which repeated count data are analyzed.

The designs specify optimal item allocation proportions across measurement occasions to achieve the most precise estimation of parameters such as those of mean abilities and commonly used contrasts. By minimizing estimator variances, these designs enhance statistical efficiency while potentially reducing participant burden.

An important direction for future research is the development of A-optimal designs for comparing multiple growth curves, as many applications focus on contrasts between trajectories, such as treatment and control groups. In addition, models imposing structured functional relationships on the evolution of the mean ability over time should be considered, particularly when data are collected at a large number of time points.