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An autoregressive growth model for longitudinal item analysis


A first-order autoregressive growth model is proposed for longitudinal binary item analysis where responses to the same items are conditionally dependent across time given the latent traits. Specifically, the item response probability for a given item at a given time depends on the latent trait as well as the response to the same item at the previous time, or the lagged response. An initial conditions problem arises because there is no lagged response at the initial time period. We handle this problem by adapting solutions proposed for dynamic models in panel data econometrics. Asymptotic and finite sample power for the autoregressive parameters are investigated. The consequences of ignoring local dependence and the initial conditions problem are also examined for data simulated from a first-order autoregressive growth model. The proposed methods are applied to longitudinal data on Korean students’ self-esteem.

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The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant R305D110027 to Educational Testing Service. The opinions expressed are those of the authors and do not represent the views of the Institute or the U.S. Department of Education.

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Correspondence to Minjeong Jeon.



Here we illustrate how to estimate the proposed model by utilizing gllamm. We use model M1 which includes a first-order lagged effect for item 1 (used in the empirical study in Sect. 7).

Let t represent years 2003 to 2008 denoted by \(t=0, 1, 2,\ldots , 5\), respectively (i.e., \(t=0\) for year 2003). We can formulate model M1 for the initial time point (\(t=0\)) as

$$\begin{aligned} g \left( \text {Pr} (y_{tis}=1 | \delta _{1s}, \gamma _{2s} ) \right)&= \beta _i' + \alpha _i' \delta _{1s} + \alpha _i' \gamma _{2s} \text {time}_{ts} + \alpha _i' \epsilon _{ts}, \end{aligned}$$

where \(\gamma _{2s} = b+ \delta _{2s}\) (with b being the mean of \(\gamma _{2s}\). See Eq. (5)). For the following time points (\(t>0\)), we formulate the model as

$$\begin{aligned} g \left( \text {Pr} (y_{tis}=1 | y_{(t-1)1s}, \delta _{1s}, \gamma _{2s} ) \right)&= \beta _i + \lambda _{1}y_{(t-1)1s}r_{i=1} + \alpha _i \delta _{1s} + \alpha _i \gamma _{2s} \text {time}_{ts} + \alpha _i \epsilon _{ts}, \end{aligned}$$

where \(\lambda _{1}\) is the parameter for the lagged response (\(y_{(t-1)1s}\)) for item 1 (here \(r_{i=1}\) is a dummy variable for item 1). We can formulate a combined model for \(t=0\) and \(t>0\) by utilizing the dummy variable \(d_{t=0,i=1}\) that indicates item 1 at the initial time point (\(t=0\)) as follows:

$$\begin{aligned} g \left( \text {Pr} (y_{tis}=1 | y_{(t-1)1s}, \delta _{1s}, \gamma _{2s} ) \right) =&\, \beta _i + \beta _1^* d_{t=0,i=1} + \lambda _{1}y_{(t-1)1s}r_{i=1} \end{aligned}$$
$$\begin{aligned}&+ \delta _{1s} (\alpha _i + \alpha _1^* d_{t=0,i=1} ) \end{aligned}$$
$$\begin{aligned}&+ \gamma _{2s} (\alpha _i \text {time}_{ts} + \alpha _1^* \text {time}_{ts} d_{t=0,i=1}) \end{aligned}$$
$$\begin{aligned}&+ \epsilon _{ts} ( \alpha _i + \alpha _1^* d_{t=0,i=1}), \end{aligned}$$

where Eq. (10) constitutes the fixed part of the model and Eqs. (11) to (13) constitute the random part of the model. In the fixed part, \(\beta _1^* =\beta _1'-\beta _1\) and in the random part \(\alpha _1^*=\alpha _1'-\alpha _1\) and \(\gamma _{2s} = b+\delta _{2s}\).


To save on computation time, users may run the model with a small number of quadrature points (e.g., 2) and use the initial estimates as starting values with a larger number of quadrature points (e.g., 5).

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Jeon, M., Rabe-Hesketh, S. An autoregressive growth model for longitudinal item analysis. Psychometrika 81, 830–850 (2016). https://doi.org/10.1007/s11336-015-9489-2

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  • autoregressive models
  • initial conditions problem
  • measurement invariance
  • serial dependence