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Suppose a longitudinal rating design to be analyzed with the HRM has
$M$ time points with ideal and observed ratings at time $m$ nested
within each $\theta_{im}$, where $\theta_{im}$ is the trait for
examinee $i$ at time point $m$, ($m=1,\ldots,M$). Now, instead of
assuming some normal distribution for the traits, we model them using
a longitudinal model, which is part trend and part autoregressive time
series model of order 1, meaning that each time point is modeled using
information only from the most previous time point. The other levels
are the same but now we have a set of ideal ratings and observed
ratings at each timepoint $m$. Item parameters and rater parameters
remain the same over time.
$$
  \begin{array}{rcl}
    \theta_{im} & \sim & \mbox{longitudinal model [AR(1) $+$ trend]}\\
    \xi_{imj}~|~\theta_{im} & \sim & \mbox{polytomous IRT model, $j=1,\ldots,J$,
      for each $i$, $m$}\\
    X_{imjr}~|~\xi_{ijm} & \sim & \mbox{polytomous signal detection model,
      $r=1,\ldots,R$, for each $i$, $j$, $m$.}
  \end{array}
	  	\eqno{(4)}
$$
In order to estimate $\theta_{im}$, let the trait for subject $i$ at
time point $m$ be a function of two quantities:
$$
  \theta_{im} = \delta_m + Z_{im}
  \eqno{(5)}
$$
Here, $\delta_m$ is the trend in $\theta_{im}$ at time $m$ and
$Z_{im}$ is the time series component of the model. We can decompose
$\delta_m$ into a function of the parameter $g$ for overall growth,
depending on the type of trend.  A linear trend is a straightforward
and reasonable approach to modeling linear relationships (whether
positive or negative). If a linear trend, for example, then $\delta_m
= g\times (m-1)/(M-1)$.  $Z_{im}$ decomposes into $Z_{im} = \rho
    Z_{i(m-1)} + \left(\sqrt{1-\rho^2}\right)\varepsilon_{im}$, where
    $Z_{i(m-1)}$ the lagged value of $Z_{im}$ is weighted by $\rho$
    the autocorrelation parameter. In the second term the random noise
    for examinee $i$ at time point m denoted is distributed as and is
    weighted by a function of the autocorrelation $\sqrt{1-\rho^2}$.
    Weighting $\varepsilon_{im}$ assures stationary variance of the
    noise (and therefore the resultant traits) across $M$ time
    points. Together, the sum of the two weighted quantities in
    $Z_{im}$ makes up the component of $\theta_{im}$ that incorporates
    the individual's trait information; the other component
    incorporates an average trend for all $N$ examinees.


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