4.4 Parameter estimation with equate groups

The Rasch model is the preferred measurement model for child development data. Chapter I, Section 4 provides an introduction of the Rasch model geared towards the D-score.

The Rasch model expresses the probability of passing an item as a logistic function of the difference between the person ability \(\beta_n\) and the item difficulty \(\delta_i\). Table 4.1 explains the symbols used in equation (4.1). Formula (4.1) defines the model as

\[\begin{equation} \pi_{ni} = \frac{\exp(\beta_n - \delta_i)}{1+\exp(\beta_n -\delta_i)} \tag{4.1} \end{equation}\]

One way to interpret the formula is as follows. The logarithm of the odds that a person with ability \(\beta_n\) passes an item of difficulty \(\delta_i\) is equal to the difference \(\beta_n-\delta_i\) (Wright and Masters 1982). See the logistic model in Chapter 1, Section 4.6.1 for more detail.

In model (4.1) every milestone \(i\) has one parameter \(\delta_i\). We extend the Rasch model by restricting the \(\delta_i\) of all items within the same equate group to the same value. We thereby effectively say that these items are interchangeable measures of child development.

Estimation of the parameter for the equate group is straightforward. Wright and Masters (1982) present a simple method for aligning two test forms with common items. There are three steps:

  1. Estimate the separate \(\delta_i\)’s per item;
  2. Combine these estimates into \(\delta_q\) by calculating their weighted average;
  3. Overwrite each \(\delta_i\) by \(\delta_q\).

Suppose that \(Q\) is the collection of items in equate group \(q\), and that \(w_i\) is the number of respondents for item \(i\). The parameter estimate \(\delta_q\) for the equate group is

\[\begin{equation} \delta_q = \frac{\sum_{i\in Q} \delta_iw_i}{\sum_{i\in Q} w_i} \tag{4.2} \end{equation}\]
Table 4.1: Overview the symbols used in equations (4.1) and (4.2).
Symbol Term Description
\(\beta_n\) Ability True (but unknown) developmental score of child \(n\)
\(\delta_i\) Difficulty True (but unknown) difficulty of item \(i\)
\(\delta_q\) Difficulty The combined difficulty of the items in equate group \(q\)
\(\pi_{ni}\) Probability Probability that child \(n\) passes item \(i\)
\(l\) The number of items in the equate group
\(w_i\) The number of respondents with an observed score on item \(i\)