Transforming Parameter Estimates to a Specified Coordinate System

Abstract

All measurement systems have some arbitrariness to them. When length is measured, the size of the unit used is arbitrary. Length units can be chosen from feet, meters, miles, light years, etc. The origin of the length measurements is usually set at true 0 and measurements using different units are typically linear transformations of each other. A common exception is astronomical measurements of distance that are taken from some arbitrary origin such as the Sun or the center of the Milky Way Galaxy. Although measurements of length are much easier to understand than the measurements that result from the applications of IRT models, some of the same concepts can be used for both types of measurements. For example, the concepts of invariance and indeterminacy that are often discussed in the IRT literature also apply to length measurement. The unit of the measurement for length is indeterminate but the length itself is invariant. The length being measured does not change because different units of measurement are used, but there is nothing in the measurement of length that indicates that one unit of measurement is more “correct” than any other unit of measurement. Some units might be more convenient than others such as using Angstroms when measuring things at the atomic level instead of miles, but that does not make them more correct.

MIRT models have the same problems of indeterminacy as other measurement systems. Because of the multidimensional nature of these models, the analogy to physical measurement is more appropriate to the way that stars are located in the night sky than to simple length measurement. At least for relatively short time spans, stars are assumed to have invariant locations and star guides give information about how to locate individual stars. This is complicated by the fact that our viewing platform, the Earth, is moving so stars seem to have different locations at different times of the year. The viewer’s location on the Earth is also important because some stars can only be seen from the Northern Hemisphere and others only from the Southern Hemisphere. Despite all of these complexities, stars can be located using a relatively simple coordinate system. The coordinates from a particular location at a particular time of year are the distance above the horizon in a specified direction. This is the coordinate system used with success by amateur star gazers.