The Random Walk Theory of Measurement

(written by Lawrence Krubner, however indented passages are often quotes)

What a great post. All of my biases tend in the direction described here:

The true score theory, or true measurement theory, postulates (and it is only a postulate) that a measurement value is the sum of a true value and a possibly random error : .

This postulate has an interesting consequence that I have rarely seen discussed. Suppose, as was the case in our pre-service teacher’s measurement process that measurements were carried out in a sequence, one following the other. This means – assuming a class size of – that we had a sequence of measurements . The true score, or true measurement, theory postulates that each . The so-called “error” terms can be thought of as random variables. Now we see that

The terms are random variables, so if they are identically distributed, as one might reasonably imagine, we can view the measurement as being obtained from the initial measurement by a random walk: successive measurements in a sequence of independent measurements are random excursions from a previous measurement.

“So what?” you may ask.

Well, the above consequence of true score, or true measurement, theory does not mention the hypothesized “true value” .

In other words, this consequence of true measurement theory, does not mention “true value” and sees repeated measurements as a random walk.

This “random walk theory” of measurement – for want of a better phrase – compels us to focus right from the beginning on the statistical aspects of measurement, rather then being side-tracked by a hypothetical “true value”.

Form this perspective, repeated measurements are a random walk, specified statistically by the nature of the “error” distribution.

That the “errors” generally follow a normal distribution is another, and deeper, story.