Change Scores

The measurement of change is fundamental in the social and behavioral sciences. Many researchers have used change scores to measure gain in ability or shift in attitude over time, or difference scores between two variables to measure a construct (e.g., self-concept vs. ideal self). This entry introduces estimation of change scores, its assumptions and applications, and at the end offers a recommendation on the use of change scores.

Let Y and X stand for the measures obtained by applying the same test to the subjects on two occasions. Observed change or difference score is D = Y – X. The true change is DT = YT – XT, where YTand XT represent the subject's true status [Page 143]at these times. The development of measuring the true change DT follows two paths, one using change score and the other using residual change score.

The only assumption to calculate change score is that Y (e.g., posttest scores) and X (e.g., pretest scores) should be on the same numerical scale; that is, the scores on posttest are comparable to scores on pretest. This only requirement does not suggest that pretest and posttest measure the same construct. Thus, such change scores can be extended to any kind of difference score between two measures (measuring the same construct or not) that are on the same numerical scale. The two measures are linked, as if the two scores are obtained from a single test or two observations are made by the same observer. The correlation between linked observations (e.g., two observations made by the same observer) will be higher than that between independent observations (e.g., two observations made by different observers). Such linkage must be considered in defining the reliability coefficient for difference scores. The reliability of change or difference scores is defined as the correlation of the scores with independently observed difference scores. The reliability for change scores produced by comparing two independent measures will most likely be smaller than that for the linked case.

Raw change or difference scores are computed with two observed measures (D = Y – X). Observed scores are systematically related to random error of measurement and thus unreliable. Conclusions based on these scores tend to be fallacious.

True change score is measured as the difference between the person's true status at posttest and pretest times, DT = YT − XT. The key is to remove the measurement error from the two observed measures. There are different ways to correct the errors in the two raw measures used to obtain raw gain scores. The first way is to correct the error in pretest scores using the reliability coefficient of X and simple regression. The second way is to correct errors in both pretest and posttest scores using the reliability coefficient of both X and Y and simple regression. The third way is the Lord procedure. With this procedure, the estimates of YT and XT are obtained by the use of a multiple regression procedure that incorporates the reliability of a measure (e.g., X) and information that can be borrowed from the other measure (e.g., Y). The estimator obtained with the Lord procedure is better than those from the previous two ways and the raw change score in that it yields a smaller mean square of deviation between the estimate and the true change

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