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# What is the relationship between the intercept and slope of scores at two time points?

The profiles pictured (in the powerpoint slide) here represent a 'typical' hypothetical scenario concerning 11 subject test scores. Each subject has baseline and retest scores. In this scenario the magnitude of the increase in score between baseline and retest (ie the difference between retest and baseline scores) is highly negatively related to the scores at both baseline (r = -0.90) and retest (r = -0.85) across subjects. The slope is also related to the average baseline-retest score (r = -0.88) which is equal to 0.5(baseline + retest) across subjects.

These negative correlations are seen visually in the above plot because the slopes in the bottom half of the plot between baseline and retest scores are greater than those in the upper half of the plot. The assumptions underlying this are that, firstly, those scoring lower at baseline will improve by a higher amount because they have more scope for improvement and, secondly, that it is more difficult to improve from a high, as opposed to low, previous score. This behaviour can typically result in asymptotic behaviour as the score plateaus out as each subject develops to the limit of their abilities.

Eventually over time the slower scoring subjects will either 'catch up' with, and even possibly subsequently pass, the initial high scorers scores or reach a level which is at a lower score than those attained by the initial high scorers. Over the early time points they will continue to lag behind, however. One characteristic of such behaviour is a decrease in the variance of the scores over time. In the above example this is the case with scores at retest having a smaller variance (variance = 2.45) than scores at baseline (variance = 3.43).

The reason that this behaviour is tenable is that the higher scoring subjects tend to be nearer the highest possible score (ceiling) and therefore need to continue to produce performances which demand an ever increasing level of excellence. This characteristic can be seen, for example, in sporting ranking tables where players who have started lower ranked at the start of a sports season can go up by a higher amount with a relatively small improvement in personal performance whereas those nearer the top of the table need to perform to a higher overall standard to show an improvement in performance and typically end up 'only' matching their previous performances with consistent form from season to season.

None: FAQ/t1t2 (last edited 2013-03-08 10:17:10 by localhost)