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

The profiles pictured [attachment:slopes.xls 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.

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

Eventually over time the slower scoring subjects will 'catch up' with the initial high scorers but over the initial time points they will continue to lag behind.

The reason for this is that the higher scoring subjects tend to be nearer the highest possible score (ceiling). 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.