Improvement score

What is positive and negative improvement?

An improvement score, IS, requires measurements at baseline, B, and after treatment, T, of a trait that increases as a positive measure of improvement. The difference T-B distinguishes three conditions:

1. T-B = 0. There is no improvement, and IS = 0.
2. T-B > 0. There is positive improvement, and IS > 0.
3. T-B < 0. There is negative improvement, and IS < 0.

When does improvement become more difficult to achieve?

The same absolute improvement, T-B, is more easily achieved when the trait has a low value at baseline (B is low). As B approaches the theoretical maximum, B_max = T_max, the scope for absolute improvement diminishes, and the same small absolute improvement becomes more difficult to achieve.

How does the improvement score behave?

1. If T=B, T-B=0, then IS=0
2. If T>B, T-B is positive; there is a positive IS.
3. If T<B, T-B is negative; there is a negative IS.
4. A positive T-B value is more difficult to achieve, the higher the initial value of B. Therefore, the same value of T-B should have a higher IS at higher values of B and T.
5. The relationship between T-B and IS should have the same proportionality for positive and negative T-B values, i.e., if the highest IS=1, then the worst is -1.
6. Positive and negative values of T-B are treated by different equations.

These criteria are met by two equations, and none of the equations is applied to T=B, when IS=0:

Positive improvement, T>B

IS = (T*T-B)/(T+B)

Negative improvement, T<B

IS = (T-B*B)/(T+B)