Corrected Treatment Effect – Nonlinear association equations.

 

An explanation of nonlinear equations is available in the following link.

Exponential Equations

Nonlinear Equations

The following text lists the corrected treatment effect for each nonlinear equation type.

Logistic, Exponential Growth, Exponential Decay (Increasing Form), Exponential Decay (Decreasing Form) all have a corrected treatment effect of a_z = x-bar/b_z. These equations have a multiplicative treatment effect.

The logarithmic equation is also multiplicative but due to the nature of the equation, the corrected treatment effect is:

\displaystyle a_z=\frac{\ln{\bar{x}}}{\ln{b_z}}

Regarding the polynomial equations for brevity, the quadratic equation is discussed. Here the corrected treatment effect is multiplied to the leading term which the x^2 term. Here the corrected treatment effect is:

\displaystyle a_z=\dfrac{\bar{x}^2+\dfrac{m_{x2}}{m_{x1}}(x-b_z)}{(b_z)^2}

The Gaussian equation has an additive treatment effect as the corrected treatment effect shifts the centre of the equation. Here the corrected treatment effect is a_z = x-bar + b_z

It is left to the reader to derive the above results as the derivation can be an exhilirating exercise.