Summary measures for learning curves

Suppose we wish to look at change in processing success (y) with ease of task (x).

We fit a cubic polynomial and obtain using multiple regression a best fitting cubic of form -0.000005x^{3 + 0.0003x}2 + 0.0026x + 0.7508. This is the curve in red in this [attachment:plot.jpeg plot.]

The asymptote occurs at a turning point when its first derivative (a quadratic) equals zero. For the curve in red this quadratic, obtained from differentiating the above cubic, is fo form -0.000015x^2 + 0.0006x - 0.0026 =0. This quadratic has a turning point in the sampled interval [10,60] at 43.94 obtained by inputting the regression coefficients into the [http://www.mathsisfun.com/quadratic-equation-solver.html: quadratic root solver here.]

The predicted learning slope, from baseline to asymptote, is obtained by subtracting the predicted response, obtained using the regression coefficients in the cubic, at the start (x=10) from that at the asymptote(x=43.94) of the learning line and dividing by the change in x (43.94-10),

So for the red curve the change from baseline to asymptote = (1.02 – 0.80)/(43.94-10)=0.0064.

Note that Howell (1995) shows that a quadratic curve with a linear component is sufficient to yield an asymptotic model. In this instance one would obtain the asymptote by differentiating the quadratic to get a linear equation with the asymptote equal to its root.

Similarly if we anticipate having three turning points we can fit predictors of form x^{4, x}3, x^2, obtain regression coefficients and differentiate the quartic to obtain a cubic curve. The roots of this cubic (obtained using a cubic root solver calculator such as [http://easycalculation.com/algebra/cubic-equation.php here]) will give the location of the three turning points.

Reference

Howell DC (1997). Statistical methods for psychology. Fourth Edition. Wadsworth, Belmont, CA.