I have been using your package to estimate germination curves and after comparing it with other dose-response packages I have realised there is a crucial difference in the formulation proposed by El-Kassby et al. (2008) and those used in other dose-response estimations. The numerator of the equation omits a crucial $-y_0$ component that is found in the classical Hill equation and its derivations. The current formula:
$y = y_0 + \frac{ax^b}{x^b + c^b}$
Should instead be:
$y = y_0 + \frac{(a-y0)x^b}{x^b + c^b}$
Without this $-y_0$, when parameter $a$ is estimated to be e.g. 95, the curve asymptote will only be placed at $ y = 95$ if and only if $y_0 = 0$. In fact, the asymptote value will always be $a + y_0$ in your current formulation, the same given by El-Kassby et al. (2008). As a result, when your package is used with fix.y0 = FALSE, if $y_0$ is estimated to be a small negative value (I see this often in my data), the estimated a parameter and derived values are actually shifted.
As an example I have seen y0 = -2 and a = 100 in one of my tests and in general I would expect this to mean that this sample has 100% germination rate at the end of the experiment. However, these parameter values, with your current formula, mean that the asymptote (the Gmax value estimated by the curve) actually sits at 98%, not 100%, leading to an overestimation of the germination. As I said, this is only impactful when y0 is estimated to be anything else than 0.
Hopefully I have interpreted your code and formulas correctly, if you think otherwise I'll be more than happy if you could share your interpretation.
I have been using your package to estimate germination curves and after comparing it with other dose-response packages I have realised there is a crucial difference in the formulation proposed by El-Kassby et al. (2008) and those used in other dose-response estimations. The numerator of the equation omits a crucial$-y_0$ component that is found in the classical Hill equation and its derivations. The current formula:
Should instead be:
Without this$-y_0$ , when parameter $a$ is estimated to be e.g. 95, the curve asymptote will only be placed at $ y = 95$ if and only if $y_0 = 0$ . In fact, the asymptote value will always be $a + y_0$ in your current formulation, the same given by El-Kassby et al. (2008). As a result, when your package is used with $y_0$ is estimated to be a small negative value (I see this often in my data), the estimated
fix.y0 = FALSE, ifaparameter and derived values are actually shifted.As an example I have seen
y0 = -2anda = 100in one of my tests and in general I would expect this to mean that this sample has 100% germination rate at the end of the experiment. However, these parameter values, with your current formula, mean that the asymptote (the Gmax value estimated by the curve) actually sits at 98%, not 100%, leading to an overestimation of the germination. As I said, this is only impactful wheny0is estimated to be anything else than 0.Hopefully I have interpreted your code and formulas correctly, if you think otherwise I'll be more than happy if you could share your interpretation.