s_{model}(s_1,b_1,\ldots, s_n,b_n),\label{eq:smodel}\\
b_{model} &= b_{model}(s_1,b_1,\ldots, s_n,b_n).\label{eq:bmodel}\end{aligned}$$ Each model is described by the shared decision function $$\begin{aligned}
P_{model} &= f\left[P_1,\ldots,P_n \right]\label{eq:pmodel},\end{aligned}$$ where $f$ is a functional. For all but two models that we investigate, $P_{model}(c) = f\left[P_1(c),\ldots,P_n(c) \right]$, that is, the dependence is pointwise (i.e., result for a given $c$ requires only knowing individual $P_i(c)$ for the same $c$).

We can obtain the effective slope and bias using straightforward formulas that involve taking the derivative of the psychometric function with respect to the stimulus: $$\begin{aligned}
s_{model} &=\left.P_{model}'(c)\right|_{c=-b_{model}} \approx \left.P_{model}'(c)\right|_{c=0} \label{eq:slopediff}\\
b_{model} & = \left[ b \mbox{ for which } P_{model}(-b)=\frac{1}{2} \right] \approx \frac{P_{model}(0)-\frac{1}{2}}{\left.P_{model}'(c)\right|_{c=0}},\label{eq:biasdiff}\end{aligned}$$ where assuming the approximation for the relative error for both $s$ and $b$ is of order $O(s^2 b^2)$ (or equivalently, $O(\frac{b^2}{\sigma^2})$), where $O(\cdot)$ stands for big $O$ notation. The derivation is in \[s:approx\]. Note that if $P_{model}(c)$ is a cumulative Gaussian function (as in ), then the formulas for slope and are equivalent. However, it can be used as a [*definition*]{} of the slope and the bias in the general case of an arbitrary communication strategy $P_{model}(c)$, even if does not hold. Bear in mind that for practical applications, we expect $P_{model}(c)$ to be close enough to the 