= Solution
A <psychometric function> specifies the <probability> of a measured response as a physical stimulus variable changes. The response might be “present,” “clockwise,” or correct in a <two-alternative forced-choice task>. These are different dependent variables. \b[Understanding the function's shape is necessary to distinguish sensory changes from decision, <attention> and measurement effects.] A reported threshold without its task, response definition and performance criterion is incomplete. For example, a visual experiment could use calibrated display <visual contrasts>, timed two-interval presentations, fixation control and button choices; the observer’s choice supplies the dependent variable.
A smooth curve need not mean that a receptor has a smoothly varying conscious threshold. Consider a noisy internal response $R=\mu(x)+\varepsilon$, with “yes” reported when $R>c$. Here $c$ is a <decision criterion>, not necessarily a biological detection limit. If $F_\varepsilon$ is the noise <cumulative distribution function>,
$$
P(\text{yes}\mid x)=1-F_\varepsilon(c-\mu(x)).
$$
With <Gaussian noise> of <standard deviation> $\sigma$, symmetry of the <normal distribution> gives a <Gaussian psychometric function>
$$
P(\text{yes}\mid x)=\Phi\left(\frac{\mu(x)-c}{\sigma}\right).
$$
The sigmoid is therefore a distribution tail swept by a changing mean response. At low means, few responses exceed the criterion; near it, many trials switch category; at high means, almost all do. The derivative is $\phi((\mu-c)/\sigma)\mu'(x)/\sigma$, so slope depends on both sensory gain and variability. Moving $c$ shifts the curve without necessarily improving sensitivity. If noise vanished and the mean crossed the criterion sharply, this simple model would approach a step rather than a graded transition.
There is no universal sigmoid in every stimulus coordinate. A nonlinear transduction function changes the curve, and a logarithmic <intensity> axis changes its appearance. As a counterexample, a toy detector responding whenever at least one event occurs in a <Poisson distribution> with mean $kx$ has $P(\text{yes}\mid x)=1-e^{-kx}$. This is concave on a linear positive-intensity axis. It follows immediately by subtracting the zero-event <probability> from one. It is a mathematical example, not evidence that human detection universally uses a one-event rule. Variation in <attention> or <decision criterion> across trials can broaden a measured curve even if the within-state sensory noise is unchanged.
Task conventions determine the endpoints. A useful <psychometric function with guessing and lapses> is
$$
p(x)=\gamma+(1-\gamma-\lambda)\Phi\left(\frac{x-\alpha}{\beta}\right),\qquad \beta>0.
$$
Here $\gamma$ is the lower asymptote and $\lambda$ the upper error asymptote. In balanced <two-alternative forced-choice> performance, $\gamma$ is commonly fixed at $1/2$. A yes/no response <probability> does not automatically have this lower bound. In this particular cumulative-Gaussian parameterization, $p(\alpha)=\gamma+(1-\gamma-\lambda)/2$; thus $\alpha$ gives 75% correct only for $\gamma=1/2$ and $\lambda=0$. Other function families use different scale parameters. Nor is $\lambda$ automatically the fraction of inattentive trials: if a fraction $r$ of trials involves random two-choice guessing, those trials contribute an upper error <probability> $r/2$.
\Image[/past-exam-of-the-perception-course-of-the-year-2-of-the-psychology-experimental-course-of-the-university-of-oxford/2022-psychometric-mechanisms.png]
{title=Original theoretical noisy-decision and psychometric-curve examples; the right panel is a phenomenological family, not observed participant data}
The relation to <signal detection theory> is equally important. Under independent equal-variance Gaussian signal and noise samples, define <sensitivity index> $d'=(\mu_s-\mu_n)/\sigma$. In an ideal two-interval comparison, the response difference has mean $\mu_s-\mu_n$ and <variance> $2\sigma^2$, giving
$$
P(\text{correct})=P(R_s-R_n>0)=\Phi(d'/\sqrt2).
$$
This <Gaussian forced-choice sensitivity relation> follows by standardizing the Gaussian difference. It is a specified decision model, not an identity for every forced-choice experiment. In a yes/no task, hits and false alarms are both needed: under the corresponding equal-variance model, $d'=\Phi^{-1}(H)-\Phi^{-1}(F)$. A criterion change can increase both rates while leaving $d'$ unchanged. The phenomenological fit above need not have parameters identical to these internal-response parameters.
These distinctions affect experiments. <Felix A. Wichmann> and <N. Jeremy Hill> used simulation and statistical modeling to show that unmodeled stimulus-independent errors can bias threshold and slope estimates, even when lapses are uncommon.<The psychometric function: I. Fitting, sampling, and goodness of fit by Felix A. Wichmann and N. Jeremy Hill>[Notation, lapse simulations, and summary; Percept Psychophys 63, pp. 1293–1313]{ref} Ignoring high-intensity errors can make a function appear shallower and falsely suggest worse sensory precision. A broad group-average curve can likewise reflect a mixture of individually sharp functions with different locations.
At predetermined stimulus levels, independent trials give counts $k_i\sim\operatorname{Binomial}(n_i,p(x_i))$. The parameter-dependent <log-likelihood> is $\sum_i[k_i\log p_i+(n_i-k_i)\log(1-p_i)]$. Fitting this <binomial likelihood>, checking goodness of fit and quantifying uncertainty are preferable to drawing a curve through noisy proportions. Easy trials help constrain lapses; levels near the transition constrain location and slope. Randomization, stable calibration and checks for <learning> or fatigue help distinguish a stationary sensory relation from drift. Adaptive procedures require inference appropriate to their sampling design.
Thus the function is both a description and a testable model. \b[Its shape determines what a threshold means, how an experiment should sample, and which claims about sensory mechanisms are warranted.]
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