Psychometric function

A function relating a stimulus variable to measured response probability or discrimination performance.

An empirical psychometric function whose core $F$ ranges from zero to one, lower asymptote is $\gamma$, and upper asymptote is $1-\lambda$. If $F(\alpha )=1/2$, then $p(\alpha )=\gamma +(1-\gamma -\lambda )/2$. A location parameter is consequently not a task-independent threshold. For a two-choice observer randomly guessing on fraction $r$ of otherwise perfect easy trials, the upper error asymptote is $r/2$, not $r$.

Let internal evidence be $R=\mu (x)+\epsilon$ with zero-mean Gaussian noise of standard deviation $\sigma$, and respond yes when $R>c$. Taking a distribution tail gives $1-\Phi ((c-\mu )/\sigma )$, which equals the displayed expression by symmetry. Its derivative is $\varphi ((\mu -c)/\sigma ){\mu }^{\prime }(x)/\sigma$. Thus sensory gain, variability and criterion all influence the observed relation, and a sigmoid in one stimulus coordinate need not remain symmetric in another.

A stimulus value corresponding to a specified performance criterion in a specified task. It is a summary of a psychometric function, not a claim that every trial below that value has no sensory response. State the response definition, chance level, lapse convention and performance criterion.