Editing Stimulus–response model (section)

== Bounded response functions ==

Since many types of response have inherent physical limitations (e.g. minimal maximal muscle contraction), it is often applicable to use a bounded function (such as the [[logistic function]]) to model the response. Similarly, a linear response function may be unrealistic as it would imply arbitrarily large responses. For binary dependent variables, statistical analysis with regression methods such as the [[probit model]] or [[logit model]], or other methods such as the Spearman–Kärber method.<ref name="HamiltonRusso1977">{{cite journal |last1= Hamilton |first1= MA |last2= Russo |first2= RC |last3= Thurston |first3= RV |title= Trimmed Spearman–Karber method for estimating median lethal concentrations in toxicity bioassays |journal= [[Environmental Science & Technology]] |volume= 11 |issue= 7 |year= 1977 |pages= 714–9 |doi= 10.1021/es60130a004|bibcode= 1977EnST...11..714H }}</ref> Empirical models based on nonlinear regression are usually preferred over the use of some transformation of the data that linearizes the stimulus-response relationship.<ref name="BatesWatts1988">{{cite book |last1= Bates |first1= Douglas M. |last2= Watts |first2= Donald G. |title= Nonlinear Regression Analysis and its Applications |year= 1988 |publisher= [[John Wiley & Sons|Wiley]] |isbn= 9780471816430 |page= 365}}</ref>

[[Logistic regression#Definition of the logistic function|One example]] of a logit model for the probability of a response to the real input (stimulus) <math>x</math>, (<math>x\in \mathbb R</math>) is

:<math>p(x) = \frac {1}{1+e^{-(\beta_0 + \beta_1 x)}}</math>

where <math>\beta_0, \beta_1</math> are the parameters of the function.

Conversely, a [[Probit model]] would be of the form

:<math>p(x) = \Phi(\beta_0 + \beta_1 x)</math>

where <math>\Phi(x)</math> is the [[Normal distribution#Cumulative distribution function|cumulative distribution function]] of the [[normal distribution]].

=== Hill equation ===

In [[biochemistry]] and [[pharmacology]], the [[Hill equation (biochemistry)|Hill equation]] refers to two closely related equations, one of which describes the response (the physiological output of the system, such as muscle contraction) to [[Drug]] or [[Toxin]], as a function of the drug's [[concentration]].<ref name = Terms>{{cite journal |title=International Union of Pharmacology Committee on Receptor Nomenclature and Drug Classification. XXXVIII. Update on Terms and Symbols in Quantitative Pharmacology|url=https://www.guidetopharmacology.org/pdfs/termsAndSymbols.pdf|last1=Neubig|first1=Richard R.|journal=Pharmacological Reviews|year=2003|volume=55|issue=4|pages=597–606|doi=10.1124/pr.55.4.4|pmid=14657418|s2cid=1729572}}</ref> The Hill equation is important in the construction of [[dose-response curves]]. The Hill equation is the following formula, where <math>E</math> is the magnitude of the response, <chem>[A]</chem> is the drug concentration (or equivalently, stimulus intensity), [[EC50|<math>\mathrm{EC}_{50}</math>]] is the drug concentration that produces a half-maximal response and <math>n</math> is the [[Hill coefficient]].

:[[File:Ivan Pavlov nobel.jpg|thumb|Ivan Pavlov]]<math>\frac{E}{E_{\mathrm{max}}}=\frac{1}{1+\left(\frac{\mathrm{EC}_{50}}{[A]}\right)^{n}}</math><ref name = Terms/>

The Hill equation rearranges to a logistic function with respect to the logarithm of the dose (similar to a logit model).