text: For a mathematical definition of a dilution assay an observation space  is defined and a function  so that the responses    are mapped to the set of real numbers. It is now assumed that a function  exists which relates the dose  to the   response
  in which  is an error term with expectation 0.  is usually assumed to be continuous and monotone. In situations where a standard preparation is included it is furthermore assumed that the test preparation    behaves like a dilution (or concentration) of the standard 
  :, for all 
  where  is the relative potency of . This is the fundamental assumption of similarity of dose-response curves which is necessary for a meaningful   and unambiguous definition of the relative potency. In many cases it is convenient to apply a power transformation  with  or a   logarithmic transformation . The latter can be shown to be a limit case of  so if  is written for the   log transformation the above equation can be redefined as
  :, for all .
  Estimates  of  are usually restricted to be member of a well-defined parametric family of functions, for example the family of linear functions characterized by an intercept and a slope. Statistical techniques such as optimization by Maximum Likelihood can be used to calculate estimates of the parameters. Of notable importance in this respect is the theory of Generalized Linear Models with which a wide range of dilution assays can be modelled. Estimates of  may describe  satisfactorily over the range of doses tested, but they do not necessarily have to describe  beyond that range. However, this does not mean that dissimilar curves can be restricted to an interval where they happen to be similar.
  In practice,  itself is rarely of interest. More of interest is an estimate of  or an estimate of the dose that induces a specific response. These   estimates involve taking ratios of statistically dependent parameter estimates. Fieller's theorem can be used to compute confidence intervals of these ratios.
  Some special cases deserve particular mention because of their widespread use: If  is linear and  this is known as a slope-ratio model. If    is linear and  this is known as a parallel line model. Another commonly applied model is the probit model where  is   the cumulative normal distribution function,  and  follows a binomial distribution.