Response modeling methodology

Response Modeling Methodology (RMM) is a general platform for modeling monotone convex relationships. RMM had been initially developed as a series of extensions to the original inverse Box-Cox transformation:$y={{(1+\lambda z)}^{1/\lambda }},$   where y is a percentile of the modeled response, Y (the modeled random variable), z is the respective percentile of a normal variate and λ is the Box-Cox parameter. Note that as λ goes to zero, the inverse Box-Cox transformation becomes: $y={{e}^{z}},$ an exponential model. Therefore, the original inverse Box-Cox transformation contains a trio of models: linear (λ=1), power (λ≠1, λ≠0) and exponential (λ=0). This implies that on estimating λ, using sample data, the final model is not determined in advance (prior to estimation) but rather as a result of estimating. In other words, data alone determine the final model.  

Extensions to the inverse Box-Cox transformation were developed by Shore (2001a) and were denoted Inverse Normalizing Transformations (INTs). They had been applied to model monotone convex relationships in various engineering areas, mostly to model physical properties of chemical compounds (Shore et al., 2001a, and references therein). Once it had been realized that INT models may be perceived as special cases of a much broader general approach for modeling non-linear monotone convex relationships, the new Response Modeling Methodology had been initiated and developed (Shore, 2005a, 2011 and references therein).

The RMM model expresses the relationship between a response, Y (the modeled random variable), and two components that deliver variation to Y:

• The linear predictor, LP (denoted η):$\eta ={{\beta }_{0}}+{{\beta }_{1}}{{X}_{1}}+{\text{ }}..{\text{ }}+{{\beta }_{k}}{{X}_{k}},$ where {X_1,.. ,X_k} are regressor-variables (“affecting factors”) that deliver systematic variation to the response;

• Normal errors, delivering random variation to the response.

The basic RMM model describes Y in terms of the LP, two possibly correlated zero-mean normal errors, ε₁ and ε₂ (with correlation ρ and standard deviations σ_ε1 and  σ_ε2, respectively) and a vector of parameters {α,λ, μ} (Shore, 2005a, 2011):

$W=\log(Y)=\mu +\left({\frac {\alpha }{\lambda }}\right)[{{(\eta +{{\varepsilon }_{1}})}^{\lambda }}-1]+{{\varepsilon }_{2}}.\,$                                                      

Note that ε₁ represents uncertainty (measurement imprecision or otherwise) in the explanatory variables (included in the LP). This is in addition to uncertainty associated with the response (ε₂). Expressing ε₁ and ε₂ in terms of standard normal variates, Z₁ and Z₂, respectively, having correlation ρ, and conditioning Z₂ | Z₁=z1 (Z₂ given that Z₁ is equal to a given value z₁), we may write in terms of a single error, ε:

 ${\begin{aligned}&{{\varepsilon }_{1}}={{\sigma }_{{\varepsilon }_{1}}}{{Z}_{1}}\,\,;\,\,{{\varepsilon }_{2}}={{\sigma }_{{\varepsilon }_{2}}}{{Z}_{2}}\,\,;\\&{{\varepsilon }_{2}}={{\sigma }_{{\varepsilon }_{2}}}\rho {{z}_{1}}+{{(1-{{\rho }^{2}})}^{(1/2)}}{{\sigma }_{{\varepsilon }_{2}}}Z\,\,=d{{z}_{1}}+\varepsilon \,,\\\end{aligned}}$

where Z is a standard normal variate, independent of both Z₁ and Z₂, ε is a zero-mean error and d is a parameter. From these relationships, the associated RMM quantile function is (Shore, 2011):

$w=\log(y)=\log({{M}_{Y}})+\left({\frac {a}{\lambda }}\right)[{{(\eta +cz)}^{\lambda }}-1]+(d)z+\varepsilon ,$

or, after re-parameterization:

$w=\log(y)=\log({{M}_{Y}})+\left({\frac {a{{\eta }^{b}}}{b}}\right)\left\{{{\left[1+\left({\frac {c}{\eta }}\right)z\right]}^{b}}-1\right\}+(d)z+\varepsilon ,$                                                

where y is the percentile of the response (Y), z is the respective standard normal percentile, ε is the model’s zero-mean normal error with constant variance, σ, {a,b,c,d} are parameters and M_Y is the response median (z=0), dependent on values of the parameters and the value of the LP, η:

  $\log({{M}_{Y}})=\mu +\left({\frac {a}{b}}\right)[{{\eta }^{b}}-1]=\log(m)+\left({\frac {a}{b}}\right)[{{\eta }^{b}}-1],$                             

where μ (or m) is an additional parameter.

If it may be assumed that cz<<η, the above model for RMM quantile function can be approximated by:

$w=\log(y)=\log({{M}_{Y}})+\left({\frac {a{{\eta }^{b}}}{b}}\right)[\exp({\frac {bcz}{\eta }})-1]+(d)z+\varepsilon .$

Note that parameter “c” cannot be “absorbed” into the parameters of the LP (η) since “c” and LP are estimated in two separate stages (as expounded below).

If the response data used to estimate the model contain values that change sign, or if the lowest response value is far from zero (for example, when data are left-truncated), a location parameter, L, may be added to the response so that the expressions for the quantile function and for the median become, respectively:

$w=\log(y-L)=\log({{M}_{Y}}-L)+\left({\frac {a{{\eta }^{b}}}{b}}\right)\left\{{{\left[1+\left({\frac {c}{\eta }}\right)z\right]}^{b}}-1\right\}+(d)z+\varepsilon \,;$

$\log({{M}_{Y}}-L)=\mu +\left({\frac {a}{b}}\right)[{{\eta }^{b}}-1].$

Major Property of RMM – Continuous Monotone Convexity (CMC)

As shown earlier, the inverse Box-Cox transformation depends on a single parameter, l, which determines the final form of the model (whether linear, power or exponential). All three models thus constitute mere points on a continuous spectrum of monotone convexity, spanned by l. This property, where different known models become mere points on a continuous spectrum, spanned by the model’s parameters, is denoted the Continuous Monotone Convexity (CMC) property. The latter characterizes all RMM models, and it allows the basic “linear-power-exponential” cycle (underlying the inverse Box-Cox transformation) to be repeated ad infinitum, allowing for ever more convex models to be derived. Examples for such models are an exponential-power model or an exponential-exponential-power model (see explicit models expounded further on). Since the final form of the model is determined by the values of RMM parameters, this implies that the data, used to estimate the parameters, determine the final form of the estimated RMM model (as with the Box-Cox inverse transformation). The  CMC property thus grant RMM models high flexibility in accommodating the data used to estimate the parameters. References given below display published results of comparisons between RMM models and existing models. These comparisons demonstrate the effectiveness of the CMC property.