Tag: normalizing transformation

This post explains the central role of Continuous Monotone Convexity (CMC) in Response Modeling Methodology (RMM).

In earlier blog entries, the unique effectiveness of the Box-Cox transformation (BCT) was addressed. I concluded that the BCT effectiveness could probably be attributed to the Continuous Monotone Convexity (CMC) property, unique to the inverse BCT (IBCT). Rather than requiring the analyst to specify a model in advance (prior to analysis), the CMC property allows the data, via parameter estimation, determine the final form of the model (linear, power or exponential). This would most likely lead to better fit of the—estimated model, as cumulative reported experience with implementation of IBCT (or BCT) clearly attest to.

In the most recent blog entry in this series, I have introduced the “Ladder of Monotone Convex Functions”, and have demonstrated that IBCT delivers only the first three “steps” of the Ladder. Furthermore, IBCT can be extended so that a single general model can represent all monotone convex functions belonging to the Ladder. This transforms monotone convexity into a continuous spectrum so that the discrete “steps” of the Ladder (the separate models) become mere points on that spectrum.

In this third entry on the subject (and Article #3, linked below), I introduce in a more comprehensive fashion (yet minimally technical) the general model from which all the Ladder functions can be derived as special cases. This model was initially conceived in the last years of the previous century (Shore, 2005, and references therein) and had since been developed into a comprehensive modeling approach, denoted Response Modeling Methodology (RMM). In the affiliated article, an axiomatic derivation of RMM basic model is outlined and specific adaptations of RMM to model systematic variation and to model random variation are addressed. Published evidence for the capability of RMM to replace current published models, previously derived within various scientific and engineering disciplines as either theoretical, empirical or semi-empirical models, is reviewed. Disciplines surveyed include chemical engineering, software quality engineering, process capability analysis, ecology and ultra-sound-based fetal-growth modeling (based on cross-sectional data).

This blog entry (with the linked article given below) was originally posted on the site of the American Statistical Association (ASA), where the linked article was visible to members only.

Haim Shore_3_ASA_Jan 2014

In a previous post in this series, I have discussed reasons for the effectiveness of the Box-Cox (BC) transformation, particularly when applied to a response variable within linear regression analysis. The final conclusion was that this effectiveness could probably be attributed to the “Continuous Monotone Convexity (CMC)” property, owned by the inverse BC transformation. It was emphasized that the latter, comprising the three most fundamental monotone convex functions, the “linear-power-exponential” trio, delivers only partial representation to a whole host of models of monotone convex relationships, which can be arranged in a hierarchy of monotone convexity. This hierarchy had been denoted the “Ladder of Monotone Convex Functions.”

In this post (and Article #2, linked below), I address in more detail the nature of the CMC property. I specify models included in the Ladder, and show how one can deliver, via a single model, representation to all models belonging to the Ladder (analogously with the inverse BC transformation, a special case of that model). Furthermore, I point to published evidence demonstrating that models of the Ladder may often substitute, with negligible loss in accuracy, published models of monotone convexity, which had been derived from theoretical discipline-specific considerations.

This blog entry (with the linked article given below) was originally posted on the site of the American Statistical Association (ASA), where the linked article was visible to members only.

Haim Shore_2_ASA_Dec 2013