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Introduction
Is Statistics, a branch of mathematics that serves central tool to investigate nature, heading in the right direction, comparable to other branches of science that explore nature?
I believe it is not.
This belief is based on my own personal experience in a recent research project, aimed to model surgery time (separately for different subcategories of surgeries). This research effort culminated in a trilogy of published articles (Shore 2020ab, 2021). The belief is also based on my life-long experience in academia. I am professor emeritus, after forty years in academia and scores of articles published in refereed professional journals, dealing both with the theory and application of Statistics. In this post, I deliver an account of my recent personal experience with modeling surgery time, and conclusions I have derived thereof, and from my own cumulative experience in the analysis of data and in data-based modeling.
The post is minimally technical, so that a layperson, with little knowledge of basic terms in Statistics, can easily understand.
We define a random phenomenon as one associated with uncertainty, for example, “Surgery”. A random variable (r.v) is any random quantitative property defined on a random phenomenon. Examples are surgery medical outcome (Success: X=1; Failure: X=0), surgery duration (X>0) or patient’s maximum blood pressure during surgery (Max. X).
In practice, an r.v is characterized by its statistical distribution. The latter delivers the probability, P (0≤P≤1), that the random variable, X, assumes a certain value (if X is discrete), or that it will fall in a specified interval (if X is continuous). For example, the probability that surgery outcome will be a success, Pr(X=1), or the probability of surgery duration (SD) to exceed one hour, Pr(X>1).
Numerous statistical distributions have been developed over the centuries, starting with Bernoulli (1713), who derived what is now known as the binomial distribution, and Gauss (1809), deriving the “astronomer’s curve of error”, nowadays known as Gauss distribution, or the normal distribution. Good accounts of the historical development of the science of probability and statistics to its present-day appear at Britannica and Wikipedia, entry History_of_statistics.
A central part of these descriptions is, naturally, the development of the concept of statistical distribution. At first, the main source of motivation was games of chance. This later transformed into the study of errors, as we may learn from the development of the normal distribution by Gauss. In more recent years, emphasis shifted to describing random variation as observed in all disciplines of science and technology, resulting, to date, in thousands of new distributions. The scope of this ongoing research effort may be appreciated by the sheer volume of the four-volume Compendium on Statistical Distributions by Johnson and Kotz (First Edition 1969–1972, updated periodically with Balakrishnan as an additional co-author).
The development of thousands of statistical distributions over the years, up to the present, is puzzling, if not bizarre. An innocent observer may wonder, how is it that in most other branches of science, the historical development shows a clear trend towards convergence, while in modeling random variation, the most basic concept to describe processes of nature, the opposite has happened, namely, divergence?
Put in more basic terms: Why in science, in general, a continuous attempt is exercised to unify, under an umbrella of a unifying theory, the “objects of enquiry” (forces in physics; properties of materials in chemistry; human characteristics in biology), while in the mathematical modelling of random variation, this has not happened? Why in Statistics, the number of “objects of enquiry”, instead of diminishing, keeps growing?
And more succinctly: Where did Statistics go wrong? And why?
I have already had the opportunity to address this issue (the miserable state-of-the-art of modelling random variation) some years ago, when I wrote (Shore, 2015):
“ “All science is either physics or stamp collecting”. This assertion, ascribed to physicist Ernest Rutherford (the discoverer of the proton, in 1919) and quoted in Kaku (1994, p. 131), intended to convey a general sentiment that the drive to converge the five fundamental forces of nature into a unifying theory, nowadays a central theme of modern physics, represents science at its best. Furthermore, this is the right approach to the scientific investigation of nature. By contrast, at least until recently, most other scientific disciplines have engaged in taxonomy (“bug collecting” or “stamp collecting”). With “stamp collecting” the scientific inquiry is restricted to the discovery and classification of the “objects of enquiry” particular to that science, however this never culminates, as in physics, in a unifying theory, from which all these objects may be deductively derived as “special cases”. Is statistics a science in a state of “stamp collecting”?”
This question remains valid today, eight years later: Why has the science of Statistics, as a central tool to describe statistically stable random phenomena of nature, has deviated so fundamentally from the general trend at unification?
In Section 2, we enumerate the errors that, we believe, triggered this departure of Statistics from the general trend in the scientific study of nature, and outline possible outlets to eliminate these errors. Section 3 is an account of the personal learning experience that I have gone through while attempting to model surgery duration and its distribution. This article is a personal account, for the naive (non-statistician) reader, of that experience. As alluded to earlier, the research effort resulted in a trilogy of articles , and in the new “Random identity paradigm”. The latter is addressed in Section 4, where new concepts, heretofore ignored by Statistics, are introduced (based on Shore, 2022). Examples are “Random identity”, “identity variation”, “losing identity” (with characterization of the process), and “Identity-full/identity-less distributions”. These concepts are underlying a new methodology to model observed variation in natural processes (as contrasted with variation of r.v.s that are mathematical function of other r.v.s).The new methodology is outlined, based on Shore, 2022. Section 5 delivers some final thoughts and conclusions.
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The historical errors embedded in current-day Statistics
Studying the history of the development of statistical distributions to date, we believe Statistics departure from the general trend, resulting in a gigantic number of “objects of enquiry” (as alluded to earlier), may be traced to three fundamental, inter-related, errors, historically committed within Statistics:
Error 1: Failure to distinguish between two categories of statistical distributions:
Category A: Distributions that describe observed random variation of natural processes;
Category B: Distributions that describe behavior of statistics, namely, of random variables that are, by definition, mathematical functions of other random variables.
The difference between the two categories is simple: Category A succumbs to certain constraints on the shape of distribution, imposed by nature, which Category B does not (the latter succumbs to other constraints, imposed by the structure of the mathematical function, describing the r.v). As we shall soon realize, a major distinction between the two sets of constraints (not the only one) is the permissible values for skewness and kurtosis. While for Category A, these fluctuate in a specified interval, confined between values of an identity-full distribution and an identity-less distribution (like the normal and the exponential, respectively; both types of distribution shall be explained soon), for Category B such constraints do not hold.
Error 2: Ignoring the real nature of error:
A necessary condition for the existence of an error, indeed a basic assumption integrated implicitly into its classic definition, is that for any observed random phenomenon, and the allied r.v, there is a typical constant, an outcome of various factors inherent to the process/system (“internal factors”), and there is error (multiplicative or additive), generated by random factors external to the system/process (“external factors”). This perception of error allows its distribution to be represented by the normal, since the latter is the only one having mean/mode (supposedly determined by “internal factors”) disconnected from the standard deviation, STD (supposedly determined by a separate set of factors, “external factors”).
A good representative of the constant, relative to which error is defined, is the raw mode or the standardized mode (raw mode divided by the STD). As perceived today, the error indeed expresses random deviations from this characteristic value (the most frequently observed value).
What happens to the error, when the mode itself ceases to be constant and becomes random? How does this affect the observed random variation or, more specifically, how is error then defined and modelled?
Statistics does not provide an answer to this quandary, except for stating that varying “internal factors”, namely, non-constant system/process factors, may produce systematic variation, and the latter may be captured and integrated into a model for variation, for example, via regression models (linear regression, nonlinear regression, generalized linear models and the like). In this case, the model ceases to represent purely random variation (as univariate statistical distributions are supposed to do). It becomes a model for systematic variation, coupled with a component of random variation (the nature of the latter may be studied by “freezing” “internal factors” at specified values). It is generally assumed in such models that a single distribution represents the component of random variation, though possibly with different parameters’ values for different values of the systematic effects, integrated into the model. Thus, implementing generalized linear models, the user is requested to specify a single distribution (not several), valid for all different sets of the effects’ values. As we shall soon learn (Error 3), “internal factors” may produce not only systematic effects, as currently wrongly assumed, but also a different component of variation, unrecognized to date. It will be addressed next as the third error.
Error 3: Failure to recognize the existence of a third type of variation (additional to random and systematic) — “Identity variation”:
System/process factors may potentially produce not only systematic variation, as currently commonly assumed, but also a third component of variation, passed under the radar, so to speak, in the science of Statistics. Ignoring this type of variation is the third historic error of Statistics. For reasons to be described soon (Sections 3 and 4), we denote this unrecognized type of variation — “Identity variation”.
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Modeling surgery duration — Personal learning experience that resulted in the new “Random identity paradigm”
I have not realized the enormity of the consequences of the above three errors, committed within Statistics to date, until a few years ago, when I have embarked on a comprehensive research effort to model the statistical distribution of surgery duration (SD), separately for each of over a hundred medically-specified subcategories of surgeries (the latter defined according to a universally accepted standard; find details in Shore 2020a). The subject (modeling SD distribution) was not new to me. I had been engaged in a similar effort years ago, in the eighties of the previous century (Shore, 1986). Then, based on analysis of available data and given the computing facilities available at the time, I divided all surgeries (except open-heart surgeries and neurosurgeries), into two broad groups: short surgeries, which were assumed to be normally distributed, and long surgeries, assumed to be exponential. There, for the first time, I have become aware of “Identity variation”, though not so defined, which resulted in modeling SD distribution differently for short surgeries (assumed to pursue a normal distribution) and long ones (assumed to be exponential). With modern available computing means, and with my own cumulative experience since publication of that paper (Shore, 1986), I thought, and felt, that a better model may be conceived, and embarked on the new project.
Probing into the available data (about ten thousand surgery times with affiliated surgery subcategories), four insights/observations were apparent:
1. It was obvious to me that different subcategories pursue different statistical distributions, beyond just differences in values of distribution’s parameters (as currently generally assumed in modeling SD distribution);
2. Given point (1), it was obvious to me that differences in distribution between subcategories should be attributed to differences in the characteristic level of work-content instability (work-content variation between surgeries within subcategory);
3. Given points (1) and (2), it was obvious to me that this instability cannot be attributed to systematic variation. Indeed, it represents a different type of variation, “identity variation”, to-date unrecognized in the Statistics literature (as alluded to earlier);
4. Given points (1) to (3), it was obvious to me that any general model of surgery time (SD) should include the normal and the exponential as exact special cases.
For the naive reader, I will explain the new concept, “identity variation”. Understanding this concept will render all of the above insights clearer.
As an industrial engineer in profession, it was obvious to me, right from the beginning of the research project, that, ignoring negligible systematic effects caused by covariates (like the surgeon performing the operation), a model for SD, representing only random variation in its classical sense, would not be adequate to deliver proper representation to the observed variation. Changes between subcategories in the type of distribution, as revealed by changes in distribution shape (from the symmetric shape of the normal to the extremely non-symmetric of the exponential, as first noticed by me in the earlier project, Shore, 1986), these changes have made it abundantly clear that the desired SD model should account for “identity loss”, occurring as we move from a repetitive process (subcategory with repetitive surgeries, having characteristic/constant work-content) to a memory-less non-repetitive process (subcategory with surgeries having no characteristic common work-content). As such, the SD model should include, as exact special cases, the exponential and the normal distributions.
What else do we know of the process of losing identity, as we move from the normal to the exponential, which account for “identity variation”?
In fact, several changes in distribution properties accompany “identity loss”. We relate again to surgeries. As work processes in general, surgeries too may be divided into three non-overlapping and exhaustive set of groups: repetitive, semi-repetitive and non-repetitive. In terms of work-content, this implies:
- Work-processes with constant work-content (only error generates variation; SD normally distributed);
- Semi-repetitive work-processes (work-content varies somewhat between surgeries, to a degree dependent on subcategory);
- Memory-less work-processes (no characteristic work-content; For example, surgeries performed within an emergency room for all types of emergency, or service performed in a pharmacy, serving customers with varying number of items on the prescription list).
Thus, work-content, however it is defined (find an example in Shore, 2020a), forms “surgery identity”, with a characteristic value, the mode, that vanishes (becomes zero) for the exponential scenario (non-repetitive work-process).
Let us delve somewhat deeper into the claim that a model for SD should include the normal and the exponential as exact special cases (not merely asymptotically, as, for example, the gamma tends to the normal).
There are four observations/properties, which put the two distributions, the identity-full normal and the identity-less exponential, apart from other distributions:
Observation 1: The mean and standard deviation are represented by different parameters for the normal distribution, and by a single parameter for the exponential. This difference is reflection of a reality, where, in the normal scenario, a set of process/system factors (“internal factors”) produces signal only, and a separate set (“external factors”) produces noise only (traditionally modelled as a zero-mean symmetrically distributed error). Moving away from the normal scenario to the exponential scenario, we witness a transition towards merging of the mean with the standard deviation, until, in the exponential scenario, both signal and noise are produced by the same set of factors — the mean and standard deviation merge to be expressed by a single parameter. The clear distinction, between “system/process factors” and “external/error factors”, typical to the normal scenario, this distinction has utterly vanished;
Observation 2: The mode, supposedly representing the typical constant on which the classical multiplicative error is defined in the normal scenario, this mode, or rather the standardized mode, shrinks, as we move away from the normal to the exponential. This movement, in reality, represents passing through semi-repetitive work-processes, with increasing degree of work-content instability. The standardized mode finally disappears (becoming zero) in the exponential scenario. What does this signify? What are the implications?
Observation 3: For both the normal and the exponential, skewness and kurtosis are non-parametric. Why is that, and what does this signify?
Observation 4: What happens to the classic error, when the r.v moves away from the normal scenario to the exponential? Can we still hold on to the classic definition of error, given that “internal factors”, assumed to generate a constant mode (signal), these factors start to produce noise? How would then error (in its classical sense) be re-specified? Can an error be defined at all?
All these considerations, as well as the need to include semi-repetitive surgeries within the desired model, brought me to the realization that we encounter here a different type of variation, heretofore unrecognized and not addressed in the literature. The instability of work-content (within subcategory), which I have traced to be the culprit for change in distribution as we move from one subcategory to another, could not possibly be regarded as cause for systematic variation. The latter is never assumed to change the shape of distribution, only at most its first two moments (mean and variance). This is evident, for example, on implementing generalized linear models, a regression methodology frequently used to model systematic variation in a non-normal environment. The user is requested to specify a single distribution (normal or otherwise), never different distributions for different sets of values of the effects being modeled (supposed delivering systematic variation). Neither can work-content variation be considered part of the classic random variation (as realized in Category A distributions) since the latter assumes existence of a single non-zero mode (for a single non-mixture univariate distribution), not zero mode or multiple modes (as, for example, with the identity-less exponential (zero mode), its allied Poisson distribution (two modes for an integer parameter), or the identity-less uniform (infinite number of modes); find details in Shore, 2022).
A new paradigm was born out of these deliberations — the “Random identity paradigm”. Under the new paradigm, observed non-systematic variation is assumed to originate in two components of variation: random variation, represented by a multiplicative normal/lognormal error, and identity variation, represented by an extended exponential distribution. A detailed technical development of this methodology, allied conjectures and their empirical support (from known theory-based results) are given in Shore (2022; A link to a pre-print is given at the References section). In the next Section 4 we deliver an outline of the “Random identity paradigm”.
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The “Random identity paradigm” — “Random identity”, “Identity variation”, “identity loss”, “identity-full/identity-less distributions” (based on Shore, 2022)
The insights, detailed earlier, have led to the development of the new “Random identity paradigm”, and its allied explanatory two-variate model for SD (Shore, 2020a). The model was designed to fulfill an a-priori specified set of requirements. Central among these is that the model includes the normal and the exponential distributions as exact special cases. After implementing the new model for various applications (as alluded to earlier), we have arrived at the realization that the model used in the article may, in fact, be expanded to introduce a new type of random variation, “random identity variation”, which served the basis for the new “Random Identity Paradigm” (Shore, 2022).
A major outcome of the new paradigm is the definition of two new types of distributions, an identity-full distribution and an identity-less distribution, and a criterion to diagnose a given distribution as “identity-full”, “identity-less”, or in between. Properties of identity-less and identity-full distributions are described, in particular, the property that such distributions have non-parametric skewness and kurtosis, namely, both types of distribution assume constant values, irrespective of values assumed by distribution parameters. Another requirement, naturally, is that the desired model includes a component of “identity variation”. However, the requirement also specifies that the allied distribution (representing “identity variation”) have support with the mode, if it exists, as its extreme left point (detailed explanation is given in Shore, 2022). As shown in Shore (2020ab, 2021, 2022), this resulted in defining the exponential distribution anew (the extended exponential distribution), adding a parameter, α, that assumes a value of α=0, for the exponential scenario (error STD becomes zero), and a value tending to 1, as the model moves towards normality (with “identity variation”, expressed in the extended exponential by parameter σi, tending to zero).
Sparing the naive reader the technical details of the complete picture, conveyed by the new “Random identity paradigm” (Shore, 2022), we outline herewith the associated model, as used in the trilogy of published paper.
The basic model is given in eq. (1):
Haim Shore_Equations_The Problem with Statistics_January 26 2022
where R is the observed response (an r.v), L and S are location and scale parameters, respectively, Y is the standardized response (L=0, S=1), {Yi ,Ye} are independent r.v.s representing internal/identity variation and external/error variation, respectively, ε is zero-mode normal disturbance (error) with standard deviation σε and Z is standard normal. The density function of the distribution of Yi in this model (the extended exponential) is eq. (2), where Yi is the component representing “identity variation” (caused by variation of system/process factors, “Internal factors”), CYi is a normalizing coefficient, and σi is a parameter representing internal/identity variation. It is easy to realize that α is the mode. At α=1, Yi becomes left-truncated normal (re-located half normal). However, it is assumed that at α=1 “identity variation” vanishes, so Yi becomes a constant, equal to the mode (1). For the exponential scenario (complete loss of identity), we obtain α=0, and the disturbance, assumed to be multiplicative, renders meaningless, namely, it vanishes (σe=0, Ye=1). Therefore, Yi and Y then both become exponential.
Let us introduce eq. (3). From (2), we obtain the pdf of Zi: (eqs. (4) and (5)). Note that the mode of Zi is zero (mode of Yi is α).
Various theorems and conjectures are articulated in Shore (2022), which deliver eye-opening insights into various regularities in the behavior of statistical distributions, previously un-noticed, and good explanation to various statistical theoretical results, heretofore considered separate and unrelated (like a logical derivation of the Central Limit Theorem from the “Random identity paradigm”).
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Conclusions
In this article, I have reported about my personal experience, which led me to the development of the new “Random identity paradigm” and allied concepts. It followed my research effort to model surgery duration, which resulted in a bi-variate explanatory model, with the extended exponential distribution as the intermediate tool, that paved a smooth way to unify, under a single umbrella model, execution times of all types of work processes/surgeries, namely, not only repetitive (normal), or non-repetitive (exponential), but also those in between (semi-repetitive processes/surgeries). To date, we are not aware of a similar unifying model that is as capable in unifying diverse phenomena as the three categories of work-processes/surgeries. Furthermore, this modeling effort has led directly to conceiving the new “Random identity paradigm” with allied new concepts (as alluded to earlier).
The new paradigm has produced three major outcomes:
First, as demonstrated in the linked pre-print, under the new paradigm virtually scores of theoretical statistical results that have formerly been derived independently and considered unrelated, are explained in a consistent and coherent manner, becoming inter-related under the unifying “Random identity paradigm”.
Secondly, various conjectures about properties of distributions are empirically verified with scores of examples/predictions from the Statistics literature. For example, the conjectures that Category B r.v.s, which are function of only identity-less r.v.s, are also identity-less, and similarly for identity-full r.v.s.
Thirdly, the new bi-variate model has been demonstrated to represent well numerous existent distributions, as has been shown for diversely-shaped distributions in Shore, 2020a (see Supplementary Materials therein).
It is hoped that the new “Random identity paradigm”, representing an initial effort at unifying distributions of natural processes (Category A distributions), this new paradigm may pave the way for Statistics to join other branches of science in a common effort to reduce, via unification mediated by unifying theories, the number of statistical distributions, the “objects of enquiry” of modeling random-variation within the science/branch-of-mathematics of Statistics.
References
[1] Shore H (1986). An approximation for the inverse distribution function of a combination of random variables, with an application to operating theatres. Journal of Statistical Computation and Simulation, 23:157-181. DOI: 10.1080/00949658608810870 .
[2] Shore H (2015). A General Model of Random Variation. Communication in Statistics- Theory and Methods, 49(9):1819-1841. DOI: 10.1080/03610926.2013.784990.
[3] Shore H (2020a). An explanatory bi-variate model for surgery-duration and its empirical validation. Communications in Statistics: Case Studies, Data Analysis and Applications, 6(2):142-166. Published online: 07 May 2020. DOI: 10.1080/23737484.2020.1740066
[4] Shore H (2020b). SPC scheme to monitor surgery duration. Quality and Reliability Engineering International. Published on line 03 December 2020. DOI: 10.1002/qre.2813
[5] Shore H (2021). Estimating operating room utilisation rate for differently distributed surgery times. International Journal of Production Research. Published on line 13 Dec 2021. DOI: 10.1080/00207543.2021.2009141.
[6] Shore H (2022). “When an error ceases to be error” — On the process of merging the mean with the standard deviation and the vanishing of the mode. Preprint.
Haim Shore_Blog_Merging of Mean with STD and Vanishing of Mode_Jan 07 2022
Haim Shore Blog 