(Related podcast: Why Predictions of Surgery-Duration are So Poor, and a Possible Remedy (Podcast) ).

Operating theatres are the most expensive resource at the disposal of hospitals. This renders optimizing scheduling of surgeries to operating rooms a top priority. A pre-condition to optimal scheduling is that accurate predictions of surgery-duration be available. Much research effort has in recent years been invested to develop methods that improve the accuracy of surgery-duration predictions. This ongoing effort includes both traditional statistical methods and newer Artificial Intelligence (AI) methods. The state-of-the-art of these methods, with relevant peer-reviewed literature, have recently been summarized by us in a new entry on Wikipedia, titled “Predictive Methods for Surgery Duation”.

Personally, I was first exposed to the problem of predicting surgery-duration over thirty years ago, when I was involved in a large-scale project encompassing all governmental hospitals in Israel (at the time). Partial results of this effort had been reported in my published paper of 1986, and further details can be found in my more recent paper of 2020. Both articles are listed in the literature section at the end of this post (for podcast listeners, this list may be found on haimshore.blog).

My second involvement in developing predictive methods for surgery-duration was in more recent years, culminating in three peer-reviewed published papers (Shore 2020, 2021 ab; see references below).

Surgery-duration is known to be very highly volatile. The larger the variability between surgeries, the less accurate the prediction may be expected to be. To reduce this variability, newly devised predictive methods for surgery-duration tend to concentrate on subsets of surgeries, classified according to some classification system. It is assumed that via this classification, prediction accuracy may be enhanced. A common method to classify surgeries, implemented worldwide, is Current Procedural Terminology (CPT®). This coding system delivers, in a hierarchical fashion, particular codes to subsets of surgeries. In doing so, variability between surgeries sharing same CPT code is expected to be reduced, allowing for better prediction accuracy.

A second effort to increase accuracy is to include, in the predictive method, certain factors, known prior to surgery, which deliver variability to surgery-duration. It is hoped that by taking account of these factors, in the predictive method, unexplained variability in surgery-duration will be reduced, thereby enhancing prediction accuracy (examples will soon be given).

A third factor that influence accuracy is the amount of reliable data, used to generate predictions. Given recent developments in our ability to process large amounts of data, commonly known as Big Data, Artificial Intelligence (AI) methods have been summoned to assist in predicting surgery times.

These new methods and others are surveyed more thoroughly in the aforementioned entry on Wikipedia.

The new methods notwithstanding, current predictive methods for surgery-duration still deliver unsatisfactory accuracy.

Why is that so?

We believe that a major factor for the poor performance of current predictive methods is lack of essential understanding of what constitute major sources of variability to surgery-duration. Based on our own personal experience, as alluded to earlier, and also on our professional background as industrial engineers, specializing in analysis of work processes (of which surgeries are an example), we believe that there are two sets of factors that generate variability in surgery-duration: A set of major factors and a set of secondary factors. We denote these Set 1 and Set 2 (henceforth, we refer only to variability between surgeries within a subset of same code):

**Set 1** — Two Major Factors:

**Factor I.**Work-content instability (possibly affected by variability in patient condition);**Factor II**. Error variability.

**Set 2** — Multiple Secondary Factors, like: patient age, professional experience and size of medical team, number of surgeries a surgeon has to perform in a shift, type of anaesthetic administered.

Let us explain why, in contrast to current practices, we believe that work-content instability has critical effect on prediction accuracy, and why accounting for it, in the predictive method, is crucial to improving current accuracy, obtained via traditional methods.

To prepare predictions for any random phenomenon, assumed to be in steady-state, the best approach is to define its statistical distribution and estimate its parameters, based on real data. Once the distribution is completely defined, various statements about the conduct of the random phenomenon (like surgery-duration) can be made.

For example:

- What is the most likely realization (given by distribution’s mode);
- What is the middle value, which delivers equal probabilities, for any realization, to be larger or smaller than that value (expressed by distribution’s median);
- What is the probability that any realization of the random phenomenon exceeds a specified value (calculated by the cumulative density function, CDF)?

Understanding that complete definition of the distribution is the best approach to predict surgery-duration, let us next explain what type of distributions can one expect in the two extreme states, regarding the two major factors of Set 1:

**State 1.** There is no variability in work-content (there is only error variability);

**State 2.** There is no error (error variability is zero; there is only work-content variability).

The two states define two different distributions for surgery-duration.

The first state, **State 1**, implies that the only source of variability is error. This incurs the normal distribution, for an additive error, or the log-normal distribution, for a multiplicative error (namely, error expressed as a percentage).

**State 2**, lack of error variability, by definition can only materialize when there is no typical value (like the mode), on which error can be defined. Since no definition of error is feasible, error variability becomes zero. For work-processes, like surgery, this can happen only when there is no typical work-content. In statistical terms, this is a state of lack-of-memory. An example is the duration of repair jobs at a car garage, relating to all types of repair. The distribution typical to such situations is the memoryless exponential.

We learn from this discussion, that any statistical model of surgery-duration, from which its distribution may be derived, needs to include, as extreme cases, both the normal/lognormal distributions and the exponential distribution.

This is a major constraint on any model for the distribution of surgery-duration. It has so far eluded individuals engaged in developing predictive methods for surgery-duration. Lack of knowledge of basic principles of industrial engineering, as well as total ignorance regarding how instability in work-content of a work process (like surgery) influences the **form of the distribution**, these probably constitute the major culprit for the poor current state-of-the-art of predicting surgery-duration.

In Shore (2020), we have developed a bi-variate model for surgery-duration, which delivers not only the distributions of surgery-duration in the extreme states (**State 1** and **State 2**), but also the distributions of intermediate states, residing between the two extreme states. The two components of the bi-variate model represent work-content and error as two multiplicative random variables, with relative variabilities (standard deviations) that gradually change as surgery-duration moves from **State 1** (normal/lognormal case) to **State 2** (exponential case).

What do we hope to achieve with publishing of this post (and the accompanying podcast)?

We hope that individuals, engaged in developing predictive methods for surgery-duration, internalize the grim reality that:

- Unless their predictive method allows for the normal/lognormal and for the exponential to serve as exact distributions of surgery-duration at the extreme states;
- Unless their predictive method allows intermediate states, spanned on a continuous spectrum between the two extreme states, to converge smoothly to these states (as in Shore, 2020),

unless these two conditions be met, the likelihood for the accuracy of predictive methods for surgery-duration to improve anytime soon, this likelihood would remain, as it is today, extremely slim.

**Literature**

[1] Shore, H (1986). An approximation for the inverse distribution function of a combination of random variables, with an application to operating theatres. *J. Statist. Com. Simul.* 1986; 23:157-81. Available on Shore’s ResearchGate page.

[2] Shore, H (2020). 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, DOI: 10.1080/23737484.2020.1740066 .

[3] Shore, H (2021a). SPC scheme to monitor surgery-duration. *Qual Reliab Eng** Int*. 37: 1561– 1577. DOI: 10.1002/qre.2813 .

[4] Shore, H (2021b). Estimating operating room utilisation rate for differently distributed surgery times. *International Journal of Production** Research*. DOI: 10.1080/00207543.2021.2009141

[5] Shore, H (2021c). “Predictive Methods for Surgery Duation”. *Wikipedia*. April 16, 2021.