ARTICLE https://doi.org/10.1057/s41599-019-0366-y OPEN Statistical reliability analysis for a most dangerous occupation: Roman emperor Joseph Homer Saleh1* ABSTRACT Popular culture associates the lives of Roman emperors with luxury, cruelty, and debauchery, sometimes rightfully so. One missing attribute in this list is, surprisingly, that this mighty office was most dangerous for its holder. Of the 69 rulers of the unified Roman d d 1234567890():,; Empire, from Augustus ( . 14 CE) to Theodosius ( . 395 CE), 62% suffered violent death. This has been known for a while, if not quantitatively at least qualitatively. What is not known, however, and has never been examined is the time-to-violent-death of Roman emperors. This work adopts the statistical tools of survival data analysis to an unlikely population, Roman emperors, and it examines a particular event in their rule, not unlike the focus of reliability engineering, but instead of their time-to-failure, their time-to-violent-death. We investigate the temporal signature of this seemingly haphazardous stochastic process that is the violent death of a Roman emperor, and we examine whether there is some structure underlying the randomness in this process or not. Nonparametric and parametric results show that: (i) emperors faced a significantly high risk of violent death in the first year of their rule, which is reminiscent of infant mortality in reliability engineering; (ii) their risk of violent death further increased after 12 years, which is reminiscent of wear-out period in reliability engineering; (iii) their failure rate displayed a bathtub-like curve, similar to that of a host of mechanical engineering items and electronic components. Results also showed that the stochastic process underlying the violent deaths of emperors is remarkably well captured by a (mixture) Weibull distribution. We discuss the interpretation and possible reasons for this uncanny result, and we propose a number of fruitful venues for future work to help better understand the deeper etiology of the spectacle of regicide of Roman emperors. 1 Georgia Institute of Technology, Atlanta, USA. *email: [email protected] PALGRAVE COMMUNICATIONS | (2019) 5:155 | https://doi.org/10.1057/s41599-019-0366-y | www.nature.com/palcomms 1 ARTICLE PALGRAVE COMMUNICATIONS | https://doi.org/10.1057/s41599-019-0366-y Introduction hat did a Roman emperor have in common with a have exclaimed, according to Suetonius, “iacta alea est” (Sueto- gladiator? The latter had better odds of surviving a nius, 1998). The expression is commonly translated as “the die is Wfi ght than the former had of avoiding a violent death. cast”, or more resolutely, “let the die be cast”, and it is taken to Popular culture traditionally associates the lives of Roman signify the passing of the point of no return in an irrevocable emperors with luxury, cruelty, and debauchery, sometimes course of action. For anyone familiar with the game of dice, which rightfully so. One missing attribute in this list is, surprisingly, that Roman soldiers and several emperors were fond of, the emphasis this mighty office was most dangerous for its holder, as the sta- of the exclamation is less on the passing of the point of no return, tistics will show. Consider the following: of the 69 rulers of the and more on the aleatory nature of the possible outcome. What is unified Roman Empire, from Augustus (d. 14 CE) to Theodosius the likelihood that the outcome will be, say, odd? Or a particular (d. 395 CE), 43 emperors suffered violent death, that is 62%, number given a four-sided or six-sided dice? What outcomes did either by assassination, the most common mode of death, suicide, Caesar contemplate? There is obviously the fate of his under- or during combat with a foreign enemy of Rome1 (Fig. 1). To put taking: will the Roman Republic endure, or is this the last nail in this statistic in context and better appreciate its magnitude, its coffin? What will his own fate be? Will he be successful, or will compare it with what is considered nowadays a seriously dan- he meet a violent death? In crossing the Rubicon, Caesar sig- gerous activity: Himalaya mountaineering. Climbers that summit nificantly narrowed down the scope of possible outcomes he above 8000 m in the Himalayas have a risk of death of about 4%, could face. a relatively consistent figure for the past 50 years (Amalberti et al., Beyond this original throw of dice, Caesar provided a 2005). Roman emperors had an order of magnitude greater risk of metaphor, not just for himself, but for every subsequent emperor: violent death than these intrepid climbers. upon confirmation by the senate, or being chosen by the legions The odds of survival for a Roman emperor were roughly or the praetorian guards, every emperor could have exclaimed, equivalent to playing the Russian roulette with a six-chambered “iacta alea est” and was in effect throwing a dice for his life. In revolver, in which the participant places not one but four bullets, retrospect, we now know that collectively the rulers of the unified spins the cylinder to randomize the outcome, and pulls the trigger Roman Empire had a 62% chance of dying a violent death. This is with the muzzle against his head. roughly the equivalent of associating four outcomes, for example This sinister comparison conjures the notion of random vari- {1, 2, 3, 4} with this gruesome end, and rolling a six-sided fair able, a central protagonist in our story. That the likelihood of dice. Only those who rolled a 5 or 6 got to die of a natural death. violent death for an emperor was high was well-known, if not This is a seemingly more benign but equally potent metaphor quantitatively at least qualitatively as far back as Edward Gibbon’s than the previous Russian roulette. publication of the first volume of his Decline and Fall of the What is not known, however, and has never been examined to Roman Empire (1776). In discussing the highly energetic emperor date is another random variable associated with these rulers, their Aurelian (ruled from 270 to 275 CE), nicknamed restitutor orbis, time-to-failure or time-to-violent-death. A brief discussion of restorer of the world or the unity of the Roman empire in the reliability engineering is in order to better understand this idea troubled third century (Watson, 1999), one can almost feel and the focus of this work. Gibbon’s disappointment when reflecting on the emperor’s murder2: Reliability engineering and Roman emperors: from time-to- failure to time-to-violent-death Such was the unhappy condition of the Roman emperors, that, whatever might be their conduct, their fate was Reliability is a popular concept that has been celebrated for years as a commendable attribute of a person or an equipment. commonly the same. A life of pleasure or virtue, of severity or mildness, of indolence or glory, alike led to an untimely Although many words in the English language have been coined by or attributed to Shakespeare, it seems that we owe the word grave; and almost every reign is closed by the same disgusting repetition of treason and murder [emphasis added]. reliability to another English author, the poet Samuel Coleridge (1772–1834). In praising a friend, Coleridge wrote (Coleridge, He then added, “the Roman senators heard, without surprise, 1983; Saleh and Marais, 2006): that another emperor had been assassinated in his camp”. fl It is worth taking a little historical detour at this point, back to He in icts none of those small pains and discomforts which the time before this track record of treason and murder Gibbon irregular men scatter about them and which in the refers to was even started, to revisit a popular landmark quotation aggregate so often become formidable obstacles both to and understand it in a new light, Julius Caesar’s “Iacta alea est”… happiness and utility; while on the contrary he bestows all the pleasures, and inspires all that ease of mind on those around him or connected with him, with perfect con- Of dice and men. And Roman emperors. As Caesar con- sistency, and (if such a word might be framed) absolute templated his decision to cross the shallow Rubicon river with his reliability. legion, a capital offense for himself and his soldiers, he is said to Since then, reliability engineering has developed into an important discipline that pervades many aspects of our modern, technologically intensive world (Hoyland and Rausand, 2009). The foundational idea in reliability engineering is that the time- to-failure, Tf, of an item is stochastic in nature, it is a random variable. Roughly speaking, reliability, S(t), is defined as the probability that an item is still operational at time t; it has not failed and is till performing its function up to this time: StðÞPrðÞTf >t ð1Þ S(t) is also known as the survival or survivor function. Reliability Fig. 1 Deaths and failure modes of the rulers of the unified Roman Empire. engineering is then concerned with quantifying this probability 2 PALGRAVE COMMUNICATIONS | (2019) 5:155 | https://doi.org/10.1057/s41599-019-0366-y | www.nature.com/palcomms PALGRAVE COMMUNICATIONS | https://doi.org/10.1057/s41599-019-0366-y ARTICLE over the life of an item. This is sufficient for our purposes. Julianus is noted for two memorable feats: the manner in which Another useful concept in reliability engineering is that of failure he became emperor, and for holding one of the shortest reigns for rate. It is less important for our story but is worth mentioning. an emperor, 66 days, before he was executed. While reliability is the complement of the cumulative distribution Another point with the data that should be mentioned is function (CDF) of the random variable time-to-failure (Tf), the censoring.