Gambling with ethics? A statistical note on the Poisson (binomial) distribution
Jaideep J. Pandit · Anaesthesia · 2008
In a recent issue of the journal, Fabregat-López et al. [1] report a study that raises both ethical and statistical issues (in addition to the specific airway management debate that may also ensue). They describe using a Proseal laryngeal mask airway (PLMA) in cases of laparoscopic emergency appendicectomy, a practice which is apparently normal in their unit and which they have practiced for some considerable time. From the UK perspective, it may seem they took an apparent gamble, but they reported no complications with its use. Fabregat-López et al. [1] wished to investigate their view that using a PLMA was safe for the circumstances they describe. They designed an observational study, that is a study with only one group of patients broadly exposed to the same technique. They reasoned that, if they observed few or no complications, then they could conclude that the PLMA was 'safe'. Is this assumption correct? How many patients would we need to observe to be reasonably confident that a technique is 'safe'? A number of articles and editorials in anaesthesia have dealt with the issue of power calculations when comparing two groups subjected to different techniques [2–4]. However, the same statistics cannot be applied to the question of observing incidence of complications in just one group of patients. The methods required for this involve the statistical techniques of the 'Poisson' or 'binomial' distribution or 'probability density functions' (all these terms are very closely related) [5]. The Appendix to this article outlines some of the mathematics involved, which indeed seem very dense. The purpose of the next section is to outline, in very simplified, practical terms, what this mathematics seeks to achieve. The Poisson function seeks to answer the following type of question. Let us suppose, as an example, that we watch birds fly past our window and we note that on average after many weeks of study, 12 birds fly past each 24 h. This data helps us to calculate that on average we will see 24 birds over two days, six birds in half a day and so on. However, such calculations alone cannot help answer two related questions of potential interest: (i) what is the probability of our seeing at least one bird if we, say, watch for just 1 h?; (ii) how long do we need watch for to be confident of seeing, say, 36 birds? Now anaesthetists (as a rule) do not watch for birds; instead we tend to look for complications (or a beneficial outcome) in populations, but the mathematics is identical. If a certain complication is thought to occur on average in 1 every 200 cases, then how many cases do we need to observe to be reasonably confident of seeing at least one complication? Conversely, if we observe 150 cases, what is the probability of our seeing at least one complication (or none)? So what the mathematics in the Appendix does is to translate the bald figures about birds and complications above into a probability distribution or density. At one extreme is the situation where no observations are made and the probability of seeing any complications (or birds) is zero; at the other extreme is when an infinite number of observations are made and the probability is 1 (i.e. we are certain to see a complication or bird if we look infinitely long or hard). In between is the probability density, which generally depends on (i) how long we look or how many observations we make and (ii) the prevalence of the thing we are looking for (i.e. how common is the event). There are a few other pertinent points about the Poisson distribution. The events being observed must be independent (i.e. the chance of seeing one event must not be influenced by the occurrence of other events). The events must be discrete and binomial (i.e. yes/no events such as complications which can either occur or be absent, or birds which can be seen or not seen; continuous variables such as blood pressure or height are not suitable for such analysis). The formula in the Appendix can be made more user-friendly with computer algorithms (e.g. http://www.stat.tamu.edu/~west/applets/binomialdemo.html or http://www.swogstat.org/stat/public/binomial_calculator.htm). As a simple rule of one extreme of this function: if, in an observational study of n observations or patients, no complications are witnessed, then the upper (95%) confidence limit for complications approximates 3/n [6, 7]. Thus for Fabregat-López's study of 102 patients, the serious complication rate using a PLMA could in fact be as high as 3%. The prevalence of most anaesthetic complications is generally low ( 5000; Table 1) to be confident of usefully examining this complication. It seems unlikely that a study of smaller size will be deemed ethically appropriate since patients will be subjected to a risk with little likelihood of the study excluding or delineating the main issue of concern. Poisson in part developed his distribution to deal with the 'martingale', a betting strategy popular in 18th–19th century France concerning predicting the repeated toss of a coin. Poisson himself, however, believed that all gambling, even apparently a 'fair' game of chance, was ruinous [16]. His work built upon, and was closely related to, his predecessor Pascal's fundamental work on probability theory. Pascal also fell in with gamblers, but experienced an epiphany when his life was saved in a horse carriage accident and thereafter devoted his life to religion [17]. Both mathematicians had to come to terms with the moral and ethical dimensions of their own probability calculations. Fabregat-López et al. took something of a gamble with a limited sample size and they observed no complications. Had they done so, in my view it would have sounded – in statistical terms – a perhaps appropriate death-knell for use of the PLMA in emergency surgery, since even one case of aspiration would have implied an unacceptably high prevalence of aspiration of ∼1% with the PLMA. Their study does not help us calculate the true prevalence, but instead simply provides us with indication that the practice they describe is commonplace. Did they have appropriate ethical approval to report an unusual technique in a novel clinical setting? Yes. Have they confirmed that using a PLMA in emergency appendicectomy is appropriate anaesthetic practice without significant risk? No. Do their results provide 'some' reassurance that, if the PLMA is used as a rescue device (e.g. after failed tracheal intubation in rapid sequence induction), the risk of aspiration is not prohibitive? Yes; 'some', but not 'a lot'. I thank Dr Devinder Sivia, Senior Scientific Officer, ISIS Data Analysis Group, Rutherford Appleton Laboratory and Lecturer, St John's College, Oxford for his helpful comments on the manuscript. N→∞, p→0, Np = constant. Strictly, a binomial distribution relates to proportions of occurrences in a population (e.g. presence or absence of disease), or a series of yes/no events (e.g. tossing a coin many times). A Poisson distribution applies to a series of events in a fixed period of time. The formulae above need further adaptation to take account of variation and confidence intervals for the prevalence.