Repeated measurements with unintended feedback: The Dutch New Herring scandals

Fengnan Gao and Richard D. Gill; 24 July 2022

Note: the present post reproduces the text of our new preprint, adding some juicy pictures. Further editing is planned, much reducing the length of this blog-post version of our story.

Summary: We analyse data from the final two years of a long-running and influential annual Dutch survey of the quality of Dutch New Herring served in large samples of consumer outlets. The data was compiled and analysed by Tilburg University econometrician Ben Vollaard, and his findings were publicized in national and international media. This led to the cessation of the survey amid allegations of bias due to a conflict of interest on the part of the leader of the herring tasting team. The survey organizers responded with accusations of failure of scientific integrity. Vollaard was acquitted of wrongdoing by the Dutch authority, whose inquiry nonetheless concluded that further research was needed. We reconstitute the data and uncover important features which throw new light on Vollaard’s findings, focussing on the issue of correlation versus causality: the sample is definitely not a random sample. Taking into account both newly discovered data features and the sampling mechanism, we conclude that there is no evidence of biased evaluation, despite the econometrician’s renewed insistence on his claim.

Keywords: Data generation mechanism, Predator-prey cycles, Feedback in sampling and measurement, Consumer surveys, Causality versus correlation, Questionable research practices, Unhealthy research stimuli., ©


In surveys intended to help consumers by regularly publishing comparisons of a particular product obtained from different consumer outlets (think of British “fish and chips” bought in a large sample of restaurants and pubs), data is often collected over a number of years and evaluated each year by a panel, which might consist of a few experts, but might also consist of a larger number of ordinary consumers. As time goes by, outlets learn what properties are most valued by the panel, and may modify their product accordingly. Also, consumers learn from the published rankings. Panels are renewed, and new members presumably learn from the past about how they are supposed to weight the different features of a product. Partly due to negative reviews, some outlets go out of business, while new outlets enter the market, and imitate the products of the “winners” of previous years’ surveys. Coming out as “best” boosts sales; coming out as “worst” can be the kiss of death.

For many years, a popular Dutch newspaper (Algemene Dagblad, in the sequel AD) published two immensely influential annual surveys of two particularly popular and typically Dutch seasonal products: the Dutch New Herring (Dutch: Hollandse Nieuwe) in June, and the Dutch “oliebol” (a kind of greasy, currant-studded, deep-fried spherical doughnut) in December. This paper will study the data published by the newspaper on its website of 2016 and 2017—the last two years of the 36 years in which the AD herring test operated. This data included not only a ranking of all participating outlets and their final scores (on a scale of 0 to 10) but also numerical and qualitative evaluations of many features of the product being offered. A position in the top ten was highly coveted. Being in the bottom ten was a disaster.

For a while, rumours had been circulated (possibly by disappointed victims of low scores!) that both tests were biased. The herring test was carried out by a team of three tasters, whose leader Aad Taal was indeed consultant to a wholesale company called Atlantic (based in Scheveningen, in the same region as Rotterdam), and who offered a popular course on herring preparation. As a director at the Dutch ministry of agriculture he had earlier successfully managed to obtain the European Union (EU) legal protection for the official designation “Dutch New Herring”. Products may only be sold under this name in the whole EU only if meticulously prepared in the circumscribed traditional way, as well as satisfying strict rules of food safety. It is nowadays indeed sold in several countries adjacent to the Netherlands. We will later add some crucial further information about what actually makes a Dutch New Herring different from the traditionally prepared herring of other countries.

Enter econometrician Dr Ben Vollaard of Tilburg University. Himself partial to a tasty Dutch New Herring, he learnt in 2017 from his local fishmonger about the complaints then circulating about the AD Herring Test. The AD is based on the city of Rotterdam, close to the main home ports of the Dutch herring fleet in past centuries. Tilburg is somewhat inland. Not surprisingly, consumers in different regions of the country seem to have developed different tastes in Dutch New Herring, and a common complaint was that the AD herring testers had a Rotterdam bias.

Vollaard decided to investigate the matter scientifically. A student helped him to manually download the data published on their website on 144 participating outlets in 2016, and 148 in 2017. An undisclosed number of outlets participated in both years, and initial reports suggested it must be a large number. Later we discovered that the overlap consisted of only 23 outlets. Next, he ran a linear regression analysis, attempting to predict the published final score for each outlet in each year, using as explanatory variables the testing team’s evaluations of the herring according to various criteria such as ripeness and cleaning, together with numerical variables such as weight, price, temperature, and laboratory measurements of fat content and microbiological contamination. Most of the numerical variables were modelled by using dummy variables after discretization into a few categories. A single indicator variable for “distance from Rotterdam’’ (greater than 30 kilometres) was used to test for regional bias.

The analysis satisfyingly showed many highly significant effects, most of which are exactly those that should have been expected. The testing team gave a high final score to fish which had a high fat content, low temperature, well-cleaned, and a little matured (not too little, not too much). More expensive and heavier fish scored better, too. Being more than 30 km from Rotterdam had a just significant negative effect, lowering the final score by about 0.5. Given the supreme importance of getting the highest possible score, 10, a loss of half a point could make a huge difference to a new outlet going all out for a top score and hence position in the “top ten” of the resulting ranking. However, just because outlets in the sample far from Rotterdam performed a little worse on average than those close to Rotterdam, can have many innocent explanations.

But Vollaard went a lot further. After comparing the actual scores to linear regression model predicted scores based on the measured characteristics of the herring, Vollaard concluded:

Everything indicates that herring sales points in Rotterdam and the surrounding area receive a higher score in the AD Herring Test than can be explained from the quality of the herring served.

That is a pretty serious allegation.

Vollaard published this analysis as a scientific paper Vollaard (2017a) on his university personal web page, and the university put out a press release. The research drew a lot of media attention. In the ensuing transition from a more or less academic study (in fact, originally just a student exercise) to a press release put out by a university publicity department, then to journalists’ newspaper articles adorned with headlines composed by desk editors, the conclusion became even more damning.

Presumably stimulated by the publicity that his work had received, Vollaard decided to go further, now following up on further criticism circulating about the AD Herring Test. He rapidly published a second analysis, Vollaard (2017b), on his university personal web page. His focus was now on the question of a conflict of interest concerning a connection between the chief herring tester and the wholesale outlet Atlantic. Presumably by contacting outlets directly, he identified 20 outlets in the sample whose herring, he believed, had been supplied by that company. Certainly, his presumed Atlantic herring outlets tended to have rather good final scores, and a few of them were regularly in the top ten.

We may surmise that Vollaard must have been disappointed and surprised to discover that his dummy variable for being supplied by Atlantic was not statistically significant when he added it to his model. His existing model (the one on the basis of which he argued that the testing team was not evaluating outlets far from Rotterdam using their own measured characteristics) predicted that Atlantic outlets should indeed, according to those characteristics, have come out exactly as well as they did! He had to come up with something different. In his second paper, he insinuated pro-Atlantic bias by comparing the amount of variance explained by what he considered to be “subjective” variables with the amount explained by the “objective” variables, and he showed that the subjective (taste and smell, visual impression) evaluations explained just as much of the variance as the objective evaluations (price, temperature, fat percentage). This change of tune represents a serious inconsistency in thinking: this is cherry-picking in order to support a pre-gone conclusion.

In itself, it does not seem unreasonable to judge a culinary delicacy by taste and smell, and not unreasonable to rely on reports of connoisseurs. However, Vollaard went much further. He hypothesized that “ripeness” and “microbiological state” were both measurements of the same variable; one subjective, the other objective. According to him, they both say how much the fish was “going off”. Since the former variable was extremely important in his model, the latter not much at all, he again accused the herring testers of bias and attributed that bias to conflict of interest. His conclusion was:

A high place in the AD Herring Test is strongly related to purchasing herring from a supplier in which the test panel has a business interest. On a scale of 0 to 10, the final mark for fishmongers with this supplier is on average 3.6 points higher than for fishmongers with another supplier.

He followed that up with the statement:

Almost half of the large difference in average final grade between outlets with and without Atlantic as supplier can be explained by a subjective assessment by the test team of how well the herring has been cleaned (very good/good/moderate/poor) and of the degree of ripening of the herring (light/medium/strong/spoiled).

The implication is that the Atlantic outlets are being given an almost 2 point advantage based on a purely subjective evaluation of ripeness.

More media attention followed, Vollaard appeared on current affairs programs on Dutch national TV, his work was even reported in The Economist,

The AD defended itself and its herring testers by pointing out that the ripeness or maturity of a Dutch new herring, evaluated by taste and smell, reflects ongoing and initially highly desirable chemical processes (protein changing to fat, fat to oil, oil becoming rancid). Degree of microbiological activity, i.e., contamination with harmful bacteria, could be correlated with that, since dangerous bacterial activity will tend to increase with time once it has started, and both processes are speeded up if the herring is not kept cold enough, but it is of a completely different nature: biological, not chemical. It is caused by carelessness in various stages of preparation of the herring, insufficient cooling, and so on. It is obviously not desirable at all. AD also pointed out that one of the Atlantic outlets must have been missed, which actually in the first of the two years had scored very badly. This could be deduced from the numbers of those outlets, and the mean score of the Atlantic-supplied outlets, both reported by Vollaard in his papers.

The newspaper AD complained first to Vollaard and then to his university. With the help of lawyers, a complaint was filed with the Tilburg University committee for scientific integrity. The committee rejected the complaint, but the newspaper took it to the national level. Their lawyers hired the second author of this paper, Richard Gill (RDG), in the hope that he would support their claims. He requested Vollaard’s data-set and also requested that the outlets in the data-set be identified, since one major methodological complaint of his was that Vollaard had not taken account of possible autocorrelation by combining samples from two subsequent years, with presumably a large overlap, but without taking any account of this. Vollaard reluctantly supplied the data but declined to identify the outlets appearing twice or even inform us how many such outlets there were. With the help of AD however, it was possible to find them, and also locate many misclassified outlets. RDG wrote an expert opinion in which he argued that the statistical analysis did not support any allegations of bias or even unreliability of the herring test.

Vollaard had repeatedly stated that he was only investigating correlations, not establishing causality, but at the same time his published statements (quoted in the media), and his spoken statements on national TV, make it clear that he considered that his analysis results were damming evidence against the test. This seemed to RDG to be unprofessional, at the very least. RDG moreover identified much statistical amateurism. Vollaard analysed his data much as any econometrician might do: he had a data-set with a variable of interest and a number of explanatory variables, he ran a linear regression making numerous modelling choices without any motivation and without any model checking. He fit a completely standard linear regression model to two samples of Dutch new herring outlets, without any thought to the data generating mechanism. How were outlets selected to appear in the sample?

According to the AD, there were actually 29 Atlantic outlets in Vollaard’s combined sample. Note, there is some difficulty in determining this number. A given outlet may obtain some fish from Atlantic, some from other suppliers, and may change their suppliers over the course of a year. So the origin of the fish actually tasted by the test team cannot be determined with certainty. We see in Table 1 (according to AD), that Vollaard “caught” only about two thirds of the Atlantic outlets, and misclassified several more.

Atlantic by VollaardNot Atlantic by Vollaard
Atlantic by AD181129
Not Atlantic by AD2261263

Table 1: Atlantic- and not Atlantic-supplied outlets tested over two years as identified by Vollaard and the AD respectively.

At the national level, the LOWI (Landelijk Orgaan Wetenschappelijk Integriteit — the Dutch national organ for investigating complaints of violation of research integrity) re-affirmed the Tilburg University scientific integrity committee’s “not guilty” verdict. Vollaard was not deliberately trying to mislead. “Guilty” verdicts have an enormous impact and imply a finding, beyond a reasonable doubt, of gross research impropriety. This generally leads to termination of university employment contracts and to retraction of publications. They did agree that Vollaard’s analyses were substandard, and they recommended further research. RDG reached out to Vollaard suggesting collaboration, but he declined. After a while, Vollaard’s (still anonymized) data sets and statistical analysis scripts (written in the proprietary Stata language) were also published on his website Vollaard (2020a, 2020b). The data was actually in the form of Stata files; fortunately, it is nowadays possible to read such files in the open source and free R system. The known errors in the classification of Atlantic outlets were not corrected, despite AD’s request. The papers and the files are no longer on Vollaard’s webpages, and he still declines collaboration with us. We have made all documents and data available on our own webpages and on the GitHub page

RDG continued his re-analyses of the data and began the job of converting his expert opinion report (English translation: into a scientific paper. It seemed wise to go back to the original sources and this meant a difficult task of extracting data from the AD’s websites. Each year’s worth of data was moreover coded differently in the underlying HTML documents. At this point he was joined by the first author Fengnan Gao (FG) of the present paper who was able to automate the data scraping and cleaning procedures — a major task. Thus, we were able to replicate the whole data gathering and analysis process and this led to a number of surprises.

Before going into that, we will explain what is so special about Dutch New Herring, and then give a little more information about the variables measured in the AD Herring Test.

Dutch New Herring, ©

Every nation around the North Sea has traditional ways of preparing North Atlantic herring. For centuries, herring has been a staple diet of the masses. It is typically caught when the North Atlantic herring population comes together at its spawning grounds, one of them being in the Skagerak, between Norway and Denmark. Just once a year there is an opportunity for fishers to catch enormous quantities of a particular extremely nutritious fish, at the height of their physical condition, about to engage in an orgy of procreation. The fishers have to preserve their catch during a long journey back to their home base; and if the fish is going to be consumed by poor people throughout a long year, further means of conservation are required. Dutch, Danish, Norwegian, British and German herring fleets (and more) all compete (or competed) for the same fish; but what people in those countries eat varies from country to country. Traditional local methods of bringing ordinary food to the tables of ordinary folk become cultural icons, tourist attractions, gastronomic specialities, and export products.

Traditionally, the Dutch herring fleet brought in the first of the new herring catch in mid-June. The separate barrels in the very first catch are auctioned and a huge price (given to charity) is paid for the very first barrel. Very soon, fishmongers, from big companies with a chain of stores and restaurants, to supermarket chains, to small businesses selling fish in local shops and street markets are offering Dutch New Herring to their customers. It’s a traditional delicacy, and nowadays, thanks to refrigeration, it can be sold the whole year long (the designation “new” should be removed in September). Nowadays, the fish arrives in refrigerated lorries from Denmark, no longer in Dutch fishing boats at Scheveningen harbour.

What makes a Dutch new herring any different from the herring brought to other North Sea and Baltic Sea harbours? The organs of the fish should be removed when they were caught, and the fish kept in lightly salted water. But two internal organs are left, a fish’s equivalent to our pancreas and kidney. The fish’s pancreas contains enzymes which slowly transform some protein into fat and this process is responsible for a special almost creamy taste which is much treasured by Dutch consumers, as well as those in neighbouring countries. See, e.g., the Wikipedia entry for soused herring for more details, According to a story still told to Dutch schoolchildren, this process was discovered in the 14th century by a Dutch fisher named Willem Beukelszoon.

The AD Herring Test

© Marco de Swart (AD),

For many years, the Rotterdam-based newspaper Algemene Dagblad (AD) carried out an annual comparison of the quality of the product offered in a sample of consumer outlets. A small team of expert herring tasters paid surprise visits to the typical small fishmonger’s shops and market stalls where customers can order portions of fish and eat them on the premises (or even just standing in a busy food market). The team evaluated how well the fish has been prepared, preferring especially that the fish have not been cleaned in advance but that they are carefully and properly prepared in front of the client. They judged the taste and checked the temperature at which it is given to the customer: by law it may not be above 7 degrees. A sample was sent to a lab for a number of measurements: weight, fat percentage, signs of microbiological contamination. They are also interested in the price (per gram). An important, though subjective, characteristic is “ripeness”. Expert tasters distinguish Dutch new herring which has not ripened (matured) at all: green. After that comes lightly matured, well matured, too much matured, and eventually rotten.

This information was all written down and evaluated subjectively by each team member, then combined. The team averaged the scores given by its three members (a senior herring expert, a younger colleague, and a journalist) to produce a score from 0 to 10, where 10 is perfection; below 5.5 is a failing grade. However, it was not just a question of averaging. Outlets which sold fish which was definitely rotten, definitely contaminated with harmful bacteria, or definitely too warm got a zero grade. The outlets which took part were then ranked. The ten highest ranking outlets were visited again, and their scores possibly adjusted. The final ranking was published in the newspaper, and put in its entirety on internet. Coming out on top was like getting a Michelin star. The outlets at the bottom of the list might as well have closed down straight away. One sees from the histogram below, Figure [fig:1], that in 2016 and 2017, more than 40% of the outlets got a failing grade; almost 10% were essentially disqualified, by being given a grade of zero. The distribution looks nicely smooth except for the peak at zero, which really means that their wares did not satisfy minimal legal health requirements.

Figure 1: Final test scores, 2016 and 2017.

It is important to understand how outlets were chosen to enter the test. To begin with, the testing team itself automatically revisited last years’ top ten. But further outlets could be nominated by individual newspaper readers, indeed, they could be self-nominated by persons close to the outlets themselves. We are not dealing with a random sample, but with a self-selecting sample, with automatically a high overlap from year to year.

Over the years, there had been more and more acrimonious criticism of the AD Herring Test. As one can imagine, it was mainly the owners of outlets who had bad scores who were unhappy about the test. Many of them, perhaps justly, were proud of their product and had many satisfied customers too. Various accusations were therefore flung around. The most serious one was that the testing team was biased and even had a conflict of interest. The lead taster gave courses on the preparation of Dutch New Herring and led the movement to have the “brand” registered with the EU. There is no doubting his expertise, but he had been hired (in order to give training sessions to their clients) by one particular wholesale business, owned by a successful businessman of Turkish origin, which as one might imagine lead to jealousy and suspicion. Especially since a number of the retail outlets of fish supplied by that particular company often (but certainly not always) appeared year by year in the top ten of the annual AD Herring Test. Other accusations were that the herring tasters favoured businesses in the neighbourhood of Rotterdam (home base of the AD). As herring cognoscenti know, people in various Dutch localities have slightly different tastes in Dutch New Herring. Amsterdammers have a different taste from Rotterdammers.

In the meantime, under the deluge of negative publicity, the AD announced that they would now stop their annual herring test. They did hire a law company who on their behalf brought an accusation of failure of scientific integrity to Tilburg University’s “Commission for Scientific Integrity”. The law firm moreover approached one of us (RDG) for expert advice. He was initially extremely hesitant to be a hired gun in an attack on a fellow academic but as he got to understand the data and the analyses and the subject, he had to agree that the AD had some good points. At the same time, various aggrieved herring sellers were following up with their own civil action against the AD; and the wholesaler whose outlets did so well in the test, also started a civil action against Tilburg University, since its own reputation was damaged by the affair.

Vollaard’s analyses

Here is the main result of Vollaard’s first report.

lm(formula = finalscore ~
                    weight + temp + fat + fresh + micro +
                    ripeness + cleaning + yr2017)
     Min      1Q  Median      3Q     Max
 -4.0611 -0.5993  0.0552  0.8095  3.9866

Residual standard error: 1.282 on 274 degrees of freedom
Multiple R-squared:  0.8268, Adjusted R-squared:  0.816
F-statistic: 76.92 on 17 and 274 DF,  p-value: < 2.2e-16

Estimate Std.Errort value Pr(>|t|) 
weight (grams)

< 7 deg0 (baseline)

7–10 deg –0.6859620.193448–3.5460.000460***

> 10 deg–1.7931390.223113–8.0372.77e–14***

< 10%0 (baseline)


> 14%0.5816020.2500332.3260.020743*
1.8170810.2003359.070< 2e–16***

very good0 (baseline)


bad–0.618397  0.448309  –1.379 0.168897–1.3790.168897



mild0 (baseline)



rotten–4.5987520.503490–9.134< 2e–16***

very good0 (baseline)



Regression model output

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1

No surprises here. The testing team prefers fatty and larger herring, properly cooled, mildly matured, freshly prepared in view of customers on-site, and well-cleaned too. We have a delightful amount of statistical significance. There are some curious features of Vollaard’s chosen model: some numerical variables (“temp” and “fat”) have been converted into categorical variables by presumably arbitrary choice of cut-off points, while “weight” is taken as numerical. Presumably, this is because one might expect the effect of temperature not to be monotone. Nowadays, one might attempt fitting low-degree spline curves with few knots. Some categories of categorical variables have been merged, without explanation. One should worry about interactions and about additivity. Certainly one should worry about model fit.

We add to the estimated regression model also R’s standard four diagnostic plots in Fig. 2. Dr Vollaard apparently did not carry out any model checking.

Figure 2a. Model validation: panel 1, residuals versus fitted values
Figure 2b. Model validation: panel 2, normal QQ plot of standardized residuals
Figure 2c. Model validation: panel 3, square root of absolute value of standardized residuals against fitted value
Figure 2d. Model validation: panel 4, standardized residuals against leverage

Model validation beyond Vollaard’s regression analysis

There are some serious statistical issues. There seem to be a couple of serious outliers. The error distribution seems to have a heavier than normal tail. But we also understand that some observations come in pairs — the same outlet evaluated in two subsequent years. The data set has been anonymized too much. Each outlet should at the least have been given a random code so that one can identify the pairs and take account of possible dependence from one year to the next, easy to do by simply estimating the correlation from the residuals, and then doing a generalized linear regression with an estimated covariance matrix of the error terms.

Inspection of the outliers led us to realize that there is a serious issue with the observations which got a final score of zero. Those outlets were essentially disqualified on grounds of gross violation of basic hygiene laws, applied by looking at just a couple of the variables: temperature above 12 degrees (the legal limit is 7), and microbiological activity (dangerous versus low or none). The model should have been split into two parts: a linear regression model for the scores of the not-disqualified outlets; and a logistic regression model, perhaps, for predicting “disqualification” from some of the other characteristics. However, at least it is possible to analyse each of the years separately, and to remove the “disqualified” outlets. That is easy to do. Analysing just the 2017 data, the analysis results look a lot cleaner; the two bad outliers have gone, the estimated standard deviation of the errors is a lot smaller, the normal Q-Q plot looks very nice.

The data-set, now as comma-separated values files and Excel spreadsheets, and with outlets identified, can be found on our already mentioned GitHub repository new- herring.

The real problem

There is another big issue with this data and these analyses which needs to be mentioned, and if possible, addressed. How did the “sample” come to be what it is? A regression model is at best a descriptive account of the correlations in a given data set. Before we should accuse the test team of bias, we should ask how the sample is taken. It is certainly not a random sample from a well-defined population!

Some retail outlets took part in the AD Herring Test year after year. The testing team automatically included last years’ top ten. Individual readers of the newspaper could nominate their own favourite fish shop to be added to the “sample”, and this actually did happen on a big scale. Fish shops which did really badly tended to drop out of future tests and, indeed, some of them stopped doing business altogether:

The “sample” evolves in time by a feedback mechanism.

Everybody could know what the qualities were that the AD testers appreciated, and they learnt from their score and their evaluation each year what they had to do better next year, if they wanted to stay in the running and to join the leaders of the pack. The notion of “how a Dutch New Herring ought to taste”, as well as how it ought to be prepared, was year by year being imprinted by the AD test team on the membership of the sample. New sales outlets joined and competed by adapting themselves to the criteria and the tastes of the test team.

The same newspaper did another annual ranking of outlets of a New Year’s Dutch traditional delicacy, actually, a kind of doughnuts (though without a hole in the middle) called oliebollen. They are indeed somewhat stodgy and oily, roughly spherical, objects, enlivened with currants and sprinkled with icing sugar. The testing panel was able to taste these objects blind. It consisted of about twenty ordinary folk and every year, part of the panel resigned and was replaced with fresh persons. Peter Grünwald of Centrum Wiskunde & Informatica, the national research institute for mathematics and computer science in the Netherlands, developed a simulation model which showed how the panel’s taste in oliebollen would vary over the years, as sales outlets tried to imitate the winners of the previous year, while the notion of what constitutes a good oliebol was not fixed. Taking the underlying quality to be one-dimensional, he demonstrated the well-known predator-prey oscillations (Angerbjorn et al., 1999). Similar lines of thinking have appeared in the study of, e.g., fashion cycles, see e.g. Acerbi et al. (2012), where the authors propose a mechanism for individual actors to imitate other actors’ cultural traits and preferences for these traits such that realistic and cyclic rise-and-fall patterns (see their Figure 4) are observed in simulated settings. A later study, Apriasz et al. (2016), divides a society into two categories of “snobs” and “followers”, where followers copy everyone else and snobs only imitate the trend of their own and go against the followers. As a result, clear recurring cyclic patterns (see their Figures 3 and 4) similar to the predator-prey cycle arise under proper parameter regimes.

The AD was again engaged in a legal dispute with disgruntled owners of low ranked sales outlets, which eventually led to this annual test being abandoned too. In fact, the AD forbade Grünwald to publish his results. We have made some initial simulation studies of a model with higher dimensional latent quality characteristics, which seems to exhibit similar but more complex behaviour.

New analyses, new insights

It turns out that the correlation between the residuals of the same outlet participating in two subsequent years is large, about 0.7. However, their number (23) is fairly small, so this has little effect on Vollaard’s findings. Taking account of it slightly increases the standard errors of estimated coefficients. However, we also knew that according to AD, many outlets were incorrectly classified by Vollaard, and since he did not wish to collaborate with us, we returned to the source of his data: the web pages of AD. This enabled us to play with the various data coding choices made by Vollaard and to try out various natural alternative model specifications. As well as this, we could use the list of outlets certified by AD and Atlantic as having actually supplied the Dutch new herring tested in 2016 and 2017.

First, it is clear from the known behaviour of the test team that a score of zero means something special. There is no reason to expect a linear model to be the right model for all participating outlets. The outlets which were given a zero score were essentially disqualified on objective public health criteria, namely temperature above 12 degrees and definitely dangerous microbiological activity. We decided to re-analyse the data while leaving out all disqualified outlets.

Next, there is the issue of correlation between outlets appearing in two subsequent years. Actually, this turned out to be a much smaller proportion than expected. So correction for autocorrelations hardly makes a difference, but on the other hand, it is easily made superfluous by dropping all outlets appearing for the second year in succession. Now we have two years of data, in the second year only of “newly nominated” outlets.

Going back to the original data published by AD, we discovered that Vollaard had made some adjustments to the published final scores. As was known, the testing team revisited the top ten scoring outlets, and ranked their product again, recording (in one of the two years) scores like 9.1, 9.2, … up to 10, in order to resolve ties. In both years there were scores registered such as  8– or 8+, meant to indicate “nearly an 8” or “a really good 8”, following Dutch traditional school and university test and exam grading. The scores “5″, “6″, “7″, “8”, “9”, “10” have familiar and conventional descriptions “unsatisfactory” or insufficient, “satisfactory” or sufficient, “good”, “very good, “excellent”. Linear regression analysis requires a numerical variable of interest. Vollaard had to convert “9–” (almost worthy of the qualification “very good”) into a number. It seems that he rounded it to 9, but one might just as well have made it 9–𝞮 for some choice of 𝞮, for instance, 𝞮 = 0.01,  0.03, or 0.1.

We compared the results obtained using various conventions for dealing with the “broken” grades, and it turned out that the choice of value of 𝞮 had major impact on the statistical significance of the “just significant” or “almost significant” variables of main interest (supplier; distance). Also, whether one followed standard strategies of model selection based on leaving out insignificant variables has a major impact on the significance of the variables of most interest (distance from Rotterdam; supplier). The size of their effects becomes a little smaller, standard errors remain large. Had Vollaard followed one of several common model selection strategies, he could have found that the effect of “Atlantic” was significant at the 5% level, supporting his prior opinion! As noted by experienced statistical practitioners such as Winship and Western (2016), in linear regression analysis where multicollinearity is present, the regression estimates are highly sensitive to small perturbations in model specification. In our data-set, what should be unimportant changes to which variables are included and which are not included, as well as unimportant changes in the quantification of the variable to be explained, keep changing the statistical significance of the variables which interested Vollaard the most — the results which led to a media circus, societal impact, and reputational damage to several big concerns, as well as to the personal reputation of the chief herring tester Aad Taal.

Having “cleaned” the data by removing the repeat tests, and removing the outlets breaking food safety regulations, and using the AD’s classification, the size of the effects of being an Atlantic-supplied outlet, and of being distant from Rotterdam, are smaller and hardly significant. By varying 𝞮, they change. On leaving out a few of the variables whose statistical significance is smallest, whether the two main variables of interest are significant changes again. The size of the effects remains about the same: Atlantic supplied outlets score a bit higher, outlets distant from Rotterdam score a bit lower, when taking account of all the other variables in the way chosen by the analyst.

By modelling the effects of so many variables by discretization, Vollaard created multicollinearity. The results depend on arbitrarily chosen cut-offs, and other arbitrary choices. For instance, “weight” was kept numerical, but “price” was made categorical. This could have been avoided by assuming additivity and smoothness and using modern statistical methodology, but in fact the data-set is simply too small for this to be meaningful. Trying to incorporate interaction between clearly important variables caused multicollinearity and failure of the standard estimation procedures. Different model selection procedures, and nonparametric approaches, end up with finding quite different models, but do not justify preferring one to another. We can come up with several excellent (and quite simple) predictors of the final score, but we cannot say anything about causality.

Vollaard’s analyses confirmed what we knew in advance (the “taste” of the testers). There is no reason whatsoever to accuse them of favouritism. The advantage of outlets supplied by Atlantic is tiny or non-existent, certainly nothing like the huge amount which Vollaard carelessly insinuated. The distant outlets are typically new entrants to the AD Herring Test. Their clients like the kind of Dutch new herring which they have been used to in their region. Vollaard’s interpretation of his own results obtained from his own data set was unjustified. He said he was only investigating correlations, but he appeared on national TV talk shows to say that his analyses made him believe that the AD Herring Test was severely biased. This caused enormous upset, financial and reputational damage, and led to a lot of money being spent on lawyers.

Everyone makes mistakes and what’s done is done, but we do all have a responsibility to learn from mistakes. The national committee for investigating accusations of violation of scientific integrity (LOWI) did not find Vollaard guilty of gross misdemeanour. They did recommend further statistical analysis. Vollaard declined to participate. No problem. We think that the statistical experiences reported here can provide valuable pedagogical material.


In our opinion, the suggestion that the AD Herring Test was in any way biased cannot be investigated by simple regression models. The “sample” is self-recruiting and much too small. The sales outlets which join the sample are doing so in the hope of getting the equivalent of a Michelin star. They can easily know in advance what are the standards by which they will be evaluated. Vollaard’s purely descriptive and correlational study confirms exactly what everyone (certainly everyone “in the business”) should know. The AD Herring Test, over the years that it operated, helped to raise standards of hygiene and presentation, and encouraged sales outlets to get hold of the best quality Dutch New Herring, and to prepare and serve it optimally. As far as subjective evaluations of taste are concerned, the test was indubitably somewhat biased toward the tastes valued by consumers in the region of Rotterdam and The Hague, and at the main “herring port” Scheveningen. But the “taste” of the herring testers was well known. Their final scores fairly represent their public, written evaluations, as far as can be determined from the available data.

The quality of the statistical analysis performed by Ben Vollaard left a great deal to be desired. To put it bluntly, from the statistical point of view it was highly amateurish. Economists who self-publish statistical reports under the flag of their university on matters of great public interest should have their work peer-reviewed and should rapidly publish their data sets. His results are extremely sensitive to minor variations in model choice and specification, to minor variations in quantifications of verbal scores, and there is not enough data to investigate his assumption of additivity. Any small effects found could as well be attributed to model misspecification as to conscious or unconscious bias on the part of the herring testers. We are reminded of Hanlon’s razor “never attribute to malice that which is adequately explained by stupidity”. In our opinion, in this case, Ben Vollaard was actually a victim of the currently huge pressure on academics to generate media interest by publishing on issues of current public interest. This leads to immature work which does not get sufficient peer review before being fed to the media. The results can cause immense damage.

Statisticians in general should not be afraid to join in societal debates. The total silence concerning this affair from the Dutch statistical society, which even has an econometric chapter, was a shame. Fortunately, the society has recently set up a new section devoted to public outreach.

A huge amount of statistical analyses are performed and published by amateurishly matching formal properties of a data-set (types of variables, the shape of the data file) to standard statistical models with no consideration at all given to model assumptions and to checks of model assumptions. Vollaard’s data-set can provide a valuable teaching resource, and we have published a version with (English language) description of the variables. We have made two versions available: Vollaard’s data-set put together by his student, but now with outlets identified, and the newly constituted data set with Atlantic-supplied outlets according to the AD, which is as well available in our GitHub repository

It would be interesting to add to the data some earlier years’ data, and investigate whether scores of repeatedly evaluated outlets tended to increase over the years. At the very least, it would be good to know which of the year 2016 outlets were repeat participants.

Just before we are about to submit this article, we become aware of Vollaard and van Ours (2021), in which Dr Ben Vollaard made the same accusations with essentially the same false arguments.

More study must be done of the feedback processes involved in consumer research panels. The man behind the herring test: journalist Paul Hovius (r), with herring taster Aad Taal (l), during the AD Herring Test in 2013. © Joost Hoving, ANP

Conflict of interest

The second author was paid by a well-known law firm for a statistical report on Vollaard’s analyses. His report, dated April 5, 2018, appeared in English translation earlier in this blog, He also reveals that the best Dutch New Herring he ever ate was at one of the retail outlets of Simonis in Scheveningen. They got their herring from the wholesaler Atlantic. He had this experience before any involvement in the Dutch New Herring scandals, topic of this paper.


Alberto Acerbi, Stefano Ghirlanda, and Magnus Enquist. The logic of fashion cycles. PloS one, 7(3):e32541, 2012.

Anders Angerbjorn, Magnus Tannerfeldt, and Sam Erlinge. Predator–prey relationships: arctic foxes and lemmings. Journal of  Animal Ecology, 68(1):34–49, 1999.

Rafał Apriasz, Tyll Krueger, Grzegorz Marcjasz, and Katarzyna Sznajd-Weron. The hunt opinion model—an agent based approach to recurring fashion cycles. PloS one, 11(11):e0166323, 2016.

The Economist. Netherlands fishmongers accuse herring-tasters of erring. The Economist, 2017, November 25.

Ben Vollaard. Gaat de AD Haringtest om meer dan de haring? 2017a.

Ben Vollaard. Gaat de AD Haringtest om meer dan de haring? een update. 2017b.

Ben Vollaard. Scores Haringtest. 2020a.

Ben Vollaard. Stata Code Haringtest. 2020b.

Ben Vollaard and Jan C van Ours. Bias in expert product reviews. 2021. Tinbergen Institute Discussion Paper 2021-042/V.

Christopher Winship and Bruce Western. Multicollinearity and model misspecification. Sociological Science, 3(27):627–649, 2016.

Het interview

Interview in “Stentor”, gepubliceerd vorige Zaterdag

Richard Gill heeft met het weerleggen van statistisch bewijs al twee medische seriemoordenaars vrij gekregen, onder wie Lucia de Berk. © Rob Voss

Deze Apeldoornse wetenschapper redt onschuldige zusters uit de gevangenis: De mens wil helaas niet in toeval geloven

Wetenschapper Richard Gill uit Apeldoorn zorgde er mede voor dat Lucia de Berk werd vrijgesproken. Datzelfde kreeg hij voor elkaar bij een vergelijkbare zaak in Italië en nu gaat hij voor de hattrick in Engeland. Wat drijft hem? 

Anne Boer 28-05-22, 08:00

Pure wetenschappelijke nieuwsgierigheid, dat is wat hem drijft, zegt de internationaal vermaarde wiskundige Richard Gill uit Apeldoorn. Als expert op het terrein van statistieken werkte Gill (70) voor het Openbaar Ministerie en het Internationaal Strafhof. Bijna zes jaar is hij gepensioneerd en staat hij te boek als emeritus professor in de statistiek aan de Universiteit Leiden.

Met zijn kennis over het gebruik van statistieken heeft hij vanuit zijn werkkamer de onschuld kunnen aantonen van twee verpleegkundigen die waren veroordeeld voor seriemoorden: de Nederlandse Lucia de Berk en de Italiaanse Daniela Poggiali. Nu zet hij zich in voor de vrijlating van verpleegkundige Ben Geen uit Engeland.

Klinkklare onzin

Alle drie zouden tijdens hun werk patiënten hebben gedood. Lucia de Berk werd zelfs veroordeeld voor zeven moorden. De bewijslast was vooral gebaseerd op statistieken. Als Lucia werkte, zouden er meer patiënten overlijden dan tijdens de diensten van haar collega’s. Het bleek klinkklare onzin, zoals Gill het fijntjes verwoordt. ,,Kwestie van roddel en achterklap, zoeken naar een zondebok om de reputatie van het ziekenhuis te redden en aannames, terwijl er helemaal geen moord is gepleegd.’’


De mens wil helaas niet in toeval geloven, we willen een oorzaak hebben. Daarom geloven we ook in duivels en goden

Statistisch bewijs speelt een grote rol in onderzoek, ook naar seriemoordenaars in de medische wereld. ,,Maar dan moet je de cijfers wel goed interpreteren’’, vindt Gill. ,,Als er ogenschijnlijk veel mensen overlijden in een ziekenhuis, moet je eerst goed kijken naar de oorzaak. Zijn er misschien meer patiënten dan anders? Zijn ze zieker dan in andere perioden? Is de methode van registreren aangepast? Zijn er wijzigingen in de staf? Als je meteen kijkt welke verpleegkundige aanwezig was, sla je bovendien de belangrijkste vragen over: is er sprake van moord of is het medisch falen of zelfs natuurlijk overlijden?’’

Dat raakt volgens Gill meteen aan een ander pijnlijk punt. ,,Een ziekenhuis is een plek waar mensen doodgaan, maar vaak is de doodsoorzaak niet duidelijk. Dat kan leiden tot clusters van verdachte overlijdens. Je moet wel weten welke doden je telt, anders zoekt de politie bewijs voor beweringen.’’


Volgens Gill moet je altijd in gedachten houden dat er een goede, onschuldige reden kan zijn voor een gebeurtenis. ,,Kijk vooral hoe vaak iemand werkt. Fulltime verpleegkundigen maken meer doden mee dan parttimers. Als iemand fulltime werkt en ook nog gepassioneerd bezig is met haar of zijn vak, is de kans nog groter dat die persoon aanwezig is als iemand overlijdt, dan iemand die een paar dagen per week werkt of strikt de uren werkt die in het rooster staan.”

Wetenschapper Richard Gill. © Rob Voss

Nooit mag je volgens hem een rare samenloop van omstandigheden uitsluiten. ,,Die gebeuren, ook zonder moord. Beroemd is het voorbeeld van een Amerikaans stel dat op één dag in twee verschillende loterijen de hoofdprijs won. Hoe groot is de kans dat zoiets gebeurt? Het gebeurde toch echt. De mens wil helaas niet in toeval geloven, we willen een oorzaak hebben en zoeken een zondebok. Daarom geloven we ook in duivels en goden.’’


Richard Gill is geboren in Engeland. Zijn vader was ook wetenschapper. De liefde brengt hem in 1974 op 23-jarige leeftijd naar Nederland. Hij is zes jaar eerder op vakantie als een blok gevallen voor een dochter van een Nederlandse vriend van zijn vader. Beide vaders werken voor Wavin uit Hardenberg. Na wat omzwervingen belandt Gill begin jaren 80 in Apeldoorn, om er nooit meer weg te gaan. Hij woont in een oud herenhuis, in een zee van weelderig groen. Dit was het ouderlijk huis van zijn vrouw. Om financieel het hoofd boven water te houden, werkt hij extra hard om snel carrière te kunnen maken.

De medische wereld komt al vroeg op zijn pad. Na een studie wiskunde in Cambridge promoveert hij op onderzoek naar de vraag hoelang kankerpatiënten bij bepaalde behandeling overleven. Zijn rekenmethode blijkt een uitkomst en wordt inmiddels ook op andere terreinen toegepast. ,,Het kwam toevallig op mijn bord. Ik had geen onderwerp en mijn promotor haalde dit onderwerp uit zijn la. Het heeft veel impact gehad en de methode wordt nog massaal gebruikt.’’


Zijn vrouw, die historicus is, wijst hem al in een vroeg stadium op de zaak Lucia de Berk, die later veroordeeld zou worden voor zeven moorden in een ziekenhuis. ,,Zij sprak van een heksenjacht en wilde dat ik ernaar keek, zeker toen het ook een heksenproces werd, zoals ze dat noemde. Ze wees me erop dat statistiek als bewijs werd gebruikt en ik er dus wel iets van zou moeten vinden. Ik wilde niet. Er waren al ervaren statistici bij betrokken, ook mensen die ik kende.’’

Lucia de Berk reageert blij na haar vrijspraak © anp

Toen er in 2006 een boek over deze zaak verscheen, ging Gill overstag. ,,Ik werd door een collega op het boek gewezen. Ik wist werkelijk niet wat ik las, was er echt ondersteboven van. Voor mij was zonneklaar dat het vonnis niet deugde en de rechters de cijfers verkeerd hadden geïnterpreteerd.’’ 

De rest is geschiedenis. Gill hielp aantonen dat de cijfers de beschuldiging niet konden onderbouwen en Lucia werd na 6,5 jaar onterechte celstraf in 2010 volledig vrijgesproken.


Als hij in 2014 over een gelijksoortige situatie in Italië leest, besluit hij direct weer in actie te komen. Dit keer wordt een verpleegkundige (Daniela Poggiali) verdacht van zestig moorden. Gill belt zijn collega Julia Mortera van de Roma Tre-universiteit en samen bieden ze hun hulp aan. Met succes, ook deze verpleegkundige is na een eerdere veroordeling tot levenslange gevangenisstraf sinds oktober op vrije voeten.

Statistiek is de wetenschap en de techniek van het verzamelen, bewerken, interpreteren en presenteren van gegevens. Statistische methoden worden gebruikt om grote hoeveelheden gegevens – bijvoorbeeld over het koopgedrag van mensen, de huizenmarkt of het aantal doden in de zorg – om te zetten in bruikbare informatie.

,,De statistiek in deze zaak was totaal amateuristisch, het deugde niet. De aanklagers beweerden dat er meer sterfgevallen waren als Daniela werkte. Tot het moment dat ze werd gearresteerd: toen daalde het plotseling. Wij ontdekten dat het sterftecijfer bij alle personeelsleden hoog was. Daniela was vaak al voor het begin van haar ingeroosterde dienst aanwezig en bleef vaak ook nog helpen nadat haar dienst voorbij was. Daardoor was ze vaker aanwezig als een patiënt stierf. Dat het aantal doden daalde nadat Daniela werd gearresteerd, is simpel te verklaren. Het nieuws over de ‘moordzuster’ was breed uitgemeten in de media. Als gevolg daarvan trok het ziekenhuis minder patiënten. Minder patiënten betekent ook minder sterfgevallen.’’

Lastige kluif

Gill doet nu onderzoek naar de zware beschuldigingen tegen de Engelse verpleegkundige Ben Geen. Dat gebeurt op verzoek van zijn advocaat. Het is vooralsnog een heel lastige kluif, vooral omdat het rechtssysteem in Engeland anders in elkaar zit. Opnieuw is Gill ervan overtuigd dat de verdachte geen moorden heeft gepleegd en dat het recht moet zegevieren.

Uit deze zaken heeft hij belangrijke lessen getrokken die hij wil overbrengen aan iedereen die wereldwijd betrokken is bij de rechtspraak, van advocaten tot rechters en van officieren van justitie tot juryleden. Samen met andere experts schrijft hij een handleiding hoe statistiek in de rechtbank kan worden gebruikt, met name bij strafprocessen tegen zogeheten seriemoordenaars in de gezondheidszorg. Dat gebeurt onder supervisie van het gezaghebbende instituut Royal Statistical Society. Het boek moet later dit jaar verschijnen.


Je kunt aannemen dat een hond vier poten heeft, maar niet dat alles met vier poten een hond is. Als op die manier naar Lucia was gekeken, was ze nooit veroordeeld

Richard Gill

De boodschap die hij heeft, is in hoofdlijnen simpel: gebruik statistische gegevens pas als je je ervan hebt verzekerd dat ze kloppen en gebruik ze goed. ,,Benoem alle factoren. Trek niet te snel conclusies. Vraag onafhankelijke experts om hulp. Onderzoek alle mogelijkheden.’’ Volgens Gill is niet alleen expertise van professionals nodig, maar ook dat rechters en advocaten worden geschoold in een goede interpretatie van statistieken.

Vier poten

Hij geeft een simpel voorbeeld. ,,Je kunt aannemen dat een hond vier poten heeft, maar niet dat alles met vier poten een hond is. Je mag aannemen dat iemand uit Peru Spaans spreekt, maar niet iedereen die Spaans spreekt komt uit Peru. Als op die manier naar Lucia was gekeken, was ze nooit veroordeeld.’’

Kenmerkend vindt Gill dat de verdachten die hij hielp, allemaal opvallende mensen zijn. Ze werkten hard, hadden een duidelijke mening, stootten daardoor waarschijnlijk ook leidinggevenden voor het hoofd en eindigden uiteindelijk als zondebok. ,,Het heeft me echt getroffen hoeveel ze gemeen hebben. Ben Geen wilde legerarts worden en was enorm gedreven in zijn werk. Hij zag zijn werk als meer dan een baan en deed veel extra als het kon. Hij botste ook met managers omdat het ziekenhuis voortdurend tegen grenzen aanliep.’’


Als expert op het terrein van statistieken werkte Gill ook voor het Openbaar Ministerie (moordzaak Tamara Wolvers) en het Internationaal Strafhof (moordaanslag president Libanon). Inmiddels is hij al bijna zes jaar gepensioneerd, maar tijd om zich te vervelen, heeft hij niet. Er ligt nog voor jaren werk op zijn bordje. Puzzels die hij graag helpt oplossen. 

Daarnaast zijn er veel onderwerpen waar hij graag in zou willen duiken, zoals de geruchtmakende Deventer moordzaak, die hem al jaren mateloos intrigeert. ,,Ik houd het nog steeds voor mogelijk dat de veroordeelde Ernest Louwes onschuldig is. Met name de dna-sporen op de blouse van de vermoorde weduwe vind ik interessant. Dna is ook statistisch bewijs en statistiek vertelt ons hoe je met onzekerheden moet omgaan. Er zijn inmiddels nieuwe moleculairbiologische methoden om veel meer uit een spoor te halen.’’


Gill helpt Kamerlid Pieter Omtzigt met het analyseren van data over uithuisplaatsingen als gevolg van het toeslagenschandaal. ,,We maken een tijdlijn om oorzaak en gevolg in beeld te krijgen. Ik heb dus eigenlijk helemaal geen tijd meer om nog meer verpleegkundigen achter de tralies vandaan te halen’’, zegt hij met een glimlach.

Kamerlid Pieter Omtzigt. © ANP

Als er toch weer een zaak van een vermeende moordzuster op zijn pad komt, zal hij waarschijnlijk moeilijk ‘nee’ kunnen zeggen. Hij geniet van het puzzelen en wil voorkomen dat het leven saai wordt. De tekst op de achterkant van zijn trui spreekt wat dat betreft misschien wel boekdelen: ‘Keep calm, en deze opa lost het wel op’. Want ja. Gill, vader van drie kinderen, is opa en zijn vijf kleinkinderen logeren graag bij hem en zijn vrouw in Apeldoorn.


Een van de vele puzzels die hem al jaren bezighoudt en soms zelfs uit zijn slaap haalt, moet hij van zichzelf oplossen: de zaak José Booij, die achttien jaar geleden werd geconfronteerd met de uithuisplaatsing van haar zes weken oude baby Julia-Lynn. 

,,Een onvoorstelbaar en afschuwelijk verhaal. Zij is vermalen door het systeem en daar compleet aan onderdoor gegaan. Ik ben het contact met José verloren, maar nog steeds in het bezit van een doos met persoonlijke spullen van haar, zoals kindertekeningen, diploma’s, dagboeken en krantenknipsels over haar strijd voor haar kind tot de hoogste rechtsorganen in Nederland en Europa aan toe. Wellicht leeft Julia-Lynn nu onder een andere naam en wellicht weet ze haar geboortenaam niet eens. Ik wil dat ze weet wie haar moeder is. Dat die nooit heeft opgegeven. Daar heeft ze recht op. Ik hoop haar ooit te vinden en de spullen van haar moeder te kunnen geven. En weet je, ook deze vrouw is een bijzonder mens, anders dan anderen.’


Published in “The Stentor” last Saturday, auto-translated from the original Dutch by Google Translate. This is the raw version. Corrections still to be made!

Richard Gill has already released two medical serial killers, including Lucia de Berk, by refuting statistical evidence. © Rob Voss

This Apeldoorn scientist saves innocent sisters from prison: Unfortunately, people do not want to believe in coincidence

Scientist Richard Gill from Apeldoorn helped ensure that Lucia de Berk was acquitted. He achieved the same in a similar case in Italy and now he is going for the hat trick in England. What drives him? 

Anne Boer 28-05-22, 08:00

Pure scientific curiosity is what drives him, says the internationally renowned mathematician Richard Gill from Apeldoorn. As an expert in statistics, Gill (70) worked for the Public Prosecution Service and the International Criminal Court. He has been retired for almost six years and is known as emeritus professor of statistics at Leiden University.

With his knowledge of the use of statistics, he was able to prove the innocence of two nurses who were convicted of serial murders from his office: the Dutch Lucia de Berk and the Italian Daniela Poggiali. Now he is campaigning for the release of nurse Ben Geen from England.

Sheer nonsense

All three are said to have killed patients on the job . Lucia de Berk was even convicted of seven murders. The burden of proof was mainly based on statistics. If Lucia worked, more patients would die than during her colleagues ‘ shifts. It turned out to be sheer nonsense, as Gill puts it delicately. “A matter of gossip and backbiting, looking for a scapegoat to save the hospital’s reputation and assumptions when no murder was committed at all. †


Unfortunately, people do not want to believe in coincidence, we want to have a cause. That’s why we also believe in devils and gods

Statistical evidence plays a major role in research, including serial killers in the medical world. ,,But then you have to interpret the figures properly ” , says Gill. “If apparently many people die in a hospital, you first have to look closely at the cause. Are there perhaps more patients than usual ? Are they sicker than in other periods? Has the registration method been adjusted? Are there changes in the staff? If you immediately look at which nurse was present, you also skip the most important questions: is it murder or is it medical failure or even natural death? †

According to Gill, that immediately touches on another sore point. “A hospital is a place where people die, but often the cause of death is not clear. This can lead to clusters of suspicious deaths. You have to know which deaths you count, otherwise the police will look for evidence for claims. †


According to Gill, you should always keep in mind that there can be a good, innocent reason for an event. “Look at how often someone works. Full-time nurses experience more deaths than part-time nurses. If someone works full-time and is passionate about their craft, they’re even more likely to be there when someone dies than someone who works a few days a week or works strictly within the schedule. †

Scientist Richard Gill. © Rob Voss

According to him, you should never rule out a strange combination of circumstances. “They happen, even without murder. A famous example is an American couple who won the top prize in two different lotteries in one day . What are the chances of something like this happening? It really happened. Unfortunately, people do not want to believe in coincidence, we want to have a cause and look for a scapegoat. That is why we also believe in devils and gods. †


Richard Gill was born in England. His father was also a scientist. Love brought him to the Netherlands in 1974 at the age of 23. Six years earlier, on holiday, he had fallen head over heels for a daughter of a Dutch friend of his father’s. Both fathers work for Wavin from Hardenberg. After some wanderings, Gill ends up in Apeldoorn in the early 1980s, never to leave. He lives in an old mansion, in a sea of lush greenery. This was his wife’s childhood home. To keep his head above water financially, he works extra hard to make a quick career .

The medical world crosses his path early on. After studying mathematics at Cambridge, he obtained his doctorate for research into the question of how long cancer patients survive with a particular treatment. His calculation method turned out to be a godsend and is now also being applied in other areas. “It just happened to be on my plate. I didn’t have a topic and my promoter took this topic out of his drawer. It has had a lot of impact and the method is still widely used. †

witch hunt

His wife, who is a historian, points him at an early stage to the case of Lucia de Berk, who would later be convicted of seven murders in a hospital. “She spoke of a witch hunt and wanted me to watch it, especially when it became a witch trial, as she called it. She pointed out to me that statistics were used as evidence and so I should have something to say about it. I did not want to. Experienced statisticians were already involved, including people I knew. †

Lucia de Berk reacts happy after her acquittal © anp

When a book about this case was published in 2006, Gill took the plunge. “I was referred to the book by a colleague. I really didn’t know what I was reading, I was really blown away by it. It was crystal clear to me that the verdict was wrong and that the judges had misinterpreted the figures . †

The rest is history. Gill helped show that the figures failed to substantiate the accusation and Lucia was fully acquitted in 2010 after 6.5 years of wrongful imprisonment.


reads about a similar situation in Italy in 2014 , he immediately decides to jump back into action. This time, a nurse (Daniela Poggiali) is suspected of sixty murders. Gill calls his colleague Julia Mortera from Roma Tre University and together they offer their help. With success, this nurse has also been free since October after a previous sentence to life imprisonment.

Statistics is the science and technology of collecting, processing, interpreting and presenting data. Statistical methods are used to convert large amounts of data – for example about people’s purchasing behaviour, the housing market or the number of deaths in care – into useful information .

,,The statistics in this case were completely amateurish, it was not good. Prosecutors claimed there were more deaths when Daniela worked. Until she was arrested: then it suddenly dropped. We found that the death rate for all staff was high. Daniela was often present before the start of her scheduled shift and often continued to help after her shift was over. As a result, she was more often present when a patient died . It is easy to explain that the number of deaths decreased after Daniela was arrested. The news about the ‘ murder sister ‘ was widely covered in the media. As a result, the hospital attracted fewer patients . Fewer patients also means fewer deaths. †

Difficult bone

Gill is now investigating the serious allegations against English nurse Ben Geen. This is done at the request of his lawyer. It is still a very difficult task, especially because the legal system in England is different. Once again, Gill is convinced that the suspect committed no murders and that justice must prevail.

He has learned important lessons from these cases that he wants to pass on to everyone involved in the justice system worldwide, from lawyers to judges and from prosecutors to jurors. Together with other experts, he is writing a manual on how to use statistics in court, especially in criminal proceedings against so-called serial killers in healthcare. This is done under the supervision of the authoritative Royal Statistical Society. The book is due out later this year.


You can assume that a dog has four legs, but not everything with four legs is a dog. If Lucia had been looked at that way, she would never have been judged

Richard Gill

The message he has is basically simple: do not use statistical data until you have ensured that they are correct and use them well. “Name all the factors. Don’t jump to conclusions too quickly. Ask independent experts for help. Explore all possibilities. ” According to Gill, not only expertise from professionals is needed, but also that judges and lawyers are trained in a good interpretation of statistics.

Four legs

He gives a simple example. “You can assume that a dog has four legs, but not that everything with four legs is a dog. You can assume that someone from Peru speaks Spanish, but not everyone who speaks Spanish is from Peru. If Lucia had been looked at that way, she would never have been convicted. †

Typically, Gill finds that the suspects he helped are all striking people. They worked hard, had strong opinions, probably offended executives as a result, and ended up as scapegoats. “It really struck me how much they have in common. Ben Geen wanted to be an army doctor and was very passionate about his work. He saw his work as more than a job and did a lot extra when he could. He also clashed with managers because the hospital was constantly running into limits. †

criminal court

As an expert in statistics, Gill also worked for the Public Prosecution Service (Tamara Wolvers murder case) and the International Criminal Court (Lebanon Presidential assassination attempt). He has now been retired for almost six years, but he has no time to get bored. There is still work on his plate for years to come. Puzzles he likes to help solve. 

In addition, there are many subjects that he would like to delve into, such as the controversial Deventer murder case, which has intrigued him immensely for years. “I still think it possible that the convicted Ernest Louwes is innocent. I find the DNA traces on the murdered widow’s blouse particularly interesting. DNA is also statistical evidence and statistics tells us how to deal with uncertainties. There are now new molecular biological methods to get much more out of a track. †


Gill helps MP Pieter Omtzigt with analyzing data about custodial placements as a result of the benefits scandal. ,,We make a timeline to get a picture of cause and effect. So I don’t really have time to get any more nurses out of prison , ” he says with a smile.

Member of Parliament Pieter Omtzigt. © ANP

If a case of an alleged murder sister comes his way, he will probably have a hard time saying ‘ no ‘ . He enjoys puzzling and wants to prevent life from getting boring. The text on the back of his sweater might speak volumes in that regard: ‘ Keep calm, and this grandpa will solve it ‘ . Because yes. Gill, a father of three, is a grandfather and his five grandchildren like to stay with him and his wife in Apeldoorn.


He has to solve one of the many puzzles that has occupied him for years and sometimes even wakes him up: the case of Jos é Booij, who was confronted eighteen years ago with the custodial placement of her six-week-old baby Julia-Lynn.

“An unbelievable and horrifying story. She was crushed by the system and completely destroyed by it. I have lost contact with José , but I still have a box with her personal items, such as children’s drawings, diplomas , diaries and newspaper clippings about her fight for her child up to the highest courts in the Netherlands and Europe. . Julia-Lynn may be living under a different name now and may not even know her birth name. I want her to know who her mother is. That he never gave up. She is entitled to that. I hope one day to find her and give her mother’s things. And you know, this woman is also a special person, different from others. †

An Italian CSI drama: social media, a broken legal system, and Micky Mouse statistics

Daniela Poggiali, on the day of her final (?) release, 25 October 2021.
Photo: ©Giampiero Corelli

The title of this blog might refer to the very, very famous trials of Amanda Knox in the case of the murder of Meredith Kercher. However, I am writing about a case that is much less known outside of Italy (neither victim nor alleged murderer was a rich American girl). This is the case of Daniela Poggiali, a nurse suspected by the media and accused by prosecution experts of having killed around 90 patients in a two-year killing spree terminated by her arrest in April 2014. She has just been exonerated after a total of three years in prison with a life sentence as well some months of pre-trial detention. This case revolved around statistics of an increased death rate during the shifts of a colourful nurse. I was a scientific expert for the defence, working with an Italian colleague, Julia Mortera (Univ. Rome Tre), later assisted by her colleague Francesco Dotto.

Piet Groeneboom and I worked together on the statistics of the case of Lucia de Berk, see our paper in Chance [Reference]. In fact, it was remarkable that the statistical community in the Netherlands got so involved in that case. A Fokke and Sukke cartoon entitled “Fokke and Sukke know it intuitively” had the exchange “The probability that almost all professors of statistics are in agreement … is obviously very small indeed”.

Fokke and Sukke do not believe that this is a coincidence.

Indeed, it wasn’t. That was one of the high points of my career. Another was Lucia’s final acquittal in 2010, at which the judges took the trouble to say out loud, in public, that the nurses had fought heroically for the lives of their patients; lives squandered, they added, by their doctors’ medical errors.

At that point, I felt we had learnt how to fight miscarriages of justice like that, of which I rapidly became involved in several. So far, however, with rather depressing results. Till a couple of months ago. This story will not have much to do with mathematics. It will have to do with simple descriptive statistics, and I will also mention the phrases “p-value” and “Bayes’ rule” a few times. One of the skills of a professional statistician is the abstraction of messy real-world problems involving chance and data. It’s not for everybody. Many mathematical statisticians prefer to prove theorems, just like any other mathematician. In fact, I often do prefer to do that myself, but I like more being able to alternate between the two modes of activity, and I do like sticking my nose into other people’s business, and learning about what goes on in, for instance, law, medicine, or anything else. Each of the two activity modes is a nice therapy for the frustrations which inevitably come with the other.

The Daniela Poggiali case began, for me, soon after the 8th of April, 2014, when it was first reported in international news media. A nurse at the Umberto I hospital in the small town of Lugo, not far from Ravenna, had been arrested and was being investigated for serial murder. She had had photos of herself taken laughing, close to the body of a deceased patient, and these “selfies” were soon plastered over the front pages of tabloid media. Pretty soon, they arrived in The Guardian and The New York Times. The newspapers sometimes suggested she had killed 93 patients, sometimes 31, sometimes it was other large numbers. It was suspected that she had used Potassium Chloride on some of those patients. An ideal murder weapon for a killer nurse since easily available in a hospital, easy to give to a patient who is already hooked up to an IV drip, kills rapidly (cardiac arrest – it is used in America for executions), and after a short time hard to detect. After death, it redistributes itself throughout the body where it becomes indistinguishable from a normal concentration of Potassium.

An IV drip. ©Stefan Schweihofer users/StefanSchweihofer

Many features of the case reminded me strongly of the case of Lucia de Berk in the Netherlands. In fact, it seemed very fishy indeed. I found the name of Daniela’s lawyer in the online Italian newspapers, Google found me an email address, and I sent a message offering support on the statistics of the case. I also got an Italian statistician colleague and good friend, Julia Mortera, interested. Daniela’s lawyer was grateful for our offer of help. The case largely hinged on a statistical analysis of the coincidence between deaths on a hospital ward and Daniela’s shifts there. We were emailed pdfs of scanned pages of a faxed report of around 50 pages containing results of statistical analyses of times of shifts of all the nurses working on the ward, and times of admission and discharge (or death) of all patients, during much of the period 2012 – 2014. There were a further 50 pages (also scanned and faxed) of appendices containing print-outs of the raw data submitted by hospital administrators to police investigators. Two huge messy Excel spreadsheets.

The authors of the report were Prof. Franco Tagliaro (Univ. Verona) and Prof. Rocco Micciolo (Univ. Trento). The two are respectively a pathologist/toxicologist and an epidemiologist. The epidemiologist Micciolo is a professor in a social science department, and member of an interfaculty collaboration for the health sciences. We found out that the senior and more influential author Tagliaro had published many papers on toxicology in the forensic science literature, usually based on empirical studies using data sets provided by forensic institutes. Occasionally, his friend Micciolo turned up in the list of authors and had supplied statistical analyses. Micciolo describes himself as a biostatistician. He has written Italian language textbooks on exploratory data-analysis with the statistical package “R” and is frequently the statistician-coauthor of papers written by scientists from his university in many different fields including medicine and psychology. They both had decent H-indices, their publications were in decent journals, their work was mainstream, useful, “normal science”. They were not amateurs. Or were they?

Daniela Poggiali worked on a very large ward with very many very old patients, many suffering terminal illnesses. Ages ranged from 50 up to 105, mostly around ninety. The ward had about 60 beds and was usually quite fully occupied. Patients tended to stay one to two weeks in the hospital, admitted to the hospital for reasons of acute illness. There was on average one death every day; some days none, some days up to four. Most patients were discharged after several weeks in the hospital to go home or to a nursing home. It was an ordinary “medium care” nursing department (i.e., not an Intensive Care unit).

The long building at the top: “Block B” of Umberto I hospital, Lugo

Some very simple statistics showed that the death rate on days when Poggiali worked was much higher than on days when she did not work. A more refined analysis compared the rate of deaths during the hours she worked with the rate of deaths during the hours she was not at work. Again, her presence “caused” a huge excess, statistically highly significant. A yet more refined analysis compared the rate of deaths while she was at work in the sectors where she was working with the rate in the opposite sectors. What does this mean? The ward was large and spread over two long wings of one floor of a large building, “Blocco B”, probably built in the sixties.

Sector B of “Blocco B” (Google Streetview). Seen from the North.

Between the two wings were central “supporting facilities” and also the main stairwell. Each wing consisted of many rooms (each room with several beds), with one long corridor through the whole building, see the floor plan below. Sector A and B rooms were in one wing, first A and then B as you you went down the corridor from the central part of the floor. Sector C and Sector D rooms were in the other wing, opposite to one another on each side of the corridor. Each nurse was usually detailed in her shifts to one sector, or occasionally to the two sectors in one wing. While working in one sector, a nurse could theoretically easily slip into a room in the adjacent sector. Anyway, the nurses often helped one another, so they often could be found in the “wrong sector”, but not often in the “wrong wing”.

Tagliaro and Micciolo (in the sequel: TM) went on to look at the death rates while Daniela was at work in different periods. They noticed that it was higher in 2013 than in 2012, even higher in the first quarter of 2014, then – after Daniela had been fired – it was much, much less. They conjectured that she was killing more and more patients as time went by, till the killing stopped dead on her suspension and arrest

TM certainly knew that, in theory, other factors might be the cause of an increased death rate on Poggiali’s shifts. They were proud of their innovative approach of relating each death that occurred while Daniela was at work to whether it occurred in Daniela’s wing or in the other. They wrote that in this way they had controlled for confounders, taking each death to provide its own “control”. (Similarly, in the case of Lucia de B., statistician Henk Elffers had come up with an innovative approach. In principle, it was not a bad idea, though all it showed was that nurses are different). TM did not control for any other confounding factors at all. In their explanation of their findings to the court, they repeatedly stated categorically that the association they had found must be causal, and Daniela’s presence was the cause. Add to this that their clumsy explanation of p-values might have misled lawyers, journalists and the public. In such a case, a p-value is the probability of what you see (more precisely, of at least what you see), assuming pure chance. That is not the same as the probability that pure chance was the cause of what you see – the fallacy of the transposed conditional, also known as “the prosecutor’s fallacy”.

Exercise to the reader: when is this fallacy not a fallacy? Hint: revise your knowledge of Bayes’ rule: posterior odds equals prior odds time likelihood ratio.

Bayes rule in odds form. p and d stand for “prosecution” and “defence” respectively, H stands for “Hypothesis”

We asked Tagliaro and Micciolo for the original Excel spreadsheets and for the “R” scripts they had used to process the data. They declined to give them to us, saying this would not be proper since they were confidential. We asked Daniela’s lawyer to ask the court to ask for those computer files on our behalf. The court declined to satisfy our request. We were finally sent just the Excel files by the hospital administration, a week before we were called to give evidence. Fortunately, with a combination of OCR and a lot of painstaking handwork, a wealthy friend of Daniela’s lawyer had already managed to help us get the data files reconstructed. We performed a lot of analyses with the help of a succession of students because extracting what we needed from those spreadsheets was an extraordinarily challenging issue. One kept finding anomalies that had to be fixed in one way or another. Even when we had “clean” spreadsheets, it still was a mess.

Next, we started looking for confounding factors that might explain the difference between Daniela and her colleagues, which certainly was striking and real. But was it perhaps entirely innocent?

Minute, hour, weekday, month of deaths

First of all, simple histograms showed that death rates on that ward varied strongly by month, with big peaks in June and again in December. (January is not high: elderly people stay home in January and keep themselves warm and safe). That is what one should expect. The humid heat and air pollution in the summer; or the damp and cold and the air pollution in the winter, exacerbated by winter flu epidemics. Perhaps Daniela worked more at bad times than at good times? No. It was clear that sectors A+B were different from C+D. Death rates were different, but also the number of beds in each wing was different. Perhaps Daniela was allocated more often to “the more difficult” sections? It was not so clear. Tagliaro and Micciolo computed death rates for the whole ward, or for each wing of the ward, but never took account of the number of patients in each wing nor of the severity of their illnesses.

Most interesting of all was what we found when we looked at the hour of the time of death of patients who died, and the minute of the time of death of patients who died. Patients tended to die at times which were whole hours, “half past” was also quite popular. There was however also a huge peak of deaths between midnight and five minutes past midnight! There were fewer deaths in a couple of hours soon after lunchtime. There were large peaks of deaths around the time of handover between shifts: 7:00 in the morning, 2:00 in the afternoon, 9:00 in the evening. The death rate is higher in the morning than in the afternoon, and higher in the afternoon than at night. When you’re dying (but not in intensive care, when it is very difficult to die at all) you do not die in your sleep at night. You die in the early morning as your vital organs start waking up for the day. Now, also not surprisingly, the number of nurses on a ward is largest in the morning when there is a huge amount of work to do; it’s much less in the afternoon and evening, and it’s even less at night. This means that a full-time nurse typically spends more time in the hospital during morning shifts than during afternoon shifts, and more time during afternoon shifts than during night shifts. The death rate shows the same pattern. Therefore, for every typical full-time nurse, the death rate while they are at work tends to be higher than when they are not at work!

Nurses aren’t authorized to officially register times of death. Only a doctor is authorized to do that. He or she is supposed to write down the time at which they have determined the patient is no longer alive. It seems that they often round that time to whole or half hours. The peak just after midnight is hard to explain. The date of death has enormous financial and legal consequences. The peak suggests that those deaths may have occurred anywhere in a huge time window. Whether or not doctors come to the wards on the dot at midnight and fill in forms for any patients who have died in the few hours before is hard to believe

What is now clear is that it is mainly around the hand-over between shifts that deaths get “processed”. Quite a few times of death are so hard to know that they are shunted to five minutes past midnight; many others are located in the hand-over period but might well have occurred earlier.

Some nurses tend to work longer shifts than others. Some conscientiously clock in as early as they are allowed, before their shift starts, and clock out as late as they can after their shift ends. Daniela was such a nurse. Her shifts were indeed statistically significantly longer than those of any of her colleagues. She very often stayed on duty several hours after the official end of the official ten-minute overlap between shifts. There was often a lot to do – one can imagine often involving taking care of the recently deceased. Not the nicest part of the job. Daniela was well known to be a rather conscientious and very hard worker, with a fiery temper, known to play pranks on colleagues or to loudly disagree with doctors for whom she had a healthy disrespect.

Incidentally, the rate of admissions to Umberto I hospital tumbled down after the news broke of a serial killer – and the news broke the day after the last day the serial killer was at work, together with the publication of the lurid “selfie”. The rate of deaths was slowly increasing over the two years up to then, as was in fact also the rate of admissions and the occupancy of the ward. A hospital getting slowly more stressed? Taking on more work?

If one finds a correlation between X and Y, it is a sound principle to suppose that it has a causal explanation. Maybe X causes Y, maybe Y causes X, … and maybe W causes both X and Y, or maybe X and Y both cause Z and there has been a selection on the basis of Z. In the case of Lucia de B., her association between inexplicable incidents and her presence on the ward was caused by her, since the definition of “unexpected and inexplicable incident” included her being there. She was already known to be a weird person, and it was already clear that there were more deaths than usual on her ward. The actual reason for that was a change of hospital policy, moving patients faster from intensive care to medium care so that they could die at home, rather than in the hospital. If she was not present, then the medical experts always could come up with an explanation for why that death, though perhaps a bit surprising at that moment, was expected to occur soon anyway. But if Lucia was there then they were inclined to believe in foul play because after all there were so many incidents in her shifts.

Julia and I are certain that the difference between Daniela’s death rates and those of other nurses is to a huge extent explainable by the anomalies in the data which we had discovered and by her long working hours.

Some residual difference could be due to the fact that a conscientious nurse actually notices when patients have died, while a lazy nurse keeps a low profile and leaves it to her colleagues to notice, at hand-over. We have been busy fitting sophisticated regression models to the data but this work will be reported in a specialist journal. It does not tell us more than what I have already said. Daniela is different from the other nurses. All the nurses are different. She is extreme in a number of ways: most hours worked, longest shifts worked. We have no idea how the hospital allocated nurses to sectors and patients to sectors. We probably won’t get to know the answer to that, ever. The medical world does not put out its dirty washing for everyone to see.

We wrote a report and gave evidence in person in Ravenna in early 2015. I did not have time to see the wonderful Byzantine mosaics though I was treated to some wonderful meals. I think my department paid for my air ticket. Julia and I worked “pro deo“. In our opinion, we totally shredded the statistical work of Tagliaro and Micciolo. The court however did not agree. “The statistical experts for the defence only offered a theoretical discourse while those of the prosecution had scientifically established hard facts”. In retrospect, we should have used stronger language in our report. Tagliaro and Micciolo stated that they had definitively proven that Daniela’s presence caused 90 or so extra deaths. They stated that this number could definitely not be explained as a chance fluctuation. They stated that, of course, the statistics did not prove that she had deliberately murdered those patients. We, on the other hand, had used careful scientific language. One begins to understand how it is that experts like Tagliaro and Micciolo are in such high demand by public prosecutors.

There was also toxicological evidence concerning one of the patients and involving K+ Cl–, but we were not involved in that. There was also the “selfie”, there was character evidence. There were allegations of thefts of patients’ personal jewellery. It all added up. Daniela was convicted of just one murder. The statistical evidence provided her motive: she just loved killing people, especially people she didn’t like. No doubt, a forensic psychologist also explained how her personality fitted so well to the actions she was alleged to have done.

Rapidly, the public prosecution started another case based largely on the same or similar evidence but now concerning another patient, with whom Daniela had had a shouting match, five years earlier. In fact, this activity was probably triggered by families of other patients starting civil cases against the hospital. It would also clearly be in the interest of the hospital authorities to get new criminal proceedings against Daniela started. However, Daniela’s lawyers appealed against her first conviction. It was successfully overturned. But then the court of cassation overturned the acquittal. Meantime, the second case led to a conviction, then acquittal on appeal, then cassation. All this time Daniela was in jail. Cassations of cassations meant that Daniela had to be tried again, by yet another appeal court, for the two alleged murders. Julia and I and her young colleague Francesco Dotto got to work again, improving our arguments and our graphics and our formulations of our findings.

At some point, triggered by some discussions with the defence experts on toxicology and pathology, Julia took a glance at Tagliaro’s quite separate report on the toxicological evidence. This led to a breakthrough, as I will now explain.

Tagliaro knew the post-mortem “vitreous humour” potassium concentration of the last patient, a woman who had died on Daniela’s last day. That death had somehow surprised the hospital doctors, or rather, as it later transpired, it didn’t surprise them at all: they had already for three months been looking at the death rates while Daniela was on duty and essentially building up a dossier against her, just waiting for a suitable “last straw”! Moreover, they already had their minds on K+ Cl-, since some had gone missing and then turned up in the wrong place. Finally, Daniela had complained to her colleagues about the really irritating behaviour of that last patient, 73-year-old Rosa Calderoni.

“Vitreous humour” is the transparent, colourless, gelatinous mass that fills your eyeballs. While you are alive, it has a relatively low concentration of potassium. After death, cell walls break down, and potassium concentration throughout the body equalises. Tagliaro had published papers in which he studied the hourly rate of increase in the concentration, using measurements on the bodies of persons who had died at a known time of causes unrelated to potassium chloride poisoning. He even had some fresh corpses on which he could make repeated measurements. His motivation was to use this concentration as a tool to determine the PMI (post-mortem interval) in cases when we have a body and a post-mortem examination but no time of death. In one paper (without Micciolo’s aid) he did a regression analysis, plotting a straight line through a cloud of points (y = concentration, x = time since death). He had about 60 observations, mostly men, mostly rather young. In a second paper, now with Micciolo, he fitted a parabola and moreover noted that there was an effect of age and of sex. The authors also observed the huge variation around that fitted straight line and concluded that the method was not reliable enough for use in determining the PMI. But this did not deter Tagliaro, when writing his toxicological report on Rosa Calderoni! He knew the potassium concentration at the time of post-mortem, he knew exactly when she died, he had a number for the natural increase per hour after death from his first, linear, regression model. With this, he calculated the concentration at death. Lo and behold: it was a concentration which would have been fatal. He had proved that she had died of potassium chloride poisoning.

Prediction of vitreous humour K+ concentration 56 hours after death without K+ poisoning

Julia and Francesco used the model of the second paper and found out that if you would assume a normal concentration at the time of death, and take account of the variability of the measurements and of the uncertainty in the value of the slope, then the concentration observed at the time of post-mortem was maybe above average, but not surprisingly large at all.

Daniela Poggiali became a free woman. I wish her a big compensation and a long and happy life. She’s quite a character.

Aside from the “couleur locale” of an Italian case, this case had incredibly much similarity with the case of Lucia de Berk. It has many similarities with quite a few other contested serial killer nurse cases, in various countries. According to a NetFlix series, in which a whole episode is devoted to Daniela, these horrific cases occur all the time. They are studied by criminologists and forensic psychologists, who have compiled a list of “red flags” intended to help warn hospital authorities. The scientific term here is “health care serial killer”, or HCSK. One of the HCSK red flags is that you have psychiatric problems. Another is that your colleagues think you are really weird. Especially when your colleagues call you an angel of death, that’s a major red flag. The list goes on. These lists are developed in scientific publications in important mainstream journals, and the results are presented in textbooks used in university criminology teaching programs. Of course, you can only scientifically study convicted HCSKs. Your sources of data are newspaper reports, judges’ summings up, the prosecution’s final summary of the case. It is clear that these red flags are the things that convince judges and jurors to deliver a guilty verdict. These are the features that will first make you a suspect, which police investigators will look for, and which will convince the court and the public of your guilt. Amusingly, one of the side effects of the case of Lucia de Berk was contributing a number of entries to this list, for instance, the Stephen King horror murder novels she had at home which were even alleged to have been stolen from the library. Her conviction for the theft of several items still stands. As does Daniela’s: this means that Daniela is not eligible for compensation. In neither case was there any real proof of thefts. Amusingly, one of the side effects of the case of Lucia de Berk was contributing a number of entries to this list. Embarrassingly, her case had to be removed from the collections of known cases after 2011, and the criminologists and forensic psychologists also now mention that statistical evidence of many deaths during the shifts of a nurse is not actually a very good red flag. They have learnt something, too.

Interesting is also the incidence of these cases: less than 1 in a million nurses killing multiple patients per year, according to these researchers. These are researchers who have the phenomenon of HCSKs as their life work, giving them opportunities to write lurid books on serial murder, appear in TV panels and TV documentaries explaining the terrible psychology of these modern-day witches, and to take the stand as prosecution witnesses. Now, that “base rate” is actually rather important, even if only known very roughly. It means that such crimes are very, very unusual. In the Netherlands, one might expect a handful of cases per century; maybe on average 100 deaths in a century. There are actually only about 100 murders altogether in the Netherlands per year. On the other hand, more than 1000 deaths every year are due to medical errors. That means that evidence against a nurse suspected of being a HCSK would be very strong indeed before it should convince a rational person that they have a new HCSK on their hands. Lawyers, judges, journalists and the public are unfortunately perhaps not rational persons. They are certainly not good with probability, and not good with Bayes’ rule. (It is not allowed to be used in a UK criminal court, because judges have ruled that jurors cannot possibly understand it).

I am still working on one UK case, Ben Geen. I believe it is yet another example of a typical innocent HCSK scare in a failing hospital leading to a typical unsafe conviction based largely on the usual red flags and a bit of bad luck. At least, I see no reason whatsoever to suppose that Ben Geen was guilty of the crimes for which he is sitting out a life sentence. Meanwhile, a new case is starting up in the UK: Lucy (!) Letby. I sincerely hope not to be involved with that one.

Time for a new generation of nosy statisticians to do some hard work.


Norman Fenton, Richard D. Gill, David Lagnado, and Martin Neil. Statistical issues in serial killer nurse cases.

Alexander R.W. Forrest. Nurses who systematically harm their patients. Medical Law International, 1(4): 411–421, 1995.

Richard D. Gill, Piet Groeneboom, and Peter de Jong. Elementary statistics on trial—the case of Lucia de Berk. CHANCE, 31(4):9–15, 2018.

Covadonga Palacio, Rossella Gottardo, Vito Cirielli, Giacomo Musile, Yvane Agard, Federica Bortolotti, and Franco Tagliaro. Simultaneous analysis of potassium and ammonium ions in the vitreous humour by capillary electrophoresis and their integrated use to infer the post mortem interval (PMI). Medicine, Science and the Law, 61(1 suppl):96–104, 2021.

Nicola Pigaiani, Anna Bertaso, Elio Franco De Palo,Federica Bortolotti, and Franco Tagliaro. Vitreous humor endogenous compounds analysis for post-mortem forensic investigation. Forensic science international, 310:110235, 2020.

Elizabeth Yardley and David Wilson. In search of the ‘angels of death’: Conceptualising the contemporary nurse healthcare serial killer. Journal of Investigative Psychology and Offender Profiling, 13(1):39–55, 2016. 1002/jip.1434

Francesco Dotto, Richard D. Gill and Julia Mortera (2022) Statistical Analyses in the case of an Italian nurse accused of murdering patients. Submitted to “Law, Probability, Risk” (Oxford University Press), accepted for publication subject to minor revision; preprint:

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