A Proper Review of the Statistics in Lucy Letby’s Trial

It is well known that the statistics used to convict Lucy Letby at her trial were fallacious. The most notorious example being the roster chart which did nothing more than show she was present at certain times when she was present, yet was used to create a sinister connotation.

What has not been done so far as we can tell at least in the public domain is a proper review of the statistics. To that end, I have recently been taking part in a project to do just that. Collaborating in this includes Peter Elston, who has done extensive work on the statistics of the case already, as well as three other collaborators (all UK based) who wish to remain anonymous for the time being.

For the exact same reasons as shown in my earlier blogpost, I was as certain as one can practically be that a proper statistical analysis of the case would in no way point towards guilt, and indeed that appears to be the case based on our preliminary findings. With that said, all involved are committed to publishing the results regardless of whether they work for or against Lucy. Our ultimate commitment is to the truth. Above all, contrary to what many pitchforkers would have you believe, if I ever found evidence pointing to a serial killer of children, I would promptly make it known to the relevant authorities. I’m not Ravi Jayaram after all!

In the rest of this blogpost, I will summarise some key preliminary findings as well as provide some formal housekeeping concerning data protection on this project as required by law.

The Rise in Mortality at CoCH:

Our preliminary findings show that, even with a very prosecution friendly model, the rise in mortality at CoCH neonatal unit from June 2015 to June 2016 was of a statistical significance one would expect to observe in around one unit per year in the country.

This comes when one accounts at least in part for the following factors:

Number of IC patient days in a given period
Uncertainty in the estimate for the mortality rate per IC patient day
Interval length can be varied to produce the most statistically significant “spike” in mortality possible

Moreover, this model operates on the assumption that the hospital continued to perform just as well even in the face of a significant rise in IC days that CoCH was forced to cope with, most likely due to the maternity crisis in North Wales.

If one accepts the common-sense belief that the hospital would have struggled to cope with this increased burden, thus causing the per IC patient day mortality rate to increase, then the chance of a rise in mortality will be substantially higher than our prosecution friendly model suggests.

Our model also does not account for other factors. For example, Thirlwall showed that the number of infants with birthweights below <2000g increased during the June 2015 to June 2016 period, which one would expect to increase the chance of a spike in mortality even higher.

How was Lucy Letby Associated with so many Deaths?

This is the current part of the research we are focusing on developing the most, but we have a few preliminary observations.

Firstly, the number of patient in intensive care is substantially higher when Lucy Letby is on shift than when she is not. A potential confounder behind this was her status as the jointly most qualified nurse in her pay bracket. This likely meant that the cheapest way to staff the rotor was to use Letby as much as possible for shifts with multiple patients in intensive care, for which she could be used with her QIS qualification, while using the lesser qualified band 5 nurses on shifts with only a single or no patient in intensive care. The fact that Letby was a young nurse and single likely meant she had greater flexibility allowing her to fit such a schedule.

Secondly, heterogeneity is a really important feature. The chance of a death occurring at a given time on the unit is not constant. It varies based on how many patients are present in intensive care at a given time, what their gestational ages and birthweights are, whether or not they are a multiple, whether or not the unit is understaffed at a given time, and any other comorbidities individual patients might have. If one assumes, even a small amount of heterogeneity, the chance of an individual nurse being present for a large number of deaths rises dramatically. This is the same phenomenon as I observed in the case of Lucia de Berk, where even a small amount of heterogeneity, combined with correcting other fallacies, meant that the true probability of de Berk being present for the deaths she was at was 1 in 50, a far cry from the 1 in 342 million claimed at her original trial.

Of course, even a 1 in 50 event still implies that a nurse was unlucky to be present at many deaths. But in a country with thousands of nurses, hundreds of thousands in the case of England, you would expect this to happen to a large number of nurses quite regularly. Just as someone is statistically almost guaranteed to win the lottery somewhere, even if their individual chances of doing so are small, it is almost guaranteed that many nurses will be present for a many deaths. It does not imply on its own that they were responsible for any wrongdoing.

We would like to develop this work further by using data such as gestational ages and birthweight to account for some of this heterogeneity and obtain a more precise estimate on the chance of a nurse being present for some minimum number of patient deaths.

Lessons for Use of Statistics:

There are several key lessons that should be learned from this story.

One key lesson is that, when the decision around what analysis to do is based on the same data one is analysing, extreme caution must be exercised. Even subconsciously, one can use the data to perform an analysis decision aimed at achieving the very outcome one wants, while ignoring analysis approaches that do not achieve that outcome.

This issue is illustrated repeatedly throughout the case of Letby including, but by no means limited to, the following:

Defining “suspicious events”, “collapses” or “collapses leading to death” on the basis of whether Lucy was present there in the first place. It is known that at least 10 instances where it was initially thought a baby had been attacked or an otherwise “suspicious” event took place were dropped from the shift chart after Letby was found to be absent from the hospital at the time.
Choosing the specific interval over which to analyse the rise in mortality on the basis of it being whether a rise in mortality happened to occur

Another key lesson is that null events are extremely important to keep in mind. The death of 10 patients out of 10 is far more suspicious than the death of 10 patients out of 1000. If an unusual event is merely the result of there being many opportunities for it to happen, and it just occurred eventually at some point, then the fact an event might be rare does not imply that a rare cause must be responsible for it.

Courts seem to be extremely poor at grasping this critical concept. This is evident when the court looked only at collapses when Lucy Letby was present while ignoring other “suspicious events” after it turns out Lucy was never there catastrophically fails to grasp this key principle. It’s rather like evaluating the performance of the Dutch National Football Team by looking at only the matches they have won, while ignoring their defeats like the 2010 World Cup final.

One potential reason courts are so poor at understanding this critical concept is courts usually handle events that are inherently rare, such as a murder where there’s no dispute a crime has occurred and the only debate is who the perpetrator was. However, determining from observational data whether an exceptionally rare cause even happened at all (such as a serial killer nurse) is a very different task. The more one looks into the case of Lucy Letby, the more one appreciates how ill equipped courts are for handling these sort of cases.

Data Protection Housekeeping:

The law on data protection in both the UK and EU requires one fulfils a number of key principles when handling personal data. One of those is transparency, and it is in that spirit that I provide some data protection housekeeping here.

Rights of Data Subjects:

Any data subject is welcome to contact me, including via email, if they want any further information or clarification regarding the project and we shall endeavour to answer all queries.

Data Subjects ordinarily have a number of rights, the full details of which can be obtained on the UK Information Commissioner’s Office website. See the following link for further details: A guide to individual rights | ICO

These rights include the following:

Right of Access: The right to access and receive a copy of one’s personal data.
Right of Rectification: One has a right for any inaccurate personal data about oneself to be corrected
Right of Restriction: One has a right to restrict how one’s personal data is used in certain circumstances
Right to Object: One has a right to object to the use of one’s personal data, although this right does not apply for processing done under article 6(1)(e) for reasons of public interest, a provision we rely on (see later for more details)
Right of Erasure: One has a right to have one’s personal data erased in certain circumstances

It is worth noting that there can be restrictions on these rights. In particular, restrictions often apply when deploying the right would substantially impede the academic research we are conducting.

Data subjects also have a right to complain both to the data controller responsible for processing the data, as well as to the appropriate regulatory, in this case the UK Information Commissioner’s Office or ICO.

However, even though such a right to complain exists, we wish to emphasise that we want to work collaboratively with data subjects as much as possible and resolve any queries that they have. If a data subject identifies as flaw in our practices, we would like to work with them and remedy it wherever possible. We also would like the data to be processed in a way that leaves subjects genuinely assured that the use of their personal data in this research project will not result in any adverse effects against them, and we are more than happy to work with them as much as possible to achieve that outcome.

General Public Interest:

There are numerous reasons why this research is so vital, and why we believe it is substantially in the public interest to do:

To help ascertain what really happened at the Countess of Chester Neonatal Unit. In particular, we hope our research can help contribute to ascertaining whether or not Lucy Letby has been the victim of a miscarriage of justice, or if she is indeed guilty of the crimes she has been convicted of.
To provide lessons in the use and misuse of statistics in the legal setting, regardless of whether Lucy Letby has been the victim of a miscarriage of justice.
If it does contribute towards showing Lucy Letby was the victim of a catastrophic miscarriage of justice, there will likely be huge implications for the criminal justice system
Thirlwall has proposed wholesale changes to the NHS, including the abolition of no blame culture which will likely make patient care worse. This would be bad enough in any event; it is even worse if it is based on something that never happened. This research can help determine whether that is indeed the case.
Given the public debate around the case of Lucy Letby is likely to continue for the foreseeable future in any event, we hope this research can help make that debate healthier and better informed.

While the practitioners of this research make no secret that we come into it with a very strong prior that Lucy Letby has been the victim of a miscarriage of justice, we do wish to emphasise our ultimate commitment is to the truth.

Both under academic standards, and human standards, we will not hesitate to publish our research if at any time it were to point against our prior belief and suggest Lucy Letby was indeed guilty of the crimes she was convicted of. We believe the public interest behind our research goes far beyond the fate of one person, and our commitment to publishing our findings no matter the result reflects that belief.

Specific Data Used:

The data used in this project will include the nurse’s shift chart, as well as admissions data for 9 out of the 13 months between June 2015 and June 2016. Unfortunately, we don’t have data for the four months where there was no indictment event. The admissions data was obtained by Peter Elston from Lucy’s defence barrister Mr Mark McDonald and, for the infants admitted to the neonatal unit in those 9 months, contains their gestational ages, birthweights, when they were in the unit and with what category of care (intensive care, high dependency care, special care, normal) on a given day.

How the Data will be Used:

The use of this data is necessary to compare the mortality at CoCH relative to gestational age and birthweight to that of other known environments. The admissions data also allows us to calculate the total IC days over a given time interval, which is also necessary for some of our modelling.

We may require using the data subsequently to develop models around the association of a particular nurse’s presence with mortality, such as discrete time survival models. Should this need arise, we will make sure not to publish anything that could theoretically be reversed engineered to obtain the original dataset.

It goes without saying this contains very sensitive data, including what GDPR regards as special category data. We therefore make clear at once that nothing will be published that could possibly be used to identify any patient. For example, any graphical displays will be scrambled appropriately to ensure they cannot be reverse engineered to obtain the original dataset. Nothing will be published pertaining to any individual patient either.

The only reason we use this data is to research the spike in mortality in the Countess, as well as the reasons behind a nurse being present for many of the deaths. In no way will the data be used to make any decision about any individual patient, and, given the summary nature of what we publish, nothing released will have any prospect of doing so either.

Legal Basis for Processing Data:

Formally, the legal bases on which we use this data are article 6(1)(e) (task in the public interest), article 6(1)(f) (legitimate interests) and article 9(2)(j) (archiving, research and statistics) together with section 1 paragraph 4 of Data Protection Act 2018 (DPA). The legitimate interests are the same as those expressed earlier as to why this research is so important, and they form the same reasons as to why we consider this research to be in the general public interest, a key requirement under article 9(2)(j).

If necessary, at certain points to ensure the completion of the project, we also rely on schedule 2, paragraph 27 of DPA (the research purposes exemption) and, if necessary, schedule 2, paragraph 26 of DPA (the academic expression exemption). In particular, with respect to the latter, because project is done with a view to publishing academic material and we believe that doing so is in the general public interest, that allows us to use the academic expression exemption whenever we reasonably believe a given provision, that the exemption is eligible to be applied to, would otherwise be incompatible with the academic purpose behind this research.

Critical Remarks on Bryan Sanctuary’s 2026 paper in Quantum Reports 2026

Anton Lorenz Vrba
Independent Researcher, UK
vrba@iow.onl

Richard D. Gill
Mathematical Institute, Leiden University
gill@math.leidenuniv.nl

29 September 2026

Abstract

We examine the logical and mathematical structure of Sanctuary’s recent paper (Quantum Rep. 2026, 8, 96), which argues that spin observables in the Bivector Standard Model (BiSM) are not jointly definable on a single Kolmogorov probability space, so that the CHSH quadruple cannot be formed and Bell’s and Fine’s theorems do not apply. The paper contains two constructions: local response functions of the analyser setting and a shared source phase, and a separate prescription that produces the correlation . We show that the first construction, as specified, meets the hypotheses of Bell’s theorem. Its outcome maps are defined on a common source domain with a setting-independent distribution, so the non-existence claim (Lemma 3, Corollary 3) does not hold for it, its product correlations obey the CHSH bound, and the paper’s own simulation of it yields the Bell triangle. The cosine is obtained instead from a scalar computed jointly from both stations’ phases and then digitised; it is not shown to be the correlation of the local outcomes. Execution of the supplementary programs further shows that the four BiSM_v2 populations do not form the partition stated in the paper, and that the doubled-angle harmonic attributed to the raw coherence data appears only after a sign convention is applied on the second half-cycle.

Introduction

Sanctuary’s paper “Spin Helicity and the Disproof of Bell’s Theorem” [3], together with [2], claimed that a bivector (quaternion) model of spin disproves Bell’s theorem. The published Comment [4] examined that claim. It concluded that the construction lies outside the hypotheses of Bell’s theorem rather than refuting it, and that the correlation is obtained by steps that “are heuristic mathematical procedures, not formal probabilistic identities”. In particular, the Comment identified the assembly of the correlation from separately computed polarisation and coherence contributions as “a modeling postulate”, for which “no general probabilistic theorem guarantees that such an additive assembly reproduces Bell-type statistics”.

In his Reply [5] the author accepted the first conclusion: “We agree that Bell’s theorem is mathematically valid within its stated assumptions. Our position is that bivector spin does not belong to the class of models Bell considered and therefore lies outside the scope of the theorem.” The Reply also defends the contextual construction of measurement domains, the statistical reconstruction of the phase relation, and a geometric account of the quantum limit. It offers no explicit probabilistic derivation resolving the objection to the split-and-recombine step, stating that “several issues raised by Vrba, particularly concerning the quantum limit and measurement structure, deserve fuller treatment than is possible in a short reply.”

Paper [1] develops that response. It does not supply the missing probabilistic derivation. Instead it isolates joint definability as a separate premise, which the model is said to fail, and it gives explicit rules: a phase uniformly distributed on , carried by local rotors to each station, and deterministic outcome rules. The present critique concerns the relation between the paper’s algebra and detector-level statistics. The explicit rules of [1] allow a sharper separation than was possible in [4] between two constructions: the local response functions, and the separate prescription that produces . The review that follows examines whether the joint-definability argument holds for the local responses (Section A), whether the correlation is derived from them or assembled separately (Section B), what the simulations compute (Section C), and which further claims rest on assertion rather than derivation (Sections D and E).

Unless stated otherwise, equation, section, definition and lemma numbers refer to [1]; the points of the present critique are numbered A1, B1, and so on. The supplementary material of [1] contains the programs BiSM_v1 and BiSM_v2 with their output files. Statements about the programs below rest on source inspection and output files.

A. The joint-definability argument

A1. Lemma 3 does not hold for the paper’s own definitions. Definition 12 introduces a context map , and Eq. (52) defines the outcome as . The composition is a function on for every setting , and likewise . Hence for every the four values

are simultaneously defined, and is a common domain of the kind Lemma 3 says does not exist. The proof of Lemma 3, whose argument is first set out in [5], rests on the sentence “observables are defined only after contextual instantiation”. That concerns which experiment is physically performed, not which functions are mathematically defined. Counterfactual definability in Bell’s sense requires only that the function be specified, not that the measurement be made.

A2. The explicit outcome rules are Bell-type local functions. The paper does not leave the outcome functions abstract. Eq. (15) gives the local phase for every and ; Eqs. (64) and (66) take the quantum-domain outcome as ; and Eq. (85) gives the polarisation outcome as , with the value assigned at zero. Each is a function of the local setting and the shared alone, which is Bell’s form . The sign-of-cosine rule is essentially Bell’s own illustrative model [6]; with a uniform source distribution and Bob’s opposite sign it gives the linear “triangle” correlation, as Figure 3 and Section 4.2.2 of [1] confirm.

A3. Geometric reality does not exempt from Bell’s framework. The paper states: “In the BiSM long–range EPR correlation originates from a common relative rotor phase established at separation and subsequently carried along the worldlines of Alice and Bob” (Section 2.1); the same description is given in [5]. Bell placed no restriction on the nature of : “It is a matter of indifference in the following whether denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous” [6]. A real, local geometric object created at the source and carried to both stations is the paradigm case of , not an exception to it. The local-response component of the model is therefore of exactly the kind Bell’s theorem addresses, and the statement that the bivector is “not a hidden variable in Bell’s sense” [5] does not hold for that component.

A4. A single Kolmogorov space is used explicitly. The paper places on with density (Eqs. (16), (70)) and uses this same space for every analyser separation: “Every segment in the simulation has the same free–flight singlet state”. The supplementary programs use a common finite grid of values for all settings, which is likewise a single space. With measurable response maps, the pushforward of under is a joint distribution reproducing the pairwise distributions of these local responses. Corollary 3 (“no global joint probability distribution exists for the CHSH quadruple”) therefore does not hold for them. The pushforward need not reproduce the separately generated pair distributions of the cosine branch (Section C).

A5. The author’s statement of Fine’s theorem, applied to the local responses. Section 5.5 states correctly that a joint distribution for the four observables “is equivalent to a deterministic Local Hidden Variable (LHV) representation”. Definition 15 asserts: “Individual measurement outcomes are real, discrete, deterministic and local.” The label alone would not settle the matter, but points A1, A2 and A4 show that the paper supplies local response functions on a common source domain and a source distribution that does not depend on the settings. For correlations formed as averages of products of these local outcomes,

for every , and integration against gives the CHSH bound; by the equivalence the author quotes, a joint distribution exists. The paper does not address the resulting alternatives:

either the reported correlations are averages of products of these local outcomes, in which case CHSH is satisfied;
or they are obtained from another quantity, in which case the locality of the response functions does not carry over to them.

The simulations (Section C) show that alternative (ii) is what occurs.

A6. Assumption 3 is not independent of Assumption 2 in the paper’s representation. The paper lists “Joint Definability” separately from “Outcome Definiteness”. Assumption 2 (“For each hidden parameter , the outcomes , are well-defined”) is stated for the settings generally; in the functional representation the paper supplies, any four settings then yield four defined values for fixed . Rejecting 3 while retaining 2 requires restricting 2 to the setting actually used, a restriction the paper does not state.

A7. The scope of Fine’s theorem. Fine’s theorem [7] concerns the scenario with two settings per party and two outcomes per setting: pairwise distributions with consistent single-observable marginals admit a joint distribution if and only if the full family of Bell–CHSH inequalities holds. A family that violates those inequalities is an instance of the theorem’s conclusion, not a case in which the theorem does not apply. Conclusion 5 of [1] states that “Bell’s theorem and Fine’s characterization remain mathematically valid within their stated premises”, which is narrower than the “inapplicability” of the title.

A8. Notation and analogy do not alter the common domain. The semicolon in is said to indicate “that is not merely another argument”; the notation does not change the domain or existence of the function. The proof of Lemma 2 moves from distinct function values, , to “distinct domains”; that step is not justified. Distinct instantiated domains need not be denied: the decisive point is that the context maps pull the observables back to the common source domain (A1). The induced-dipole analogy (Remark 1) illustrates this: a polarisability tensor fixes the response to every field direction simultaneously, so the counterfactual dipoles are jointly defined.

A9. The common-domain structure is already present in Bell’s representation. The paper states that “Bell did not isolate this common probability–space structure as a separate premise under the name of joint definability; it entered tacitly through the hidden–variable representation”. Bell wrote , and explicitly [6]. Joint definability is a consequence of that representation rather than an additional assumption, which is why it required no separate statement.

B. The correlation derivation

B1. The cosine is the scalar part of a joint product. Of the ellipsis in Eq. (13) the paper states: “We added the dots to emphasize that the two rotors do not cancel but are carried as independent phases by the separated EPR pair. The dots have no mathematical meaning expressing only locality by the product.” Eq. (14) nevertheless evaluates as the scalar part of the product of Alice’s and Bob’s rotors, in which cancels and results. The paper’s own second line of Eq. (14) shows the alternative: the product of the separate scalar parts, , which it states “obeys the CHSH bound”. The cosine is thus obtained from the scalar part of a product formed from both stations’ rotors, not from the product of the two locally generated Boolean detector outcomes. Joint processing is not itself the issue: ordinary coincidence analysis also compares records at a common later stage. The issue is which function of the records is computed.

B2. The cosine holds event by event. Eq. (20) gives the relative phase for every , so the scalar of Eq. (71) equals for every pair. Section 4.2.2 states that the correlation “emerges only statistically from the ensemble of rotor phases ”. The source phase affects the individual local quantities, but averaging over it is not what produces the cosine in : that cosine already holds event by event. The required identification of this scalar with the correlation of local Boolean outcomes is not supplied by the cancellation.

B3. The singlet operator is the identity. Eqs. (9), (13) and (72) define . Eq. (73) then writes

Inserting the identity between and yields directly. The target correlation thus enters through this algebraic representation, and no argument is given that the representation describes the detector statistics. Eq. (73) also combines matrix transposition with geometric-algebra products and uses , both as angles and as vectors; a clarification of the representation would help.

B4. The plotted mustache is a residual. Eq. (71) is the trigonometric addition identity,

The legend of Figure 3 reads verbatim “Mustache = Quaternion – Bell”, and the caption calls the mustache “Their difference”. The plotted mustache is therefore the residual between the target and the triangle; it is neither the continuous coherence term of Eq. (71) nor an independently derived contribution to a detector correlation. The cosine itself enters through the identity applied to the jointly held phases (B1, C1). The Comment [4] had described the corresponding split-and-recombine step as “a modeling postulate”.

B5. The split-and-recombine step is not connected to detector statistics. The Comment [4] identified the assembly of the correlation as a sum of separately computed polarisation and coherence contributions, , as “a modeling postulate” for which “no general probabilistic theorem guarantees that such an additive assembly reproduces Bell-type statistics”. Neither the Reply [5] nor [1] supplies the missing connection. The operators do not appear in [1], and how they relate to the continuous terms of Eq. (71) and to the populations of Section 4.3 is not stated. In [1] the recombination appears as Eq. (71), , with the two terms described as “complementary projections of a single underlying quaternion rotor” (Section 4.1). The algebraic sum is well defined and follows from the chosen rotor product. What is not established is its identification with the product of the recorded local Boolean outcomes, or with that product’s expectation. Describing and as complementary sectors does not supply this measurement relation, and the subsequent digitisation (C2) samples a new coincidence variable with mean rather than showing that the local detector products have that mean.

B6. The term “polarisation” is used for three different quantities. The term is used for: (Eq. (14), second line), which is not rotation-invariant; (Eq. (71)), which averages over to ; and the digitised triangle (Figure 3, Section 4.3). Moreover, Section 4.3.3 states that “The symmetric combination Equation (71) reconstructs the scalar invariant and produces the piecewise linear correlation”, whereas Section 4.2 states that the same equation yields . These two statements contradict each other.

B7. The Boolean benchmark is one member of the Bell-local class. The second line of Eq. (14) is said to follow “from Equation (11) by dropping the off–diagonal terms in the scalar product, ”, which “represents complete removal of the bivector phase structure”. The paper does not say which physical operation this prescription represents, or how it relates to the detector rule it specifies. The resulting product, , is then used as the Boolean benchmark that obeys CHSH. It is one example of a Bell-local correlation. Bell’s theorem, however, concerns the whole class of local response functions under a common preparation, so a comparison with this single product does not establish how the model stands in relation to that class.

B8. The rotor scalar is not shown to be a detector expectation. For a binary coincidence variable , and normalisation give , so Eq. (22) is a legitimate identity once the rotor scalar is known to be such an expectation (with the paper’s convention ). That identification is the step not supplied. The accompanying statement (“Since they are constructed from a product state of two independently propagating local rotors their use is consistent with locality”) asserts the locality of the construction rather than showing it.

C. The simulations

C1. The abstract’s description of BiSM_v1 differs from its construction. The abstract states: “Numerical simulations reproduce this correlation using only local detector events.” Section 4.2 states of : “It is not a local detector click and is never used to generate either Alice’s or Bob’s local Boolean outcome”, and Section 4.2.2 adds: “The scalar is not Bell’s response function .” is computed from both stations’ continuous phases together (Eq. (71)). The quoted statement holds for the local clicks of the Bell control, but the per-station outcomes of the cosine branch are generated from (C3).

C2. The digitisation step reproduces any prescribed mean. Eqs. (74)–(75) set whenever , with one auxiliary uniform random number per pair. Then

exactly, for any function with values in . Any correlation function with is “reproduced” in this way, including one whose four-setting CHSH combination exceeds the Tsirelson bound . The procedure samples a prescribed mean; it is not a test of whether that mean has a Bell-local realisation. The source of BiSM_v1 states that “No Malus probability P_eq=(1+R)/2 is formed or used anywhere in event generation”; the uniform threshold is nonetheless mathematically identical to sampling with that probability. The reported maximum error of (Eq. (78)) verifies the trigonometric identity in floating point.

C3. In the cosine branch the station outcomes are assigned after pairing. The paper describes the digitisation of into the coincidence variable (Eqs. (74)–(76)), but not how outcomes for Alice and Bob are then obtained. The source of BiSM_v1 shows this step: after is drawn, a further random number assigns an equal event to or , and an unequal event to or (“The equal pair is ++ or – with equal marginal weight”). The station-resolved counts in the output file are generated in this way. Alice’s recorded outcome in this branch is therefore not a function of her setting and : whether it agrees with Bob’s is decided by , a function of both settings. The marginals remain uniform, but the joint law is not of Bell’s form ; the program’s own Bell control, which is of that form, gives the triangle. Since identically (Eq. (20)), the retained phases carry exactly the difference of the two settings. Any experiment that records its analyser settings, as all Bell tests do, could generate the same coincidence data by this procedure.

C4. The paper’s local-outcome analysis is consistent with Bell’s theorem. Section 4.2.2 states that analysing the local records gives “the conventional Bell result of the piecewise–linear correlation”; executing BiSM_v1 confirms this, its Bell control giving at and at . Eqs. (80)–(81) present the two computations as analyses of “the same experimental results”. Both use the same enlarged simulated record, but they apply different functions to it, and the cosine branch also generates additional random pair labels. The enlarged record contains continuous phases and “retained local rotor geometry” (Eqs. (67)–(68)), and the paper does not establish that these are accessible detector records. The statement that “The same digitized coincidence statistics observed experimentally are thereby recovered” therefore compares a simulation using additional, jointly processed quantities with experiments that record only outcomes.

C5. The BiSM_v2 rules are not derived, and Eq. (88) does not describe the program. Several event rules are given without derivation from the bivector dynamics: the phase offsets in Eq. (84), which the program describes as implementing singlet preparation by antipodal source orientations; the factor in Eq. (85); and the “larger signed contribution” rule of Eqs. (86)–(87). Eq. (88) states that “Each event contributes to one of the four populations”, whose counts sum to . The supplementary program instead evaluates both the polarisation and the coherence rule for every event; its own comments state that “EQP+NEQP = 1 and EQC+NEQC = 1 separately” and that “the four fractions are NOT intended to sum to one”. Execution confirms that the four fractions sum to 2 at every angle. Eq. (88) therefore does not describe the implemented algorithm.

As implemented, is an average of products of local coherence outcomes, additionally averaged over absolute analyser orientation. Being generated by local functions of under a common preparation, such averages are subject to the CHSH bound (A2, A5); a harmonic shape in this angular summary does not by itself indicate a violation, which would require four consistently defined setting pairs. The program’s run report compares with on the interval only; its behaviour over the full cycle is taken up in C6.

C6. The doubled-angle harmonic appears only after a sign convention. Section 4.3.4 states that “the coherence imbalance exhibits the smooth doubled–angle, two–period harmonic”, and Section 4.3.6 that it “exhibits a doubled angular dependence and therefore has period in the analyzer separation”, from which the double-cover interpretation is drawn (E8). The implemented coherence rule does not produce this. Under the doubled-angle terms that select the blade are unchanged, while both single-angle phases change sign; away from threshold ties the coherence outcome therefore reverses, and . Execution of BiSM_v2 confirms this: , whereas . Over the full cycle the Fourier coefficient of is ; its dominant component is the first harmonic (coefficient of about ). On the data do follow , consistent with Eq. (94), , when is read as the separation folded into ; that function is not -periodic in .

The program then defines, for its portrait, an “oriented” coordinate for , describing this as a convention that “does not alter EQC or NEQC”. Only this coordinate carries a component (coefficient ) and is -periodic. The paper describes the portrait as the ordered pair of Eq. (95) and does not mention the sign reversal. The figure-eight thus contains an imposed orientation convention, and the doubled harmonic on which Section 4.3.6 relies is a property of that convention rather than of the raw populations. The lemniscate of Gerono, , illustrates the shape obtained from these two harmonics; the implemented polarisation coordinate is approximately triangular rather than exactly . The experiment proposed in Section 5.1.2 would require an operational means of controlling , resolving the two sectors, and deriving their observable statistics under that controlled preparation; the paper does not yet supply these.

D. Framing of the result

The introduction states that “Section 3 questions Bell’s locality theorem” and that “Failure of either locality or joint definability leads to failure of Bell’s locality theorem”. It also states that the work “does not challenge Bell’s derivation”, and Conclusion 5, like [5], calls both theorems “mathematically valid”. These positions are compatible only if “failure” means that the theorem’s hypotheses are not met; a theorem whose hypotheses are not met does not thereby fail.

The introduction states that “The CHSH bound applies to correlations constructed solely from Boolean outcomes ”. The bound is ; is the per- value (Eq. (62)). The same passage attributes the violation to “additional geometric effects” before any argument is given.

The paper states that “Entanglement is not required since the quaternion carries their common phase”, and that the bivector singlet of Eq. (9) “is not entangled”. Whether such a construction yields a Bell-local model depends on the conditions set out in A5: local response functions, a common setting-independent distribution, and correlations formed from products of outcomes. For the model’s local responses these conditions are met (A1, A2, A4); for the cosine they are not shown (B1, C1).

E. Physical claims made without derivation

E1. follows from the chosen function. It is obtained by maximising , which the paper calls “the simplest symmetric measure”. The maximum at is a property of that chosen function, not a consequence of symmetry as such (the equally symmetric has a minimum there), and no evolution or stability analysis links the extremum to a dynamical bifurcation. The quantum limit thus remains what [4] called a “structural postulate”.

E2. The Dirac-equation step is asserted. The coupled equation of Eq. (42) does not by itself imply the separate equalities and of Eq. (43). These require an independent restriction on the operators or their domains, which is asserted on the grounds that and act in opposite parity sectors but is not shown; a parity label is not an annihilation property. The statement “This occurs, as we show, at ” is not supported by Eqs. (37)–(44), in which does not appear; any dependence on would need to be exhibited.

E3. and Noether. The identification of with the angular momentum of the spin-1 state is asserted, without stating whether a component or a magnitude is meant. The statement that spin “does not become classical as unlike continuous variables governed by Noether’s theorem” attributes to Noether’s theorem a content it does not have: the theorem connects continuous symmetries of the dynamics with conservation laws, and does not classify discrete observables as exempt from classical limits. A fixed-spin limit and a large-spin semiclassical limit are also different and should be distinguished.

E4. No quantitative account of beta decay. Section 5.2.1 states that “A cleaved and free blade is indistinguishable from a neutrino”; Section 5.2.3 states that “There can be no neutrinos produced in the BiSM with its intact blades”. The paper distinguishes cleaved from intact blades, so these statements need not contradict each other, and it states that “no detailed BiSM reaction or assignment of charge is proposed here”. Until such a model is given, the proposal does not account quantitatively for the continuous beta spectrum and the energy and angular-momentum balance of the decay, which the neutrino hypothesis was introduced to explain.

E5. Assigned parity does not establish conservation in the decay. “The BiSM electron is odd to party [sic] while the Fermi electron of the SM is even to parity. This difference demonstrates that parity is not violated in the BiSM.” Parity assigned to one state does not show that the decay dynamics conserve parity or reproduce the asymmetry measured in the Wu experiment.

E6. Rhetorical claims supply no evidence for the alternative. Section 5.5.7 states that developments in quantum information “have no rational explanation”. The paper elsewhere acknowledges the experimental success of quantum mechanics and questions its interpretation; the statement expresses a philosophical position and supplies no mathematical evidence for the proposed alternative. Similarly, the assertion that “control of these new parameters restores determinism” (Section 5.1.1) would require preparation procedures and detector-response predictions for and , which are not given.

E7. The coherence analogy does not establish Bell-local correlations. Section 5.1.4 likens EPR correlations to “lasers, superconductors, superfluids, and Bose–Einstein condensates where collective behaviour emerges through long–range phase coherence”, and Section 4.3.5 to the double-slit experiment. The ordinary double-slit pattern invoked there is a single-particle distribution, whereas Bell’s theorem concerns joint statistics at two space-like separated stations with freely chosen settings. Macroscopic coherent systems are described quantitatively by quantum mechanics, and the analogy does not by itself supply a local model of two-station correlations.

E8. The double-cover claim is not established by a frequency ratio. Section 4.3.6 states that the original blade configuration is restored only after “a total rotation of of the blades”, and calls this “a geometric realization of the double cover ”. Unit quaternions do carry the standard double cover: with acting on vectors by , and induce the same rotation, and a physical rotation through lifts from to . Establishing that the proposed blade dynamics realise this structure requires showing how rotations about different axes, and their compositions, lift; a fixed frequency ratio about one axis does not establish it. Moreover, the period- dependence offered as its statistical signature is not present in the raw coherence data (C6). The same claim appears in [5].

F. Procedural note

The paper cites the two Quantum Reports articles [2, 3] addressed in the published Comment [4] and the author’s Reply [5], but does not cite either, or explicitly identify its responses to their specific objections, although both precede it and bear directly on its Sections 3.5, 3.6 and 5.5. The legend of Figure 3 defines the plotted mustache as the difference between the quaternion and Bell curves, the construction the Comment described as a modeling postulate. The paper’s central joint-definability argument restates, in new terminology, the argument of the Reply. The difference is that the paper now specifies explicit local response functions and a common source distribution (Eqs. (15), (16), (85), Definition 12). It is these, rather than the labels “local” and “deterministic”, that bring the local-response component within the scope of Bell’s and Fine’s theorems (A5).

G. Summary

The analysis establishes three related findings:

The local response functions of the model are defined on a common source domain with a setting-independent distribution, so Lemma 3 and Corollary 3 do not hold for them (A1–A4, A8).
Their product correlations therefore satisfy the CHSH bound, and by the equivalence the author quotes a joint distribution exists; the paper’s own simulation of these outcomes yields the Bell triangle (A5, C4).
The correlation is obtained from a scalar computed jointly from both stations’ phases and then digitised; it is not shown to be the correlation of the local outcomes (B1, B5, C1–C3). The supplementary programs further show that the BiSM_v2 populations do not form the stated partition, and that the doubled-angle harmonic appears only after a sign convention (C5, C6).

Declaration on the use of generative AI

This document was written with the assistance of Anthropic’s Claude Opus 5.5, under the direction of the authors, using material from the cited publications and the supplementary programs in [1]. Drafts were also reviewed for consistency and accuracy using OpenAI models (ChatGPT and Codex), and resulting suggestions were incorporated following manual review. Every claim in this document has been independently verified by both authors.

References

B. Sanctuary, Joint Definability, Context–Instantiated Geometry, and the Inapplicability of Fine’s Theorem, Quantum Rep. 2026, 8, 96. doi:10.3390/quantum8030096 (including Supplementary Data: quantumrep-08-00096-s001.zip).
B. Sanctuary, EPR Correlations Using Quaternion Spin, Quantum Rep. 2024, 6, 409–425. doi:10.3390/quantum6030026
B. Sanctuary, Spin Helicity and the Disproof of Bell’s Theorem, Quantum Rep. 2024, 6, 436–441. doi:10.3390/quantum6030028
A. L. Vrba, A Collective Comment on “Spin Helicity and the Disproof of Bell’s Theorem” and Sanctuary’s Bivector Spin Framework (2023–2025), Quantum Rep. 2026, 8(2), 56. doi:10.3390/quantum8020056
B. Sanctuary, Reply to Vrba, A.L. A Collective Comment on “Sanctuary, B. ‘Spin Helicity and the Disproof of Bell’s Theorem’ and Sanctuary’s Bivector Spin Framework (2023–2025)”, Quantum Rep. 2026, 8, 57. doi:10.3390/quantum8030057
J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika 1964, 1, 195–200.
A. Fine, Hidden variables, joint probability, and the Bell inequalities, Phys. Rev. Lett. 1982, 48, 291.

Why I am more than 99.99% certain that Lucy Letby is innocent

I use Bayes theorem: posterior odds equals prior odds times likelihood ratio. For an introduction, please read this nice blog post https://entropicthoughts.com/bayes-rule-odds-form

I use this rule, Bayes’ rule, repeatedly, each time taking account of another part of the evidence. It is named for Thomas Bayes, a presbyterian minister and mathematician, who was interested in using it to find a mathematical proof of the existence of God. https://en.wikipedia.org/wiki/Thomas_Bayes

The likelihood ratio for the question at hand, based on some part of the evidence, is the ratio of the probabilities of that part of the evidence under the two competing hypotheses. More precisely, one uses the conditional probabilities of that fact given previously incorporated evidence. We have to start somewhere and we start by describing two alternative hypotheses and our probabilities or degrees of belief or personal betting odds for those two hypotheses, before further evidence is taken into account.

Let’s start with the news reports of a police investigation of a possible killer nurse at a neonatology unit in the UK; the investigation being triggered by a disturbing spike in the death rate on that unit.

I think that in the last fifty years there simply hasn’t been been a case in the UK of a killer nurse on a neonatal ward, except possibly the case of Beverley Allitt. One might argue that there do exist doubts as to the safety of her conviction, or one might argue that there can have been serial killer nurses who completely evaded detection. Did Alittit work in an intensive care unit? I also think that in recent years, every year has seen a scandalous calamity in a UK neonatal ward, leading to avoidable deaths of quite a few babies. So a priori: the relative chances of a killer nurse being responsible for the spike, or simply poor care, is in my estimation 50:1 in favour of poor care in a failing hospital unit rather than activity of a killer. If you disagree, give me your arguments for both those rates and hence their ratio. If you would like to take a different starting point, try that. Eg, what is the chance a random nurse is a serial killer? At some point one will have to use the information that this was a neonatal unit and one will have to take account of the “normal” rate of deaths on the unit. I think my choice is reasonably specific. One could argue that the prior odds should be 10 to 1, or 100 to 1, instead of 50 to 1. I expect that most people will at least agree that killer nurses on neonatal units are very rare, disastrously poor care on a neonatal unit in the UK is not rare at all.

So we are back in 2017 and hear the news and rightly we should be sceptical that there really is a case here. But clearly there are grounds to investigate what is the cause of that spike, and maybe there is more information which the police already have.

Then, many years go by. A particular nurse is detained for questioning in two successive years; and finally arrested in a third year. Two more years go by (Corona). At last, a trial begins. It turns out that roughly seven years of police investigation has uncovered no direct evidence at all (neither medical evidence, toxicological evidence, witness testimony or CCTV recordings, finger prints or DNA) of unlawful action by the nurse who has been under intensive investigation all that time. And not just no evidence against that nurse – no direct strong evidence of malevolent activity by anyone.

One might want to argue that the insulin evidence is strong toxicological evidence. We could argue about that for a long time. Even if one or two babies were given unauthorised doses of insulin there is no direct proof that Lucy Letby did that herself. There is the possibility of accidental administration (twins in adjacent cots). The argument that Lucy did administer insulin seems to have been that we know at some point she carried out other murderous attacks and it is unlikely that there were two murderous nurses working in the unit. But why do we believe there are murderous nurses working on the unit? This argument can only be made after hearing all the other evidence in the case.

So we have to estimate the probability of a 7 year police hunt for evidence of murder by a particular nurse finding no direct evidence of any malevolent activity at all by anyone, if Lucy Letby actually was innocent, and if she truly was a serial killer. In my opinion ,what we actually observed is much more likely under the innocence hypothesis than under the guilty hypothesis. If she truly is innocent the chance of finding powerful directly incriminating evidence must be rather small; if she truly is a serial killer then it must be unlikely that that no baby can be definitely proven to have been murdered or attacked. I guess the two probabilities of no hard evidence to be 95% and 5% respectively. These are probabilities of 19/20 and 1/20 respectively, so a likelihood ratio of 19. I’ll be a bit more cautious and call it 10.

We already had odds of 50:1 in favour of innocence. We have a likelihood ratio of 10:1 in favour of innocence, having learnt that police investigation uncovered no strong and direct proof of malevolent harm to any baby. The odds on Lucy being innocent are therefore now 50 times 10, or 500 to 1.

Let’s now bring in the evidence from psychology. Are there reasons to believe Lucy is a psychopath? Which surely she must be, if she is a serial killer of babies in her care. It seems there is no reason at all to suspect she is a psychopath. I think that there very likely would be strong independent signs of psychopathy in her life history if she really is a serial killer, but obviously not so likely if she is completely innocent. [Clearly she could be both a psychopath but did not actually harm or try to harm any baby. I don’t think this is an interesting hypothesis to explore. I will also not pay attention to the Munchhausen by proxy idea, that she was trying to attract the attention of an older male doctor. All the evidence says that he was more romantically interested in her, than vice versa.]

Put the likelihood ratio at 2, ie twice as likely to see no evidence for psychopathy if innocent, than if a serial killer. Actually I think it should be closer to 10. We should ask some psychologists. Lucy Letby did not sadistically kill little animals when she was a child. By all accounts, she was a dedicated nurse and cared deeply for her work.

We were at 500 to 1 for innocence. Factor in a likelihood ratio of 2 for psychological evidence. Now it’s 1000 to 1. But we are not done yet.

Next, I would like to take account of the statistical evidence that the spike in deaths is quite adequately explained by the acuity of the patients being treated in those 18 months. I would say that this is exactly what we would expect if Lucy is innocent but very unlikely if she’s a serial killer. I think this hypothesis is very adequately supported by published MBRRACE-UK statistics, and what we know about the acuity of the babies in the case. We know why acuity went up in around 2014 and we know why it went down midway in 2017. The spike seems to have been caused by hospital policy which was being made and implemented by the consultants on that unit. They should have expected it.

Say a likelihood ratio of 10. That brings us to 10,000 to 1 she’s innocent; a posterior probability of 99.99%. I haven’t yet brought in the facts of an investigation driven by tunnel vision and coached by doctors who, as we now know, were making quite a few deadly mistakes themselves. I haven’t brought in yet the innocent explanation of the post-it note. In my opinion, the post-it note is powerful evidence for innocence; it makes absolutely no sense under the hypothesis of guilt. The irrelevance of the handover notes and the notations in her diary. Facebook searches? Her alleged lies (about what she was wearing when she was arrested). Anything else?

Anyway, I am now well above 99.99% sure that Lucy is innocent and since the press conference and the report of Shoo Lee and his colleagues, we can all be even more sure that that is the case.

My LinkedIn conversation with Dewi Evans

This LinkedIn conversation started with me asking Dewi Evans to connect to me. I was amazed that he accepted. I guess my request contained a brief message too, but this is not recorded in my LinkedIn account. I suppose Dewi has it in an email sent to him from LinkedIn. It would be nice to see it.

Dewi – are you there, reading this?

Feb 3, 2024
Dewi Evans sent the following message at 11:46 AM
11:46 AM Dear Richard. I’ve read your comments re the importance of statistics in court cases. I can’t comment on specific cases currently because of reporting restrictions.
I would welcome discussing this with you, as it’s a matter that is worth exploring.
Richard Gill sent the following messages at 12:30 PM
12:30 PM I agree! I am looking forward to the reporting restrictions being lifted. Hope to talk to you within a year from now…
12:32 PM Do take a look at the case of Lucia de Berk. The case is horrifically similar to that of Lucy Letby. I helped get her out of jail. Also an Italian nurse, Daniela Poggiali. I acted as expert on the applications to the CCRC of Ben Geen. I am sure he is innocent but UK criminal justice is nowadays badly tilted in favour of the prosecution.
Dewi Evans sent the following message at 12:36 PM
12:36 PM With a bit of luck reporting restrictions will be lifted after the end of the retrial due in June.
Richard Gill sent the following messages at 12:44 PM
12:44 PM👍
6:44 PM We have both been interviewed by Raj Persaud! https://rajpersaud.libsyn.com/Raj Persaud in conversation – the podcastsrajpersaud.libsyn.com
Feb 6, 2024
7:03 AM I’m visiting Liverpool to give a lecture next week, dept of statistics. Will probably also check out Chester. Would you like to meet? I’m not interested in reporting restrictions. They are unfair and immoral. Science must not be stopped. Chester police sent Dutch police to my door in the night to intimidate me. This only made me speak out more loudly.
7:21 AM Do you have an email address? I’d like to send you some links and materials
8:11 AM By the way, reporting restrictions means reporters cannot write in newspapers about evidence supporting Lucy’s innocence. However, it allows the Daily Mail to publish week by week horrible stories about how evil she is, her cushy life in jail, her friendship with another killer … You and I are not reporters. There is no law against us exchanging information. You can tell me about medicine, I can tell you about forensic science. I’m sorry for you that you live in a police state. In the Netherlands there are also disturbing developments. The state is eroding civil rights. In the UK the process has got much further.
8:12 AM Fortunately, many investigative reporters are working on the case and many scientists are working on the case. The dam is starting to crack and it won’t be long before it crashes down.
Dewi Evans sent the following message at 9:37 AM
9:37 AM No problem having a private discussion. But reporting restrictions are reporting restrictions and all that. I’ve no wish to contaminate due process. I’ve no idea re Daily Mail articles. Never read it (apart from the one where they covered the Letby story after speaking to me. My easy access email is xxxx@xxxxx.xxx
Richard Gill sent the following message at 10:48 AM
10:48 AM Thanks for the email address! I do not wish to contaminate due process either. I wish to ensure due process. I can promise in advance of any discussion with you total confidentiality.
Dewi Evans sent the following message at 10:48 AM
10:48 AM Thanks Richard
Richard Gill sent the following messages at 11:37 AM
11:37 AM Interesting development: the Mirror uses a *nice* photograph of #LucyLetby. Doesn’t call her a sadistic killer. And a leading barrister calls for “open justice”. Geoffrey Robertson KC said “Open justice is the principle that makes British courts the best in the world and judges should be more vigilant in protecting it”. They used to be the best in the world. Right now they are among the worst in the developed/free world. https://www.mirror.co.uk/news/uk-news/lucy-letby-anger-cowardly-doctors-32045213 Fury as ‘cowardly’ docs and nurses who worked with Lucy Letby keep names secretmirror.co.uk
Feb 22, 2024
6:56 PM You asked for all the deaths and all the collapses in the period January 2015 to July 2016. They gave you all the deaths but only collapses when Lucy was there. There must have been at least 50 collapses when she wasn’t on duty, given the acuity of those infants. You were lied to, you were used.
Feb 23, 2024
Dewi Evans sent the following message at 10:31 AM
10:31 AM That is incorrect. I received information re numerous collapses. I separated them into those that were explained by the common causes- infection, haemorrhage etc and those that were not explained, ie suspicious. The name Lucy Letby was not known to me at the time.
As for “at least 50 collapses” I don’t know where you got that figure from.
Richard Gill sent the following message at 10:54 AM
10:54 AM Interesting. Your story does not match the story one gets from other sources (for instance, the police themselves). I got my figure from several neonatologists and a similar figure from nurses with experience in neonatal intensive care. Secondly, “not explained” is not synonymous with “suspicious”. This confusion of words in the minds of on experts was exactly what led to the conviction of Lucia de Berk. I recommend you study it carefully!
Dewi Evans sent the following message at 11:17 AM
11:17 AM Away this weekend. Back Tuesday.
Content to engage post the appeal and retrial. Information from the police was all disclosed to the Defence presumably. Those are the rules.
No idea which other neonatologists involved. 2 gave reports for the Defence. They were not called. That’s a Defence issue.
Read the Lucia de Berk story via Wikipedia weeks ago.
Richard Gill sent the following messages at 11:35 AM
11:35 AM I know, the defence was useless. Scandalous. The newspapers were appalling. Social media too. This was not a fair trial.
11:37 AM Lucy Letby was a whistleblower and got crushed by the NHS. Much better to put the blame on a killer nurse than on lax consultants and poor management. Focussed on cost cutting at the expense of patient care.
11:51 AM Have a. nice weekend! I just had a great visit to Liverpool and to Chester. Wonderful to see Welsh mountains in the distance from the city walls of Chester.
Feb 24, 2024
7:31 AM All deaths in the period when Lucy was fully qualified and full time (with very much overtime) at CoCH, and 15 non-fatal collapses *selected by the gang of four* and exclusively at times when Lucy was on duty. How much of the time do you suppose she was in the ward? “Since the start of our enquiries and, as the information gathering process has continued, the scope of the investigation has now widened. We are now currently investigating the deaths of 17 babies and 15 non-fatal collapses between the period of March 2015 and July 2016” https://www.chesterstandard.co.uk/news/16329278.healthcare-worker-countess-chester-hospital-arrested-suspicion-murdering-eight-babies/
8:46 AM Interestingly in this case a judge blames previous judges https://www.judiciaryni.uk/sites/judiciary/files/decisions/Re%20A%20and%20B%20(Children%20Injury%20Proof%20Suspicion%20Speculation).pdf. The root of the problem is lack of understanding of science of judges and barristers and police. They ask scientists and experts and doctors questions which those persons should not be asked, because those experts are not supposed to judge, not supposed to give their opinion given *everything* they know. This is almost impossible for a doctor who, in his or her practice, does have to judge all the time! 2015 NIFam 14judiciaryni.uk1 Neutral Citation No. [2015] NIFam 14 Ref: OHA9745 Judgment: approved by the Court for handing down Delivered: 23/09/2015 (subject to ed…
Jul 7, 2024
9:25 AM Hi Dewi, maybe it’s time we had a chat? I don’t want to blame you. I blame NHS underfunding. Really bad police work, “experts” who don’t follow the rules (and apparently don’t know the relevant science either). The jury system, the contempt of court rules, a biased judge, a weak defence. The farce of her appeal being rejected but the CPS appeal accepted. The CCRC is utterly unfit for purpose and the next stage is going to take five to ten years. This certainly is the biggest miscarriage of justice since those big famous ones which led to the setting up of the CCRC. Of course in public you will presumably, for a while, go on saying you believe Lucy is guilty. I know all about obstinacy! Anyway: my suggestion is we do a Zoom chat, not recorded, Chatham house rules for just us two. Clear the air. I’ll tell you some things you don’t know yet, and vice versa. Win win.
Jul 8, 2024
Dewi Evans sent the following message at 10:42 AM
10:42 AM Currently getting over Covid, so back to normal next week.
Afraid I don’t agree with you re the verdict. Letby was as guilty as they come. And to date, I’ve not seen a single comment from a suitably qualified person or institution that offers a reasoned defence. As for social media – best to give it a big ignoral. CCRC?
As a witness of course one has to work within the system, but recognise its limitations.
As for other cases, I expect that the police are investigating them. They involve displacement of breathing tubes for no apparent reason. And of course Lucy Letby was the nurse looking after the baby at the time. I’m not in touch with the police or the investigation any longer, so it will be interesting to find the outcome. My hunch is that there are quite a lot of other cases out there.
[Interesting that since she was suspended in July 2016 (and I knew nothing about that before the trial) there have been few deaths at Chester apparently, and no ‘suspicious’ events.]
Jul 25, 2024
Richard Gill sent the following message at 7:27 PM
7:27 PM You would say that, wouldn’t you. Sorry, you are going to end up on the wrong side of history. “No apparent reason” does not equal “murder”. There are plenty of reasons breathing tubes get displaced. A person who gives expert evidence must be completely neutral and mention alternative explanations and margins of error. They must be fully qualified too. I’m sorry, but you are going to be in deep trouble.
Dewi Evans sent the following message at 8:07 PM
8:07 PM Richard. Just asking.
Have you seen the clinical records of the babies? Have you seen / read the statements of the local medics and nurses? Have you read the statements of the parents? Were you at the trial? Does your medical experience extend beyond knowing which side of the bandaid you put on the wound?
Cheshire Police are reviewing the notes of all babies at Chester, with the aid of an experienced neonatologist – I’m not involved and unaware of the findings. Don’t know if they are employing a statistician.
So, Richard. Stick to your opinion. I’m still waiting for your evidence. But, for the record. None of my reports was based on “statistics”. It was based on Evidence. Look up its meaning. Her arrest, her charging, her guilty verdict had nothing to do with “statistics”. It was based on the Evidence of 6 independent experienced doctors, and the evidence of numerous local nurses and doctors. Evidence dear boy, Evidence!
As for the Defence. It was their decision not to call evidence from independent clinicians and pathologists. Why do you think that was? Apart from the local plumber of course. Does one therefore assume that they consider your opinion less useful than that of a local plumber.
Finally. Please do NOT make accusations about my evidence, or alleging that my evidence is not impartial. Half of my reports in criminal cases are at the request of the Defence. My most recent report for the Defence, just a few weeks ago, led to the Prosecution withdrawing the allegations within 2 hours of the opening of the trial.
As for breathing tubes getting displaced for a number of reasons. For once you are correct. If you had listened to my evidence you would have known that I did not allege deliberate displacement of a breathing tube in any of the cases where I gave evidence.
So, I’ll stick with the facts. Endorsed so far by judge and jury, Appeal Judge and 3 Appeal Court Judges. Not too bad I suppose.
I look forward to your answers to my questions. I’m sorry if the facts (that word Evidence once more) get in the way of your opinion. But there you are. That’s a problem for you to address.
Don’t see much point in engaging in a continuing dialogue. But thought that I should respond once.
Jul 26, 2024
Richard Gill sent the following messages at 11:08 AM
11:08 AM Thanks. I hope you will keep your mind open to new evidence as it comes to light, like a real scientist always does. I hope you will also bear in mind the growing criticism against our judges and courts. UK criminal justice is just as broken as the NHS. I too stick to the facts. The fact is that Brearey and Jayaram ran to the police when they realised that they were in deep shit due to the RCPCH findings. The fact is that you violated the duty of a scientific expert, but of course, as you said, you are not really. a scientist.
11:11 AM PS thanks for keeping the line of contact open! I still think that you have a fantastic opportunity to display wisdom and integrity. Do not let yourself be led by pride and vanity. Read the RSS report with great care. I talk to enough doctors who are better qualified than you to interpret those clinical notes, and many of them have access to those notes now. The arguments of appeal judges just show what big fools those people are. Also puffed up with vanity.
Dewi Evans sent the following message at 2:31 PM
2:31 PM Not seen the RSS report! You are welcome to send it, or send me the link. But as I said, my evidence had nothing to do with statistics.
Richard Gill sent the following messages at 4:07 PM
4:07 PM https://rss.org.uk/news-publication/news-publications/2022/section-group-reports/rss-publishes-report-on-dealing-with-uncertainty-i/. This was two years in the writing. Two of the five authors are lawyers RSS publishes report on dealing with uncertainty in medical “murder” casesrss.org.uk
4:13 PM The defence didn’t understand medicine, didn’t understand science, didn’t understand statistics. Yeah – lawyers. Defence scored an own-goal by not disputing the interpretation of the immunoassay results. I just talked to a prominent UK professor of paediatrics. I asked him what he thought about the insulin? He basically said it was just bollocks and he explained why. This means that the evidence of Hindmarsh, Milan and Wark was just bollocks too. It may take 10 years, knowing UK criminal justice, but Lucy is going to walk free, you mark my words. Cheshire Constabulary have made enormous fools of themselves, costing the UK taxpayer millions, and by digging themselves in they are only making the debacle for themselves worse.
Dewi Evans sent the following messages at 4:54 PM
4:54 PM Who’s this professor of paediatrics Richard?
4:55 PM I’ll read the 64 page report over the weekend.
Jul 27, 2024
Richard Gill sent the following messages at 6:18 AM
6:18 AM I can’t tell you that professor’s name, sorry. Enjoy your weekend! Let me know if you have any questions about the report. It went pre-trial to defence, to prosecution, and in fact to all concerned parties. The man from the CPS said to me (at a pre-publication try out at the Newton Institute in Cambridge) “we are not using any statistics in the Letby case. They only makes people confused”. The defence too just did not understand the statistical issues, which are issues of scientific research methodology. And of forensic scientific investigation methodology. They got comprehensive advice from a very competent statistician but did not understand a word of it.
6:23 AM Sarrita Adams and I sent her medical analyses to the court during the trial, as an amicus brief. It was intercepted by Cheshire Police who sent Dutch police to my door in the night to deliver an intimidating letter. It threatened arrest next time I visited UK, two years in jail, and the cost of re-running the whole trial. The witch-burning mob doxed by mother’s address in a care home in Marlow, Bucks, and planned a demonstration outside. My Mother was 97. They also disrupted a lecture I gave at Liverpool University. The Mirror wrote that I was a sick and deluded conspiracy theorist and that I was attempting to corrupt the youth of England at the university, so the university authorities were evil too. Yep. All in a day’s work when you stand up against a witch hunt. Just so you know you are not the only guy getting publicly attacked in the media.
5:47 PM A friend – professor of mathematics – recently asked me: “I can’t understand why medical experts seem to often be so dishonest”. Here is my answer: “They are not scientists. They are trained to be rapid judges and executioners. That’s one thing. The other thing is the medical hierarchy, and clan forming. The paediatricians are more concerned with the paediatricians than with their patients. The youngest and least paid always have to get the blame to protect the reputation and earning power of the older and most highly paid.” These are unfortunately easily verifiable true facts! Think of all those who kept on supporting Prof Sir Roy Meadow. Maybe he was a good children’s doctor. He was however a lousy statistician and a lousy psychologist and this caused enormous disasters. No doubt he was a charming and amicable man. I always invoke Hanlon’s razor. Stupidity is a more likely cause of despicable behaviour than malice.
9:12 AM Here’s a name for you. Dr Svilena Dimitrova, NHS consultant neonatologist. Neonatologists against the paediatricians? CoCH had no neonatologist. Your friend well-known bully Dr Stephen Brearey “had an interest in neonatology” but was not a neonatologist. RCPCH (paediatricians!) told CoCH to recruit a neonatologist immediately.
9:14 AM I’m not saying it’s all your fault. Police lie and cheat. This has been proven again and again. Cheshire Constabulary have an especially bad reputation in this respect. I think you should cut all links with the bad guys and change sides as fast as possible, if only to save your own skin.
9:16 AM The momentum is growing. The Lucy Letby case is the biggest miscarriage of justice in the UK since the Guildford 4 and the Liverpool 6 (or were they 7 or were the numbers the other way round?). Just like the Lucia de Berk case in the Netherlands. The two cases are carbon copies of one another, but things panned out much worse in every respect in the UK. Utterly failed NHS, utterly failed CJS. Appalling gutter press. You should give an interview to a quality newspaper for a change.
Sep 13, 2024
7:07 PM Intelligent life outside the M25: what do you think of me then?
7:11 PM Babies do just suddenly drop dead. The Lucia de Berk case made that clear.
7:17 PM Unexplained and unexpected actually does happen all the time.
7:24 PM How did you exclude infection?
7:38 PM There is not one unrecordable explanation of a low C-peptide level. One of the explanations is that it is too high!!!!
7:43 PM What about the hook effect?
Sep 24, 2024
4:30 PM And could we discuss the meaning of “accuracy” and “reliability” of immunoassays to determine insulin and C-peptide concentrations? How about specificity and sensitivity? Statistical concepts, I know. Would you be interested in a public debate? Online? https://en.wikipedia.org/wiki/Sensitivity_and_specificity Sensitivity and specificity – Wikipediaen.wikipedia.org
Oct 1, 2024
9:37 AM Hi Dewi, yet again, I want to suggest you change sides! Become a hero. I know you do have the guts to do it, you are not a scared little man. I hope you’ve now studied Marks and Wark (2013) https://pubmed.ncbi.nlm.nih.gov/23751444/. It has a great summary of recommendations at the end. You see, Lucy Letby is innocent, 100% (ie, 99.99% certain, at least). Shall I show you the calculation? Science thrives on the clash of theories. As Niels Bohr once said “now we have a contradiction, at last we can make progress”. I honestly wish you well and know you are a good guy at heart. Forensic aspects of insulin – PubMedpubmed.ncbi.nlm.nih.gov
9:45 AM This is also worth re-reading. https://www.researchgate.net/publication/15580088_Practical_concerns_about_the_diagnosis_of_Munchausen_syndrome_by_proxy (PDF) Practical concerns about the diagnosis of Munchausen syndrome by proxyresearchgate.net
Oct 2, 2024
11:28 PM So you made a nice start, changing your expert medical opinion on three cases! But you are still convinced Lucy Letby was responsible. Surely that can only be by a statistical argument? But where are your statistical calculations and statistical qualifications? As far as I know neither police nor prosecution used testimony from a statistician. Lucy was often present at unpleasant events, but she worked the most hours of any nurse on that unit, and eagerly took the hardest shifts. Lucy’s defence team let her down badly, the judge was disgustingly biased. The whole disaster was not your fault. NHS managers and lawyers have a lot to answer for. Many reforms are needed. But they can only come if the system admits its failings.
11:32 PM I offered my services to defence, prosecution, and to the court. But no one wanted to know. Cheshire constabulary threatened me and sent Dutch police to my door in the night, to deliver a letter in person which I’d already received by email. Very intimidating. They needed legal proof I’d received their warning.

What went wrong with the NHS went badly wrong at CoCH and it’s not a coincidence

I just recently became aware of a deep connection between Countess of Chester hospital and radical restructuring of the NHS in the early 90’s, which brought in new layers of bureaucracy and internal competition. Don’t coordinate and distribute. Instead, let hospitals compete, survival of the fittest, dynamic leadership,and innovation! We’ll end up with better health care for less money.

The connection is Sir Duncan Nichol, former chairman of the Countess of Chester hospital trust. That’s a higher management level than the hospital executive board. Side effect of the innovations was more managers with even bigger top salaries. But Nichol is not just any manager. He’s a former NHS chief-executive, “part tycoon, part mandarin”. Read all about Sir Duncan’s innovations here: https://www.managementtoday.co.uk/uk-profile-sir-duncan-nichol-nhs-chief-executive/article/409550

Source of the table: https://www.coch.nhs.uk/media/204393/BOD-March-2015.pdf, https://www.coch.nhs.uk/corporate-information/board-of-directors/board-of-directors-meeting-packs/archive.aspx. Sorry for all the misprints. CoCH management and more generally NHS management produced expensive glossy annual reports and other publicity material but it seems nobody ever bothered to check the text for spelling errors.

Here are some more quotes from Management Today, emphasis added by myself.

“Nichol helped oversee the greatest shake-up in the health service since the war. Out went the old-style consensus management where low-grade administrators charged round trying to keep high-grade doctors happy; in came a whole raft of modern business nostrums: greater pressure on cost-efficiency and customer satisfaction, the introduction of ‘internal markets’, the separation of key functions like purchasing and service provision, and, of course, the increasing use of snappy titles like chief executive and general manager.”

“Some, still seething at the enforced changes, argue that you cannot apply market doctrines to the basic tenets of caring and curing. Others, especially those working in conventional businesses, remain unconvinced that any amount of fancy tinkering will change the nature of the beast. The last few years, they note, have still been dotted with high profile examples of cash squandering on a massive scale.”

Time for a new post … on insulin

I would recommend everyone interested in the Lucy Letby case to carefully study the 58 page appeal judgement – a board of three judges refused Lucy’s application to appeal. It is only 58 pages, it is well written and carefully argued… It is just built on heaps of false assumptions. What it says about the insulin evidence is particularly significant.

R V Letby Final Judgment 20240702

Three points specifically concern the insulin babies, babies F and L: points 14, 30 and 104 [and another on the evidence of Prof Hindmarsh, which I won’t go into right now.] Here they are in italics; my underlining.

14. A proposed ground 4 (that the jury were wrongly directed on evidence relating to the persistence of insulin in the bloodstream) was withdrawn following the refusal of leave to appeal by the single judge.

30. At trial, the integrity of the blood samples and reliability of the biochemical testing was challenged by Mr Myers. However, in her evidence at trial, the applicant [Lucy Letby!] admitted that both babies had been poisoned by insulin, but denied that she was the poisoner. The prosecution relied upon the unlikelihood of there being two poisoners at work on the unit. As the judge expressed it shortly before the jury retired to consider their verdicts: “If you are sure that two of the babies…had Actrapid, manufactured insulin, inserted into the infusion bag that were set up for them 8 months apart in August 2015 and April 2016 respectively, and you are sure that that was done deliberately, you then have to consider whether that may have been a coincidence, two different people independently acting in that way or were they the acts of the one person and, if so, who.”

104. The prosecution made some general points to rebut the allegations of bias and unreliability, including that almost every opinion given by Dr Evans was corroborated by another expert. In addition, it was pointed out that Dr Evans was the person who had identified that two of the babies had been poisoned by insulin (Baby F and Baby L). This was a matter which had eluded the treating medics and went to prove that someone was committing serious offences against babies in the unit; and it was particularly important independent evidence, bolstering Dr Evans’ credibility and reliability. Further, when Dr Evans reached his conclusions, he did so without knowing about other circumstantial evidence relied on by the prosecution in establishing guilt, including the applicant’s Facebook searches, the shift pattern evidence, and the “confession” in the note recovered from the applicant’s home on 3 July 2018

Richard’s comment on point 30: the applicant was told during the trial that it had been proven that two babies had been poisoned. Her reply was not that she admitted this fact, it was more like “well if you say it is proven it must be true. But I didn’t do it”. Notice that for the prosecution, six experts all say that this was true: Evans, Bohin, Hindmarsh, Milan, Wark; the hospital doctors had concurred (Gibbs, in particular). The defence had apparently not even looked for an expert on the insulin matter. They had raised problems about the reliability of the tests but these are ignored because Lucy herself agrees that the babies were poisoned with hospital insulin.

How can a young nurse agree or disagree with a deduction made by half a dozen senior medics that the immunoassay results proved deliberate insulin poisoning? All doctors and nurses have been taught in doctors’ or nurses’ school something about insulin metabolism and know that the ratio of insulin to C-peptide (after someone has been fasting for three hours) should be about 1 to 6. They are not taught anything about the forensic determination of deliberate insulin poisoning, which tells us that an anomalous ratio is a warning sign for something that might have been happened but it should be followed up with completely different tests which exclude a numerous sets of artefacts which can each also cause an anomalous ratio of immunoassays’ numbers. The test result comes back from the lab with a warning note printed in red with exactly this information. It was ignored by hospital doctors. The specialists weren’t called in and never told about it. The babies were nicely recovering from their hypoglycaemia and were rapidly transferred elsewhere.

Note that according to point 14 the defence had submitted and withdrawn a “ground for appeal” concerning the insulin. [I don’t know what it was, someone should find out!]. Note that the insulin is seen by all concerned as proof of presence of a murderer on the unit. The court of appeal sees it as ludicrous to suggest there are two murderers on the unit! They believe that the other evidence (air embolism etc etc) proves Lucy is a murderer. The insulin evidence shows someone tried to murder babies F and L. There can’t be two murderers. Therefore Lucy also tried to murder babies F and L.

The logic of the argument is impeccable. Just the premises of the argument are wrong.

These points do show the utter incompetence of the defence team.

One of the prosecution experts (Gwen Wark) has even published a paper Marks and Wark (2013) with a concluding list of recommendations which state that the immunoassay only suggests a possibility; in order to prove it other tests must be done. https://pubmed.ncbi.nlm.nih.gov/23751444/ They were not done. No sample was saved so they never can be done. Both babies F and L are alive and well to this day though one has cerebral palsy … linked to problems experienced during birth.

At the trial Dr. Wark confirmed the other experts’ claim that insulin poisoning had been proven. How could she? What had she written in any written testimony provided pre trial to police or CPS? Richard Thomas (Lucy’s solicitor) says that the defence team can’t say whether or not any documents exist and whether or not they saw them. The jury is not shown any such documents.

People say “the jury was given so much more, information…”. That is quite simply not true. The jury in a UK criminal trial hears what is said and it observes the body language. That’s all. It does not receive copies of written testimony of scientific experts. (This is a consequence of “open justice” with a trial by a jury of your peers).

What was the fifth ground in the original application for an appeal? The one which a single judge scrapped and the defence team then scrapped too?

Maybe some journalist should chase after that. I’m afraid the defence team is unlikely to tell us. They are not obliged to, and they can only do it if Lucy instructs them to do so. And nobody is able to talk to her.

From a correspondent

I first became interested in the Lucy Letby (LL) case when my wife referred me to a 10 hour podcast entitled: “The Case of LL; The Facts – Crime Scene 2 Court Room”, https://www.youtube.com/watch?v=_OA0ukO7D7c. Since, I have searched for further background to this case. Richard Gill’s website raises issues around imbalance between the prosecution and the defence, or lack of it! Also, there was toxic atmosphere at the Countess of Chester (CC) neonatal unit and LL reported problems. I worked for over 25 years anaesthetizing children down to 500g, in addition to adult anaesthesia, as well as expert witness experience. I also spent 6 months attached to a Neonatal Unit. The question of whether LL committed the alleged crimes is a difficult one to answer as (a) no one actually witnessed her doing the alleged crimes, (b) there is no obvious motive, (c) the actions would be very hard to achieve and (iv) there are other alternative explanations that were not explored by the court case, or at least the 10 hour transcript.

The Countess of Chester baby unit: The main purpose of the unit was a nursey to look after and feed babies too small and fragile to leave hospital, born at the CC. There was a small 4 bedded-neonatal unit in addition to the 4 nursery rooms. LL probably wanted to gain neonatal experience hence her involvement with the 17 cited cases.

Neonates and vulnerability: Small preterm babies easily deteriorate and die. Their organs are still developing and without the advent of neonatal units in the 1980s most would die. It is only today that a baby born prematurely before 28 to 32 weeks has a good chance of survival.

Staffing levels, staff experience and standard of care: LL was only 25 years old at the time and had only been a neonatal nurse for a few years. That is not very long and she lacked experience! She still needed further training in Liverpool to advance her career. Yet, she seemed to be one of the most senior neonatal nurses (band 5) on the unit and nowhere in the transcript do we find an older, more senior or experienced colleague other than a charge nurse who managed the duties and was not hands on. Similarly, it appears that medical cover was by paediatricians who also covered the wards and there was no doctor solely on duty for the unit. Therefore, when compared to other bigger units (i.e. Alder-Hey) the level of care was limited, so it would not be surprising if a baby deteriorates, and that happens with “prems”, outcomes are not as good. So, the evidence suggests that the CC was not up to standard, and it was an overflow unit for Alder-Hey. The CC neonatal unit has since been closed down. So were the cited incidents and deaths really due to LL or a result of a poorly supported / under-funded unit looking after sick neonates that should have been elsewhere?

The prosecution case focused on a number of methods of harming babies allegedly used by LL: (i) Distending the stomach by giving too much feed or (ii) injecting air into the stomach, (iii) injecting air into the circulation causing sudden collapse, (iv) traumatising the airway and causing bleeding, (v) dislodged tracheal and chest tubes, and (vi) adding insulin to the intravenous feed. The discussion of the pathophysiology of these mechanisms was disjointed and difficult to follow. However, the connection of LL to the sudden deteriorations and deaths in 7 seemed very compelling. However, I have to take issue with a number of the prosecution’s assertions.

(i) Over-distending the stomach with feed to an extent to cause collapse and projectile vomiting. I don’t have any experience of tube feeding prems, but projectile vomiting can be a reaction to bad / infected milk? Was LL in hurry to feed the baby? I find it hard to believe this was an attempt at murder.

(ii) Most obvious was the air in the stomach and intestines at post mortem. LL must have injected air via the gastric feeding tube, or how else did it get there? Well anyone who works in theatre or resuscitation knows during mask ventilation, which all these babies had (i.e. Neopuff), that it is very easy to blow up the stomach and intestines with air / anaesthetic gases, especially if one’s technique is not perfect. I regularly had to pass a suction catheter to empty the stomach of gas at the start of surgery to deflate the stomach and improve ventilation. I even did a study on the carbon dioxide levels that often reached the level of expired gas. However, the role of the Neopuff as a potential cause was never mentioned. So what is more likely, LL injected the air or the air got there through resuscitative efforts by stressed staff.

(iii) Some of the babies suddenly collapsed and developed a strange rash on the abdomen. Some recovered rapidly. This was said to be due to LL injecting air into the circulation. Air was found to be in the blood vessels at post mortem in some deaths. The premature baby can revert to a foetal circulation (by passing the lungs) when they become unstable and this can take time to treat (revert back). Sometimes “persistent foetal circulation” manifests itself during anaesthesia until the ductus closes fully. Point not mentioned in the case but could explain the above. Also, chest compression would cause significant sucking in of air to the heart if intravenous access lines were left open to air during the resuscitation after injecting a drug (adrenaline). LL sent a Datex about a line being left uncapped by one of the doctors. So there are other explanations and mechanisms by which air could have entered the circulation.

(iv) One of the cases had trauma to oral airway and significant blood loss, I think this was one of the twins with Haemophilia, a blood clotting disorder. LL was accused of traumatising the airway. I cannot imagine how. The likely explanation would be repeated intubation attempts, not an attempt to murder the baby by LL. I recall up to seven attempts as the neonate was difficult to intubate!

(v) LL was also accused of dislodging an endotracheal tube and a chest drain which lead to deterioration in two patients. Preterm babies are very small, endotracheal tubes can easily move and become dislodged however carefully one secures them, particularly if the neck is flexed or extended! Similarly, with chest drains, the baby had bilateral drain presumably as a result of premature lungs, and one drain became dislodged / was not working and a third drain was needed. These things happen so just because LL was present does not automatically mean it was her fault. Then there was the incident with deliberate liver injury, which equally could have occurred during chest compressions by someone else?

(vi) The addition of insulin to intravenous feeds has already been mentioned by Gill from a biochemistry and reliability of blood test perspective. I don’t fully understand this one. One baby was receiving regular intravenous nutrition made up in sealed bags from pharmacy. The baby had unexplained hypoglycaemia. For three bags it persisted and when LL was not on duty the hypoglycaemia resolved. Blood were analysed insulin and C peptide. Hypoglycaemia is common in preterm babies because their mechanisms to maintain blood glucose levels are immature (i.e. glycogen stores in the liver). The child may have had an infection as Gill says there was virus circulating. That being said it difficult to image how LL managed to injected the correct and same amount of insulin into a sealed bag on three separate occasions. There is a rigid nursing protocol involving two nurses when a new bag is put up to maintain sterility. Then there was a further insulin contaminated Dextrose infusion in a second baby. BTW, LL was not the only nurse present for both these cases.

Hence, I find it very difficult to accept the verdict that Lucy Letby was responsible for all 7 deaths and a further 6 attempts at murder. I think that her case needs to be reviewed by someone with a better understanding of neonatal medicine and how a premature baby unit is run.

Signed Lester; 20.10.2023

Letter to the BMJ

Rapid response to:

John Launer: Thinking the unthinkable on Lucy Letby

BMJ 2023; 382 doi: https://doi.org/10.1136/bmj.p2197, published 26 September 2023, cite as: BMJ 2023;382:p2197

Dear Editor

I am a coauthor of the report of the Royal Statistical Society https://rss.org.uk/news-publication/news-publications/2022/section-group-reports/rss-publishes-report-on-dealing-with-uncertainty-i/. It is deeply distressing that the police investigation into the case of Lucy Letby and the subsequent trial made all of the mistakes in our book. The jury was never told how the police investigation arrived at that list of “suspicious” events and how it was further narrowed down to the list of charges. This is a case in which a target was painted around a suspect by investigators. We call it confirmation bias, in statistics. It is also often referred to as the Texas sharpshooter paradox.

Thanks to amateurs who report their work on Twitter and YouTube, we now know how the list of charges in the Lucy Letby case evolved. It is utterly scandalous that this history was not revealed to the court. Here is the broad picture.

Doctors reported Lucy to the police, against the wishes of the hospital board.

They told the police the exact period she had been on the ward and gave them the files on all deaths in that period and on some of the incidents: namely, exactly and only those “arrests” at which Lucy had been present.

What qualifies as an incident, what is an arrest?

There is no medical category “arrest, resuscitation” under which such events are logged in hospital administration. Probably there were about five times as many such events when Lucy was not on duty, but nobody has ever looked. There is no medical definition of such an event. No formal criteria.

“Unexpected, unexplained, sudden” are also not defined in any formal way. Nor is “stable”.

Next the absolutely unqualified, long retired, paediatrician Dewi Evans, who has a business helping out in civil child custody cases, went through those medical files looking for anomalies about which he could fantasise a murder or murder attack. His ideas that milk was injected into the stomach or air into the veins were far fetched, and later not confirmed by any other evidence. On the contrary, the actual evidence certainly contradicts the idea that Lucy Letby actually attacked any child. He never gave alternative medical explanations, as would have been the obligation of a forensic scientist. All the deaths had had a post-mortem and a coroner’s report. Every single event on the charge sheet has absolutely normal explanation. Lucy was never seen doing anything wrong.

The medical experts for the prosecution merely confirmed Evans’ diagnosis, they also did not do the job of a forensic scientist.

The defence had no experts. They had brought in one paediatrician. But at the pre-trial hearing he said he wasn’t qualified in endocrinology, toxicology, etc etc etc.

This was Texas sharpshooter, big time. Plus utterly incompetent defence.

Richard Gill

Member of Royal Dutch Academy of Sciences

Past president of Dutch statistical society.

The Lucy Letby case

Lucy Letby enjoys prosecco at a small new year party with friends and family at her home (?).

Newest Note: [14 May] And then Rachel Aviv’s splendid article in the New Yorker came out, here is the link: https://www.newyorker.com/magazine/2024/05/20/lucy-letby-was-found-guilty-of-killing-seven-babies-did-she-do-it. I think this really marks “the end of the beginning” to quote a famous British statesman. Amusingly, The New Yorker has kindly blocked people with UK based IP address from reading it online, in order to comply with current reporting restrictions. One must not interfere with UK criminal justice; the next trial of Lucy – for killing baby K – must not be influenced. Some links to internet archive copies of the article can be found here: https://mephitis.co/new-yorker-magazine-lucy-letby-bombshell/. Hopefully readers in the UK will be able to get over these hurdles and find out what a lot of people outside the UK have already known for a year.

New Note: [10 May 2024] By now this post is even more terribly incomplete. Fortunately, Lucy Letby has more supporters (and support is growing by the day, to judge by social media) and this has resulted in a two splendid websites each with many articles concerning many aspects of the case, and a podcast with a fantastic series of conversations about the trial, and a YouTube channel. Here are links to them:

https://www.lucyletby.press “Vaudeville: Junk science and the Trial of Lucy Letby” is a website run by a retired medical practitioner from Ireland calling himself James Egan. Its blog contains numerous articles. Today it seems to be offline, I hope it will be back soon … and two days later it is back with a different address: https://jameganx.notepin.co/. Clearly in the process of redesign of the website, as well as moving to a new URL.

https://mephitis.co is run by Peter Elston, whose “chimpinvestor” site was one of the first to criticise the prosecution case against Lucy Letby. Peter has moved articles about the Letby case on that site to this new one. “Mephitis mephitis” is the scientific name of the animal we all know as the skunk.

https://podcasts.apple.com/dk/podcast/we-need-to-talk-about-lucy-letby/id1736761161 is a podcast series in which Peter talks about the case with another retired doctor, Michael McConville.

Finally I want to recommend Mark Mayes’ YouTube channel https://www.youtube.com/@ThePersecutionofLucyLetby (“The Persecution of Lucy Letby”). I also highly recommend Ceri Morrice’s videos, such as this one: https://www.youtube.com/watch?v=HFTSV_qh_Ik. And while I’m at it I’ll mention my own YouTube production (two hours, sorry): a scientific lecture focussing on the spreadsheet, the green post-it note, and the insulin evidence https://www.youtube.com/watch?v=RxmFLKTlim8 “A tale of two Lucy’s”. I need to update part of that talk since I have since realised that the police were given medical notes (selected by the consultant paediatricians) on 32 babies, not on 32 events. And I’ve also learnt a great deal more about insulin. I expect the next version will be a set of three one-hour lectures (or maybe four).

Note: [20 August 2023] This post is incomplete. It needs a prequel: the history of medical investigations into two “unexplained clusters” of deaths at the neonatal ward of the Countess of Chester Hospital. It needs many sequels: statistical evidence; how the cases were selected (the Texas sharpshooter paradox) and the origin of suspicions that a particular nurse might be a serial killer; the post-it note; the alleged insulin poisonings; the trouble with sewage backflow and the evidence of the plumber; the euthanasias. For the medical material, the site to visit is the magnificent https://rexvlucyletby2023.com/.

This is how the post originally started:

Lucy Letby, a young nurse, has been tried at Manchester Crown Court for 7 murders and 15 murder attempts on 17 newborn children in the neonatal ward at Countess of Chester Hospital, Chester, UK, in 2015 and 2016.

She was found:– Guilty of 7 counts of murder (against 7 babies)
– Guilty of 7 counts of attempted murder (against 6 babies)
– Not guilty on 2 counts of attempted murder (against 2 of the 6 babies she *was* found guilty of attempting to murder). No decision was reached on 6 counts of attempted murder against 6 different babies. However, 2 of those 6 she was also found guilty of a different count of attempted murder. [Thanks to the commenter who corrected my numbers.]

The prosecution dropped one further murder charge just before the trial started, on the instruction of the judge. Several groups of alleged murders and murder attempts concern the same child, or twin or triplet siblings. All but one child was born pre-term. Several of them, extremely pre-term.

I’m not saying that I know that Lucy Letby is innocent. As a scientist, I am saying that this case is a major miscarriage of justice. Lucy did not have a fair trial. The similarities with the famous case of Lucia de Berk in the Netherlands are deeply disturbing.

BTW [16 May 2024], I still don’t *know* that she is innocent but I am increasingly certain that she is.

The image below summarizes findings concerning the medical evidence. This was not my research. The graphic was given to me by a person who wishes to remain anonymous, in order to disseminate the research now fully documented on https://rexvlucyletby2023.com/, whose author and owner wishes to remain anonymous. Note that the defence has not called any expert witnesses at all (except for one person: the plumber). Possibly, they had not enough funds for this. Crowd-sourcing might be a smart way of getting the necessary work done for free, to be used at a subsequent appeal. That’s a dangerous tactic, and it seems to me that the defence has already taken a foolish step: they admitted that two babies received unauthorised doses of insulin, and their client was obliged to believe that too.

This blog post started in May 2023 as a first attempt by myself to blog about a case which I have been following for a long time. The information I report here was uncovered by others and is discussed on various internet fora. Links and sources are given below, some lead to yet more excellent sources. Everything here was communicated to the defence, but they declined to use it in court. Maybe they felt their hands were bound by pre-trial agreement between the trial parties as to what evidence would be brought to the attention of the jury, which witnesses, etc.

An extraordinary feature of UK criminal prosecution law is that if exculpatory evidence is in the possession of the defence, but not used in court, then it should not be used at a subsequent appeal, whether by the same defence team or a new one. This might explain why the defence team would not even inform their client of their knowledge of the existence of evidence which exonerated her. Even though, it is also against the law that they did not, as far as we know, disclose evidence which they had which was in her favour. The UK law on criminal court procedure is case law. New judges can always decide to depart from past judges’ rulings.

A very important issue is that the rules of use of expert evidence is that all expert evidence must be introduced before the trial starts. It is strictly forbidden to introduce new expert evidence once the trial is underway.

UK criminal trials are tightly scripted theatre. The jury is of course incommunicado, very close to its verdict, and I do not aim to influence the jury or their verdict. I aim to stimulate discussion of the case in advance of a likely appeal against a likely guilty verdict. I wish to support that small part of the UK population who are deeply concerned that this trial is going to end in an unjustified guilty verdict. Probably it will, but that will not be the end. So much information has come out in the 9 months of the trial so far, that a serious fight on behalf of Lucy Letby is now possible. Public opinion crystallised long ago against Lucy. It can be made fluid again, and maybe it can even be reversed, and this is what must happen if she is to get a fair re-trial.

As a concerned scientist who perceives a miscarriage of justice in the making, I attempted to communicate information not only to the defence but also to the prosecution, to the judge (via the clerk of the court), and to the Director of Public Prosecutions. That was a Kafkaesque experience which I will write about on another occasion. Personally, I tend to think that Lucy is innocent. That was however not my reason for attempting to contact the authorities. As a scientist, it was manifestly clear to me that she was not getting a fair trial. Science was being abused. I tried to communicate with the appropriate authorities. I failed to get any response. Therefore I had to “go public”.

Here is a short list of key medical/scientific issues, originally copied from an early version of the incredible and amazing website https://rexvlucyletby2023.com/, with occasional slight rephrasing and some small, hopefully correct, additions by myself. That site presents full scientific documentation and argumentation for all of the claims made there.

Air embolism cannot be determined by imaging, and can only be determined soon after death, and requires the extraction of air from the circulatory system, and analysis of the composition of the air using gas chromatography.
The coroner found a cause of death in 5 out of 7 of the alleged murder cases. Two of them appeared to be, in part, related to aggressive CPR, two appeared to be due to undiagnosed hypoxic-ischemic encephalopathy and myocarditis, one of the infants received no autopsy, and the other infant was determined to have died due to prematurity. It is highly unusual for the cause of death to be altered years after the fact and using methodology that is not supported by the coroner’s office.
The two claims of insulin poisoning are not supported by the testing conducted, and the infants (who are still alive and well) did not have dangerously low or dangerously high blood glucose levels for any period of time. There are many physiological reasons that could explain their low blood glucose during the whole period. In one of the two cases, assumptions are being made on the basis of one test taken at a single time point, clearly inconsistent with the other medical readings, and contravening the manufacturer’s own instructions for use (see image below). The report detailing the conclusions from that single test violates the code of practice of the forensic science regulator. Moreover, it appears that some numerical error has been made in the necessary calculation, resulting in an outcome which is physiologically impossible (or the person responsible did not know about the so-called “hook effect”). The mismatch between C-peptide and insulin concentration does not prove that the excess insulin found must have been synthetic insulin. There are many other biological explanations for a mismatch. No testing was done to determine the origin of the insulin. Similarly, there are many innocent explanations for the detection of some insulin in a feeding bag.
The air embolism hypothesis is confusing because it fails to explain why some children apparently perished and others did not, and it has not been supported by the minimal necessary measurements.
In at least one case, Lucy is blamed with causing white matter brain injury. This claim is utterly dishonest. The infant who experienced this brain injury was born at 23 weeks gestation, and white matter brain injury is associated with such early births. Further, there is sufficient evidence that demonstrates that enterovirus and parechovirus infection has been linked to white matter brain injury in neonates, resulting in cerebral palsy.
At the time of the collapses and deaths of the infants, enterovirus and parechovirus had been reported in other hospitals. There is a history of outbreaks of these viruses in neonatal wards in hospitals around the world. They especially harm preterm infants who do not yet have a functioning immune system. It is reported that many parents of the infants were concerned that their ward had a virus (as was Lucy) and that Dr Gibbs denied this was so. To date we have seen no evidence to show they did any viral testing, and if they did what the results were.

Then a fact pertaining to my own scientific competence.

Both prosecution and defence were warned long ago about the statistical issues in such cases. Both have responded that they are not going to use any statistics. They are also not using the services of any statistician. Seems the RSS report https://rss.org.uk/news-publication/news-publications/2022/section-group-reports/rss-publishes-report-on-dealing-with-uncertainty-i/ has had the opposite effect to that intended. Amusingly, the same thing happened in the case of Lucia de Berk. At the appeal the prosecution stopped using statistics. She was convicted solely on the grounds of “irrefutable medical scientific evidence”. (Here, I’m quoting from the words both spoken by the judges and written down on the first page of their > 100 page report of the reasons and reasoning which had led to their unshakable conviction that Lucia de Berk was guilty. The longest judge’s summing up in Dutch legal history). I was one of the five coauthors of the RSS report. We were a “task force”, formally commissioned by the “Statistics and the Law” section of the society. I consider it the most important scientific work of my career. It took us two years to put together. We started the work in 2020; we had seen the Lucy Letby trial on the horizon since 2017 when police investigations started and the suspect being investigated was already common knowledge.

The UK does not have anything like that because a jury of ordinary folk are the ones who (legally) determine guilt or innocence. This is a clever device which makes fighting a conviction very difficult; no one can know what arguments the jury had in their mind, no one knows what, if anything, was the key fact that convinced them of guilt. Ordinary people are convinced by what seems to be a smoking gun, they then see all the other evidence through a filter. This is called “confirmation bias”. In the Lucy Letby case, the smoking gun was probably the post-it note, and the insulin then seems to clinch the matter. The prosecution cross-examination convinces those who already believe Lucy is guilty that she moreover is constantly lying. More on all this in later posts, I hope.

Back to the insulin. Here are the instructions on the insulin testing kit used for the trial, taken from this website http://pathlabs.rlbuht.nhs.uk/ccfram.htm, the actual file is http://pathlabs.rlbuht.nhs.uk/insulin.pdf. Notice the warning printed in red. Yes, it was printed in red, that was not something I changed later. (All this is not my discovery; the person who uncovered these facts wishes to remain anonymous).

The toxicological evidence used in the trial violates the code of practice of the UK’s Forensic Science Regulator (see link below). It should have been deemed inadmissible. Instead, the defence has not disputed it, and thereby obliged their own client Lucy to agree that there must have been a killer on the ward. The jury are instructed to believe that two babies were given insulin without authorization, endangering their lives. (The two babies in question are still very much alive, to this day. Probably now at primary school.)

The defence stated to me that they cannot inform Lucy of the alternative analysis of the insulin question. It appears to me that this violates their own code of practice. Do they feel bound by the weird rules of UK’s criminal prosecution practice? Their client, Lucy Letby, is herself essentially merely a piece of evidence, seized by the police from what they believe is a scene of crime. No one may tamper with it during the duration of her own trial, which is lasting 10 months! I think this constitutes an appalling violation of basic human rights. The UK laws on contempt of court are meant to guarantee a fair trial. But in the case of a 10-month trial on 22 charges of murder and attempted murder, they are guaranteeing an unfair trial.

Lucy’s solicitor refused to pass on a friendly personal letter of support to Lucy or to her parents because she had not instructed him to do so. Should one laugh or cry about that excuse? I have the impression that he is not very bright and that he may have been convinced she is guilty. If so, I hope he is changing his mind. In the UK, the solicitor does all the legwork and communication between the client and the defence team. The barrister does the cross-examinations and the court theatrics, but probably never builds up a personal relationship with his client. Lucy has been all this time prison, in pre-trial detention, far from Manchester or Hereford. This might explain the extraordinarily weak defence which has been put up so far. But it might be deliberate.

One must take into account the fact that funding for legal support is meagre. The prosecution has been working on the case for 6 or so years, with unlimited resources. The defence has had a relatively very short time, with very limited resources. Probably the solicitor and the barrister already put in many more hours than they are paid for. There are no funds for expensive scientific witnesses. It is very possible that the defence team well understands that they cannot put up a serious defence during the 9 to 10 months of the trial, but that precisely this time period, with a huge number of revelations being made outside the trial, material for a serious defence during an appeal has been “crowd-sourced”. It seems to me that this mass of high-quality independent scientific work provides plenty of grounds for an appeal, in the case that the jury hands down a guilty verdict.

Some links:

Sarrita Adams’ Science on Trial website

scienceontrial.com

Formerly: https://rexvlucyletby2023.com/

Scott McLachlan’s Law Health and Tech blog

LL Part 0: Scepticism in Action: Reflections on evidence presented in the Lucy Letby trial. https://lawhealthandtech.substack.com/p/scepticism-in-action

LL Part 1: Hospital Wastewater https://lawhealthandtech.substack.com/p/ll-part-1-hospital-wastewater

LL Part 2: An ‘Association’ https://lawhealthandtech.substack.com/p/ll-part-2-an-association

LL Part 3: Death already lived in the NICU Environment, https://lawhealthandtech.substack.com/p/ll-part-3-death-already-lived-in

LL Part 4: Outbreak in a New NICU: Build it and the pathogens will come…https://lawhealthandtech.substack.com/p/ll-part-4-outbreak-in-a-new-nicu

LL Part 5: The Demise of Child A https://lawhealthandtech.substack.com/p/ll-part-5-the-demise-of-child-a

LL Part 6: The Incredible Dr Dewi Evans https://lawhealthandtech.substack.com/p/ll-part-6-the-incredible-dr-dewi

LL Part 7: The Demise of Child C. https://lawhealthandtech.substack.com/p/ll-part-7-the-demise-of-child-c

LL Part 8: The Death of Child D. Had she been left or resumed on CPAP, she might still be alive today. https://lawhealthandtech.substack.com/p/ll-part-8-the-death-of-child-d

Peter Elston’s “Chimpinvestor” blog

Do Statistics Prove Accused Nurse Lucy Letby Innocent? https://www.chimpinvestor.com/post/do-statistics-prove-accused-nurse-lucy-letby-innocent This splendid and comprehensive blog post also has a large list of links to reports and data sets. Yet more data analysis can and should be done. This site gives anyone who wants to a quick-start. And after that, two more outstanding posts…

https://www.chimpinvestor.com/post/more-remarkable-statistics-in-the-lucy-letby-case

https://www.chimpinvestor.com/post/the-travesty-of-the-lucy-letby-verdicts

Data obtained from FOI requests

FOI requests provided some fantastic data sets https://www.whatdotheyknow.com/request/neonatal_deaths_and_fois#incoming-1255362 see especially https://www.whatdotheyknow.com/request/521287/response/1265224/attach/2/FOI%204568×1.xlsx?cookie_passthrough=1

How forensic science should be reported in court

Forensic Science Regulator: statutory code of practice https://www.gov.uk/government/publications/statutory-code-of-practice-for-forensic-science-activities

One of numerous enterovirus and parechovirus epidemics in neonatal wards

Cluster of human parechovirus infections as the predominant cause of sepsis in neonates and infants, Leicester, United Kingdom, 8 May to 2 August 2016 https://www.eurosurveillance.org/content/10.2807/1560-7917.ES.2016.21.34.30326

Someone commissioned a pretrial statistical and risk analysis – results not used in the trial

Lucy Letby Trial, Statistical and Risk Analysis Expert Input. Who commissioned this analysis, and what did it yield? (I can give you the answer after the verdict has come out). https://www.oldfieldconsultancy.co.uk/lucy-letby-trial-statistical-and-risk-analysis-expert-input/

The RSS (statistics and law section) report – not used in the trial

Royal Statistical Society: “Healthcare serial killer or coincidence?
Statistical issues in investigation of suspected medical misconduct” by the RSS Statistics and the Law Section, September 2022 https://rss.org.uk/news-publication/news-publications/2022/section-group-reports/rss-publishes-report-on-dealing-with-uncertainty-i/

At a pre-publication meeting of stake-holders held to gain feedback on our report, a senior West Midlands police inspector told me “we are not using statistics because they only make people confused”. Lucy’s sollicitor and barrister knew well in advance of our report, were even given names of excellent UK experts whom they could consult, but did not bother to contact one of them. No statistics in our courts please, we are British! Yet the UK has the best applied statisticians and epidemiologists in the world.

Article in “Science” about my work on serial killer nurses

Unlucky Numbers: Richard Gill is fighting the shoddy statistics that put nurses in prison for serial murder. Science, Vol 379, Issue 6629, 2022. https://www.science.org/content/article/unlucky-numbers-fighting-murder-convictions-rest-shoddy-stats

Two subreddits on the Lucy Letby case

https://www.reddit.com/r/scienceLucyLetby/ (the Lucy Letby Science subreddit)

https://www.reddit.com/r/lucyletby/ (general)

Medical Ethics

John Gibbs, recently retired Consultant Paediatrician at the Countess of Chester
Hospital, defined Medical Ethics as “Playing God with Life and Death decisions.” See article “Medical Ethics” on page 6 of The Messenger, Monthly Newsletter of St Michael’s, Plas Newton, Chester) – reporting on talk by Dr John Gibbs, retiring paediatrician at CoCH. https://stmichaelschester.com/wp-content/uploads/2019/04/Messenger-April-2020.pdf. Audio: https://stmichaelschester.com/sermons/encounter-medical-ethics/

The state of forensic science in the UK

https://www.bbc.co.uk/sounds/play/m001k7vt?partner=uk.co.bbc&origin=share-mobile “The UK’s forensic science used to be considered the gold standard, but no longer. The risk of miscarriages of justice is growing. And now a new Westminster Commission is trying to find out what went wrong. Joshua talks to its co-chair, leading forensic scientist Dr Angela Gallop CBE, and to criminal defence barrister Katy Thorne KC.”

Criminal Procedure Rules and Criminal Practice Directions

Revised rules came out earlier this year, so maybe they do not apply to a trial which started earlier. Still, they express what the Lord Chief Justice of England and Wales presently wants to promote. https://www.judiciary.uk/guidance-and-resources/message-from-lord-burnett-lord-chief-justice-of-england-and-wales-new-criminal-practice-directions-2023/ . See especially Section 7 of his “Criminal Practice Directions (2023)” https://www.judiciary.uk/wp-content/uploads/2023/04/Criminal-Practice-Directions-2023-1-3.pdf

New expert evidence cannot be admitted once a trial is in progress

“The courts have indicated that they are prepared to refuse leave to the Defence to call expert evidence where they have failed to comply with CrimPR; for example by serving reports late in the proceedings, which raise new issues (Writtle v DPP [2009] EWHC 236). See also: R v Ensor [2010] 1 Cr. App. R.18 and Reed, Reed & Garmson[2009] EWCA Crim. 2698″. This quote comes from https://www.cps.gov.uk/legal-guidance/expert-evidence. Note, a judge is always allowed to break with precedence. The rule is not actually a permanent rule, it is merely a description of current practice. Current practice evolves when and if a new judge sees fit to break with precedence. Obviously, he would have to come up with good legal reasons why he believes he has to do that. It’s his prerogative, his free choice. That’s the essence of case law, aka common law.

BOAS, Breed, CFR

Relationship between incidence of breathing obstruction and degree of muzzle shortness in pedigree dogs

The little dog in front of van Eyck’s Arnolfini’s is a “Griffon Bruxellois”.
[Arnolfini Portrait. (2022, July 27). In *Wikipedia*. https://en.wikipedia.org/wiki/Arnolfini_Portrait%5D

This blog post is the result of rapid conversion from a preprint, typeset with LaTeX, posted on arXiv.org as https://arxiv.org/abs/2209.08934, and submitted to the journal PLoS ONE. I used pandoc to convert LaTeX to Word, then simply copy-pasted the content of the Word document into WordPress. After that, a few mathematical symbols and the numerical contents of the tables needed to be fixed by hand. I have now given up on PLoS ONE and posted an official report on Zenodo: https://doi.org/10.5281/zenodo.7543812. I am soliciting post publication peer reviews on PubPeer: https://pubpeer.com/publications/78DF9F8EF0214BA758B2FFDED160E1

Abstract

There has been much concern about health issues associated with the breeding of short-muzzled pedigree dogs. The Dutch government commissioned a scientific report Fokken met Kortsnuitige Honden (Breeding of short-muzzled dogs), van Hagen (2019), and based on it rather stringent legislation, restricting breeding primarily on the basis of a single simple measurement of brachycephaly, the CFR: cranial-facial ratio. Van Hagen’s work is a literature study and it draws heavily on statistical results obtained in three publications: Njikam (2009), Packer et al. (2015), and Liu et al. (2017). In this paper, I discuss some serious shortcomings of those three studies and in particular, show that Packer et al. have drawn unwarranted conclusions from their study. In fact, new analyses using their data lead to an entirely different conclusion.

Introduction

The present work was commissioned by “Stichting Ras en Recht” (SRR; Foundation Justice for Pedigree dogs) and focuses on the statistical research results of earlier papers summarized in the literature study Fokken met Kortsnuitige Honden (Breeding of short-muzzled – brachycephalic – dogs) by dr M. van Hagen (2019). That report is the final outcome of a study commissioned by the Netherlands Ministry of Agriculture, Nature, and Food Quality. It was used by the ministry to justify legislation restricting breeding of animals with extreme brachycephaly as measured by a low CFR, cranial-facial ratio.

An important part of van Hagen’s report is based on statistical analyses in three key papers: Njikam et al. (2009), Packer et al. (2015), and Liu et al. (2017). Notice: the paper Packer et al. (2015) reports results from two separate studies, called by the authors Study 1 and Study 2. The data analysed in Packer et al. (2015) study 1 was previously collected and analysed for other purposes in an earlier paper Packer et al. (2013) which does not need to be discussed here.

In this paper, I will focus on these statistical issues. My conclusion is the cited papers have many serious statistical shortcomings, which were not recognised by van Hagen (2019). In fact, a reanalysis of the Study 2 data investigated in Packer et al. (2015) leads to conclusions completely opposite to those drawn by Packer et al., and completely opposite to the conclusions drawn by van Hagen. I come to the conclusion that the Packer et al. study 2 badly needs updating with a much larger replication study.

A very important question is just how generalisable are the results of those papers. There is no word on that issue in van Hagen (2019). I will start by discussing the paper which is most relevant to our question: Packer et al. (2015).

An important preparatory remark should be made concerning the term “BOAS”, brachycephalic obstructive airway syndrome. It is a syndrome, which means: a name for some associated characteristics. “Obstructed airways” means: difficulty in breathing. “Brachycephalic” means: having a (relatively) short muzzle. Having difficulty in breathing is a symptom sometimes caused by having obstructed airways; it is certainly the case that the medical condition is often associated with having a short muzzle. That does not mean that having a short muzzle causes the medical condition. In the past, dog breeders have selected dogs with a view to accentuating certain features, such as a short muzzle: unfortunately, at the same time, they have sometimes selected dogs with other, less favourable characteristics at the same time. The two features of dogs’ anatomies are associated, but one is not the cause of the other. “BOAS” really means: having obstructed airways and a short muzzle.

Packer et al. (2015): an exploratory and flawed paper

Packer et al. (2015) reports findings from two studies. The sample for the first study, “Study 1”, 700 animals, consisted of almost all dogs referred to the Royal Veterinary College Small Animal Referral Hospital (RVC-SAH) in a certain period in 2012. Exclusions were based on a small list of sensible criteria such as the dog being too sick to be moved or too aggressive to be handled. However, this is not the end of the story. In the next stage, those dogs who actually were diagnosed to have BOAS (brachycephalic obstructive airway syndrome) were singled out, together with all dogs whose owners reported respiratory difficulties, except when such difficulties could be explained by respiratory or cardiac disorders. This resulted in a small group of only 70 dogs considered by the researchers to have BOAS, and it involved dogs of 12 breeds only. Finally, all the other dogs of those breeds were added to the 70, ending up with 152 dogs of 13 (!) breeds. (The paper contains many other instances of carelessness).

To continue with the Packer et al. (2015) Study 1 reduced sample of 152 dogs, this sample is a sample of dogs with health problems so serious that they are referred to a specialist veterinary hospital. One might find a relation between BOAS and CFR (craniofacial ratio) in that special population which is not the same as the relation in general. Moreover, the overall risk of BOAS in this special population is by its construction higher than in general. Breeders of pedigree dogs generally exclude already sick dogs from their breeding programmes.

That first study was justly characterised by the authors as exploratory. They had originally used the big sample of 700 dogs for a quite different investigation, Packer et al. (2013). It is exploratory in the sense that they investigated a number of possible risk factors for BOAS besides CFR, and actually used the study to choose CFR as appearing to be the most influential risk factor, when each is taken on its own, according to a certain statistical analysis method, in which already a large number of prior assumptions had been built in. As I will repeat a few more times, the sample is too small to check those assumptions. I do not know if they also tried various simple transformations of the risk factors. Who knows, maybe the logarithm of a different variable would have done better than CFR.

In the second study (“Study 2”), they sampled anew, this time recruiting animals directly mainly from breeders but also from general practice. A critical selection criterium was a CFR smaller than 0.5, that number being the biggest CFR of a dog with BOAS from Study 1. They especially targeted breeders of breeds with low CFR, especially those which had been poorly represented in the first study. Apparently, the Affenpinscher and Griffon Bruxellois are not often so sick that they get referred to the RVC-SAH; of the 700 dogs entering Study 1, there was, for instance, just 1 Affenpinscher and only 2 Griffon Bruxellois. Of course, these are also relatively rare breeds. Anyway, in Study 2, those numbers became 31 and 20. So: the second study population is not so badly biased towards sick animals as the first. Unfortunately, the sample is much, much smaller, and per breed, very small indeed, despite the augmentation of rarer breeds.

Figure 1: Figure 2 from Packer et al. (2015). Predicted probability of brachycephalic dog breeds being affected by brachycephalic obstructive airway syndrome (BOAS) across relevant craniofacial ratio (CFR) and neck girth ranges. The risks across the CFR spectrum are calculated by breed using GLMM equations based on (a) Study 1 referral population data and (b) Study 2 non-referral population data. For each breed, the estimates are only plotted within the CFR ranges observed in the study populations. Dotted lines show breeds represented by <10 individuals. The breed mean neck girth is used for each breed (as stated in Table 2). In (b), the body condition score (BCS) = 5 (ideal bodyweight) and neuter status = neutered

Now it is important to turn to technical comments concerning what perhaps seems to speak most clearly to the non-statistically schooled reader, namely, Figure 2 of Packer et al., which I reproduce here, together with the figure’s original caption.

In the abstract of their paper, they write “we show […] that BOAS risk increases sharply in a non-linear manner”. They do no such thing! They assume that the log odds of BOAS risk , that is: log(p/(1 – p)), depends exactly linearly on CFR and moreover with the same slope for all breeds. The small size of these studies forced them to make such an assumption. It is a conventional “convenience” assumption. Indeed, this is an exploratory analysis, moreover, the authors’ declared aim was to come up with a single risk factor for BOAS. They were forced to extrapolate from breeds which are represented in larger numbers to breeds of which they had seen many less animals. They use the whole sample to estimate just one number, namely the slope of log(p/(1 – p)) as an assumed linear function of CFR. Each small group of animals of each breed then moves that linear function up or down, which corresponds to moving the curves to the right or to the left. Those are not findings of the paper. They are conventional model assumptions imposed by the authors from the start for statistical convenience and statistical necessity and completely in tune with their motivations.

One indeed sees in the graphs that all those beautiful curves are essentially segments of the same curve, shifted horizontally. This has not been shown in the paper to be true. It was assumed by the authors of the paper to be true. Apparently, that assumption worked better for CFR than for the other possible criteria which they considered: that was demonstrated by the exploratory (the author’s own characterisation!) Study 1. When one goes from Study 1 to Study 2, the curves shift a bit: it is definitely a different population now.

There are strange features in the colour codes. Breeds which should be there are missing, and breeds which shouldn’t be there are. The authors have exchanged graphs (a) and (b)! This can be seen by comparing the minimum and maximum predicted risks from their Table 2.

Notice that these curves represent predictions for neutered dogs with breed mean neck girth, breed ideal body condition score (breed ideal body weight). I don’t know whose definition of ideal is being used here. The graphs are not graphs of probabilities for dog breeds, but model predictions for particular classes of dogs of various breeds. They depend strongly on whether or not the model assumptions are correct. The authors did not (and could not) check the model assumptions: the sample sizes are much too small.

By the way, breeders’ dogs are generally not neutered. Still, one-third of the dogs in the sample were neutered, so the “baseline” does represent a lot of animals. Notice that there is no indication whatsoever of statistical uncertainty in those graphics. The authors apparently did not find it necessary to add error bars or confidence bands to their plots. Had they done so, the pictures would have given a very, very different impression.

In their discussion, the authors write “Our results confirm that brachycephaly is a risk factor for BOAS and for the first time quantitatively demonstrate that more extreme brachycephalic conformations are at higher risk of BOAS than more moderate morphologies; BOAS risk increases sharply in a non-linear manner as relative muzzle length shortens”. I disagree strongly with their appraisal. The vaunted non-linearity was just a conventional and convenience (untested) assumption of linearity in the much more sensible log-odds scale. They did not test this assumption and most importantly, they did not test whether it held for each breed considered separately. They could not do that, because both of their studies were much, much too small. Notice that they themselves write, “we found some exceptional individuals that were unaffected by BOAS despite extreme brachycephaly” and it is clear that these exceptions were found in specific breeds. But they do not tell us which.

They also tell us that other predictors are important next to CFR. Once CFR and breed have been taken into account (in the way that they take it into account!), neck girth (NG) becomes very important.

They also write, “if society wanted to eliminate BOAS from the domestic dog population entirely then based on these data a quantitative limit of CFR no less than 0.5 would need to be imposed”. They point out that it is unlikely that society would accept this, and moreover, it would destroy many breeds which do not have problems with BOAS at all! They mention, “several approaches could be used towards breeding towards more moderate, lower-risk morphologies, each of which may have strengths and weaknesses and may be differentially supported by stakeholders involved in this issue”.

This paper definitely does not support imposing a single simple criterion for all dog breeds, much as its authors might have initially hoped that CFR could supply such a criterion.

In a separate section, I will test their model assumptions, and investigate the statistical reliability of their findings.

Liu et al. (2017): an excellent study, but of only three breeds

Now I turn to the other key paper, Liu et al. (2017). In this 8-author paper, the last and senior author, Jane Ladlow, is a very well-known authority in the field. This paper is based on a study involving 604 dogs of only three breeds, and those are the three breeds which are already known to be most severely affected by BOAS: bulldogs, French bulldogs, and pugs. They use a similar statistical methodology to Packer et al., but now they allow each breed to have a different shaped dependence on CFR. Interestingly, the effects of CFR on BOAS risk for pugs, bulldogs and French bulldogs are not statistically significant. Whether or not they are the same across those three breeds becomes, from the statistical point of view, an academic question.

The statistical competence and sophistication of this group of authors can be seen at a glance to be immeasurably higher than that of the group of authors of Packer et al. They do include indications of statistical uncertainty in their graphical illustrations. They state, “in our study with large numbers of dogs of the three breeds, we obtained supportive data on NGR (neck girth ratio: neck girth/chest girth), but only a weak association of BOAS status with CFR in a single breed.” Of course, part of that could be due to the fact that, in their study, CFR did not vary much within each of those three breeds, as they themselves point out. I did not yet re-analyse their data to check this. CFR was certainly highly variable in these three breeds in both of Packer et al.’s studies, see the figures above, and again in Liu et al. as is apparent from my Figure 2 below. But Liu et al. also point out that anyway, “anatomically, the CFR measurement cannot determine the main internal BOAS lesions along the upper airway”.

Another of their concluding remarks is the rather interesting “overall, the conformational and external factors as measured here contribute less than 50% of the variance that is seen in BOAS”. In other words, BOAS is not very well predicted by these shape factors. They conclude, “breeding toward [my emphasis] extreme brachycephalic features should be strictly avoided”. I should hope that nowadays, no recognised breeders deliberately try to make known risk features even more pronounced.

Liu et al. studied only bulldogs, French bulldogs and pugs. The CFRs of these breeds do show within breed statistical variation. The study showed that a different anatomical measure was an excellent predictor of BOAS. Liu et al. moreover explain anatomically and medically why one should not expect CFR to be relevant for the health problems of those races of dogs.

It is absolutely not true that almost all of the animals in that study have BOAS. The study does not investigate BOS. The study was set up in order to investigate the exploratory findings and hypotheses of Packer et al. and it rejects them, as far as the three races they considered were concerned. Packer et al. hoped to find a simple relationship between CFR and BOAS for all brachycephalic dogs but their two studies are both much too small to verify their assumptions. Liu et al. show that for the three races studied, the relationship between measurements of body structure and ill health associated with them, varies between races.

Figure 2: Supplementary material Fig S1 from Liu et al. (2017.) Boxplots show the distribution of the five conformation ratios against BOAS functional grades. The x-axis is BOAS functional grade; the y-axis is the ratios in percentage. CFR, craniofacial ratio; EWR, eye with ratio; SI, skull index; NGR, neck girth ratio; NLR, neck length ratio.

In contradiction to the opinion of van Hagen (2019), there are no “contradictions” between the studies of Packer et al. and Liu et al. The first comes up with some guesses, based on tiny samples from each breed. The second investigates those guesses but discovers that they are wrong for the three races most afflicted with BOAS. Study 1 of Packer et al. is a study of sick animals, but Study 2 is a study of animals from the general population. Liu et al. is a study of animals from the general population. (To complicate matters, Njikam et al., Packer et al. and Liu et al. all use slightly different definitions or categorisations of BOAS.)

Njikam et al. (2009), like the later researchers in the field, fit logistic regression models. They exhibit various associations between illness and risk factors per breed. They do not quantify brachycephaly by CFR but by a similar measure, BRA, the ratio of width to length of the skull. CFR and BRA are approximately non-linear one-to-one functions of one another (this would be exact if skull length equalled skull width plus muzzle length, i.e., assuming a spherical cranium), so a threshold criterium in terms of one can be roughly translated into a threshold criterium in terms of the other. Their samples are again, unfortunately, very small (the title of their paper is very misleading).

Their main interest is in genetic factors associated with BOAS apart from the genetic factors behind CFR, and indeed they find such factors! In other words, this study shows that BOAS is very complex. Its causes are multifactorial. They have no data at all on the breeds of primary interest to SRR: these breeds are not much afflicted by BOAS! It seems that van Hagen again has a reading of Njikam et al. which is not justified by that paper’s content.

Packer et al. (2015) Study. 2, revisited

Fortunately, the data sets used by the publications in PLoS ONE are available as “supplementary material” on the journal’s web pages. First of all, I would like to show a rather simple statistical graphic which shows that the relation between BOAS and CFR in Packer et al.’s Study 2 data does not look at all as the authors hypothesized. First, here are the numbers: a table of numbers of animals with and without BOAS in groups split according to CFR as a percentage, in steps of 5%. The authors recruited animals mainly from breeders, with CFR less than 50%. It seems there were none in their sample with a CFR between 45% and 50%.

BOAS versus CFR group

BOAS	(0,5]	(5,10]	(10,15]	(15,20]	(20,25]	(25,30]	(30,35]	(35,40]	(40,45]
0	1	4	12	12	22	13	12	4	15
1	9	11	19	5	5	4	1	2	3

Table 1: BOAS versus CFR group

This next figure is a simple “pyramid plot” of percentages with and without BOAS per CFR group. I am not taking into account the breed of these dogs, nor of other possible explanatory factors. However, as we will see, the suggestion given by the plot seems to be confirmed by more sophisticated analyses. And that suggestion is: BOAS has a roughly constant incidence of about 20% among dogs with a CFR between 20% and 45%. Below that level, BOAS incidence increases more or less linearly as CFR further decreases.

Be aware that the sample sizes on which these percentages are based are very, very small.

Figure 3: Pyramid plot, data from Packer et al. Study 2

Could it be that the pattern shown in Figure 3 is caused by other important characteristics of the dogs, in particular, breed? In order to investigate this question, I, first of all, fitted a linear logistic regression model with only CFR, and then a smooth logistic regression model with only CFR. In the latter, the effect of CFR on BOAS is allowed to be any smooth function of CFR – not a function of a particular shape. The two fitted curves are seen in Figure 4. The solid line is the smooth, the dashed line is the fitted logistic curve.

Figure 4. BOAS vs CFR, linear logistic regression and smooth logistic regression

This analysis confirms the impression of the pyramid plot. However, the next results which I obtained were dramatic. I added to the smooth model also Breed and Neutered-status, and also investigated some of the other variables which turned up in the papers I have cited. It turned out that “Breed” is not a useful explanatory factor. CFR is hardly significant. Possibly, just one particular breed is important: the Pug. The differences between the others are negligible (once we have taken account of CFR). The variable “neutered” remains somewhat important.

Here (Table 2) is the best model which I found. As far as I can see, the Pug is a rather different animal from all the others. On the logistic scale, even taking account of CFR, Neckgirth and Neuter status, being a Pug increases the log odds ratio for BOAS by 2.5. Below a CFR of 20%, each 5% decrease in CFR increases the log odds ratio for BOAS by 1, so is associated with an increase in incidence by a factor of close to 3. In the appendix can be seen what happens when we allow each breed to have its own effect. We can no longer separate the influence of Breed from CFR and we cannot say anything about any individual breeds, except for one.

	Model 1
(Intercept)	–3.86***	(0.97)
(CFRpct – 20) * (CFRpct < 20)	–0.20***	(0.05)
Breed == “Pug”:TRUE	2.48***	(0.71)
NECKGIRTH	0.06*	(0.03)
NEUTER:Neutered	1.00*	(0.50)
AIC	144.19
BIC	153.37
Log Likelihood	–67.09
Deviance	134.19
Num. obs.	154
* p < 0.001; p < 0.01; * p < 0.05

Table 2: A very simple model (GLM, logistic regression)

The pug is in a bad way. But we knew that before. Packer Study 2 data:

	W.out BOAS	With BOAS
Not Pug	92	30
Pug	3	29

Table 3: The Pug almost always has BOAS. The majority of non-Pugs don’t.

The graphs of Packer et al. in Figure 1 are a fantasy. Reanalysis of their data shows that their model assumptions are wrong. We already knew that BOAS incidence, Breed, and CFR are closely related and naturally they see that again in their data. But the actual possibly Breed-wise relation between CFR and BOAS is completely different from what their fitted model suggests. In fact, the relation between CFR and BOAS seems to be much the same for all breeds, except possibly for the Pug.

Final remarks

The paper Packer et al. (2015) is rightly described by its authors as exploratory. This means: it generates interesting suggestions for further research. The later paper by Liu et al. (2017) is excellent follow-up research. It follows up on the suggestions of Packer et al., but in fact it does not find confirmation of their hypotheses. On the contrary, it gives strong evidence that they were false. Unfortunately, it only studies three breeds, and those breeds are breeds where we already know action should be taken. But already on the basis of a study of just those three breeds, it comes out strongly against taking one single simple criterion, the same for all breeds, as the basis for legislation on breeding.

Further research based on a reanalysis of the data of Packer et al. (2015) shows that the main assumptions of those authors were wrong and that, had they made more reasonable assumptions, completely different conclusions would have been drawn from their study.

The conclusion to be drawn from the works I have discussed is that it is unreasonable to suppose that a single simple criterion, the same for all breeds, can be a sound basis for legislation on breeding. Packer et al. clearly hoped to find support for this but failed: Liu et al. scuppered that dream. Reanalysis of their data with more sophisticated statistical tools shows that they should already have seen that they were betting on the wrong horse.

Below a CFR of 20%, a further decrease in CFR is associated with a higher incidence of BOAS. There is not enough data on every breed to see if this relationship is the same for all breeds. For Pugs, things are much worse. For some breeds, it might not be so bad.

Study 2 of Packer et al. (2015) needs to be replicated, with much larger sample sizes.

References

van Hagen MAE (2019) Fokken met Kortsnuitige Honden. Criteria ter handhaving van art. 3.4. Besluit Houders van dieren Fokken met Gezelschapsdieren. Departement Dier in Wetenschap en Maatschappij en het Expertisecentrum Genetica Gezelschapsdieren, Universiteit Utrecht. https://dspace.library.uu.nl/handle/1874/391544; English translation: https://www.uu.nl/sites/default/files/eng_breeding_short-muzzled_dogs_in_the_netherlands_expertisecentre_genetics_of_companionanimals_2019_translation_from_dutch.pdf

Liu N-C, Troconis EL, Kalmar L, Price DJ, Wright HE, Adams VJ, Sargan DR, Ladlow JF (2017) Conformational risk factors of brachycephalic obstructive airway syndrome (BOAS) in pugs, French bulldogs, and bulldogs. PLoS ONE 12 (8): e0181928. https://doi.org/10.1371/journal.pone.0181928

Njikam IN, Huault M, Pirson V, Detilleux J (2009) The influence of phylogenic origin on the occurrence of brachycephalic airway obstruction syndrome in a large retrospective study. International Journal of Applied Research in Veterinary Medicine 7(3) 138–143. http://www.jarvm.com/articles/Vol7Iss3/Nijkam%20138-143.pdf

Packer RMA, Hendricks A, Volk HA, Shihab NK, Burn CC (2013) How Long and Low Can You Go? Effect of Conformation on the Risk of Thoracolumbar Intervertebral Disc Extrusion in Domestic Dogs. PLoS ONE 8 (7): e69650. https://doi.org/10.1371/journal.pone.0069650

Packer RMA, Hendricks A, Tivers MS, Burn CC (2015) Impact of Facial Conformation on Canine Health: Brachycephalic Obstructive Airway Syndrome. PLoS ONE 10 (10): e0137496. https://doi.org/10.1371/journal.pone.0137496

Appendix: what happens when we try to separate“Breed” from “CFR”

	Model 2
(Intercept)	–3.73*	(1.65)
Breed:American Bulldog	–43.51	(67108864.00)
Breed:Bolognese	–40.45	(47453132.81)
Breed:Boston Terrier	0.35	(1.84)
Breed:Boxer	1.23	(1.72)
Breed:Bulldog	1.04	(1.68)
Breed:Cavalier King Charles Spaniel	0.82	(1.37)
Breed:Chihuahua	–42.77	(38745320.70)
Breed:Dogue de Bordeaux	–43.35	(67108864.00)
Breed:French Bulldog	2.36	(1.59)
Breed:Griffon Bruxellois	–0.97	(1.18)
Breed:Japanese Chin	1.70	(1.46)
Breed:Lhasa Apso	1.75	(1.63)
Breed:Mastiff cross	1.97	(2.60)
Breed:Pekingese	–45.60	(38745320.70)
Breed:Pug	2.69*	(1.26)
Breed:Pug cross	–44.79	(47453132.81)
Breed:Rottweiler	–43.29	(47453132.81)
Breed:Shih Tzu	0.16	(1.23)
Breed:Staffordshire Bull Terrier	–43.37	(47453132.81)
Breed:Staffordshire Bull Terrier Cross	2.36	(2.07)
Breed:Tibetan Spaniel	–44.14	(67108864.00)
Breed:Victorian Bulldog	–43.16	(67108864.00)
NECKGIRTH	0.06	(0.06)
NEUTER:Neutered	1.80*	(0.84)
EDF: s(CFRpct)	1.00	(1.00)
AIC	158.59
BIC	237.55
Log Likelihood	–53.29
Deviance	106.459
Deviance explained	0.48
Dispersion	1.00
R^2	0.46
GCV score	0.03
Num. obs.	154
Num. smooth terms	1
* p < 0.001; p < 0.01; * p < 0.05

Table 4: A more complex model (GAM, logistic regression)

The above model (Table 4) allowing each breed to have its own separate “fixed” effect is not a success. That certainly was presumably the motivation to make “Breed” a random, not a fixed, effect in the Packer et al. publication, because treating breed effects as drawn from a normal distribution and assuming the same effect of CFR for all breeds disguises the multicollinearity and lack of information in the data. Many breeds, most of them contributing only one or two animals, enabled the authors’ statistical software to compute an overall estimate of “variability between breeds” but the result is pretty meaningless.

Further inspection shows that many breeds are only represented by 1or 2 animals in the study. Only five are in something a bit like reasonable numbers. These five are the Affenpinscher, Cavalier King Charles Spaniel, Griffon Bruxellois, Japanese Chin and Pug; in numbers 31, 11, 20, 10, 32. I fitted a GLM (logistic regression) trying to explain BOAS in these 105 animals and their breed together with variables CFR, BCR, and so on. Still then, the multicollinearity between all these variables is so strong that the best model did not include CFR at all. In fact: once BCS (Body Condition Score) was included, no other variable could be added without almost everything becoming statistically insignificant. Not surprisingly, it is good to have a good BCS. Being a Pug or a Japanese Chin is disastrous. Cavalier King Charles Spaniel is intermediate. Affenpinscher and Griffon Bruxellois have the least BOAS (and about the same amount, namely an incidence of 10%), even though the mean CFRs of these two species seem somewhat different (0.25, 0.15).

Had the authors presented p-values and error bars the paper would probably never have been published. The study should be repeated with a sample 10 times larger.

Acknowledgments

This work was partly funded by “Stichting Ras en Recht” (SRR; Foundation Justice for Pedigree dogs). The author accepted the commission by SSR to review statistical aspects of MAE van Hagen’s report “Breeding of short-muzzled dogs” under the condition that he would report his honest professional and scientific opinion on van Hagen’s literature study and its sources.

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