Saturday 12 October 2019

Microbes that cause Coronary Heart Disease

Micro-organisms and Coronary Heart Disease

Today on the coast of North Wales
I have suggested previously that coronary heart disease is due to  micro-organisms, several causing the endemic form of the disease, and this might have been a feature of life since the beginning of Homo Sapiens. There is a suggestion of this from embalmed Egyptian mummies, but otherwise there has been no preservation of human soft tissues, only bones. The well-described 20th century pandemic of coronary heart disease must have been the result of one specific micro-organism.

The basis for this is that coronary heart disease clearly has an environmental cause. A possible chemical cause occurring simultaneously in all continents in temperate zones would be unlikely. A dietary factor (effectively chemical) has been assumed, but despite half a century of investigation, no robust evidence has been identified. We must conclude that coronary heart disease is not due to diet.

The epidemic

The epidemic nature of coronary heart disease in the 20th century clearly indicates that it is not genetic: it is not possible for a genetic disease to create an epidemic because genetic factors within a society change extremely slowly. However coronary heart disease has a strong familial pattern, but it is not Mendelian; it would appear to be non-genetic inheritance. 

It is important to appreciate that we inherit 
from our parents more than our genes. We also inherit place of residence, type of housing, money, religion, culture, education, and also micro-organisms. It is important to appreciate that microbial diseases can spread within a household, within and between generations. We also inherit immunity to disease, and this appears to be a dominant genetic pattern, being acquired by all offspring (except those with very rare disorders of immunity).

In the 1960s the UK cardiologist Dr Maurice Campbell, studying national statistics,  recognised that coronary heart disease appeared after 1924 as an apparently new disease with an exponential increase in death rate. This suggested an epidemic, but it was only with the decline of the death rate after 1970 that the true pattern of an epidemic could be confirmed. 

Campbell M. Death rate from diseases of the heart: 1876 to 1959. British Medical Journal 1963: August 31st.p 528.
Campbell M. The main cause of increased death rate from diseases of the heart: 1920 to 1959. British Medical Journal 1963: September 21st. page 712.
A similar pattern emerged in the USA and elsewhere. The USA data source was mainly from health insurance companies, the information being compiled by Professor William G Rothstein.

Suggestions of infection

It was suggested in 1983 that the epidemic pattern of coronary heart disease was very suggestive of a viral epidemic, leading to the suggestion that it could be the result of a micro-organism. The authors were EP Benditt, T Barrett and JK McDougall, writing in the Proceedings of the National Academy of Science, USA. They stated:

‘The rise and fall of coronary heart disease (CHD) in a number of Western countries in this century has been dramatic….The shape of the CHD mortality curve reminds one of the rise and fall of epidemic diseases due to microbial agents.’

They also stated:

‘If we accept that aberrations in cholesterol metabolism provide the whole etiological basis for coronary and other vascular occlusive disease, we need hunt no further. On the other hand, we think there is still a great deal to be explained and that viruses may well hold some, if not all, of the missing pieces of the puzzle.’ 

Benditt EP, Barrett T, McDougall JK. Viruses in the etiology of atherosclerosis. Proc Natl Acad Sci USA. 1983; 80: 6386-9.

36 years later we know very well that there are indeed serious aberrations in the cholesterol story. A microbiological causation of coronary heart disease seems to provide a much more plausible explanation. But is it plausible that coronary heart disease might be due to a microbe? To answer this question it is not essential to identify a microbe, but it is the principle that needs to be clarified. Firstly, is there a better explanation? The answer to this is ‘No’, but conventional wisdom is unfortunately still firmly entrenched in the diet-cholesterol-heart theory.

Figure 1. The epidemic of coronary heart disease in USA men
Observations of infection

In 1977 it was noticed that: ‘A considerable proportion of the patients admitted to the coronary-care unit of the King Edward VII Hospital, Midhurst, Kent, gave an unsolicited history of an influenza-like illness before admission.”

Nicholls AC, Thomas M. Coxsackie virus infection in acute myocardial infarction. Lancet 1977; 309: 883-884. 

This led to a structured research project, the result of which was the observation that 10 out of 38 patients admitted with an acute myocardial infarction had positive antibodies indicating recent infection with Coxsackie virus. It was already known that Coxsackie B virus could cause myocarditis, an infection of the heart muscle. Myocardial infarction is a different disease, although there can be some clinical confusion between the two in individual cases. 

Griffiths PD, Hannington G, Booth JC. Coxsackie B virus infections and myocardial infarction. Lancet 1980; 315: 1387-1389.

Houston, Texas, has been a pioneering centre of vascular surgery, led by the charismatic surgeon Michael DeBakey. A study from his department found high levels of antibodies to Cytomegalovirus in 57% of his surgical patients, compared to 26% in a control group.

Adam E, Melnick JL, Probtsfield JL, et al. High levels of cytomegalovirus antibody in patients requiring vascular surgery for atherosclerosis. Lancet 1987; 330: 291-293.

Coronary heart disease is a particular problem in a transplanted heart, for reasons that are not clear but which are likely to be the result of therapeutic suppression of immunity. Cytomegalovirus has been detected in 62% of patients with coronary artery disease developing after heart transplantation compared to its detection in only 25% of those without coronary heart disease. 

McDonald K, Rector TS, Braunlin EA, et al. Association of coronary artery disease in cardiac transplant recipients with cytomegalovirus infection. Am J Cardiol 1989; 64: 359-362.

A study from Helsinki investigated 54 consecutive patients with stroke, all aged less than 50 years, and compared them with 54 community-based age and sex-matched controls. 19 patients had evidence of a febrile illness during the previous month compared with just 3 controls. In 80% of cases, the infection was respiratory, and mainly bacterial. This study again indicates an infection preceding an atherosclerotic event.

Syrjänen J, Valtonen V, Iivavainen M, et al. Preceding infection as an important risk factor for ischaemic brain infarction in young and middle aged patients. BMJ 1988; 296: 1156-1160.

It is interesting to note that there has also been an association identified between Helicobacter pylori and coronary heart disease. The plausibility of a significant direct causative effect is uncertain, even though it has been detected in atheromatous tissue. However it is important to remember how readily Helicobacter pylori  became accepted as the cause of peptic ulcer. Previously diet and stress were considered to be the cause.

Mendall MA, Goggin PM, Molineaux N, et al. Relation of Helicobacter pylori infection and coronary Heart disease. Br Heart J 1994; 71: 437-439.

Ameriso SF, Fridman EA, Leiguarda RC, Sevlever GE. Detection of Helicobacter pylori in human carotid atherosclerotic plaques. Stroke 2001; 32: 385-391.

Earlier this year it has been reported that there is a significant association between coronary heart disease and Hepatitis C Virus (HCV). The nature of this association is uncertain, but the main significance is the fact that HCV is so widespread worldwide.

Lee KK, Stella D, Bing R, Anwar M et al. Global burden of atherosclerotic cardiovascular disease in people with hepatitis C virus infection: a systematic review, meta-analysis, and modelling study. Lancet Gastroenterolog/Hepatology 2019. Published Online  July 31, 2019 S2468-1253(19)30227-4

The discovery of Chlamydia pneumoniae

Chlamydia pneumoniae 
Chlamydia pneumoniae was first identified in 1950 in Taiwan, in a man with conjunctivitis. A previously unidentified micro-organism, it initially became known as Taiwan Acute Respiratory agent (TWAR). It has since been noted to be a common respiratory pathogen. There are no reports of it being detected in retrospective investigations from patients from before 1950.

The association between coronary heart disease and preceding respiratory infection stimulated a further study from Helsinki. It was published in 1988 and it looked at 40 men who survived acute myocardial infarction, 30 with previous coronary heart disease and 41 matched controls. They were all tested for antibody evidence of infection with the newly-discovered organism Chlamydia TWAR, subsequently reclassified as Chlamydia pneumoniae.

Generally, bacteria are identified by microscopy and culture, whereas viruses are identified by the appearance of antibodies in the blood. Chlamydia pneumoniae is a tiny bacterium that grows only within host cells and it can be cultured only with great difficulty. It can be detected by rising antibody titres and now by specific DNA detection using PCR techniques. The organism has been isolated in recent years and cultured from atherosclerotic tissue.

Figure 2. Positive serology to Chlamydia pneumoniae

Saikku P, Leinonen M, Mattila K, et al. Serological evidence of an association of a novel Chlamydia, TWAR, with chronic coronary heart disease and myocardial infarction. Lancet 1988; 332: 983-985.

We can see in Figure 2 that in the 1988 Helsinki study, 68% of the sample with recent myocardial infarction had positive antibodies (serology) to Chlamydia pneumoniae (TWAR). 50% of those with known coronary heart disease had positive antibodies.  A lower proportion of 17% of controls had positive antibodies. 

Infection with this organism appeared to be quite common. It is not known if the 17% of controls had coronary heart disease, just that so far in their lives they had no clinical manifestations of it.  If Chlamydia pneumoniae does have a causative role in coronary heart disease, then progression from the initial respiratory infection to myocardial infarction does not always occur. This might depend on susceptibility of individuals to the infection., or it might depend on repeated low-grade infection.

A group from Johannesburg, South Africa, looked at atherosclerotic tissue obtained at autopsy and in about half the cases they were able to identify Chlamydia pneumoniae by the technologically advanced methods of immuno-histochemistry and polymerase chain reaction (PCR). Many similar studies have been undertaken since then, a total of 27 by 1998. These studies have been undertaken in a number of countries and in only one study, in Brooklyn, New York, was there a failure to detective presence of Chlamydia pneumoniae in atheromatous tissue. 

Kuo C, Shor A, Campbell LA, et al. Demonstration of Chlamydia pneumoniae in atherosclerotic lesions of coronary arteries. J Infectious Dis 1993; 167: 841-849.

Chlamydia pneumoniae has been detected in atherosclerotic coronary arteries and also in carotid, femoral, popliteal arteries and in the aorta. It has not been detected in normal arterial tissue. 

Antibiotic therapy

If coronary heart disease is due a bacterium it might be anticipated that antibiotic therapy would help. This has never been extensive tested, as an infective cause has been considered only by a few individuals. But there is some suggestion of benefit, mainly by the antibiotic azithromycin. In Figure 3 we can see a positive benefit in those with coronary heart disease treated with a short course of azithromycin. Follow up was for just one month following a coronary event, and azithromycin appeared to give a significant advantage.

Figure 3. The effect of azithromycin on coronary heart disease

Gurfinkel E, Bozowich G, Daroca A, et al, for the ROXIS study group. Randomised trial of roxithromycin in non-q-wave coronary syndromes: ROXIS pilot study. Lancet 1997; 350: 404-407.

This was a preliminary study, but subsequent experience did not confirm these findings.

Cercek B, Shah PK, Noc M, et al, for AZACS investigators. Effect of short-term treatment with azithromycin on recurrent ischaemic events if patients with acute coronary syndrome in the Azithromycin in Acute Coronary Syndrome (AZACS) trial: a randomised controlled trial. Lancet 2003; 361: 809-813.

However, before abandoning the prospect of antibiotics having a role in the treatment of coronary heart disease, we must remember from a previous Blog Post that statin drugs were initially developed as antibiotics. In 1971 the microbiologist Professor Akira Endo in Japan discovered a natural inhibitor of the pathway in a broth of the mould of Penicillium citrinum. The compound, which was named compactin or mevastatin, inhibited the enzyme HMG-CoA reductase. The first statin had been identified, but it had a significant unexpected effect: it reduced blood levels of cholesterol. Serendipity led to the highly successful marketing of statins as cholesterol-lowering agents rather than as antibiotics. 

But as I have pointed out previously, it is clear that the benefit of statins is not related to the cholesterol-lowering property. The observation that statins are of the greatest benefit at the moment of a myocardial infarction, when the infective inflammatory process is at it greatest, indicates an immediate antibacterial effect.

Plausibility of an infective cause of coronary heart disease

Clearly the story is not yet complete. Unfortunately the diet-cholesterol-heart story has monopolised research into coronary heart disease, and as the epidemic is now effectively over, the opportunities to investigate and explain the epidemic are very much constrained. How many people with coronary heart disease have had a micro-biological investigation?

We know that the three major heart diseases in the early years of the 20th century were due to micro-organisms. The first is the sexually transmitted disease syphilis. It damages in particular the aorta and the aortic valve. The causative micro-organism Treponema palladium (identified 400 years after the disease was recognised) invades the arterial wall, as happens in coronary heart disease. The second is rheumatic fever. This is due to a throat infection with Streptococcus pyogenes, and resulting damage to the the valves of the heart, mainly the mitral valve. The third is endocarditis. This is due to a variety of micro-organisms, mainly the Streptococcus viridans group of bacteria, which enter the blood circulation via diseased gums, or tooth sockets following dental extraction. The infection further damages already abnormal heart valves. Heart valves are liable to damage as the result of poor blood supply.

The most plausible explanation of the cause of atherosclerosis and coronary heart disease is that they are due to bacterial invasion of the arterial wall. Cholesterol is present in the tissues as part of the body’s defence mechanism. What we should regard as endemic disease with low mortality has probably always been present and will continue in the future. A variety of micro-organisms will be responsible.

On the other hand, the epidemic form of the 20th century with a very high mortality rate and onset of sub-clinical disease at an early age must have been the result of a new micro-organism to which humankind had no immunity at the time of its appearance at the beginning of the 20th century. 

The rapid rise of death rate was followed by an equally rapid fall due to the development of immunity during three generations. The new micro-organism was most likely to have been Chlamydia pneumoniae, although no substantial study has been undertaken (in contrast to the search for a new micro-organism following the appearance of AIDS toward the end of the 20th century).

Saturday 31 August 2019

The Polypill - statistical spin and amplification of benefit

Many people in the world are taking medications of various sorts to improve their health and duration of life, most of these "medications" being futile. Life is at present longer and healthier than has ever been known. We are increasingly free of disease and as a result old age is the expectation for most of us.

But food enthusiasts of various sorts try to persuade us that changing diet can make our lives even better and longer. And there are of course many others who can make money out of our almost universal health pessimism, and our remarkable optimism in the power of medicines and special diets. Although the 20th century epidemic of high mortality coronary heart disease has now come to an end, we are still under pressure to take medications to prevent it.

Many people who are well and disease-free take statins. Many take aspirin. High blood pressure is constantly redefined so as to make more of us have hypertension. Pharmaceutical consumption without need increases, although of course it is necessary for economic growth. 

The polypill
If vast numbers of people are taking four tablets per day (statin, aspirin and two to lower blood pressure) it might make life easier for these medications to be combined into just one tablet. This “polypill” has been developed, containing aspirin, atorvastatin, hydrochlorothiazide, and either enalapril or valsartan. Even though the component medications might have been approved, combinations require a separate approval by the pharmaceutical regulating authority, and this requires additional clinical trials. 

The polypill trial
The clinical trial of the polypill has now been undertaken in Iran, and it has rather cleverly been called  the PolyIran study. The results were reported very recently in The Lancet, which appears to give special priority to articles promoting cholesterol and statins.  

The study involved 3417 subjects allocated to advice care only,  and 3421 subjects advice plus polypill. The subjects were aged between 50 and 75 years and the duration of follow-up was five years. 

There were 222 total deaths in the advice only group, this being  6.5% of the 3417 subjects.

There were 202 total deaths in the advice plus polypill group, this being 5.9% of the 3421 subjects

The difference is 0.6%, meaning that almost 200 (precisely 167) people would need to take a daily polypill for five years to prevent one death during this time. The polypill was effective in preventing death, but it is doubtful that a person prescribed it for five years for the purpose of preventing death would take it knowing that there is a 99.4% chance of it not preventing death.

Cardiovascular events
During follow-up, there were: 

301 (8.8%) non-fatal major cardiovascular events in those who received advice only, and 

202 (5.9%) non-fatal major cardiovascular events in those receiving advice plus polypill. 

The difference is 2.9%, the absolute reduction of event rates, meaning that the polypill was effective in the prevention of major cardiovascular events.

The spin
The absolute reduction of events by 2.9% does not sound to be very dramatic, and so statistical manipulation follows. 2.9% must be expressed as a proportion of 8.8%.

Once again we can see statistical spin, using the dubious method of taking percentages of percentages. 

(8.8-5.9)/8.8 expressed as a percentage = 33%. This sounds much better and this is what was appeared in the press, for example in The Guardian August 22nd 2019.

This spin amplifies the benefit of the polypill from 2.9% to 33%, by a factor of more than ten.

A useful expression of effectiveness is the NNT, the Number Needed to Treat in order to achieve one clinical benefit, We see in respect of the polypill that for death this is 167.

The paper expressed the NNT for the prevention of cardiovascular events as being 34.5 (100/2.9). This is correct and it means that if 34.5 people take the polypill for five years, one will not experience a major cardiovascular event. The value of this in clinical practice requires an informed discussion between doctor and patient, in this case a disease-free normal person.

Primary prevention only: 
By this is meant subjects without a history of cardiovascular disease. Within the discussion section we find: 

“The PolyIran study showed that the use of polypill resulted in around a 40% reduction in the risk of major cardiovascular events in individuals without a history of cardiovascular disease”.

It is not easy to identify the origin of the 40%, but it can be obtained from Table 2. 

737 of the subjects had a history of pre-existing cardiovascular disease, but the majority 6101 did not have. 

Of those without a history of cardiovascular disease (primary prevention) and receiving advice only, 229 out of 3068 experienced a cardiovascular event during the study, that is 7.5 %. 

Of those without a history of cardiovascular disease and receiving advice plus polypill, 136 out of 3033 experienced a cardiovascular event, 4.5%.

Thus the absolute benefit was a reduction of 7.5-4.5 = 3% in cardiovascular events in those receding the polypill.

More spin
A 3% reduction in events is not very dramatic. How can this number be increased?

The statistical manipulation becomes the expression of 3% as proportion of 7.5%. This is (7.5-4.5)/7.5 expressed as a percentage = 40%. 

40% reduction sound much better than 3%, amplification of effect by a factor of 13.

As I have expressed in a previous Blog post, statistical spin of this sort is disgraceful manipulation designed to mislead the general public and their medical advisors. The expression of proportionate over absolute benefit should be stopped.

Gholamreza Roshandel, Masoud Khoshnia, Hossein Poustchi et al.  Effectiveness of polypill for primary and secondary prevention of cardiovascular diseases (PolyIran): a pragmatic, cluster-randomised trial.  Lancet 2019; 394: 672–83.

Tuesday 11 June 2019

Food and Health – Statistics and Lies

Is highly processed food bad for us?
Can we believe the statistics that are presented to us?

Shopping: a large range of ready-made meals
We live in an era of “dietism”. There are many books and articles on how various aspects of our food are responsible for all sorts of illnesses and threats to our longevity. I have pointed out previously that the past fifty years, since 1970, have seen a major and apparently spontaneous (that is "natural") decline in deaths from coronary heart disease, the end of the greatest epidemic of the 20th century. However this occurred at a time during which we have seen the emergence of “fast food”, that many people regard as “junk food”, and also our supermarkets have become stocked with  food that has been prepared in factories. The food is imaginative and it tastes good. But is it bad for us? Despite such allegations, how is it that the proportion of our population living beyond the 90th birthday continues to increase? Are we not in reality healthier than ever?

Statin trial

Shepherd J, Cobbe SM, Ford I, Isles CG, Lorimer AR, Macfarlane PW,
McKillop JH, Packard CJ, for the West of Scotland Coronary Prevention
Study Group. Prevention of coronary heart disease with pravastatin in
men with hypercholesterolemia. N Engl J Med. 1995;333:1301–1307.

We are also in an era of distorted statistics. I have illustrated previously that we are told from the WOSCOPS study (reference above) that pravastatin reduces the risk of death at eight years by 25%, but the reality is that the reduction was just 1%. The absolute reduction in death rates was from 4% to 3%. The 25-fold amplification of the result is manipulative, quoting proportionate reduction in death rate in place of absolute reduction. This means presenting the numerator without reference to the denominator. 

The absolute reduction of 1% tells us clearly that if 100 high-risk men take pravastatin daily for eight years, then the life of one of them will be prolonged. The proportionate reduction of 25% gives no useful information as the denominator is ignored. It is a manipulation to enable sales.

Let me illustrate this manipulation further.

Diet study from Paris

On May 30th 2019 there was a press release in the UK, appearing in many national newspapers and on radio transmissions. It stated that:

The press release indicted an increase of cardiovascular disease by 12% when the proportion of highly processed food in the diet of 105,000 individuals rose from 10% to 20%. Incidentally, is this an increase of 10% or of 100%? 

Read on for further understanding.

It is necessary to look at the original data, some of which was provided in an expansion of the press release in The Guardian, and it is of course found in the original paper in the British Medical Journal.

The study was of 105,000 adults age 18 or greater, and the follow-up was for five years. For the purpose of analysis the population studied was divided into four groups based on the proportion of highly processed food in the diet.


The cardiovascular disease rate in the highest quartile of consumption of highly processed food was 277 per 100,000. This is the usual way of presenting health statistics.

For the lowest quartile of highly processed food consumption the disease rate was 242 per 100,000.

There was thus a higher disease rate in those consuming a diet relatively high in processed food. But how much higher?

Let us note the simple calculation:

277 - 242 = 35

This would appear to be non-controversial. It indicates that if 100,000 people consume a high processed food diet, then 35 more of them will develop cardiovascular disease compared to a similar group with a lower consumption of highly processed food. 

35 out of 100,000  = 0.035 out of 100  =  0.035%

If you are one of the 100,000 would an increased risk of 0.035% make you change your diet? The answer is probably “No”.

If this were the stated result of the study, it would almost certainly be ignored by the media. There would be no press release and the authors would remain in obscurity with no fame and little prospect of future research grants. The cost of the study was not stated but the paper had 13 authors and no doubt many unrecorded assistants and clerical staff employed over 10 years. This would suggest a cost of at least £/€/$ 1.5 million.

The Spin

Turning bad news into good news is a process called “spin”, frequently used by politicians. Enter statistical manipulation.

I have presented absolute proportions, and these enable us to understand the real world, how the results of the study affect "me" as an individual. We now need to look as to how the result is translated into “proportionate” changes.

The increase in disease rate is from 242 to 277, a difference of 35. This number is presented as proportion of 242.

35 / 242 = 0.145

and this represents an increase of 14.5% of cardiovascular disease by consuming a diet high in highly processed food.

0.035% represents a change. But using a proportionate result mean that we can use an increase from the lower number or a decrease from the higher number. 

And so alternatively:  35 / 277 = 0.127

which is a 12.7% decrease by having a diet low in highly processed food realise to a high level diet.

Using proportionate figures of 14.5% of 12.7% is much more dramatic than 0.035%, hence widespread newspaper headlines.

Expressing the result of this study as a proportionate rather than an absolute result amplified the diet effect from 0.035% to 12.7%. This is an amplification factor of about 350.

0.035% tells us about the real world, how the health of us as individuals is influenced by our diet. 25% tells us nothing.

This spin is common in medical papers and it ought to stop or be stopped.

Dubious mathematics - changing denominators

Disease incidence per 100,000
We encounter the mathematical effect of manipulating numerators with no reference to the denominator. In this case the numerators were 277 and 242, and the denominator was 100,000. You will see that the manipulation above took no account of the denominator.

Disease incidence per 1000
Let us assume that the denominator was different by a factor of 100:
High consumption group  277 per 1000
Low consumption group   242 per 1000.

The difference of 35 is now per 1000.

35 per 1000 = 3.5 per 100 = 3.5 %.
This is important, a hypothetical effect 100 times greater than the real study result.

But the proportionate calculation would remain just the same:
35 / 242 = 0.145 = 14.5%  or  35 / 277 = 0.127 = 12.7%

Disease incidence per 10,000,000
Let us suppose that the disease incidence is much lower, by a factor of 100. The numerators remain the same but are now found to be per 10,000,000, a much much lower incidence and a therefore a much larger denominator. The results would be:

277 per 10,000,000  and 242 per 10,000,000. 

The difference (numerator) would remain at 35, but now out of 10,000,000.
The absolute difference would be 0.00035%. This is a tiny population effect, but if the sample size is big enough it could achieve statistical significance, even though of no health significance.

However proportionate change manipulation still leaves us with 
35 / 242 = 0.145 = 14.5%  or 35 / 277 = 0.127 = 12.7%

Whatever the disease incidence, the answer is always 12.7% (or 14.5%). 

We can therefore see that proportionate results are meaningless in the real world and give no indication of real risk to the individual who is making a choice of what to eat.

Expressing the proportionate result implies a much higher risk than is really the case. It is intended to frighten the public (and their doctors) into taking tablets or altering their life-styles or to produce an impact much greater than it should be.

Is it acceptable to use numerators without reference to denominators?

This type of manipulation is completely outrageous. It should not be allowed in papers published by reputable journals, or any other form of publication. Numerators should not be separated from their denominators. 

I am no mathematician but I learned about numerators, denominators and percentages when I was in my primary school. I was taught that if the numerator is increased or reduced, then the denominator should be changed by the same proportion:

 3/4 = 6/8, not 6/4.  3/4 = 12/16, not 12/4.

The manipulations that we have seen would result in an examination "fail".

Are my salami sandwiches a form of suicide?


Srour B, Fezeu LK, Kesse-Guyot E et al. Ultra-processed food intake and risk of cardiovascular disease: prospective cohort study (NutriNet-Santé). BMJ 2019;365:l1451