7. Basic Maths in Medical Research and Decision Making

The aim here is to explain the mathematics and statistics in medical decision making in a relatively simple way.

This chapter contains the following sections:

If you wish to go directly to one of these headings, click on the blue underlined title above.

If you can manage GCSE or ‘O’ level maths, you should understand this

Mathematics and statistics in medical research and decision-making may seem daunting and it can be quite difficult. This section aims to keep it simple so that people without great mathematical skills can understand it. It will concentrate on principles rather than statistics.

The mathematics involved in biological science can be very complex and many academic medical departments employ a professional statistician. I was going to call this section “Maths for meerkats” as the intention is to keep it “simples”, as they say on a heavily advertised price comparison website. The mathematics presented here should be readily comprehensible to anyone with an ‘O’ level pass in maths or GCSE at grade C or above. Do not be put off. The reason for this chapter is to give some insight into some of the calculations and terms used without getting too complicated.

It will also give insight into how NICE makes decisions about the effectiveness and cost-effectiveness of treatments.

Normal Distribution; Mean, Median and Mode

Suppose that you measure the height of 100 adult men to the nearest centimetre and plot it on a graph with height on the abscissa (x-axis) and the number of men at that height on the ordinate (y-axis). The graph that results would be shaped like a bell and symmetrical both sides. There are certain characteristics of these graphs. It is called a normal or Gaussian distribution. Add up all the heights and divide it by the number of men to give the mean or arithmetical average.

There is a parameter called standard deviation which is important, but it is unnecessary to know how it is calculated. For a Gaussian distribution, one standard deviation (SD) from the mean in each direction includes about two thirds of all the results. Two standard deviations from the mean contain around 95% of results and three standard deviations include approximately 99%. The figure below illustrates this, giving the numbers more precisely. For most purposes, two thirds and 95% for one and two standard deviations from the mean is satisfactory.

To be more accurate (or pedantic) this shows how many observations fall within each SD for a normal or Gaussian distribution.

If someone was to ask the height of an adult man, you could not give a single figure that is correct. Mass produced cars of a certain make are all the same size but in the biological world we are all different. On the basis of that graph you could give the average figure and the standard deviation to show the range that would include 66.6% and the range that would include 95% but not everyone. I am assuming that all the men were normal and healthy and that none had any condition that would affect their stature. Otherwise this would skew the graph.

People vary

You might then do a number of blood tests. Perhaps a blood count, blood chemistry and some enzyme tests as in liver function tests. These are all normal, healthy men. What is the normal level for haemoglobin, blood sodium, potassium or calcium? What is the normal level for albumin or for enzymes? What is the blood pressure? What is their IQ? All these parameters would not give a single figure but a Gaussian distribution, the same as for height. Therefore, when a doctor takes a blood test and gets the result back, what should he take as normal? Hospital laboratories do not give a single figure for normal but a reference range. Normally this is taken as two standard deviations from the mean. This will include 95% of normal people. Therefore, if we have done a number of tests on these 100 men, we would expect to find that for each test around five of them (5%) have a result that is outside the range of “normal”.

They are not ill. They are all fit and healthy but for each test, on average 2.5 will have a result that it too high and 2.5 will have a result that it too low. If you take a fit, healthy person and do 20 tests, it is likely that at least one result will fall outside the reference range. Then what should the doctor do when he gets back a result that is “abnormal”? He should look at how far outside the reference range it is. One in 100 results would be expected to fall at three standard deviations from the mean. The further outside the range it falls the more likely it is to be significant. The doctor may decide to repeat the test as some of these figures show slight variation from day to day or there may be laboratory error. Perhaps the next time it will come back normal. It is important to be aware of this normal distribution and if there is no reason to suspect that a result outside the reference range represents disease, it is important not to investigate too far. Look at the patient and ask if he is ill. If not, leave him alone.

If you were to measure the height of 100 boys of 8 years old the result would be a bell-shaped Gaussian curve similar to the one for men but if you compare them, there would be little or no overlap. Therefore, it is clear that 8 years old boys are smaller than adult men. However, suppose that you were to measure 100 adult women. The two curves would look something like this with the heavy line for women and the lighter, dotted line for men.

The darker line represents the line for women and the lighter line the line for men. There is considerable overlap but we can still see than overall, men tend to be taller than women.

On average men are taller than women but many women are taller than many men

This shows that on average men are taller than women but there are a substantial number of women who are taller than a substantial number of men. So, are the two groups really different groups or could the difference observed between the two have occurred by chance? This would require a statistical analysis which would then give a figure to show the chance that these two results would be obtained if the groups were really similar. This was discussed with the results from randomised controlled trials. Perhaps the one set of numbers represents the blood cholesterol before treatment and the other set after treatment. Is it significantly different? Has the treatment worked?

If instead of measuring the height of the 100 men or women you had measured their weight, the resultant graph would be asymmetrical and skewed. The figure below shows the two types of skewed distribution along with a normal distribution. The right or positive skew may occur with weight or income. Some people are more than twice the mean but no one is less than zero. A left or negative skew may occur with length of life. Perhaps the average is around 80 years. A few die in infancy or childhood but no one exceeds twice the average and lives to be 160 years. In fact 120 years is almost unheard of in verified cases.

The left shows a left or negative skew, the centre is a normal distribution and the right shows a right or positive skew. Note the mean, median and mode in each.

With the Gaussian or normal curve, we had the mean or arithmetical average. With skewed curves we have this but also two parameters called median and mode. The median is the line where there is an equal number of readings both above and below it. The mode is the reading with the highest number of that figure. It is the peak of the curve. To remember which is which, à la mode means in fashion and the mode is the most popular or fashionable reading. A Gaussian distribution also has a median and mode but for this symmetrical curve, the mean, median and mode are all the same.

It is essential to have the right tool for the job

I am sticking to concepts and avoiding the actual statistics. Gaussian distributions are called parametric. Skewed are non-parametric. Parametric and non-parametric distributions require different statistical techniques. It tends to be easier to show significant difference for two parametric curves than for non-parametric ones. You may come across a test called student’s t for parametric and chi squared or χ2 for non-parametric distributions. Chi is pronounced k-eye. I shall not go further into the mathematics of theses calculations. There are a number of other tests that can also be used.

Confidence Intervals

Statistics is about likelihood or chance. You have done the trial and you have calculated numbers but how reliable are those numbers? Again the mathematics of how to do so will be ignored but it is possible to calculate 95% confidence intervals. This means that whereas your result is a specific number or point on a graph, there is a 95% chance that the true figure lies somewhere between the two confidence limits. Thus you may have a result of 30% but the statistics show that there is a 95% chance that the true figure lies 5% either side of that point. This is usually presented as 30±5%. It means that there is a 95% chance that the true value lies somewhere between 25% and 35%. This also means that there is a 5% chance that it lies further out either way. If the figure was 12 with a 95% chance of it being ±4, a common way of presenting this graphically is like this:

The confidence intervals for 12±4 can be presented like this

It shows that the point plotted from the figures was 12 but the two lines with T-bars at the end give the 95% confidence limits, which in this case is ±4. Therefore there is a 95% chance that the true value is between 8 and 16. The larger the sample and the less spread the results; the closer will be the confidence limits. Similarly, a number of observations with a wide span of numbers will produce a very wide confidence limit. In other words, it is not very reliable.

One way of looking at statistical significance is to see if confidence limits overlap. Perhaps, in a randomised controlled trial (RCT), the results of the test group were 30±3% and in the control group 40±3%. This means that there is a 95% chance that the true value for the test group lies between 27% and 33% whilst for the control it lies between 37% and 43%. The highest figure for the test group is 33%. The lowest figure for the control group is 37%. There is no overlap and so the two figures may be seen as different. The intervention works. It makes a difference. However, suppose that the respective figures were 33±7% and 43±7%. The confidence figures lie between 26% and 40% and between 36% and 50%. Therefore there is overlap. Therefore the groups cannot be said to be different. The value of the intervention is unproven.

The confidence limits for the three points overlap

There are many other ways to present confidence limits and these illustrations are to help to identify what they are when you see them.

Numbers Needed to Treat or Harm

Imagine that the trial of a new cancer drug has shown that it reduces the death rate at 5 years for a certain type of cancer at a certain stage from 30% to 20% but it costs £10,000 for each course of treatment. How much does it cost to save a life? This is an important calculation in health economics and the sort that NICE often makes. We need to ask not only if the treatment works but if it represents good value. The NHS spends well over £110 billion a year but resources are finite and money spent in one way is not available to spend elsewhere.

The immediate temptation is to say that as the course costs £10,000, that is the cost to save a life. This is wrong as not everyone who is treated will have a better outcome as a result of the treatment. The death rate had fallen from 30% to 20% which means from 3 in 10 to 2 in 10. Therefore, for every 10 people given the new treatment, 7 will survive who would have survived without the new treatment. The two who have died would also presumably have died with the old treatment. Just one person has survived with the new treatment who would have died with the old treatment. Only 1 in 10 has a different outcome. Therefore it takes 10 people to be treated to save one life. Hence the number needed to treat (NNT) is 10. The cost of saving one life is 10 x £10,000 =£100,000.

This illustrates how drop from 30% to 20% affects just 1 person in 10

How much is a life worth? It is impossible to put a figure on it but not all lives are equal. It may be argued that there is no such thing as saving lives, just delaying death. We all have to die of something at some time. Giving an extra 6 months of life of poor quality to an elderly person with advanced cancer is totally different from curing a much younger person and giving very many more years of active life. Not all NHS activity is directed towards saving lives and a little later we shall explore some quality of life issues. Hip or knee replacements do not save lives, but they improve the quality of life significantly.

Decisions must be based on evidence, not pressure groups

Whenever NICE decides that a treatment does not represent good value for money there is always an outcry from groups with a vested interest. An excellent example of this was when NICE decided against the use of trastuzumab, better known by its trade name of Herceptin, for early breast cancer. I remember such sound bites on the television as, “£20,000 is not a lot to save a life”. I had remarkable difficulty getting even reasonable information but this figure of £20,000 to save a life was nowhere near the truth.

As far as I could see the cost of treatment was about £20,000 per year and the length of treatment was 2 or 3 years. It reduced the recurrence rate in early breast cancer from just less than 20% to around 12%. As we have seen above, if it had reduced it from 20% to 10% the NNT would have been 10. NICE used a NNT of 13.3 which seems reasonable. If we say that the cost of a course of treatment is £50,000 (2½ years at £20,000 a year) and it is necessary to treat 13.3 on average to prevent one recurrence, then the cost of preventing this one recurrence is £665,000 (£50,000 x 13.3). I do not know the death rate in recurrence. If it is 100% then the cost of saving a life is £665,000. If it is 70%, the cost is £950,000. These figures are very much higher than the cost of a single course of treatment and they are nowhere near the figure of £20,000. Even if the cost of a course is £40,000, the cost of preventing one recurrence is more than £½million. The cost of saving a life will be higher.

After a great public outcry without the true figures in the debate and much political pressure, NICE decided to approve the drug. This drug is used for breast cancer that is positive for the HER2 receptor and this usually occurs in women before the menopause. Therefore, rather than looking at the elderly who were perhaps in generally poor health before their cancer, we are looking at women in their 40s or even 30s. They may have dependent children in their teens or younger and if cured, they may have many years of active life ahead.

Aspirin has benefits and risks

Some drugs provide benefit but they also have significant side effects that may cause harm. Here we need to look at the risk: benefit ratio. A good example is aspirin. It protects against heart disease and strokes but occasionally it leads to massive haemorrhage from the upper gut and this can be fatal. Therefore, as well as looking at numbers needed to treat (NNT) to prevent heart attack or stroke, we need to examine numbers needed to harm (NNH) by giving the patient a massive, potentially fatal haemorrhage. If a group of people are at high risk of heart disease or stroke the NNT is much lower than for people with a low risk. Low risk people do get benefit but at a lower rate. However, the risk of harm may not be much different between the two groups. Here I am going to use numbers purely as a means of illustration and these numbers should not be taken as the actual figures.

It is a balance of benefits and risks

Let us suppose that for people at high risk of heart attack and stroke the NNT for aspirin is 50 but for people at low risk it is 1,000. Let us also suppose that the NNH for bleeding is 200 for both groups. For people at high risk the chance of benefit from aspirin is 1 in 50 whilst the chance of harm is 1 in 200. For people at low risk the chance of benefit from aspirin is 1 in 1,000 whilst the chance of harm is still 1 in 200. Therefore, it seems that for high risk people the chance of benefit significantly exceeds the risk of harm whereas for low risk people the risk of harm outweighs possible benefit. Hence aspirin is recommended for people at high risk of heart attack or stroke, but it is not recommended for people at low risk.

In early 2019 a systematic review calculated a NNH for aspirin as 210.1Aspirin Use for Primary Prevention With Cardiovascular Events and Bleeding Events This means that 1 person in 210 can expect to have a problem which I think it is quite a low number and high risk.

Prevention of heart disease in people who have no evidence of existing coronary artery disease is called primary prevention whilst secondary prevention is for those who are known to have disease, usually because they have angina or have had a previous heart attack. There may also have been a stroke, which is also the result of arterial disease. Secondary prevention is prevention for people with much higher risk than primary prevention. The risk factors for heart attack and stroke are similar except that for stroke, cholesterol is less important whilst blood pressure is more important. The role of aspirin in secondary prevention is well established.2NICE. Myocardial infarction: cardiac rehabilitation

Three different trials assessed aspirin in the different scenarios

However, the role of aspirin in primary prevention has been much less clear. The ARRIVE trial has made the case for primary prevention much less sustainable.3ARRIVE trial Those who design trials love acronyms and, in this case, ARRIVE means “aspirin to reduce risk of initial vascular events”. ARRIVE was a randomised, double-blind, placebo-controlled, multicentre study done in seven countries and recruiting 12,546 patients. The results showed no real benefit for aspirin in primary prevention.

I had been taking aspirin since the 1970s when the low-dose aspirin story began. As I had suffered no apparent ill-effects, I continued when the question of benefit versus harm was first considered. However, ARRIVE convinced me to stop.

Diabetics are at very high risk of heart disease. I used to say that there were three levels of risk; low risk, high risk and diabetic. We were taught that primary prevention in diabetics should be treated the same as secondary prevention in others. The ASCEND trial showed some benefit for aspirin in diabetics but concluded that this was counterbalanced by adverse effects, especially bleeding from the upper intestinal tract.4ASCEND trial

Numbers give meaning and
proportion to risk.

Older people are more likely to bleed and have less resilience for surviving massive haemorrhage. The ASPREE Trial specifically looked at older people and concluded that “The use of low-dose aspirin as a primary prevention strategy in older adults resulted in a significantly higher risk of major haemorrhage and did not result in a significantly lower risk of cardiovascular disease than placebo.” This study recruited more than 19,000 people.

Aspirin may be of value in more than just slowing arterial disease that causes heart attacks and strokes. A synthesis of the evidence shows that it may also offer some protection against cancer,5Aspirin for the Prevention of Cancer especially cancer of the colon and rectum. 6Aspirin for the Prevention of Colorectal Cancer

Quality of Life

Not all potentially life-saving interventions are equal. One may give just a few months more to an elderly person with advanced malignancy and a poor quality of life. Another may produce a full cure in a young person and give a full expectation of life. A person may have severe osteoarthritis of the hip or knee. Mobility is limited and pain is constant and it disturbs sleep. If that person has a hip or knee replacement it is not a life-saving procedure but it is a life-changing intervention. That person may have the next 12 to 15 years without pain, with mobility restored and with sleep undisturbed. The NNT for such operations is a little more than one as most are successful. This compares well with most life-saving interventions such as cancer treatments where the NNT is usually in double figures. Health economists need a way to try to put a value on interventions as well as a cost to be able to assess value for money.

The unit employed is called a QALY. This stands for quality-adjusted life year. It takes into account the number of years of life ahead and the expected quality of that life. A year of perfect health gives a multiple of one so that 20 years in perfect health is 20 x 1= 20 QALYs.7Bandolier. QALY Less than perfect health gives a figure of less than one, depending upon how poor it is. Death gives a figure of zero. Some states of health are so poor that they have have negative scores. This is saying that they are a fate worse than death. If two interventions both offer an additional four years of life but one at a health state of 0.75 and the other at a score of 0.5, these two interventions provide 3 and 2 QALYs respectively. The health state examines several factors including pain, ability at self-care and mental state such as anxiety or dementia.

Quality of life is at least as important as quantity

These figures are combined with the costs of the interventions to give a cost-utility ratio. This indicates the additional costs required to produce a year of perfect health or one QALY. It gives an objective way of comparing interventions and enables health economists to establish priorities. Some interventions give good value based on a low cost per QALY. Others are relatively expensive with a high cost per QALY. No one would pretend that this technique is perfect but it does offer an objective measure when many different pressure groups are demanding resources for their field. Life expectancy can be objective but putting figures on quality of life is a much more contentious issue.

Some examples of QALYs based on figures from 1990 are:

These are rather old figures as I was unable to get any newer, but they give an indication of value.

NICE uses QALY calculations to assess value for money but they seem to have removed the overview of this from their website. Suppose that a patient has a serious life-threatening disease. With standard treatment he will live for a year and his quality of life is 0.4. With a new drug his life expectation is 15 months with a quality of 0.6. Hence the standard treatment gives 0.4 x 1 = 0.4 QALY whilst the new treatment gives 0.6 x 1.25= 0.75 QALY. The new treatment offers an extra 0.35 QALY. Suppose that the standard treatment costs £3,000 but the new treatment costs £10,000. It costs an extra £7,000 to add an extra 0.35 QALY. Hence the cost per QALY is £7,000/0.35 = £20,000.

NICE generally accepts about £20,000 to £30,000 per QALY but there are exceptions.8How much is a year of life worth? BBC

There is also a measure called the Disability Adjusted Life Year or DALY.9WHO. Disability-Adjusted Life Year (DALY) It is similar to the QALY but is aimed more at assessing the burden of disability. The World Bank and World Health Organisation use it as a combination of morbidity (illness) and mortality (death).

Assumptions and Extrapolations

Extrapolation is making assumptions from outside the range of the data

Suppose that we want to know the average blood pressure at various ages. We may measure the blood pressure of 100 people aged 20 and calculate the mean. We may do the same with 100 people aged 30, 40, 50, 60 and 70 years old. We can then plot a graph of age versus mean blood pressure and draw a line to join the points. This is always called a curve, even if it is a straight line. Suppose that we want to know the mean blood pressure at age 45. We may go to the line and read off mean blood pressure for that age. As it is between measured points this is called interpolation and is probably a valid assumption.

If we only had points for age 20 and 70 this would not be valid. However, suppose we want to know the mean blood pressure at age 80 years. To get this we have to extend the line beyond what has actually been measured. This is called extrapolation and it is a much more dangerous assumption than interpolation. Usually interpolation is safe but extrapolation may severely mislead.

Living with Risk

Time and again the point has been made that there is no such thing as absolute certainty in life. Life is about balancing risks. If we look both ways before crossing the road we are more likely to reach the other side safely. If we just dash across we shall probably get there safely but there is a much greater chance that we shall not. I can advise you how you can reduce your risk of coronary heart disease or cancer but I cannot tell you how to be absolutely certain that you will never get it. A field in which there is often failure to appreciate the balance of risk is pregnancy and childbirth. It is not an illness. It is a normal physiological event. It is also fraught with danger.

Pregnancy is not a disease but it carries significant risk for both mother and baby

We cannot tell for certain where problems will arise in pregnancy but we can stratify risk. Diabetes, high blood pressure and multiple pregnancies (twins or more) represent high risk. So does a previous problem. Obesity is a significant problem. In April 2007 the Confidential Enquiry into Maternal and Child Health (CEMACH) reported on perinatal mortality for 2005 and found that 30% of stillbirths and neonatal deaths occurred with mothers who were obese.

Perinatal mortality is defined as the number of stillbirths and deaths within the first week of life per 1,000 live births. What percentage of mothers fall into the category of obese, they do not disclose. A systematic review and meta-analysis has found that there is a small but significant increase in congenital abnormalities in babies of overweight and obese mothers.10Maternal overweight and obesity and the risk of congenital anomalies

A first pregnancy is uncharted territory. In the past I have heard the dictum “a low risk pregnancy is a diagnosis that can only be made retrospectively” (after the event). This is an argument for intensive monitoring of every labour. The statement is untrue as risk is not certainty. Risk can be assessed beforehand. Outcome can only be determined after.

Another argument is that 50% of complications occur in the low risk group. Of course this group is much larger than the high risk group but it does emphasise that there is no such thing as a zero risk group. At the other extreme there are those who argue that having babies is a natural event and doctors should not be involved. However, I fail to see how being surrounded by whale noises and being immersed in a tub of water is natural. This is pretentious anti-science that does no favour to either mother or baby. What is needed is a balance so that there is not excessive interference in low risk labours but even low risk labours can go wrong and some sort of monitoring is required to identify problems at an early stage.

A baby born dead or damaged is a terrible tragedy. Mothers still die too. The monitoring may just be intermittent listening to the baby’s heart and does not have to be continuous electronic monitoring. Everyone likes their local maternity unit, but if it does not have expertise and equipment at hand, it may be necessary to make a transfer in advanced labour to a distant site and this increases risk.

Get the balance right between unnecessary interference and neglect

Getting the correct balance between avoiding unnecessary interference and putting mothers and babies at risk is difficult. As we move ever more in the American direction of a litigation driven system, the number of caesarean sections rises. Is this justified? Defensive medicine is not good medicine.11Defensive Medicine Nearly 1 in 4 pregnancies in the UK is now delivered by caesarean section and many believe that this is too many. A paper examining figures from 2008 showed considerable variation between hospital trusts.12Variation in rates of caesarean section among English NHS trusts The overall rate for single births was 23.8%. Unadjusted rates of caesarean section among the NHS trusts ranged from 13.6% to 31.9%. Trusts differed in their patient populations, but adjusted rates still ranged from 14.9% to 32.1%. Some hospitals have more than twice the rate of others. Women were more likely to have a caesarean section if they had had one previously (70.8%) or had a baby with breech presentation (89.8%). A breech delivery is more dangerous for the baby but a caesarean is more dangerous for the mother. The art of breech delivery has almost been lost.

Uncertainty is a fact of life and we must learn to live with it

One of the problems of intensive monitoring of low risk individuals is that it is a screening procedure and screening can produce false positive results. What this means is that intervention may occur in response to a false alarm. This is an unnecessary intervention that has risk attached. Hence, intensive screening of low risk labours may possibly increase risk by causing more unnecessary interventions than by detecting problems that would otherwise have gone unnoticed.

What level of risk do you accept?

This has been a brief and rather superficial look at the problems of risk benefit analysis in maternity care but the principle permeates throughout all of medical practice. It also affects every aspect of life. If you wish to overtake the car in front, the benefit is that you can go faster and reach your destination sooner, but the manoeuvre has risk. What level of risk you will accept is dependent upon personality, but other factors too. If you are in a hurry, you will accept a greater risk than if you have plenty of time. This has shifted your risk benefit analysis. What do you regard as an acceptable level of risk?

In 2002 the Criminal Records Bureau was so inundated with work that it was unable to process checks on all teachers due to start at new schools before the start of the new school year in September. As soon as the magic words “child protection” are uttered, any vestige of common sense is thrown to the wind, and not just by politicians. It was announced initially that teachers would not be able to start work until they had been cleared. Most teachers starting at a new school have had an enhanced check at the old school and if they had received a relevant criminal record whilst there, their employers would have been notified. The rest have probably been at teacher training college and will also have been checked. How many people would be applying for jobs as teachers despite having a criminal record that would bar them, knowing that an enhanced CRB check will be performed and they will be found out? If there is a single one, what is the risk that he or she will start work and will harm a child or children before the check comes through? Balance that against the benefit to children of having a teacher in their classroom from the beginning of the school year or the risk of being without a teacher.

Are CRB checks really of any value?

The school term started and, according to the BBC News, thousands of children were sent home because their teacher had not yet received clearance. According to the BBC, “Some education authorities had defied the Department for Education’s strict guidelines and let people start work provided they were supervised.” On 5th September 2002 the BBC announced that Estelle Morris, Minister of Education, had granted discretionary powers to head teachers to use staff whose CRB check was not completed, after finding out that they would not be completed for weeks.13Teachers start work without checks. BBC The following month she resigned, admitting that she was not up to the job, but her inability to tell the difference between significant risk and political expediency was not an issue.14Estelle Morris resigns. Daily Telegraph None of her colleagues knew the difference either. Shortly after, she was awarded a life peerage.

Great emphasis has been laid on the concept that there is no such thing as absolute certainty. Even in nuclear physics scientists are looking at probabilities rather than absolutes, especially in quantum theory. However, rather than emphasising the term uncertainties it would be better to use the word tolerance. This is a term often used in engineering. For example a gap may have to be 10 microns (µm) with a tolerance of 5%. This means that it must be 10±0.5µm or between 9.5 and 10.5µm wide. A gap that is 9.7 µm is satisfactory. A gap of 9.2 µm is not. Similarly, confidence interval should be seen not as uncertainty but as tolerance limits.

Tolerance does not mean that anything goes

Tolerance is also a term that is used in politics, religion and social expectations. As in science and engineering it does not mean “anything goes”. There is a degree of laxity and permissiveness but it is not infinite. Some people like absolutes in political or religious dogma or in social expectations. They tend to be people of limited ability and very little imagination. The great thing about totalitarianism is that you do not have to think for yourself. The Americans use the term, “Cut me some slack”, meaning, “Show some tolerance please”. This is a good term as a slack rope will permit some movement but not unlimited movement as when the slack has been taken up it is taut and firm. Tolerating variation is not uncertainty.

Genetics

Much of genetics is about chance or statistics, so it merits a place in this section on maths. Again, it is kept simple with the emphasis on principles rather than complex calculations.

Charles Darwin proposed the idea of natural selection but he did not understand how features were handed down from one generation to the next. Lamarck’s idea that traits that were acquired in life are passed on is untrue. Mendel’s work was published in an obscure journal and went unrecognised until about 1900. We now understand chromosomes and DNA and so we have a good idea of how inheritance works.

Understanding DNA has helped to make sense of
genetics

Each human has 46 chromosomes of which 23 are received from each parent. There are 22 pairs called autosomes and one pair of sex chromosomes.15National Human Genome Research Unit. The pairs of autosomes control the same features so there are two genes for each feature. The chromosomes are labelled 1 to 22 in decreasing order of size. The sex chromosomes control the gender of the person but also a number of other features. They are called X and Y because they look rather like those letters when stained and seen under a microscope. A person with two X chromosomes is female. A person with an X and a Y is male. In butterflies and moths it is the other way round. The X chromosome is rather larger than the Y and so contains rather more genetic material.

For reproduction the 23 pairs split into 23 single chromosomes in each ovum or sperm. An ovum has 22 autosomes and an X chromosome. A sperm has 22 autosomes and either an X or Y chromosome. It is also possible to get some change of genetic material with a piece of a chromosome being swapped to its partner. This cross-over is called chiasma. Historically it has always been the woman who has been blamed for failing to produce the statutory son but the male determines the sex of the offspring. Perhaps her ova should have been more discriminating about which spermatozoa they accepted.

For a given chromosome or even just a gene, the mother has two which we may call A and B and the father has two which we may call C and D. They pass on one of each to each offspring. There are four possible combinations as shown in the figure. It also shows the transmission of the X and Y sex chromosomes

Mendel realised that features may be dominant or recessive. A dominant gene is one that causes its effect to be displayed even if the other chromosome of the pair does not have it. A recessive gene requires both chromosomes to have this feature for it to be manifest. If a person has one abnormal and one normal gene this is called heterozygous and if there are two abnormal genes this is called homozygous. This comes from the Greek where homo means same and hetero means different.

A number of times in this section there is reference to OMIM. This stands for Online Inheritance In Man and comes from the same resource as PubMed. It may be found at https://www.ncbi.nlm.nih.gov/omim/

An example of a disease due to a dominant gene is Huntington’s chorea, also called Huntington’s Disease or HD.16OMIM Huntington disease. It is a very unpleasant degeneration of the brain that causes involuntary movements and dementia and eventually death. It does not show itself until between 30 and 40 years of age and so the person may well have completed the family before they know that their children may be affected.

If a person has just one of those genes he will have the disease. For each child he may pass on either the affected gene or the normal one. Therefore there is a 1 in 2 chance that any child will be affected.

The gene D is replaced by H meaning Huntington’s chorea, originating in the mother in this case. Half the offspring are affected.

Again we must look at statistics and realise that this means chances, not certainties. Thus, we may “expect” half the offspring of a person with this condition to be affected but, depending on luck, they may be more or less than half in any instance.

An example of a recessive genetic disease is cystic fibrosis.17OMIM Cystic Fibrosis This is the most common of the serious genetic diseases. For the disease to be manifest the person must carry two genes for the condition. This can only be achieved if both parents are carriers. However, because they have only one gene each they seem normal and are probably unaware of carrying the condition until they have a child with it. The diagram is repeated but with C to represent the cystic fibrosis gene and N to represent the normal gene.

If a person has one abnormal and one normal gene this is called heterozygous and if there are two abnormal genes this is called homozygous. Heterozygotes are carriers. Homozygotes are affected.

This shows that both parents are carriers, 1 child in 4 is affected, having both genes (CC), 2 in 4 are carriers like the parents with one cystic fibrosis gene and one normal gene (CN, NC) whilst 1 in 4 is completely normal with two normal genes (NN).

The frequency of the cystic fibrosis gene in the British population is about 1 in 25. Assuming no inbreeding, the chance of two parents both having the genes and being carriers is 1 in 25 x 25= 1 in 625. The chance of any single offspring being affected is 1 in 4 and so the frequency of the disease in the general population is 1 in 625 x 4 = 1 in 2,500.

Ancient Egyptian attempts to keep royal blood lines pure led to interbreeding, especially the Ptolemies

Where reproduction occurs with someone related, as in cousin marriages, the risk of genetic disorders is much increased. It is also more common in inbred communities. For example, Tay-Sachs disease is prevalent in Ashkenazi Jews.18OMIM Tay-Sachs Disease By and large, the more serious genetic conditions are recessive as if they were dominant they would have killed the affected carrier before he or she reproduced. Huntington’s chorea is an exception as they usually reproduce before the disease is manifest. It is probably the noted risk of abnormalities in the offspring of related parents that has led to incest being taboo in almost all cultures. However, Egyptian pharaohs often called their consorts “my sister wife” and this was probably true.19Impact of Royal Inbreeding They married their sisters to keep the pure royal line. The Ptolemy dynasty, which included Cleopatra, was probably the worst offender.

Colour-blindness is much more common in men than women

The third type of inheritance to examine is sex-linked, often called X-linked as it relates to the X chromosome. There are some genetic disorders that have been linked to the Y chromosome but they are few and not so important. The significance of X-linked conditions is that the male has only one X chromosome whilst the female has two. The commonest X-linked condition is red-green colour blindness.20OMIM partial colour blindness It affects about 7% of men but only about 1 in 200 women. If the gene frequency in the general population is 7 in 100, the expected number of men to be affected is 7 in 100 or 7%. Women may have one affected X chromosome, in which case they are carriers but with normal vision or two affected chromosomes which would make them colour blind. The chance of having two affected chromosomes is 7 in 100 x 7 in 100 = 49 in 10,000 which is 0.49%. This is compatible with the observed 1 in 200 or 0.5%.

If a man has colour blindness, all his daughters will be carriers as he will pass on his one X-chromosome to them. They will only be affected if their mother passes on an affected gene too. Provided that the mother is not a carrier, all sons will be normal as they receive the X chromosome from the mother and the Y chromosome from the father.

Queen Victoria was the source of many cases of haemophilia across many royal houses in Europe

Colour blindness is not a serious condition and does not affect life expectancy. Many sufferers are probably never diagnosed. However, a famous condition that is X-linked is haemophilia.21OMIM Haemophilia A This disease probably had Queen Victoria as a source and caused disaster to many of the royal families of Europe. The most notable was in Russia. It is sometimes called haemophilia A to distinguish it from another X-linked bleeding disorder called haemophilia B or Christmas disease.22OMIM Haemophilia B Haemophilia A is a deficiency of clotting factor VIII whilst Christmas disease affects clotting factor IX and is usually slightly less severe. The mode of inheritance of both is the same as for colour blindness but it is a much more serious disease that until recently would probably have killed the sufferer before he reproduced. Females with true haemophilia are very rare as they require a haemophiliac father and a carrier mother. There are a number of genetic disorders causing bleeding which do affect females too, but they are not actually haemophilia.

The concept of genes being either fully dominant or recessive is slightly simplistic. Sometimes there are implications for being heterozygous for a recessive gene beyond possibly passing it on to offspring. An example of this is sickle cell disease, also called sickle cell anaemia.23OMIM Sickle Cell Anaemia

If a person is heterozygous for sickle cell anaemia he has an abnormal type of haemoglobin in his red blood cells called haemoglobin S and the red blood cells are liable to distort to a sickle shape. The blood cells have a much shorter life than the usual 120 days, causing anaemia.

If oxygen levels fall, the red blood cells with the abnormal haemoglobin will sickle

Sickling may also activate the clotting system causing blockage of blood vessels. This leads to damage to various organs and great pain when it happens. The spleen may be totally destroyed. The problems of sickle cell disease and sickle cell crises will not be discussed but much more about the nature of the disease is given by Patient UK.24Patient UK Sickle cell disease and sickle cell anaemia People who are heterozygous may appear to be unaffected although they can have problems during anaesthesia if the oxygen level is allowed to fall. The interesting point about being heterozygous for sickle cell disease is that whilst it appears to cause no immediate problem, it does confer some protection against malaria. Presumably the infected cells tend to sickle and are taken out by the spleen and the malaria parasites are destroyed. Most people with the sickle cell gene have a racial origin from the West coast of Africa. There are many carriers of the gene in the UK, having come directly from West Africa or possibly their ancestors were taken in the slave trade many generations ago and they came from the West Indies. It is a policy in British hospitals that any black person who may need a general anaesthetic should be screened for sickle cell disease.

The diagram below is very similar to that for the inheritance of cystic fibrosis but instead it gives the genes as A for normal haemoglobin A and S for the haemoglobin S. As the diagram shows, for every four children born to parents who are both sickle cell carriers, we would expect one to have full blown sickle cell anaemia as a homozygote, we would expect two to be heterozygous carriers and we would expect one to be normal with just haemoglobin A. The significance of this is that in West Africa, the one who is homozygous would die of sickle cell disease. The one who is completely normal may die of malaria. The two who are heterozygous have a survival advantage with resistance to malaria but without overt sickle cell anaemia.

A is normal haemoglobin and S is the gene for the sickle cell type

This is an excellent example of where an apparently disadvantageous gene can confer benefits and so survives. Not all genetic disease can be attributed to past generations. Certain types of diseases can occur spontaneously due to genetic mutation.25Direct estimate of the haemophilia B mutation rate Errors occur during DNA replication and some types of errors are more common than others. The role of parental age in determining the risk of mutation is unclear. In autistic spectrum disorder it seems to be related to paternal age but differentiating maternal and paternal age is difficult as they tend to get older together.26Parental Age and the Risk of Autism Down’s syndrome is more common with older mothers.27Down’s Syndrome. NHS Choices It is trisomy 21 which means that instead of having two number 21 chromosomes the person has three.

The terms genetic and congenital are often used as if synonymous but they are not. Genetic means that it occurs from the genes whilst congenital means “born with”. Thus Huntington’s chorea is a genetic disease but it is not congenital as it does not manifest itself for 30 to 40 years although it may be argued that the genes are still present. Congenital rubella is not a genetic disease but the baby is born with the condition having contracted rubella before birth.

The examples of genetic disease and its inheritance, as given above, is unfortunately rather simplistic as only about 2% of genetic features are monogenic, meaning controlled by just one gene. Eye colour was thought to be controlled by three genes but now it is thought to be at least 20.

The DNA in our bodies contains about 3 billion base pairs but there are probably no more than around 23,000 genes. Hence only about 5% of our DNA is involved in coding. The other 95% was called “junk DNA” as there was no known function for it, but it is unwise to think that we carry 95% garbage in our DNA. A much safer term is non-coding DNA.

We share 99% of our DNA with chimpanzees

We share 99% of our DNA with chimpanzees despite splitting in evolution about 6 million years ago, 90% with mice from whom we spilt 100 million years ago and 31% with yeast despite a split in evolution about 1.5 billion years ago.

We saw in A Very Brief History of Science And Medicine that genes are carried as Mendel explained and that Lamarck’s ideas of acquired traits being passed on to the next generation were wrong. However, it seems that Lamarck was not entirely wrong. All the cells in our bodies carry all our DNA with the exception of the gametes (ova and sperms) which carry just half and erythrocytes (red blood cells) and platelets which do not have a nucleus although they had one during development. Our cells develop into muscles, nerve cells, liver, kidneys and much more but they have the theoretical potential to become anything. Most of the genes in a cell are suppressed. Only the relevant ones are active. Stem cells are those which can still develop into any type of cell. Features in our cells may be activated or suppressed by a system called epigenetics from the Greek for “around genetics”. It means that some acquired traits may be passed on to the next generation.

We are a mixture of our genes and our environment, called nature and nurture but how much is each? The best way to examine this is with twin studies. About a third of twins are identical and have identical DNA. Two thirds are fraternal and share just half their DNA, the same as other brothers and sisters. We may examine a condition or trait and find that if one identical twin has it, the chance of the other twin having it too is 80%, but for fraternal twins the chance is only 50%. This is called concordance. We take the difference between the two figures and multiply by two to give the heritability of the condition. Thus, if the figures are 80% and 50% the heritability is (80% – 50%) x 2 = 60%.28(Identically Different by Professor Tim Spector)

Back in 2000, Bill Clinton and Tony Blair announced the first draft of the Human Genome Project and it was thought that before long, all would be revealed. Life is never that simple. I have not even mentioned RNA and mitochondrial DNA.

What’s in a Name?- Medical Eponyms

There are many eponymous diseases in medicine and especially in genetics. In 1872, George Huntington of Pomeroy, Ohio, described the disease that bears his name. The affected family was traced back to Suffolk but the story that they originated from Huntingdon in Cambridgeshire does not seem to be substantiated. There is also the difference in spelling between the surname Huntington and the place Huntingdon.

Thomas Addison

Christmas disease was named after the affected family rather than the doctor and it was reported in the Christmas 1953 edition of the British Medical Journal, now BMJ. It is unusual for the disease to be named after the patient rather than the doctor or doctors. Why give the patient the kudos and immortality when the doctor can claim it? Diseases are usually named with the doctor’s surname. For example, underactive adrenal glands is called Addison’s disease after Thomas Addison, a physician at Guy’s Hospital in the 19th century. Overactive adrenals, classically due to a pituitary tumour is called Cushing’s syndrome after Harvey Cushing, an American neurosurgeon in the early 20th century. The Americans do not use the personal apostrophe but call it Addison disease or Cushing syndrome.

A syndrome is simply a disease with several components. For example, Ménière’s disease is a syndrome consisting of nerve deafness, vertigo and tinnitus. It requires two of the three features to make the diagnosis. Vertigo means a spinning feeling of rotation. It does not mean fear of height.

Georges Giles de la Tourette

Normally a syndrome name is the surname or surnames of the eponymous person or people, but there are two exceptions. Cornelia de Lange syndrome was named after the paediatrician of that name but Cornelia was her Christian name and so it should have been de Lange syndrome. The other exception is Tourette’s syndrome, named after Georges Gilles de la Tourette. It used to be called Gilles de la Tourette syndrome as that was his surname, but it seems that too many people found that too much of a mouth-full, so it is usually called Tourette’s syndrome these days.

Down’s syndrome has been mentioned. We used to call it Mongolism as it was said that the facial features of those affected resembled the appearance of Mongol people. However, this was felt to be racist and so Down’s syndrome is preferred. It was named after John Down, not Downs, and so the apostrophe comes before the s. An overactive thyroid gland is named after Robert Graves and so it is called Graves’ disease with the apostrophe at the end.

Not a smile to be seen on any child’s face

Asperger’s syndrome is mild autism and is named after the Austrian paediatrician Hans Asperger. However, there are some who wish to remove the eponym from him as it transpires that he was not as kind and caring a doctor as we may wish. He allied himself so closely with the Nazi ideology that he frequently referred children to the Am Spiegelgrund clinic, which was set up as a collecting point for children who failed to conform to the regime’s criteria of “worthy to live”. Nearly 800 children died at the clinic between 1940 and 1945, many of whom were murdered under the notorious child “euthanasia” scheme.29Hans Asperger aided and supported Nazi programme, study says

Sir James Paget was rather greedy in having a disease of bone as well as a breast disease named after him. The hospital in Gorleston, near Great Yarmouth, near where he lived also bears his name.

An interesting dictionary of medical eponyms can be found at http://www.whonamedit.com

2 + 2 = 5. It Can Do!

We have seen that numbers are not absolute figures but there is uncertainty in their accuracy. We have examined 95% confidence limits. We have put figures on “touchy feely” matters such as quality of life. We have accepted uncertainty and we have looked at risks and benefits.

A basic principle that applies as much to physics and chemistry as to biological science is that the final result cannot be more accurate than the least accurate reading. Suppose that a researcher measures blood pressure several times and then concludes, on the basis of simple arithmetic, that the average blood pressure is 125.75/84.25. This is nonsense. Blood pressure cannot be measured to 2 places of decimals. A figure with 2 places of decimals implies that it is correct ±0.005. Figures presented to the nearest integer (whole number) imply an accuracy of ±0.5. There is a basic principle that the final result cannot be more accurate than the least accurate reading.

If an actual sum is 2.3+2.4=4.7, then presented to the nearest integer this will be written as 2+2=5. If 2 really means 2±0.5 then 2+2 is (2±0.5) + (2±0.5) = 4±1. Note that the error has multiplied as measurements increase.

Here is another challenge to consider. We have been taught that the sum of the internal angles of a triangle equal 180°. Imagine that you are at the North Pole and you go south along the Greenwich meridian until you reach the equator. Turn left 90° to go east along the equator until you reach 90° east. Turn left through 90° again and follow that meridian to the North Pole. Turn left through 90° again, taking you back on the Greenwich meridian. You have travelled through a triangle with three angles, all of 90°, making a total of 270°. How can this be if the angles of a triangle total 180°?

Simples!

The answer is that the dictum that the angles of a triangle add up to 180° is based on Euclidian or plain geometry. This triangle was on the surface of a sphere and so the rules are different. Always make sure that the rules are applicable to the circumstances.

Further Resources

References

  1. Zeng SL, Roddick AJ. Association of Aspirin Use for Primary Prevention With Cardiovascular Events and Bleeding Events. A Systematic Review and Meta-analysis. JAMA. 2019;321(3):277-287
    https://jamanetwork.com/journals/jama/article-abstract/2721178
  2. NICE. Myocardial infarction: cardiac rehabilitation and prevention of further cardiovascular disease. Clinical guideline [CG172] Published date: November 2013. https://www.nice.org.uk/guidance/cg172
  3. Gaziano JM, Brotons C, Coppolecchia R, Cricelli C, Darius H, Gorelick PB, Howard G, Pearson TA, Rothwell PM, Ruilope LM, Tendera M, Tognoni G; ARRIVE Executive Committee. Use of aspirin to reduce risk of initial vascular events in patients at moderate risk of cardiovascular disease (ARRIVE): a randomised, double-blind, placebo-controlled trial. Lancet. 2018 Aug 24. https://www.ncbi.nlm.nih.gov/pubmed/30158069
  4. ASCEND Study Collaborative Group. Effects of Aspirin for Primary Prevention in Persons with Diabetes Mellitus. N Engl J Med. 2018 Aug 26. https://www.ncbi.nlm.nih.gov/pubmed/30146931
  5. Whitlock EP, Williams SB, Burda BU, Feightner A, Beil T. Aspirin Use in Adults: Cancer, All-Cause Mortality, and Harms: A Systematic Evidence Review for the U.S. Preventive Services Task Force. September 2015 [full text] https://www.ncbi.nlm.nih.gov/pubmedhealth/PMH0079333/
  6. Chubak J, Kamineni A, Buist, Anderson,Whitlock EP. Agency for Healthcare Research and Quality (US); 2015 Sep. Report No.: 15-05228-EF-1. Aspirin Use for the Prevention of Colorectal Cancer. An Updated Systematic Evidence Review No. 133.
    https://www.ncbi.nlm.nih.gov/pubmedhealth/PMH0079342/
  7. Malek M. Implementing QALYs. Bandolier March 2001.
    http://www.bandolier.org.uk/Conflicts%20Folder/ImplementQALYs.pdf
  8. How much is a year of life worth? BBC 29th August 2014. https://www.bbc.co.uk/news/health-28983924
  9. World Health Organisation. Metrics: Disability-Adjusted Life Year (DALY)
    http://www.who.int/healthinfo/global_burden_disease/metrics_daly/en/
  10. Stothard KJ, Tennant PW, Bell R, Rankin J. Maternal overweight and obesity and the risk of congenital anomalies: a systematic review and meta-analysis. JAMA. 2009;301:636–650.
    http://www.ncbi.nlm.nih.gov/pubmed/19211471
  11. Hermer LD, Brody H. 18. Defensive medicine, cost containment, and reform. J Gen Intern Med. 2010 May;25(5):470-3. Epub 2010 Feb 9. Review. [full text] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2855004
  12. Bragg F, Cromwell DA, Edozien LC, Gurol-Urganci I, Mahmood TA, Templeton A, van der Meulen JH. Variation in rates of caesarean section among English NHS trusts after accounting for maternal and clinical risk: cross sectional study. BMJ 2010; 341:c5065 doi: 10.1136/bmj.c5065 (Published 6 October 2010)
    http://www.bmj.com/content/341/bmj.c5065.full
  13. Teachers start work without checks. BBC News 5th September 2002 http://news.bbc.co.uk/1/hi/education/2237534.stm
  14. Daily Telegraph, 23rd October 2002. Estelle Morris resigns. http://www.telegraph.co.uk/news/1411041/Estelle-Morris-resigns.html
  15. National Human Genome Research Unit. Education. 2011.
    http://www.genome.gov/11508982
  16. Online Mendelian Inheritance in Man (OMIM). Huntington’s disease. http://www.ncbi.nlm.nih.gov/omim/143100
  17. Online Mendelian Inheritance in Man (OMIM). Cystic fibrosis.http://www.ncbi.nlm.nih.gov/omim/219700
  18. Online Mendelian Inheritance in Man (OMIM). Tay-Sachs Disease http://www.ncbi.nlm.nih.gov/omim/272800
  19. Impact of Royal Inbreeding: Part III. Medical Bag. 5th June 2014. https://www.medicalbag.com/grey-matter/impact-of-royal-inbreeding-part-iii/article/472635/
  20. Online Mendelian Inheritance in Man (OMIM). Colorblindness partial. http://www.ncbi.nlm.nih.gov/omim/303800 /
  21. Online Mendelian Inheritance in Man (OMIM). Hemophilia A http://www.ncbi.nlm.nih.gov/omim/306700
  22. Online Mendelian Inheritance in Man (OMIM). Hemophilia B http://www.ncbi.nlm.nih.gov/omim/306900
  23. Online Mendelian Inheritance in Man (OMIM). Sickle cell anaemia. http://www.ncbi.nlm.nih.gov/omim/603903
  24. Patient UK Sickle cell disease and sickle cell anaemia. https://patient.info/health/sickle-cell-disease-sickle-cell-anaemia
  25. Montandon AJ, Green PM, Bentley DR, et al; Direct estimate of the haemophilia B (factor IX deficiency) mutation rate and of the ratio of the sex-specific mutation rates in Sweden.; Hum Genet. 1992 May;89(3):319-22. . http://www.ncbi.nlm.nih.gov/pubmed?term=1601423
  26. Durkin MS, Maenner MJ, Newschaffer CJ, Li-Ching Lee, Cunniff CM, Daniels JL, et al. Schieve Advanced Parental Age and the Risk of Autism Spectrum Disorder Am J Epidemiol. 2008 December 1; 168(11): 1268–1276. Published online 2008 October 21. doi: 10.1093/aje/kwn250 [full text] http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2638544/
  27. Down’s syndrome. NHS Choices.
    https://www.nhs.uk/conditions/downs-syndrome/
  28. Identically Different by Professor Tim Spector. Available from Amazon as paperback or Kindle e-book
  29. Hans Asperger aided and supported Nazi programme, study says. The Guardian. 19th April 2018.
    https://www.theguardian.com/world/2018/apr/19/hans-asperger-aided-and-supported-nazi-programme-study-says

Site Index

This website is under construction although nearing completion. The following list shows the sections that are planned but, so far, only the ones in blue have been completed. Just click on the topic in blue to go to that part of the site.

1 Introduction
2 A Very Brief History of Science And Medicine
  Fundamentals of Medical Science
3 Finding Good Medical Advice and Evidence Based Medicine
4 Randomised Controlled Trials
5 Cohort or Longitudinal and Epidemiological Studies
6 Qualitative Research
7 Basic Maths in Medical Research and Decision Making
8 How Good is the Evidence?
9 Ethics in Practice and Research
  Public Health Issues
10 Screening Programmes
11 Fake News and Vaccine Scares
12 Electronic Cigarettes (E-Cigarettes)
13 Motor Vehicle Emissions, Air Pollution and Health
14 COVID-19. What You Need to Know
15 What we Must Learn from the COVID-19 Pandemic
  Nutrition
16 Basics of Nutrition
17 Exercise, Obesity and Diets for Weight Loss
18 Diets and Nutrition for Health and Fitness
19 Supplements
  Complementary and Alternative Medicine
20 Introduction to Alternative Healthcare
21 Homeopathy
22 Acupuncture
23 Manipulation of the Spine
24 Reflexology
25 Herbal Remedies
26 Other Natural Products
27 Chelation Therapy
28 Hypnosis
29 Other Modalities of Complementary and Alternative Medicine
  Some Controversial Diseases
30 Fibromyalgia
31 Chronic Fatigue Syndrome (CFS) or Myalgic Encephalitis (ME)
32 Systemic Candidiasis and Leaky Gut Syndrome
33 Mobile Phones, Masts, Wi-Fi and Electro-sensitivity
  The Environment
34 Global Warming and Climate Change
35 Alternative Energy
  Some Final Thoughts
36 Still Searching for the Age of Reason