Statistics at Square One

Michael J. Campbell, Michael J. Campbell

• Publisher: Wiley-Blackwell
• Year: 2021
• ISBN:
  9781119402343
• Edition: 12
• Topic: Medicine
• Subtopic: Medical Theory, Practice & Reference

STATISTICS AT SQUARE ONE

The new edition of the popular introduction to the world of statistics for health care professionals and medical students

First published nearly three decades ago, Statistics at Square One remains one of the most popular introductions to medical statistics. Now in its twelfth edition, this international bestseller continues to be a must-have resource for anyone in need of a thorough introduction to statistics in the health sciences. Clear and accessible chapters help students with no previous background in the subject understand fundamental topics including summary statistics for quantitative and binary data, diagnostic and screening tests, populations and samples, survival analysis, correlation and regression, study design, computer modeling, and more.

This edition reflects contemporary understanding of medical statistics and emphasizes the importance of statistics in public health, including extensively updated coverage of diagnostic tests and new COVID-related examples. All figures and examples now include code to reproduce them in the R statistical software. New chapters cover the basics for understanding numbers and introduce the use of models in medical statistical analysis. Based on the author's many years of experience teaching medical and health science students, the latest edition of this classic textbook:

• Highlights the connections between different medical statistics methods
• Emphasizes the proper use of p-values in testing
• Features practical examples from recent literature
• Contains end-of-chapter exercises with answers, some of which are based on the Royal College of General Practitioners (RCGP) Advanced Knowledge Test

Statistics at Square One is required reading for all medical and health care practitioners and students wanting to understand the use and value of statistical analysis in the health sciences.

CHAPTER 1
Understanding basic numbers

Numbers are not necessarily easy to understand and, notwithstanding stories of grandmothers and teaching them to suck eggs, this chapter will try to cover some of the basics for understanding numbers. The chapter warns about ‘orphan’ numbers and how percentage changes are difficult.

When is a number large?

Consider the following examples:

1. On 6 May 2020 there were 30,000 deaths due to COVID‐19 in the UK, 75,000 in the USA and 265,000 in the world.¹
2. There were about 634,000 deaths in the UK population, 2,909,000 in the USA and 58 million deaths in the world in 2018.²
3. The UK Government stated in 2018: ‘We have invested an extra £1 billion in the NHS [National Health Service] this year.’³
4. The UK sent £350 million to the European Union every week.⁴
5. The Global Burden of Disease Report (GBDR) on sepsis estimated that there were 48.9 million cases in 2017, and 11 million deaths, across 195 countries and territories.⁵

Are these large amounts? They certainly sound like large amounts, but how do we come to terms with what they mean? Large numbers are often quoted on their own by people in authority, to try to impress the public with how big the numbers are. (A useful term might be ‘orphan’ numbers because they are not related to other numbers.) However, there is an old joke that if you ask a statistician how well they are, they will reply ‘Compared with whom?’ Likewise, numbers on their own are by and large meaningless; it is only with comparisons that we can extract a meaning. In example 1, the COVID‐19 deaths are, on their own, just large numbers. However, we can employ an analogy to give them some meaning. The first number of deaths roughly equates to the same number of people at an average Premier League football club in the UK, whereas the second is closer in number to a capacity crowd at Old Trafford, home ground of Manchester United. The third is the size of an average town in the UK (e.g. Southampton). These analogies put the number of deaths into a very human perspective. However, to get a better understanding of these numbers we need more specific comparisons.

A helpful basis for comparison is knowing the approximate size of the populations to which each statistic is referring. In 2019, the population of the UK was 67 million, that of the USA was 330 million and that of the world 7.7 billion (7700 million).⁶ We can then calculate the ratio of the number of deaths to the size of the population. A ratio is simply one number (numerator) divided by another (denominator). In this case, since the numerator is a subset of the denominator, we have proportions. The deaths per head of population are 0.044%, 0.023% and 0.003% for the UK, USA and the world, respectively. These percentages lead to another comparison: that between countries. The UK appears to be doing worse than the USA, which is doing worse than the rest of the world. Is this a reasonable conclusion? Cause of death is often very unreliably reported. Completion of a death certificate is often assigned to a junior doctor with little training. In an elderly patient with multiple diseases, it can be especially difficult to ascribe one main cause. So in this example we should consider how we know the person died of COVID‐19. Presumably the patient was tested before they died or they had symptoms similar to COVID‐19. However, testing rates have varied widely between countries and diagnosing symptoms of COVID‐19 is very subjective. Thus, these numbers for death rates due to COVID‐19 are not at all reliable and a reliable comparison is therefore difficult.

In contrast, deaths (from any cause) are reliably reported in the UK and the USA and probably well reported for the rest of the world. In example 2, again the numbers by themselves are not meaningful, but compared to the size of the relevant populations we can extract some meaning. A quick calculation reveals that 0.95% of the UK population dies every year, compared to 0.88% in the USA and 0.76% in the world. These numbers on their own are interesting. In the UK about 1 person in 100 dies each year. This brings the numbers down to something we can appreciate. Again, we can compare the proportions dying by country, and once more it appears that the UK is so much less healthy than the USA, and both countries are less healthy than the rest of the world. This may lead to further investigations.

In example 3, we could compare the extra sum invested in the NHS to the annual budget for the NHS, which is about £130 billion, so this extra £1 billion is less than 1% of the total. Another way to look at this is to consider that we now know there are about 67 million people in the UK, so £1 billion equates to about £15 for every person in a year, roughly the cost of five pints of beer (at current UK prices outside of London). It doesn’t sound so big now, does it?

In example 4, it is worth knowing that the UK economy was worth £8.8 trillion a year in 2016 (a trillion is 1000 billion).⁷ The £350 million a week given to the EU is £18.2 billion a year, so the amount the UK sends to Europe is

of the UK economy. Again, it doesn’t sound so big now, does it?

If we combine the information from example 5 with the worldwide death data in example 2, we would deduce that approximately 1 in 5 deaths worldwide is due to sepsis. This certainly is a large number! However, all unusual numbers should be subjected to a little scrutiny. As a quick reality check, you might start by asking yourself whether of the people you know who died recently, did 1 in 5 die of sepsis? One would expect the answer to be no. Thus, we might query whether the GBDR is right. One issue is that sepsis can be difficult to diagnose and the rate of diagnosis varies hugely from one country to another, so local experience may be misleading in that in another country sepsis might be more readily diagnosed.

When you hear a number given that you believe the presenter wants to sound big, it is always worth applying reality checks such as those described in Box 1.1. A light‐hearted example has been provided in a video from the Sheffield Methods Research Institute⁸ concerning a news report that stated that floods in New Zealand had caused 30,000 pigs to be washed down a river. This was then reported uncritically by other news outlets, until someone thought: ‘30,000? That’s an enormous number, is it believable? How many pigs are likely to fall into a river at any one time?’ Going back to the original broadcast, it turned out that the reporter had in fact said 30 sows and pigs, but owing to their New Zealand accent, this got ‘misheard’ and repeated uncritically to the wider media.

A further question about a large number is to ask what period of time the number refers to. By expanding or contracting the time scale, a presenter can make a number look big or small. When a large sum of money is promised, one should ask: How much does this equate to per year? In example 2 above, 634,000 deaths sounds large, whereas 0.95% sounds small. However, if one stated that approximately 1650 people die every day, or about 1 every minute, it may sound even bigger, since in our everyday experience people are not dying every minute! It is worse when reports state the ‘risk of death’ or ‘lives saved’ without stating a time period. The risk of death in the long run is one!

Definitions of the quantities discussed in this chapter are given in the Glossary. Ways of questioning numbers are given in Box 1.1.

Ratios

As we have shown, a number on its own is difficult to comprehend, but when compared to another number it can be given meaning. The simplest way to compare two numbers is to divide one by the other. A ratio is simply one number (numerator) divided by another (denominator). Ratios of continuous variables are often used to ‘adjust’ the numerator by the denominator. Possibly the most commonly used ratio in medicine is the Body Mass Index (BMI), which is a person’s weight in kilograms divided by their height in metres squared (kg/m²). The idea is that tall people are naturally heavier than small people because they are bigger, but that doesn’t make all tall people overweight! To decide whether someone is overweight, you can’t just compare weight, you need to make some allowance for height. The idea of dividing by the square of height is credited to Adolphe Quetelet (1796–1874), who observed in a cross‐sectional study that weight increased as the square of height....