Introduction

By now you are likely to have a very clear idea about how science is done. Science is the process of rational enquiry, which seeks explanations for natural phenomena. Scientific method was discussed in a very prescriptive way in Chapter 2 as the proposal of an hypothesis from which predictions are made and tested by doing experiments. Depending on the results, which may have to be analysed statistically, the decision is made to either retain or reject the hypothesis. This process of knowledge by disproof advances our understanding of the natural world and seems extremely impartial and hard to fault.

Unfortunately, this is not necessarily the case, because science is done by human beings who sometimes do not behave responsibly or ethically. For example, some scientists fail to give credit to those who have helped propose a new hypothesis. Others make up, change, ignore or delete results so their hypothesis is not rejected, omit details to prevent the detection of poor experimental design or deal unfairly with the work of others. Most scientists are not taught about responsible behaviour and are supposed to learn a code of conduct by example, but this does not seem to be a good strategy considering the number of cases of scientific irresponsibility that have recently been exposed. This chapter is about the importance of behaving responsibly and ethically when doing science.

Introduction

This chapter explains how some parametric tests for comparing the means of one and two samples actually work. The first test is for comparing a single sample mean to a known population mean. The second is for comparing a single sample mean to an hypothesised value. These are followed by a test for comparing two related samples and a test for two independent samples.

The 95% confidence interval and 95% confidence limits

In Chapter 8 it was described how 95% of the means of samples of a particular size, n, taken from a population with a known mean, μ, and standard deviation, σ, would be expected to occur within the range of μ ± 1.96 × SEM. This range is called the 95% confidence interval, and the actual numbers that show the limits of that range (μ ± 1.96 × SEM) are called the 95% confidence limits.

Introduction

Every time you make a decision to retain or reject an hypothesis on the basis of the probability of a particular result, there is a risk that this decision is wrong. There are two sorts of mistakes you can make and these are called Type 1 error and Type 2 error.

Type 1 error

A Type 1 error or false positive occurs when you decide the null hypothesis is false when in reality it is not. Imagine you have taken a sample from a population with known statistics of μ and σ and exposed the sample to a particular experimental treatment. Because the population statistics are known, you could test whether the sample mean was significantly different to the population mean by doing a Z test (Section 9.3).

Introduction

Before starting on experimental design and statistics, it is important to be familiar with how science is done. This is a summary of a very conventional view of scientific method.

Basic scientific method

These are the essential features of the ‘hypothetico-deductive’ view of scientific method (see Popper, 1968).

Preface

If you mention ‘statistics’ or ‘biostatistics’ to life scientists, they often look nervous. Many fear or dislike mathematics, but an understanding of statistics and experimental design is essential for graduates, postgraduates and researchers in the biological, biochemical, health and human movement sciences.

Since this understanding is so important, life science students are usually made to take some compulsory undergraduate statistics courses. Nevertheless, I found that a lot of graduates (and postgraduates) were unsure about designing experiments and had difficulty knowing which statistical test to use (and which ones not to!) when analysing their results. Some even told me they had found statistics courses ‘boring, irrelevant and hard to understand’.

Introduction

Parametric analysis of variance assumes the data are from normally distributed populations with the same variance and there is independence, both within and among treatments. If these assumptions are not met, an ANOVA may give you an unrealistic F statistic and therefore an unrealistic probability that several sample means are from the same population. Therefore, it is important to know how robust ANOVA is to violations of these assumptions and what to do if they are not met as in some cases it may be possible to transform the data to make variances more homogeneous or give distributions that are better approximations to the normal curve.

This chapter discusses the assumptions of ANOVA, followed by three frequently used transformations. Finally, there are descriptions of two tests for the homogeneity of variances.

Introduction

Often life scientists collect samples of multivariate data – where more than two variables have been measured on each sample – because univariate or bivariate data are unlikely to give enough detail to realistically describe the object, location or ecosystem being investigated. For example, when comparing three polluted and three unpolluted lakes, it is best to collect data for as many species of plants and animals as possible, because the pollutant may affect some species but not others. Data for only a few species are unlikely to realistically estimate the effect upon an aquatic community of several hundred species.

If all six lakes were relatively similar in terms of the species present and their abundance, it suggests the pollutant has had little or no effect. In contrast, if the three polluted lakes were relatively similar to each other but very different to the three unpolluted ones, it suggests the pollutant has had an effect. It would be very useful to have a way of assessing similarity (or its converse dissimilarity) among samples of multivariate data.

Why do life scientists need to know about experimental design and statistics?

If you work on living things, it is usually impossible to get data from every individual of the group or species in question. Imagine trying to measure the length of every anchovy in the Pacific Ocean, the haemoglobin count of every adult in the USA, the diameter of every pine tree in a plantation of 200 000 or the individual protein content of 10 000 prawns in a large aquaculture pond.

The total number of individuals of a particular species present in a defined area is often called the population. But because a researcher usually cannot measure every individual in the population (unless they are studying the few remaining members of an endangered species), they have to work with a very carefully selected subset containing several individuals (often called sampling units or experimental units) that they hope is a representative sample from which they can infer the characteristics of the population. You can also think of a population as the total number of artificial sampling units possible (e.g. the total number of 1m2 plots that would cover a whole coral reef) and your sample being the subset (e.g. 20 plots) you have to work upon.

Introduction

Parametric tests are designed for analysing data from a known distribution, and most of these tests assume the distribution is normal. Although parametric tests are quite robust to departures from normality and a transformation can often be used to normalise data, there are some cases where the population is so grossly non-normal that parametric testing is unwise. In these cases, a powerful analysis can often still be done by using a non-parametric test.

Non-parametric tests are not just alternatives to the parametric procedures for analysing ratio, interval and ordinal data described in Chapters 9 to 18. Often life scientists measure data on a nominal scale. For example, Table 3.4 gave the numbers of basal cell carcinomas detected and removed from different areas of the human body. This is a sample containing frequencies in several discrete and mutually exclusive categories and there are non-parametric tests for analysing this type of data (Chapter 20).

Introduction

So far, this book has only covered tests for one and two samples. But often you are likely to have univariate data for three or more samples and need to test whether there are differences among them.

For example, you might have data for the concentration of cholesterol in the blood of seven adult humans in each of five different dietary treatments and need to test whether there are differences among these treatments. The null hypothesis is that these five treatment samples have come from the same population. You could test it by doing a lot of independent samples t tests (Chapter 9) to compare all of the possible pairs of means (e.g. mean 1 compared to mean 2, mean 1 compared to mean 3, mean 2 compared to mean 3 etc.), but this causes a problem. Every time you do a two-sample test and the null hypothesis applies (so the samples are from the same population), you run a 5% risk of a Type 1 error. As you do more and more tests on two samples from the same population, the risk of a Type 1 error rises rapidly.

Introduction

When you use a single-factor ANOVA to compare the means in an experiment with three or more treatments, a significant result only indicates that one or more appear to be from different populations. It does not identify which particular treatments appear to be the same or different.

For example, a significant difference among the means of three treatments, A, B and C can occur in several ways. Mean A may be greater (or less) than B and C, mean B may be greater (or less) than A and C, mean C may be greater (or less) than A and B and finally means A, B and C may all be different to each other.

Introduction

Often life scientists obtain data for two or more variables measured on the same set of subjects or experimental units because they are interested in whether these variables are related and, if so, the type of functional relationship between them.

If two variables are related, they vary together – as the value of one variable increases or decreases, the other also changes in a consistent way.

Introduction

Statisticians and life scientists who teach statistics are often visited in their offices by a researcher or student they may never have met before, who is clutching a thick pile of paper and perhaps a couple of flash drives or CDs with labels such as ‘Experiment 1’ or ‘Trial 2’. The visitor drops everything heavily on the desk and says ‘Here are my results. What stats do I need?’

This is not a good thing to do. First, the person whose advice you are seeking may not have the time to work out exactly what you have done, so they may give you bad advice. Second, the answer can be a very nasty surprise like ‘There are problems with your experimental design.’

Overview

Thermodynamics is unquestionably the most powerful and most elegant of the engineering sciences. Its power arises from the fact that it can be applied to any discipline, technology, application, or process. The origins of thermodynamics can be traced to the development of the steam engine in the 1700's, and thermodynamic principles do govern the performance of these types of machines. However, the power of thermodynamics lies in its generality. Thermodynamics is used to understand the energy exchanges accompanying a wide range of mechanical, chemical, and biological processes that bear little resemblance to the engines that gave birth to the discipline. Thermodynamics has even been used to study the energy exchanges that are involved in nuclear phenomena and it has been helpful in identifying sub-atomic particles. The elegance of thermodynamics is the simplicity of its basic postulates. There are two primary ‘laws’ of thermodynamics, the First Law and the Second Law, and they always apply with no exceptions. No other engineering science achieves such a broad range of applicability based on such a simple set of postulates.

So, what is thermodynamics? We can begin to answer this question by dissecting the word into its roots: ‘thermo’ and ‘dynamics’. The term ‘thermo’ originates from a Greek word meaning warm or hot, which is related to temperature. This suggests a concept that is related to temperature and referred to as heat. The concept of heat will receive much attention in this text. ‘Dynamics’ suggests motion or movement. Thus the term ‘thermodynamics’ may be loosely interpreted as ‘heat motion’. This interpretation of the word reflects the origins of the science. Thermodynamics was developed in order to explain how heat, usually generated from combusting a fuel, can be provided to a machine in order to generate mechanical power or ‘motion’. However, as noted above, thermodynamics has since matured into a more general science that can be applied to a wide range of situations, including those for which heat is not involved at all. The term thermodynamics is sometimes criticized because the science of thermodynamics is ordinarily limited to systems that are in equilibrium. Systems in equilibrium are not ‘dynamic’. This fact has prompted some to suggest that the science would be better named ‘thermostatics’ (Tribus, 1961).