1
Basic Design Considerations

SUMMARY

This chapter reviews the reasons why sample size considerations are important when planning a clinical study of any type. The basic elements underlying this process include the null and alternative study hypotheses, effect size, statistical significance level and power, each of which are described. We introduce the notation to distinguish the population parameters we are trying to estimate with the study, from their anticipated value at the planning stages and also from their estimated value once the study has been completed. We emphasise for comparative studies that, whenever feasible, it is important to randomise the allocation of subjects to respective groups.

The basic properties of the standardised Normal distribution are described. Also discussed is how, once the effect size, statistical significance level and power for a comparative study using a continuous outcome are specified, the Fundamental Equation (which essentially plays a role in most sample size calculations for comparative studies) is derived.

The Student's t-distribution and the Non-central t-distribution are also described. In addition the Binomial, Poisson, Negative-Binomial, Beta and Exponential statistical distributions are defined. In particular, the circumstances (essentially large study sizes) in which the Binomial and Poisson distributions have an approximately Normal shape are described. Methods for calculating confidence intervals for a population mean are indicated together with (suitably modified) how they can be used for a proportion or a rate in larger studies. For the Binomial situation, formulae are also provided where the sample size is not large. Finally, a note concerning numerical accuracy of the calculations in the illustrative examples of later chapters is included.

1.1 Why Sample Size Calculations?

To motivate the statistical issues relevant to sample size calculations, we will assume that we are planning a two-group clinical trial in which subjects are allocated at random to one of two alternative treatments for a particular medical condition and that a single endpoint measure has been specified in advance. However, it should be emphasised that the basic principles described, the formulae, sample size tables and associated software included in this book are equally relevant to a wide range of design types covering all areas of medical research ranging from the epidemiological to clinical and laboratory-based studies.

Whatever the field of inquiry the investigators associated with a well-designed study will have considered the research questions posed carefully, formally estimated the required sample size (the particular focus for us in this book), and recorded the supporting reasons for their choice. Awareness of the importance of these has led to the major medical and related journals demanding that a detailed justification of the study size be included in any submitted article as it is a key component for peer reviewers to consider when assessing the scientific credibility of the work undertaken. For example, the General Statistical Checklist of the British Medical Journal asks statistical reviewers of their submitted papers 'Was a pre-study calculation of study size reported?' Similarly, many research grant funding agencies such as the Singapore National Medical Research Council now also have such requirements in place.

In any event, at a more mundane level, investigators, grant-awarding bodies and medical product development companies will all wish to know how much a study is likely to 'cost' both in terms of time and resources consumed as well as monetary terms. The projected study size will be a key component in this 'cost'. They would also like to be reassured that the allocated resource will be well spent by assessing the likelihood that the study will give unequivocal results. In particular for clinical trials, the regulatory authorities, including the Committee for Proprietary Medicinal Products (CPMP, 1995) in the European Union and the Food and Drug Administration (FDA, 1988 and 1996) in the USA, require information on planned study size. These are encapsulated in the guidelines of the International Conference on Harmonisation of Technical Requirements for Registration of Pharmaceuticals for Human Use (1998) ICH Topic E9.

If too few subjects are involved, the study is potentially a misuse of time because realistic differences of scientific or clinical importance are unlikely to be distinguished from chance variation. Too large a study can be a waste of important resources. Further, it may be argued that ethical considerations also enter into sample size calculations. Thus a small clinical trial with no chance of detecting a clinically useful difference between treatments is unfair to all the patients put to the (possible) risk and discomfort of the trial processes. A trial that is too large may be unfair if one treatment could have been 'proven' to be more effective with fewer patients as a larger than necessary number of them has received the (now known) inferior treatment.

Providing a sample size for a study is not simply a matter of providing a single number from a set of statistical tables. It is, and should be, a several-stage process. At the preliminary stages, what is required are 'ball-park' figures that enable the investigators to judge whether or not to start the detailed planning of the study. If a decision is made to proceed, then the later stages are used to refine the supporting evidence for the preliminary calculations until they make a persuasive case for the final patient numbers chosen. Once decided this is then included (and justified) in the final study protocol.

After the final sample size is determined and the protocol is prepared and approved by the relevant bodies, it is incumbent on the research team to expedite the recruitment processes as much as possible, ensure the study is conducted to the highest of standards possible, and ensure that it is eventually reported comprehensively.

1.2 Statistical Significance

Notation

In very brief terms the (statistical) objective of any study is to estimate from a sample the value of a population parameter. For example, if we were interested in the mean birth weight of babies born in a certain locality, then we may record the weight of a selected sample of N babies and their mean weight is taken as our estimate of the population mean birth weight denoted ?Pop. The Greek ? distinguishes the population value from its estimate, the Roman . When planning a study, we are clearly ignorant of ?Pop and neither do we have the data to calculate . As we shall see later, when planning a study the investigators will usually need to provide some value for what ?Pop may turn out to be. This anticipated value is denoted ?Plan. This value then forms (part of) the basis for subsequent sample size calculations.

Outcomes

In any study, it is necessary to define an outcome (endpoint) which may be, for example, the birth weight of the babies concerned, as determined by the objectives of the investigation. In other situations this outcome may be a measure of blood pressure, wound healing time, degree of palliation, a patient reported outcome (PRO) that indicates the level of some aspect of their Quality of Life (QoL) or any other relevant and measureable outcome of interest.

The Effect Size

Consider, as an example, a proposed randomised trial of a placebo (control, C) against acupuncture (A) for the relief of pain in patients with a particular diagnosis. The patients are randomised to receive either A or C (how placebo acupuncture can be administered is clearly an important consideration). In addition, we assume that pain relief is assessed at a fixed time after randomisation and is defined in such a way as to be unambiguously evaluable for each patient as either 'success' or 'failure'. We assume the aim of the trial is to estimate the true difference dPop between the true success rate pPopA of A and the true success rate pPopC of C. Thus the key (population) parameter of interest is dPop which is a composite of the two (population) parameters pPopA and pPopC.

At the completion of the trial the A patients yield a treatment success rate pA which is an estimate of pPopA and for C the corresponding items are pC and pPopC. Thus, the observed difference, d = pA - pC, provides an estimate of the true difference (the effect size) dPop = pPopA - pPopC.

Significance Tests

In a clinical trial, two or more forms of therapy or intervention may be compared. However, patients themselves vary both in their baseline characteristics at diagnosis and in their response to subsequent therapy. Hence in a clinical trial, an apparent difference in treatments may be observed due to chance alone, that is, we may observe a difference but it may be explained by the intrinsic characteristics of the patients themselves rather than 'caused' by the different treatments given. As a consequence, it is customary to use a 'significance...