Practical Statistics for Nursing and Health Care

1

INTRODUCTION

1.1 What do we mean by statistics?

Statistics are a familiar and accepted part of the modern world, and already intrude into the life of every nurse and health care worker. We have statistics in the form of patients registered at a GP practice or outpatient clinic; hospital measurements and records of temperature, blood pressure and pulse rate; data collected from various surveys, censuses and clinical trials, to name but a few. It is impossible to imagine life without some form of statistical information being readily at hand.

The word statistics is used in two senses. It refers to collections of quantitative information, and methods of handling that sort of data. A hospital’s database listing the names, addresses and medical history records of all its registered patients, is an example of the first sense in which the word is used. Statistics also refers to the drawing of inferences about large groups on the basis of observations made on smaller ones. Estimating the relationship between smoking and the incidence of lung cancer illustrates the second sense in which the word is used.

Statistics looks at ways of organizing, summarizing and describing quantifiable data, and methods of drawing inferences and generalizing upon them.

1.2 Why is statistics necessary?

There are two reasons why some knowledge of statistics is an important part of the competence of every health care worker. First, statistical literacy is necessary if they are to read and evaluate reports and other literature critically and intelligently. Statements like ‘there was a significant improvement in mean scores on the depression scale over time, F(2,92) = 13.99, P < 0.001’ (Cimprich & Ronis 2001), enable the reader to decide the justification of the claims made by the particular author.

A second reason why statistical literacy is important to health care workers is if they are going to undertake an investigation that involves the collection, processing and analysis of data on their own account. If the results are to be presented in a form that will be authoritative, then a grasp of statistical principles and methods is essential. Indeed, a programme of work should be planned anticipating the statistical methods that are appropriate to the eventual analysis of the data. Attaching some statistical treatment as an afterthought to make a survey seem more ‘respectable’ is unlikely to be convincing.

1.3 The limitations of statistics

Statistics can help an investigator describe data, design experiments, and test hunches about relationships among things or events of interest. Statistics is a tool that helps acceptance or rejection of the hunches within recognized degrees of confidence. They help to answer questions like, ‘If my assertion is challenged, can I offer a reasonable defence?’

It should be noted that statistics never prove anything. Rather, they indicate the likelihood of the results of an investigation being the product of chance.

1.4 Calculators and computers in statistics

A hand calculator that has the capacity to calculate a mean and standard deviation (typically referred to as a ‘scientific’ calculator) from a single input of a set of data is indispensable. All the calculations and worked examples in this book were first worked out using such a calculator. Because different makes of calculator operate somewhat differently, we have not attempted to offer guidance about the use of individual calculators: we suggest that you study the instruction booklet that comes with your calculator.

In a modern world, computer packages are readily available and easy to use. However, we suggest caution against jumping straight into computer packages without first understanding the underlying background and principles of a particular statistical technique. Computers undertake any analysis that you ask of it, but can not provide the intelligent reasoning about whether the test is appropriate for the kind of data you are using. Moreover, a ‘print-out’ of the analysis can be ambiguous and confusing if you do not understand the underlying principles. We feel this is best achieved by first familiarizing yourself with the techniques ‘long hand’, working through our own examples and applying them to your own data. In due course, support from a computer package will become a natural extension of your analysis.

1.5 The purpose of this text

The objectives of this text stem from the points made in Section 1.2. First, the text aims to provide nurses and health care workers with sufficient grounding in statistical principles and methods to enable them to read survey reports, journals and other literature. Secondly, the text aims to present them with a variety of the most appropriate statistical tests for their problems. Thirdly, guidance is offered on ways of presenting the statistical analyses, once completed.

Full details of references and other material that we suggest for further reading are listed in full in the Bibliography in Appendix 13. For assistance in cross-referencing, we classify items according to chapter. Thus, Section 9.1, Figure 9.1, Table 9.1 and Example 9.1 are all to be found in Chapter 9.

2

HEALTH CARE INVESTIGATIONS: MEASUREMENT AND SAMPLING CONCEPTS

2.1 Introduction

A health care investigation is typically a five-stage process: identifying objectives; planning; data collection; analysis; and, finally, reporting. The methodologies frequently used are sample surveys, clinical trials and epidemiological studies that are the subject of this and subsequent chapters. However, we must first be clear about the definitions of some basic terms. Many of the terms used in statistics also have usage in daily life, where the meaning might be quite different. The word ‘population’ may conjure images of ‘people’, whilst ‘sample’ might mean a ‘free sample’ of cream offered by a pharmaceutical company, or a ‘sample’ requested by a doctor for urine analysis. In statistics, however, these words have much more precise meanings.

2.2 Populations, samples and observations

In statistics, the term ‘population’ is extended to mean any collection of individual items or units that are the subject of investigation. Characteristics of a population that differ from individual to individual are called variables. Length, age, weight, temperature, number of heart beats, to name but a few, are examples of variables to which numbers or values can be assigned. Once numbers or values have been assigned to the variables, they can be measured.

Because it is rarely practicable to obtain measures of a particular variable from all the units in a population, the investigator has to collect information from a smaller group or sub-set that represents the group as a whole. This sub-set is called a sample. Each unit in the sample provides a record, such as a measurement, which is called an observation. The relationship between the terms we have introduced is summarized below:

Note that the biological or demographic population would include babies of both sexes, and indeed, all individuals of whatever age or sex in a particular community.

2.3 Counting things – the sampling unit

We sometimes wish to count the number of items or objects in a group or collection. If the number is to be meaningful, the dimensions of the collection have to be specified.

For example ‘the number of patients admitted to an accident and emergency department’ has little meaning unless we know the time scale over which the count was made. A collection with specified dimensions is called a sampling unit. An observation is, of course, the number of objects or items counted in a sampling unit. Thus, if 52 patients are admitted to a particular A & E department in a 24 hr period, the sampling unit is ‘one A. & E. 24-hour period’ and the observation is 52. The sample is the number of such 24-hour periods that were included in the survey. However, the definition of the ‘population’ requires care. It might be tempting to think that the population under investigation is something to do with patients, but this is not the case when they are being counted. The statistical population comprises the same ‘thing’ as the sample units that comprise the sample. In this case, the statistical population is a rather abstract concept, and represents all possible ‘A & E department 24 hour periods’ that could have been included in the survey.

It is very important to be able to identify correctly the population under investigation, because this is essential in formulating a ‘null hypothesis’ when undertaking statistical tests. This is the subject of Chapter 11.

2.4 Sampling strategy

As we said above, it is not always possible or practicable to sample every single individual or unit in a particular population either due to its size, or constraints on available resources (for example, cost, time, manpower). The solution is to take a sample from the population of interest and use the sample information to make inferences about the population.

A common, but misguided, approach to sampling is to first decide what data to collect, then undertake the survey, and finally, decide what analyses should be done. However, without initial thought being given to the aims of the survey, the information or data may not be appropriate (e.g. wrong data collected, or data collected on wrong subjects, or insufficient data collected). As a result, the desired analysis may not be possible or effective.

The key to good sampling is to:

1. Formulate the aims of the study.

2. Decide what analysis is required to satisfy these aims.

3. Decide what data are required to facilitate the analysis.

4. Collect the data required by the survey.

The crucial point relates to the sequence. For example, if the aim of a study is to identify the effectiveness of asthmatic care within a single GP practice, suitable measures of effectiveness need to be defined. One measure could be based on the number of acute asthma exacerbations (deteriorations) in the preceding 12 months, and this number could be compared with that for the previous 12 months. Other measures might assess the number of patients who have had their inhaler technique checked or are using peak flow meters at home. Most of this information can be obtained from practice records, although crosschecking with hospital records may be required to validate the assessment based on acute exacerbations.

2.5 Target and study populations

We have to distinguish between the target and study populations. The target population in the asthma example above is the number of patients registered with the GP practice who have asthma. The study population consists of all patients who could actually be selected to form the sample, i.e. those who are known to have asthma. For example, a proportion of the target population may not know they have asthma, will not therefore be registered, and thus will not form part of the study population. Ideally, the ‘target’ and ‘study’ populations coincide.

2.6 Sample designs

Once the study population has been defined, the next task is to decide which subjects from the population should form the sample. The following list is not exhaustive, but gives a selection of sample designs pertinent to audit:

simple random sampling

systematic sampling

stratified sampling

quota sampling

cluster sampling.

The first three designs can be applied to sampling from finite populations, i.e. situations where every member of the study population can be identified. Such is the case in our asthmatic care example (Section 2.4), where a list of all asthmatic patients registered with the GP practice is available or can easily be obtained prior to the study. Quota and cluster sampling are used when it is not possible or practicable to enumerate every member of the study population.

2.7 Simple random sampling

In a simple random sampling design, every individual in the study population has an equal chance of being included in the sample. That is to say, steps are taken to avoid bias in the sampling. In our asthma example above, the population being sampled is all patients registered with the GP practice who are known to have asthma (say, 800). To select a simple random sample of size n = 20, each patient (‘sampling unit’) is assigned a unique number: 1, 2, 3, and so on, until all 800 patients have been numbered. Then 20 numbers in the range 1 to 800 are selected at random, and the patients (sampling units) corresponding to these numbers represent the sample.

There are two usual ways of obtaining random numbers. First, many calculators and pocket computers have a facility for generating random numbers. These are often in the form of a fraction, e.g. 0.2771459. You may use this to provide a set of integers, 2, 7, 7, 1,…; or 27, 71, 45,…; or 277, 145; or 2.7, 7.1; and so on, according to your needs, keying in a new number when more digits are required.

Secondly, use may be made of random number tables. Appendix 1 is such a table. The numbers are arranged in groups of five in rows and columns, but this arrangement is arbitrary. Starting at the top left corner, you may read: 2, 3, 1, 5, 7, 5, 4…; or 23, 15, 75, 48,…; or 231, 575, 485…; or 23.1, 57.5, 48.5, 90.1,…; and so on, according to your needs. When you have obtained the numbers you need for your investigation, mark the place in pencil. Next time, carry on where you left off. It is possible that a random number will prescribe a subject (sampling unit) that has already been drawn. In this event, ignore the number and take the next random number. The purpose is to eliminate your prejudice as to which items should be selected for measurement. Unfortunately, observer bias, conscious or unconscious, is notoriously difficult to avoid when gathering data in support of a particular hunch!

Random sampling is the preferred approach to sampling. Although it does not guarantee that a representative sample is taken from the study population (due to sampling error, described in Section 10.1), it gives a better chance than any other method of achieving this.

2.8 Systematic sampling

Systematic sampling has similarities with simple random sampling, in that the first subject in the sample is chosen at random and then every subsequent tenth or twentieth patient (for example) is chosen to cover the entire range of the population.

Example 2.1

What interval is required to select a systematic sample of size 20 from a population of 800?

The required fixed interval is:

Therefore, the first patient (‘sampling unit’) is selected at random (as described in Section 2.8) from among patients numbered 1 to 40. Suppose number 23 is selected. The sample then comprises patients 23, 63, 103, 143,…., 783.

A disadvantage of systematic sampling occurs when the patients are listed in the population in some sort of periodic order, and thus we might inadvertently systematically exclude a subgroup of the population. For example, given a population of 800 patients listed by ‘first attendance’ at the clinic, and that over a 20 week period, 40 patients registered per week, 20 during the daytime and 20 during the evening surgeries. If these patients were listed in the following order: Week 1 daytime patients, Week 1 evening patients, Week 2 daytime patients,…., Week 10 evening patients, then selecting patients 23, 63,…., 783 would result in a sample of evening clinic patients, and exclude all the daytime patients. It is possible that this could generate a biased, or unrepresentative, sample.

An argument in favour of systematic sampling occurs when patients are listed in the population in chronological order, say, by date of first attendance at the GP practice. A systematic