Main article An introduction to medical statistics for health care professionals: Describing and presenting data

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1 218 Musculoskeletal Care Volume 2 Number 4 Whurr Publishers 2004 Main article An introduction to medical statistics for health care professionals: Describing and presenting data Elaine Thomas PhD MSc BSc Lecturer in Biostatistics, Primary Care Sciences Research Centre, Keele University, North Staffordshire, UK Abstract This article is the first in a series of three that will give health care professionals a sound and helpful introduction to medical statistics. This article covers three main areas: description of the different types of data available, appropriate summary measures used to describe different data types, and methods used to present these data. Key words: statistics, data types, data presentation Introduction The need to use statistical principles in a project depends on the type of study being carried out. A presentation of a patient with a clinically interesting history (a case study) may be presented with few, if any, statistical ideas, whereas a project comparing patients with a disease and healthy subjects (case-control study) to assess specific risk factors or outcome of different treatments is likely to need the input of a statistician. However, it is important that all health care professionals (HCPs) have some insight into statistical principles to enable them to feel confident about carrying out a critique of published literature in their own field. There is an abundance of basic statistical texts available. For those new to statistics Jordan et al. (1998) or Swinscow (1998) offer an excellent introduction to medical statistics. For those looking for something more advanced Altman (1994) or Bland (2000) both cover a wider area in breadth and depth, but still retain the readability necessary for non-experts in this field. Understanding data types The first step, before any calculations or plotting of data, is to determine what type of data has been collected. In any particular study, several pieces of information will

2 Medical statistics: Describing and presenting data 219 TABLE 1: Data types and examples Numerical data Continuous Height (cms) Blood pressure (mmhg) Urinary lead concentration (μmol/24h) Categorical data Ordinal (Ordered categories) Social class (I, II, IIIM, etc) Apgar score (0 10) Pain severity (mild, moderate, severe) Discrete Number of adults consulting for shoulder pain Number of children in family Number of hand nodes Nominal (Un-ordered categories) Marital status (married, single, divorced) Blood group (A, O, B, AB) Ethnicity (European, Afro-Caribbean, Asian) be collected on the participating subjects and this information will vary across the subjects, and, therefore, are known as variables. The type of data and the scales used to measure these variables are crucial when deciding on the method of data analysis and presentation. The two main types of data are numerical and categorical, both of which can, in turn, be further sub-divided (Table 1). Numerical variables fall into two main subtypes: discrete data and continuous data. Discrete data increase in whole steps (e.g., 1, 2, 3, 4 ) and hence can take only a finite number of values. A good example is, How many people were newly diagnosed with rheumatoid arthritis in the Midlands region in 2003? The answer to this query will be a discrete number, such as 127, as you cannot have people. In contrast, continuous data can take any value between two numbers and are only limited by the accuracy of the method used for measuring the value. For example, body height is measured on a continuous scale, but the value recorded will depend on whether your measurement method is accurate to, say, centimetres (i.e. 158 cms) or millimetres (i.e mm or cms). Categorical variables are used to classify or categorize data where each observation will fall into one (or more) particular category. Data for which there is no ordering to the categories are termed nominal data, e.g. marital status. However when there is an inherent or natural ordering to the categories, such as pain severity, the data are ordinal. With a nominal variable the category labels are almost exclusively adjectival (e.g. marital status: single, married, divorced ) but for ordinal variable the category labels may either be adjectival (e.g. pain severity: mild, moderate, severe ) or numerical (e.g. pain score: 0 10). Most numerical ordinal data arise from measuring scales and normally do not represent true numbers. A well-known example of a numerical ordinal measure is the Apgar score for evaluating the well-being of newborn babies (Apgar, 1953). To calculate the score, five areas are assessed, each on a scale 0 2 and a total score is derived from the sum of these individual scores (range: 0 10) in which high scores indicate better health. Using this scale we cannot say that

3 220 Thomas a baby with an Apgar score of 6 is exactly twice as healthy as a baby with a score of 3, or that the difference in health between any two points on the scale that are the same distance apart, i.e. 5 and 6 compared to 6 and 7, is the same. We can only say that a higher score indicates better health. Hence, for either ordinal or nominal variables, the normal rules of mathematics do not apply. An important option available is to convert numerical data into categorical data by using cut-off points. As an example, a diastolic blood pressure measurement could be classified as hypertension if it is greater than 90 mmhg or normotension if less than or equal to 90 mmhg. The choice of what values to use as the cut-off points can be difficult. If there are clinically meaningful cut-off points, such as those defined by the World Health Organization for obesity (World Health Organization, 2000), or definitions used for research purposes, such as those applied to the Hospital Anxiety and Depression Scale (HADS) to define cases and non-cases (Zigmond and Snaith, 1983), then these should be applied. If standard cut-offs are not available then the HCP needs to consider how best to categorize the data for the research. It is important to remember, however, that although numerical data can be converted into categorical data, the reverse is not true. Numerical data Summary measures for data Measures of average The main objective when describing a set of numerical data is to derive useful summary measures that give an accurate reflection of the individual values recorded. The first summary measure derived should summarize the distribution of the variable under investigation, i.e. an average score. To complicate matters, there are a number of possible measures of average that can be calculated on a set of data, although this article will concentrate on the three that appear most frequently: the mean, the median, and the mode. 1. Mean: the sum of the individual values divided by the number of values. 2. Median: the middle value when all values are arranged in numerical order. 3. Mode: the most frequently occurring value. Below is a small data set of the ages (in years) of nine individuals representing a larger population: In order to ease the calculation of the mode and the median, the data have been

4 Medical statistics: Describing and presenting data 221 TABLE 2: Example of calculating the three measures of average on a set of numerical data Observation Age (years) Median (middle) age = 42 years Modal (most common) age = 24 years Total 380 Mean = 380 / 9 = 42.2 years rearranged in numerical order, i.e. ranked in numerical order, and are presented in Table 2, together with the three measures of average for this dataset. Important calculations on this data set include: 1. The sample size: the number of subjects who contribute information, so here this is 9 and is generally written as n = The mean: calculated by adding up each of the individual values and dividing by the total number of values, i.e. mean = ( ) / 9 = 42.2 years. 3. The median: to determine which is the middle observation, rank the observations, add 1 to the sample size and then divide by 2, i.e. (1 + n) / 2. The median is then the data value for this observation. For the current dataset of 9 ages, the median is the 5th observation, i.e. (1 + 9) / 2, which has the value of 42 years. If the dataset had contained 10 observations the middle observation will be the 5.5th, hence the median value would be halfway between the 5th and 6th observation. 4. The mode: the value 24 years appears twice in the dataset and all other values appear only once, hence it is the mode of the dataset. The mean is determined by totalling the individual values and hence is influenced by extreme observations, known as outliers. If in the above dataset of ages the oldest person was actually aged 104 years, the mean age would now be 46.7 years. However, because the median is based on the rank of the values, the median is not influenced by outliers and the median would remain at 42 years. Measures of variability In addition to calculating a measure of average it is important also to describe the variability in the distribution. Two different datasets can have the same mean but

5 222 Thomas TABLE 3: Datasets with the same mean value but very different distributions Dataset Range IQR SD A , , B , , IQR = inter-quartile range, SD = standard deviation can be made up of very different values. For example, the two small datasets in Table 3 have the same mean value (100) but have very different distributions. The simplest method of measuring the variability in a distribution is by presenting the range, i.e. the lowest and the highest values. However, the range depends on the extreme values and hence can vary widely from dataset to dataset. By specifying two values that encompass most of the data values we can overcome this problem. This is done by calculating percentiles, i.e. the value below which a given percentage of the observations occur. The most frequently used percentile is the 50th percentile, also known as the median, as 50% of the data lie below it (and consequently 50% above it). Other percentiles can be calculated such as the 25th percentile (the lower quartile 25% of the data lie below this value) and the 75th percentile (the upper quartile 75% of the data lie below this value). From this we can determine the inter-quartile range (IQR) which is represented by the 25th and 75th percentiles and encompasses the middle 50% of the data. A more numerically complex method for determining variability is the standard deviation (SD). This is calculated by determining how far from the mean value each individual value is and summarizing this difference across all observations. Hence, observations that are far from the mean (dataset A in Table 3) will result in large differences from the mean and hence in a large SD, which implies greater variability. Conversely, values that are close to the mean (dataset B) will result in smaller differences from the mean, hence a small SD, which implies less variability. Appropriate measures of average and variation All of the measures described can be calculated for any numerical variable. It is important to determine which are the most appropriate to present for any particular variable. Many of the statistical tests that will be presented in a subsequent article have a number of assumptions about the type of distribution the data have come from. Moreover, when presenting summary data, either in text or tables, it is important to present both a measure of average and variability. Many distributions occurring naturally, such as human height, follow the normal distribution (Figure 1). The distribution is characterized by a bell-shaped curve in which the most common values are in the middle of the distribution with

6 Medical statistics: Describing and presenting data Frequency Height Height (cms) (cms) FIGURE 1: Example of a normal distribution: body height in females. the values being rarer the further we move away from the midpoint. The centre of the distribution is determined by the mean (which for the normal distribution is also the same as the median and mode) and the shape of the curve is determined by the standard deviation. The larger the standard deviation the more the data values are spread out from the mean and hence the curve is flatter. Conversely, the smaller the standard deviation, the more the values are close to the mean and hence the curve is more peaked. For data that follow a normal distribution, the appropriate summary measures are the mean and the standard deviation. Not all numerical variables follow the normal distribution. Commonly, distributions not following the normal distribution are skewed (Figure 2). Data that are positively skewed have most values at the lower end of the scale with few higher values (Figure 2a). This figure shows AIMS-2 hand and finger function scores (Meenan et al., 1992) from a sample of 278 hand pain sufferers based in the general population data. Conversely, data that are mainly in the higher end of the scale with a few smaller values are negatively skewed (Figure 2b). Figure 2b shows scores from the EuroQol EQ-5D questionnaire (EuroQol Group, 1990) from 201 participants in a primary care-based randomized clinical trial assessing the effectiveness of common therapies for shoulder pain. Skewed data, together with other non-normal distributions, are best summarized by the median and inter-quartile range. Categorical data When assessing categorical data the first step is to examine the frequency distribution of the measure. This is simply all the possible values for the variable in question together with how frequently each category occurs. Associated with this is

7 224 Thomas 100 (a) positively skewed Frequency AIMS-2 Hand and and finger finger function function 60 (b) negatively skewed Frequency EuroQol score at recruitment EuroQol score at recruitment FIGURE 2: Examples of skewed data distributions. the relative frequency of each of the individual categories, which is the proportion of subjects in that particular category. Say we had a sample of 80 subjects and we have recorded their marital status. Of this sample 36 reported that they were married, 24 were single and 20 were divorced. These three figures (36, 24, 20) represent the frequency distribution in this sample and the relative frequencies for the categories are: married = 45% (36/80), single = 30% (24/80), divorced = 25% (20/80). It was stated earlier in the article that the normal rules of mathematics do not apply to categorical data. One major consequence is that it is not correct to calculate the mean of a set of numerical ordinal scores as it has no meaning. A more appropriate measure of average for this form of data would be the category containing the most observations, i.e. the modal category.

8 Medical statistics: Describing and presenting data Males Females 180 Height (cms) Weight, (kg) kgs FIGURE 3: Scatterplot of weight (kgs) versus height (cms): Separately for males and females. Appropriate data presentation The dissemination of findings from study data, through a research article, dissertation or conference presentation, will benefit from the use of a range of presentation methods, such as tables and graphs. Summary data should also be presented in the main body of the text of a research article or dissertation, although careful consideration must be given to the amount of data presented in this form. Too much data or data that are inappropriately summarized can make the text difficult to read. When using either tables or graphs to present study data the figure should have a clear, concise title and the data given in the figure should be able to be understood without needing to refer to the accompanying text. In tables, columns and rows should be suitably labelled and where symbols have been used, footnotes should be given at the bottom of the table to describe what the symbols represent. For graphs, both the vertical and the horizontal axes should be labelled and a legend should be presented when different symbols or lines are used to represent groups of participants (Figure 3). The decision to present data in a table or a graph is a personal choice. A graphical representation offers the opportunity to show more detail than can be presented in a table, and should be used when presentation of the data in a table is

9 226 Thomas TABLE 4: Characteristics of patients presenting to primary care with shoulder pain, according to treatment allocation Physiotherapy Injection (n = 72) (n = 69) Gender, n (%) Male 27 (37.5%) 30 (43.5%) Female 45 (62.5%) 39 (56.5%) Age in years, mean (SD) 55.4 (13.4) 54.4 (14.1) Employment status, n(%) Full-time employed 45 (62.5%) 43 (62.3%) Part-time employed 15 (20.8%) 17 (24.6%) Not working due to ill health 7 (9.7%) 5 (7.2%) Retired 2 (2.8%) 1 (1.4%) Other 3 (4.2%) 3 (4.3%) Shoulder disability score 1, median (IQR) 6 (3, 8) 5 (2, 8) Restriction in active external rotation 2, n(%) Present 15 (20.8%) 11 (15.9%) Absent 57 (79.2%) 58 (84.1%) SD = standard deviation, IQR = inter-quartile range 1 Shoulder disability score (0 23, where 23 indicates severe disability)(croft et al., 1994) 2 Restriction of greater than 50% in affected compared to unaffected arm difficult. Moreover, graphs are preferable when presenting individual level data rather than summary statistics. Table 4 presents recruitment data from a clinical trial that compared the effectiveness of physiotherapy and steroid injection for treating patients with shoulder pain who presented to primary care. The demographic information collected was gender, age and employment status. To determine the severity of the shoulder complaint, participants completed a shoulder disability questionnaire (Croft et al., 1994) on which scores ranged from 0 to 23; 23 indicates severe disability. Participants also underwent a clinical examination in which restriction in active external rotation of the involved shoulder was determined as either present or absent. For categorical data, i.e. gender, employment status and shoulder restriction, two measures are presented. The frequency of each category is given followed by the percentage of the whole dataset that category represents. From the table we can see that in the physiotherapy group, there are 45 female patients which represents 62.5% of the whole physiotherapy group. In both groups the majority of patients were in full-time employment (Physiotherapy = 62.5%, Injection = 62.3%) with the minority being in retirement (Physiotherapy = 2.8%, Injection = 1.4%). Approximately one in five patients in each treatment group (Physiotherapy = 20.8%, Injection = 15.9%) had a restriction in active external rotation.

10 Medical statistics: Describing and presenting data 227 The distribution of the two numerical measures in the trial data, i.e. age and shoulder disability score, were assessed to determine the appropriate summary measures. Age was determined to be normally distributed and so the mean and standard deviation are presented. The physiotherapy group was, on average, 1 year older than the injection group (Physiotherapy: mean age = 55.4 years, Injection: mean age = 54.4 years) and because the standard deviations in the two groups were similar (Physiotherapy: SD = 13.4 years, Injection: SD = 14.1 years), the distribution of the ages in both groups is alike. The distribution of the shoulder disability score was positively skewed and hence the median and inter-quartile are presented. For the physiotherapy group the median score was 6 with the middle 50% of the observations, i.e. the inter-quartile range, lying between scores of 3 and 8. The distribution of the disability scores was similar in the injection group (median = 5; IQR 2, 8). A common error when presenting summary data either in a table or directly in the text is to present the ± sign after a mean without specifying what the figure after the sign represents. As an example, the following phrase may be given in a research article, The mean age of the physiotherapy group was 55.4 ± 13.4 years. The reader is unsure as to what the 13.4 represents. A more correct presentation of the same data would be, The mean age of the physiotherapy group was 55.4 years (standard deviation 13.4 years). This latter representation is unambiguous and avoids the incorrect implications that the standard deviation can take a negative value. This practice is now less common and many journals do not allow the use of this symbol. The precision of the original data should be borne in mind when presenting results. Hence it would be inappropriate to present a mean age of years if the individual measure was recorded to the nearest year. There are no definite rules regarding numerical precision, but most measures need only be presented to a maximum of two decimal places, hence the above figure would be better presented as years. It is important to remember to present data with their appropriate unit such as cm for height and mmhg for blood pressure. Conclusion The aim of this first article was to present a clear guide to the different types of data that occur, and how best to summarize and present them. The two subsequent articles in the series will build upon the concepts presented here. The aim of the series is to give health care professionals a greater understanding of the use of statistics in health research, so that they may feel confident when reading the literature in their area of interest. It is hoped that this introduction to statistics will encourage health care professionals to be more aware of statistics and less fearful.

11 228 Thomas Acknowledgements I would like to thank Sarah Ryan for asking me to write this series, the two anonymous reviewers for their constructive comments on an earlier draft, and Professor Peter Croft for allowing me the time to complete this series. References Altman DG (1994). Practical Statistics for Medical Research. London: Chapman and Hall. Apgar V (1953). Proposal for a new method of evaluation of newborn infants. Anesthesia and Analgesia 32: Bland JM (2000). An Introduction to Medical Statistics (3rd edn.) Oxford: Oxford University Press. Croft P, Pope D, Zonca M, O'Neill T, Silman A (1994). Measurement of shoulder related disability: Results of a validation study. Annals of the Rheumatic Diseases 53: EuroQol Group (1990). Euro-Qol a new facility for the measurement of health-related quality of life. Health Policy 16: Jordan K, Ong BN, Croft P (1998). Mastering Statistics: A Guide for Health Service Professionals and Researchers. Cheltenham: Stanley Thornes (Publishers) Ltd. Meenan RF, Mason JH, Anderson JJ, Guccione AA, Kazis LE (1992). AIMS2: The content and properties of a revised and expanded Arthritis Impact Measurement Scales health questionnaire. Arthritis and Rheumatism 35: Swinscow TDV (1998). Statistics at Square One (9th edn.) London: BMJ Publishing Group. World Health Organization (2000). Obesity: Preventing and Managing the Global Epidemic. WHO Technical Report Series 894. Geneva: WHO. Zigmond AS, Snaith RP (1983). The Hospital Anxiety and Depression Scale. Acta Psychiatrica Scandinavica 67: Address correspondence to Elaine Thomas, Primary Care Sciences Research Centre, Keele University, North Staffordshire, ST5 5BG. Tel: , Fax: , e.thomas@keele.ac.uk Received date January 2004 Accepted date May 2004

Biostatistics. Donna Kritz-Silverstein, Ph.D. Professor Department of Family & Preventive Medicine University of California, San Diego

Biostatistics. Donna Kritz-Silverstein, Ph.D. Professor Department of Family & Preventive Medicine University of California, San Diego Biostatistics Donna Kritz-Silverstein, Ph.D. Professor Department of Family & Preventive Medicine University of California, San Diego (858) 534-1818 dsilverstein@ucsd.edu Introduction Overview of statistical