Medical Statistics Made Easy, fourth edition

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Mode 15 You may see reference to a bi-modal distribution. Generally when this is mentioned in papers it is as a concept rather than from calculating the actual values, e.g. The data appear to follow a bi-modal distribution. See Fig. 5 for an example of where there are two peaks to the data, i.e. a bi-modal distribution Number of patients Ages of patients Fig. 5. Graph of ages of patients with asthma in a practice. The arrows point to the modes at ages and Bi-modal data may suggest that two populations are present that are mixed together, so an average is not a suitable measure for the distribution. Montalban X, Hauser SL, Kappos L, Arnold DL, Bar-Or A, Comi G et al. Ocrelizumab vs placebo in primary progressive multiple sclerosis. New England Journal of Medicine 2017;376(3):209-220. Normal distributions can be entirely defined in terms of the mean and standard deviation in the data set.

What does it mean? The P value gives the probability of any observed difference having happened by chance. P = 0.5 means that the probability of the difference Standard Deviation 17 EXAMPLE Let us say that a group of patients enrolling for a trial had a normal distribution for weight. The mean weight of the patients was 80 kg. For this group, the SD was calculated to be 5 kg. 1 SD below the average is 80 5 = 75 kg. 1 SD above the average is = 85 kg. ±1 SD will include 68.2% of the subjects, so 68.2% of patients will weigh between 75 and 85 kg. 95.4% will weigh between 70 and 90 kg (±2 SD). 99.7% of patients will weigh between 65 and 95 kg (±3 SD). See how this relates to the graph of the data in Fig Number of patients ±1 SD (68.2%) 2 ±2 SD (95.4%) ±3 SD (99.7%) Weight (kg) Fig. 6. Graph showing normal distribution of weights of patients enrolling in a trial with mean 80 kg, SD 5 kg.

Let’s say we are trying to establish whether there is a link between smoking and lung cancer. Our sample can be split into four groups: exposed with cancer, exposed without cancer, unexposed with cancer and unexposed without cancer (table 5). Attrition bias: when those patients who are lost to follow-up differ in a systematic way to those who did return for assessment or clinic. Medical Statistics Made Easy EXAMPLE Data were collected on 80 patients referred for heart transplantation. The researcher wanted to compare their ages. The data for age were put in decade bands and are shown in Table 1. Table 1. Ages of 80 patients referred for heart transplantation Years a Frequency b Percentage c Total a Years = decade bands; b Frequency = number of patients referred; c Percentage = percentage of patients in each decade band. For example, in the age band there were 14 patients and we know the ages of 80 patients, 14 so 100 = 17.5%. 80 Watch out for... Authors can use percentages to hide the true size of the data. To say that 50% of a sample has a certain condition when there are only four people in the sample is clearly not providing the same level of information as 50% of a sample based on 400 people. So, percentages should be used as an additional help for the reader rather than replacing the actual data.

P VALUES How important is it? A really important concept, P values are given in more than four out of five papers. How easy is it to understand? LLL Not easy, but worth persevering as it is used so frequently. It is not important to know how the P value is derived just to be able to interpret the result. When is it used? The P (probability) value is used when we wish to see how likely it is that a hypothesis is true. The hypothesis is usually that there is no difference between two treatments, known as the null hypothesis. What does it mean? The P value gives the probability of any observed difference having happened by chance. P = 0.5 means that the probability of the difference having happened by chance is 0.5 in 1, or 50:50. P = 0.05 means that the probability of the difference having happened by chance is 0.05 in 1, i.e. 1 in 20. Risk ratios (or relative risks) are one way of comparing two proportions. They are calculated with the following equation: Medical Statistics Made Easy those studies by showing the mean changes and 95% CIs in a chart. An example is given in Fig. 8. Study A Study B Study C Study D Study E Combined estimate Change in BP (mmhg) Fig. 8. Plot of 5 studies of a new antihypertensive drug. See how the results of studies A and B above are shown by the top two lines, i.e. 20 mmhg, 95% CI for study A and 20 mmhg, 95% CI -5 to +45 for study B. The vertical axis does not have a scale. It is simply used to show the zero point on each CI line. The statistician has combined the results of all five studies and calculated that the overall mean reduction in BP is 14 mmhg, CI This is shown by the combined estimate diamond. See how combining a number of studies reduces the CI, giving a more accurate estimate of the true treatment effect. The chart shown in Fig. 8 is called a Forest plot or, more colloquially, a blobbogram. Standard deviation and confidence intervals what is the difference? Standard deviation tells us about the variability (spread) in a sample. The CI tells us the range in which the true value (the mean if the sample were infinitely large) is likely to be. Biases are systematic differences between the data that has been collected and the reality in the population. There are numerous types of bias to be aware of, some of which are listed below:

Medical Statistics Made Easy RRR is the proportion by which the intervention reduces the event rate. EXAMPLES One hundred women with vaginal candida were given an oral antifungal, 100 were given placebo. They were reviewed 3 days later. The results are given in Table 4. Table 4. Results of placebo-controlled trial of oral antifungal agent Given antifungal Given placebo Improved No improvement Improved No improvement ARR = improvement rate in the intervention group improvement rate in the control group = 80% 60% = 20% NNT = = =5 ARR 20 So five women have to be treated for one to get benefit. The incidence of candidiasis was reduced from 40% with placebo to 20% with treatment, i.e. by half. Thus, the RRR is 50%. In another trial young men were treated with an expensive lipid-lowering agent. Five years later the death rate from ischaemic heart disease (IHD) is recorded. See Table 5 for the results. Table 5. Results of placebo-controlled trial of Cleverstatin Given Cleverstatin Given placebo Survived Died Survived Died 998 (99.8%) 2 (0.2%) 996 (99.6%) 4 (0.4%) ARR = improvement rate in the intervention group improvement rate in the control group = 99.8% 99.6% = 0.2% P Values 25 It is the figure frequently quoted as being statistically significant, i.e. unlikely to have happened by chance and therefore important. However, this is an arbitrary figure. If we look at 20 studies, even if none of the treatments work, one of the studies is likely to have a P value of 0.05 and so appear significant! The lower the P value, the less likely it is that the difference happened by chance and so the higher the significance of the finding. P = 0.01 is often considered to be highly significant. It means that the difference will only have happened by chance 1 in 100 times. This is unlikely, but still possible. P = means the difference will have happened by chance 1 in 1000 times, even less likely, but still just possible. It is usually considered to be very highly significant. A difficult concept, but one where a small amount of understanding will get you by without having to worry about the details. Let’s now apply the information in table 6 to a theoretical population of 5,000 stroke patients. We can see that without the drug we could expect 2,500 of these patients to die (), but with the new treatment, this would be reduced to 2,000 (). In other words, for every 10 patients who are treated with the new drug, there is 1 patient whose life will be saved who would otherwise have died. In a population of 5,000, this means 500 lives will be saved (Table 6). An exam question may give a chart similar to that in Fig. 8 and ask you to summarize the findings. Consider: • Which study showed the greatest change? • Did all the studies show change in favour of the intervention? • Were the changes statistically significant? In the example above, study D showed the greatest change, with a mean BP drop of 25 mmHg. Study C resulted in a mean increase in BP, though with a wide CI. The wide CI could be due to a low number of patients in the study. The combined estimate of a mean BP reduction of 14 mmHg, 95% CI 12–16, is statistically significant.