Confidence Interval

(To our junior colleagues and students)

Confidence interval is a statistical term frequently encountered in papers describing various kinds of medical research. It is one of the terms that postgraduate medical students need to know.

The following examples serve to explain it.

If you want to find the mean weight of a group of 30 men, you measure the weight of each one, add the figures and divide by 30. The result is the mean weight of the group.

If you want to find the mean body weight of men residents of a big city, it is not practical to do the same because of the large number involved. Statisticians get around this by taking a random sample of the men in question. They calculate the mean weight of the men in the sample and consider it a satisfactory representation of the required mean of the total. They may take e.g. a thousand men, chosen randomly from various districts of the city, measure the weight of each and then add and divide to find the mean. They consider this representative of the mean body weight of the men of that city. Now imagine yourself to be the person who requested that mean because you wanted to assess the nutrition status of the people in the city and you asked a statistician to do it for you. Imagine also that the mean weight of the thousand men was 60 kg. You may then have the following dialogue with the statistician:

·         Are you sure the figure you gave me is exactly the same as the figure you would have obtained had you taken all the men in the city?

·         No, most probably it is not, but it is very near that figure and is sufficient for your purpose.

·         How near is it? How much is the difference?

·       I cannot tell you the exact difference because I do not know the true figure of all the men in the city. We statisticians usually deal with probabilities. I can tell you the probability of the difference being of a certain magnitude. I can work out from the data of the thousand men a figure we call the Standard Error (SE). In fact I have already done that and found the standard error of the mean of the sample to be 2. We know from statistics laws and rules that the probability of the difference being not more than one standard error (1SE) is approximately 67% and being not more than 2SE approximately 95%. In other words I can tell you that I am practically 95% confident that the true figure of the mean weight of all men in the city is within 2SE above and below the figure of 60 i.e. between 56 and 64 kg. That is what I mean when I say the mean body weight of the sample of men is 60 kg. and its 95% Confidence Interval (CI) is 56 - 64.

The 95% probability (or confidence) becomes approximately 67% if you choose 1SE above and below the mean as the limits of your confidence interval and approximately 99% if you choose 2.5 SE. The 95% (i.e. mean ± 2SE) is commonly used and if the percentage is not written it usually means 95%.

I used the mean as an example to explain the confidence interval. The same applies to other parameters like proportions when samples are used instead of the total.

The use of ‘normal’ values

To judge a finding in a patient, be it body weight or height, a blood test like white cell count or serum potassium etc. we usually compare it with so called ‘normal’ values (better called reference values or range). These values are obtained by studying the frequency distribution of the finding (variable) in question in a sample of healthy people and determining its mean and standard deviation (SD). The normal range is taken as the range between + 2 standard deviations and -2 standard deviations.

(Normal frequency distribution curve with number of standard deviations on the horizontal axis)

We know from statistics that this will include 95% of people which means that 5% of healthy people have values outside the normal range. We have to remember then that if a patient’s measurement is slightly outside the normal range it still can be normal. Conversely if it is within the normal range it still can be abnormal for that particular patient because it is significantly different from his usual figure. For example a patient whose white cells count is 4000/c.mm. in normal circumstances the number may rise to 8000/c.mm. when he develops an infection. It is still in the normal range though it is raised compared to his count during health.

Like every thing else in medical practice data have not to be taken blindly and in isolation but have to be interpreted with caution taking all other findings to form an overall picture of the situation.

Relative risk and absolute risk

Statistics can be deceiving if one is not careful. We learn of things that double our risk of developing this or that disease or increase it by say 25 or 50%. We also learn of things that decrease our risk of developing a disease by a certain percentage. It may sound very important to have a risk doubled or reduced to half but unless we know how much it originally was, can we really tell how much the change is? If you get a job and your boss tells you that your salary will be double the salary of Mohammed you immediately ask: How much is the salary of Mohammed? We do not do the same when we learn about doubling or halving the risk of a certain condition.  We do not bother to know how much the original risk was. Unless we translate a relative risk (i.e. a risk expressed as a ratio of another risk) into an absolute risk (i.e. the chance of developing an event regardless of how it compares with another risk) we cannot judge its magnitude and importance. For example, if you know that smoking increases your risk of developing a cardiovascular event in the next 10 years by say 50% and your absolute risk is already 20%, the increase is 10% which is important and worth avoiding smoking (leaving aside other harms of smoking). On the other hand if someone tells you that using your mobile phone increases your chance of developing an acoustic nerve tumour by 100% and your statistical chance of developing this disease (i.e. your absolute risk) is 1/100,000 then using the mobile phone will increase it to 2/100,000 i.e. an increase of 1/100,000. Most people will consider this increase, even if it is true, insufficient to make them stop using mobile phones. I can mention other examples of things that decrease the risk and the same thing applies.

We should always remember that it is the absolute risk that counts.