25.05.2014
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A thermometer with mercury bursting through the glass, and the words Confidence Level, symbolizing a positive attitude. Every science research starts with a detailed analysis of the field of interest, defining the clear aim and hypothesis, by thorough planning of the research design and the way of data collecting and data analysis, as well as the reporting of the results. Let us presume, for example, that we want to know what the average cholesterol concentration in the population is.
In statistics, for any statistical measure, a confidence interval presents a range of possible values within which, with some certainty, we can find the statistical measure of the population.
As such, confidence interval is a realistic estimate of (in)precision and sample size of certain research (3). Only the studies with a large sample will give a very narrow confidence interval, which points to high estimate accuracy with a high confidence level. First we have to determine the confidence level for estimating the mean of a parameter in a population. The lower margin of a confidence interval is calculated by deducting the previous formula result from the mean.
In our sample example, with the mean cholesterol concentration in the population, the confidence interval would be calculated by using the Z value because of the large size of the sample (N=121) that is normally distributed. Due to the fact that today there are a lot of statistical softwares that calculate and provide confidence intervals for the majority of statistical indicators, we shall rarely calculate a confidence interval manually. Confidence interval can be calculated for difference or ratio between any two statistical indicators, so we could examine if this difference or ratio is of any statistical significance.
How do we define the confidence interval when it is related to a ratio as, for example, in OR? To answer this question we select a sample for which we believe that represents the population and in this sample we calculate the mean of cholesterol concentration. By random selection we have chosen one sample of 121 individuals and on this sample we have calculated the mean cholesterol concentration. Therefore, we can consider confidence interval also as a measure of a sample and research quality. Depending on the confidence level that we choose, the interval margins of error also change. In other words: if we select randomly hundred times a sample of 121 individuals and calculate the mean cholesterol concentration and the confidence interval of that estimate in the sample, then in ten out of these hundred samples the confidence interval will not include the actual mean of the population. It should be kept in mind that a confidence interval is accurate only for the samples that follow a normal distribution, whereas it is approximately accurate for large samples that are not distributed normally.
However, it is important to know the input based on which the confidence interval is calculated, so we could better understand its meaning and interpretation.
They describe the same thing, but in two different ways and are complementary to each other. Let us go back to our example of cholesterol concentration in the population to see how the confidence interval can be used for estimating statistical significance of the difference between two means.
Let us remember what the definition of confidence interval was: it defines margins of error within which we can expect the actual value with 95% confidence. Let us imagine that we have assessed the degree of carotid artery stenosis for all our individuals from the sample (N = 121) using echocolour Doppler sonography analysis. In the last twenty years, every day there are more journals that require reporting of the confidence intervals for each of their key results. Therefore, we can consider confidence interval also as a measure of the sample and research quality.
Many journals therefore require providing the key results with respective confidence intervals (4,5).
The most used confidence intervals in the biomedical literature are the 90%, 95%, 99% and not so often 99.9% one.
The larger the confidence interval is, the higher is the possibility that this interval includes also the mean of cholesterol concentration in the population. Although there are some other ways of calculating it, the confidence interval is generally and most frequently calculated using standard error. Most often we decide for the 95% confidence what means that we will allow that only in 5% cases the actual mean of population does not fall into our interval.
For small samples (N < 30) the t value should be used instead of the Z value in the formula for confidence interval, with N1 degrees of freedom (9). The P value describes probability that the observed phenomenon (deviation) occurred by chance, whereas the confidence interval provides margins of error within which it is possible to expect the value of that phenomenon. Do women in our sample have indeed a lower cholesterol concentration then men, or did the observed difference only occur by chance?
Our confidence interval contains also a zero (0) meaning that it is quite possible that the actual value of the difference will equal zero, namely, that there is no difference between the cholesterol concentration between men and women. It means that the odds of cholesterol concentration being and not being a risk factor for carotid artery stenosis are even.
Reporting of this confidence interval provides additional information about the sample and the results. By analysing some sample characteristics we actually want to have an overview of the situation in the population. In this case we use the arithmetic mean as a point estimate of cholesterol concentration in the population. We repeat sampling for 100 times and each time we calculate a certain arithmetical mean of cholesterol concentration. Standard error is a standard deviation of sample means, calculated out of hundred random population samples (8). We are interested whether those two groups are different in average cholesterol concentration and is the cholesterol concentration a risk factor for development of carotid artery stenosis. That is, for any cholesterol concentration the odds of that person having or not having carotid artery stenosis are the same.
It is, moreover, above all useful and irreplaceable supplement to a classical hypothesis testing and to the generally accepted P value.
Depending on the confidence level that we choose, the interval margins of error and respective range also change. The thing we want to know is the answer to the question: can we consider this calculated mean as a good estimate for cholesterol concentration in the population.
For small samples the t value is higher than the Z value what logically means that the confidence interval for smaller samples with the same confidence level is larger.
The P value calculated by the ttest is 0.426 proving that the difference of the cholesterol concentration between men and women is not statistically significant.
This also confirms the percentage of correctly classified individuals considering the cholesterol concentration (50.41%).
It should become a standard of all scientific journals to report key results with respective confidence intervals because it enables better understanding of the data to an interested reader. Therefore, based on what we find out about our sample using the descriptive analysis we make conclusions for the population as a whole (2).
Out of a total of hundred related 95%confidence intervals, 95% of them will contain the actual arithmetical mean of the population (?). There is a rule for same sized samples: the smaller the confidence level is, the higher is the estimate accuracy.
Only half of the individuals are correctly categorised in an adequate group – so the selection does not depend on cholesterol concentration but on a pure chance. Let us now see how the range of confidence interval and its margins of error change depending on the confidence level in our example of cholesterol concentration estimate in the population (Figure 1). Many statistical textbooks contain tables with t values for matching confidence level and different degrees of freedom (1). What is hereby understood is the fact that everything we conclude about the sample is reliably applicable to the whole population. Only the studies with a large sample will give a very small confidence interval, which points to high estimate accuracy with a high confidence level. The P value describes probability that the observed phenomenon (difference) occurred by chance, whereas the confidence interval provides margins of error within which it is possible to expect the value of that phenomenon.
In the last twenty years, increasing number of journals require reporting of the confidence intervals for each of their key results. It is, moreover, very useful and irreplaceable supplement to a classical hypothesis testing and to the generally accepted P value. It should become a standard of all scientific journals to report key results with respective confidence intervals because it enables better understanding to the interested reader.
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