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? Our New BMJ website does not support IE6 please upgrade your browser to the latest version or use alternative browsers suggested below.

Previously we have considered how to test the null hypothesis that there is no difference between the mean of a sample and the population mean, and no difference between the means of two samples. With small samples, where more chance variation must be allowed for, these ratios are not entirely accurate because the uncertainty in estimating the standard error has been ignored. The application of the t distribution to the following four types of problem will now be considered. The mean and standard deviation of a sample are calculated and a value is postulated for the mean of the population. In each case the problem is essentially the same - namely, to establish multiples of standard errors to which probabilities can be attached. A rare congenital disease, Everley's syndrome, generally causes a reduction in concentration of blood sodium.

To find the 95% confidence interval above and below the mean we now have to find a multiple of the standard error.

Since the size of the sample influences the value of t, the size of the sample is taken into account in relating the value of t to probabilities in the table. Here we apply a modified procedure for finding the standard error of the difference between two means and testing the size of the difference by this standard error (see Chapter 5 for large samples). The addition of bran to the diet has been reported to benefit patients with diverticulosis. The table of the tdistribution Table B (appendix) which gives two sided P values is entered at degrees of freedom. If the standard deviations in the two groups are markedly different, for example if the ratio of the larger to the smaller is greater than two, then one of the assumptions of the ttest (that the two samples come from populations with the same standard deviation) is unlikely to hold.

Although this may look very complicated, it can be evaluated very easily on a calculator without having to write down intermediate steps (see below). When the effects of two alternative treatments or experiments are compared, for example in cross over trials, randomised trials in which randomisation is between matched pairs, or matched case control studies (see Chapter 13 ), it is sometimes possible to make comparisons in pairs.

Let us use as an example the studies of bran in the treatment of diverticulosis discussed earlier. In calculating t on the paired observations we work with the difference, d, between the members of each pair. This is quite wide, so we cannot really conclude that the two preparations are equivalent, and should look to a larger study. The second case of a paired comparison to consider is when two samples are chosen and each member of sample 1 is paired with one member of sample 2, as in a matched case control study. It would seem logical that, because the t test assumes Normality, one should test for Normality first. Pairing provides information about an experiment, and the more information that can be provided in the analysis the more sensitive the test. What is the difference between the mean levels in the two wards, and what is its significance? 7.4 A new treatment for varicose ulcer is compared with a standard treatment on ten matched pairs of patients, where treatment between pairs is decided using random numbers.

What are the mean difference in the healing time, the value of t, the number of degrees of freedom, and the probability? Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. I'm doing a research and trying to calculate and put error bars (95% confidence) on my excel chart. I then multiplied SE with 1.96 to get he values, however, the lower limit goes to the negative.

My initial thinking was that I should calculate standard errors for each groups separately, which would give different error bars. Discouraged from giving A's, but students are doing excellent work, what can a professor do? Find the probability of getting one diamond and one spade in a five-card hand, using binomial coefficients.

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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 echo-colour 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. We obtained the difference between the means by subtraction, and then divided this difference by the standard error of the difference. Some modification of the procedure of dividing the difference by its standard error is needed, and the technique to use is the t test. What is the significance of the difference between the means of the two sets of observations? This is thought to provide a useful diagnostic sign as well as a clue to the efficacy of treatment.

Assuming that blood sodium concentration is Normally distributed what is the 95% confidence interval within which the mean of the total population of such cases may be expected to lie?

For large samples we used the standard deviation of each sample, computed separately, to calculate the standard error of the difference between the means. Several different bran preparations are available, and a clinician wants to test the efficacy of two of them on patients, since favourable claims have been made for each. By random allocation the clinician selects two groups of patients aged 40-64 with diverticulosis of comparable severity.

An approximate test, due to Sattherwaite, and described by Armitage and Berry, (1)which allows for unequal standard deviations, is as follows.

It can produce a degree of freedom which is not an integer, and so not available in the tables. Whether treatment A or treatment B is given first or second to each member of the sample should be determined by the use of the table of random numbers Table F (Appendix).

The clinician wonders whether transit time would be shorter if bran is given in the same dosage in three meals during the day (treatment A) or in one meal (treatment B). As the aim is to test the difference, if any, between two types of treatment, the choice of members for each pair is designed to make them as alike as possible. It is not valid to compare each treatment with each other treatment using t tests because the overall type I error rate will be bigger than the conventional level set for each individual test. What is the 95% confidence interval within which the mean of the population of such cases whose specimens come to the same laboratory may be expected to lie? However, I have trouble understanding what exactly do I have to calculate in order to present my data correctly.

Should I use only the control and treatment groups and calculate the interval between them? 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 N-1 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.

If the difference is 196 times its standard error, or more, it is likely to occur by chance with a frequency of only 1 in 20, or less. Its foundations were laid by WS Gosset, writing under the pseudonym "Student" so that it is sometimes known as Student's t test. Little is known about the subject, but the director of a dermatological department in a London teaching hospital is known to be interested in the disease and has seen more cases than anyone else. Whether it should be regarded clinically as abnormally high is something that needs to be considered separately by the physician in charge of that case. Among the consequences of administering bran that requires testing is the transit time through the alimentary canal.

Sample 1 contains 15 patients who are given treatment A, and sample 2 contains 12 patients who are given treatment B. The null hypothesis that there is no difference between the means is therefore somewhat unlikely. In this way any effect of one treatment on the other, even indirectly through the patient's attitude to treatment, for instance, can be minimised.

A random sample of patients with disease of comparable severity and aged 20-44 is chosen and the two treatments administered on two successive occasions, the order of the treatments also being determined from the table of random numbers. The null hypothesis is that there is no difference between the mean transit times on these two forms of treatment. The more alike they are, the more apparent will be any differences due to treatment, because they will not be confused with differences in the results caused by disparities between members of the pair. With a small sample a non-significant result does not mean that the data come from a Normal distribution. A rule of thumb is that if the ratio of the larger to smaller standard deviation is greater than two, then the unequal variance test should be used.

By repeating measures within subjects, each subject acts as its own control, and the between subjects variability is removed.

One doctor is responsible for treatment and a second doctor assesses healing without knowing which treatment each patient had. 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. The procedure does not differ greatly from the one used for large samples, but is preferable when the number of observations is less than 60, and certainly when they amount to 30 or less.

With small samples these multiples are larger, and the smaller the sample the larger they become.

The transit times of food through the gut are measured by a standard technique with marked pellets and the results are recorded, in order of increasing time, in Table 7.1 .

Since it is possible for the difference in mean transit times for A-B to be positive or negative, we will employ a two sided test.

Many statistical packages now carry out this test as the default, and to get the equal variances I statistic one has to specifically ask for it. Occasionally it is possible to give both treatments simultaneously, as in the treatment of a skin disease by applying a remedy to the skin on opposite sides of the body. On the other hand, with a large sample, a significant result does not mean that we could not use the t test, because the t test is robust to moderate departures from Normality - that is, the P value obtained can be validly interpreted.

With a computer one can easily do both the equal and unequal variance t test and see if the answers differ.

In general this means that if there is a true difference between the pairs the paired test is more likely to pick it up: it is more powerful. What I wanted to estimate is to see how many (%) people are affected in certain age groups and how groups 1 and 2 compare to each other. 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 t-test 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. Conversely, as the sample becomes larger t becomes smaller and approaches the values given in table A, reaching them for infinitely large samples.

The unequal variance t test tends to be less powerful than the usual t test if the variances are in fact the same, since it uses fewer assumptions. For instance, in a test for a drug reducing blood pressure the colour of the patients' eyes would probably be irrelevant, but their resting diastolic blood pressure could well provide a basis for selecting the pairs. There is something illogical about using one significance test conditional on the results of another significance test. 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.

In practice the degrees of freedom amount in these circumstances to one less than the number of observations in the sample. However, it should not be used indiscriminantly because, if the standard deviations are different, how can we interpret a nonsignificant difference in means, for example? Another (perhaps related) basis is the prognosis for the disease in patients: in general, patients with a similar prognosis are best paired. 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).

Often a better strategy is to try a data transformation, such as taking logarithms as described in Chapter 2.

Whatever criteria are chosen, it is essential that the pairs are constructed before the treatment is given, for the pairing must be uninfluenced by knowledge of the effects of treatment.

One can "eyeball" the data and if the distributions are not extremely skewed, and particularly if (for the two sample t test) the numbers of observations are similar in the two groups, then the t test will be valid. What is hereby understood is the fact that everything we conclude about the sample is reliably applicable to the whole population. This is because only 17 observations plus the total number of observations are needed to specify the sample, the 18th being determined by subtraction. Transformations that render distributions closer to Normality often also make the standard deviations similar. The main problem is often that outliers will inflate the standard deviations and render the test less sensitive.

Also, it is not generally appreciated that if the data originate from a randomised controlled trial, then the process of randomisation will ensure the validity of the I test, irrespective of the original distribution of the data. 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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