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Health Sciences M.Sc. Programme
Applied Biostatistics
Week 5: Standard Error and Confidence Intervals
Sampling
Most research data come from subjects we think of as samples drawn from a larger population. The sample tells us something about the population. The notion of sampling is a familiar one in health care. For example, if I want to measure a subject’s blood glucose, I do not take all the blood. I draw a sample. One drop of blood is then used to represent all the blood in the body. I did this three times, from the same subject (myself) and got three measurements: 6.0, 5.9, and 5.8 mmol/L. Which of these was correct? The answer is that none of them were; they were all estimates of the same quantity. We do not know which of them was actually closest.
In research, we collect data on our research subjects so we can draw conclusions about some larger population. For example, in a randomised controlled trial comparing two obstetric regimes, the proportion of women in the active management of labour group who had a Caesarean section was 0.97 times the proportion of women in the routine management group who had sections (Sadler et al ., 2000). (We call this ratio the relative risk.) This trial was carried out in one obstetric unit in New Zealand, but we are not specifically interested in this unit or in these patients. We are interested in what they can tell us about what would happen if we treated future patients with active management of labour rather than routine management. We want know, not the relative risk for these particular women, but the relative risk for all women.
The trial subjects form a sample, which we use to draw some conclusions about the population of such patients in other clinical centres, in New Zealand and other countries, now and in the future. The observed relative risk of Caesarean section, 0.97, provides an estimate of the relative risk we would expect to see in this wider population. If we were to repeat the trial, we would not get exactly the same point estimate. Other similar trials cited by Sadler et al. (2000) have reported different relative risks: 0.75, 1.01, and 0.64. Each of these trials represents a different sample of patients and clinicians and there is bound to be some variation between samples. Hence we cannot conclude that the relative risk in the population will be the same as that found in our particular trial sample. The relative risk which we get in any particular sample would be compatible with a range of possible differences in the population.
When we draw a sample from a population, it is just one of the many samples we could take. If we calculate a statistic from the sample, such as a mean or proportion, this will vary from sample to sample. The means or proportions from all the possible samples form the sampling distribution. To illustrate this with a simple example, we could put lots numbered 1 to 9 into a hat and sample by drawing one out, replacing it, drawing another out, and so on. Each number would have the same chance of being chosen each time and the sampling distribution would be as in Figure 1(a). Now we change the procedure, draw out two lots at a time and calculate the average. There are 36 possible pairs, and some pairs will have the same average (e.g. 1 and 9, 4 and 6 both have average 5.0). The sampling distribution of this average is shown in Figure 1(b).
Relative frequency
Digits 1 to 9
(a) Single digit
Relative frequency
Mean of two digits
(b) Mean of two digits
Figure 1. Sampling distribution for a single digit drawn at random and for the mean of two digits drawn together
There are three things which we should notice about Figure 1(b).
- The mean of the distribution remains the same, 5.
- The sampling distribution of the mean is not so widely spread as the parent distribution. It has a smaller variance and standard deviation.
- It has a different shape to Figure 1(a). The sampling distribution of a statistic does not necessarily have the same shape as the distribution of the observations themselves, which we call the parent distribution. In this case, as so often, it looks closer to a Normal distribution than does the distribution of the observations themselves.
If we know the sampling distribution it can help us draw conclusions about the population from the sample, using confidence intervals and significance tests. We often use our sample statistic as an estimate of the corresponding value in population, for example using the sample mean to estimate the population mean. The sampling distribution tells us how far from the population value the sample statistic is likely to be. Any statistic which is calculated from a sample, such as a mean, proportion, median, or standard deviation, will have a sampling distribution.
Standard error
If the sample statistic is used as an estimate, we call the standard deviation of the sampling distribution the standard error. Rather confusingly, we use this term both for the unknown standard deviation of the sampling distribution and for the estimate of this standard deviation found from the data.
Figure 2. Simulation of 20 samples of 100 observations from a Standard Normal distribution, mean = 0 and SD =1.0, SE = 0.10.
Random variable
Sample number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-.
0
.
In the FEV1 example, we have a fairly large sample, and so we can assume that the observed mean is from a Normal Distribution. For this illustration we shall also assume that the standard error is a good estimate of the standard deviation of this Normal distribution. (We shall return to this in Week 5.) We therefore expect about 95% of such means to be within 1.96 standard errors of the population mean. Hence, for about 95% of all possible samples, the population mean must be greater than the sample mean minus 1.96 standard errors and less than the sample mean plus 1. standard errors.
If we calculated mean minus 1.96 standard errors and mean plus 1.96 standard errors for all possible samples, 95% of such intervals would contain the population mean. In this case these limits are 4.062 – 1.96 × 0.089 to 4.062 + 1.96 × 0.089 which gives 3.89 to 4.24, or 3.9 to 4.2 litres, rounding to 2 significant figures. 3.9 and 4.2 are called the 95% confidence limits for the estimate, and the set of values between 3. and 4.2 is called the 95% confidence interval. The confidence limits are the ends of the confidence interval.
Strictly speaking, it is incorrect to say that there is a probability of 0.95 that the population mean lies between 3.9 and 4.2, though sloppy thinkers often put it that way. The population mean is a number, not a random variable, and has no probability. We sometimes say that we are 95% confident that the mean lies between these limits, but this doesn’t help us understand what a confidence interval is. The important thing is: we use a sample to estimate something about a population. The 95% confidence interval is chosen so that 95% of such intervals will include the population value.
Confidence intervals do not always include the population value. If 95% of 95% confidence intervals include it, it follows that 5% must exclude it. In practice, we cannot tell whether our confidence interval is one of the 95% or the 5%.
Figure 2 shows confidence intervals for the mean for 20 random samples of 100 observations from the Standard Normal Distribution. The population mean is, of course, 0.0, shown by the horizontal line. Some sample means are close to 0.0, some further away, some above and some below. The population mean is contained by 19
of the 20 confidence intervals. Thus, for 95% of confidence intervals, it will be true to say that the population value lies within the interval. We just don't know which 95%.
We expect to see 5% of the intervals having the population value outside the interval and 95% having the population value inside the interval. This is not the same as saying that 95% of further samples will have estimates within the interval. For example, if we look at the first interval in Figure 2, we can see that samples 8, 10, and 20 all have point estimates outside this interval. In fact, we expect about 83% of further samples to have their point estimates within a 95% confidence interval chosen at random.
The confidence interval need not have a probability of 95%. For example, we can also calculate 99% confidence limits. The upper 0.5% point of the Standard Normal Distribution is 2.58, so the probability of a Standard Normal deviate being above 2. or below –2.58 is 1% and the probability of being within these limits is 99%. The 99% confidence limits for the mean FEV1 are therefore 4.062 – 2.58 × 0.089 and 4.062 + 2.58 × 0.089, i.e. 3.8 and 4.3 litres. These give a wider interval than the 95% limits, as we would expect since we are more confident that the mean will be included. We could also calculate a 90% confidence interval, which is 3.916 to 4.208, narrower than the 95% confidence interval. However, only 90% of such intervals will include the population value, 10% will not.
The probability we choose for a confidence interval is thus a compromise between the desire to include the estimated population value and the desire to avoid parts of scale where there is a low probability that the mean will be found. For most purposes, 95% confidence intervals have been found to be satisfactory and this is what is usually quoted in health research.
For the trial comparing active management of labour with routine management (Sadler et al ., 2000), the relative risk for Caesarean section was 0.97. Sadler et al. quoted the 95% confidence interval for the relative risk as 0.60 to 1.56. Hence we estimate that in the population which these subjects represent, the proportion of women undergoing Caesarean section when undergoing active management of labour is between 0.60 and 1.56 times the proportion who would have Caesarean section with routine management.
Significance tests and confidence intervals
Significance tests and confidence intervals often involve similar calculations. For example, we can test the null hypothesis that two groups have the same mean and we can find a confidence interval for the difference between the means. If the 95% confidence interval for the difference does not include the null hypothesis value, the difference is significant at the 5% level. If the 95% confidence interval for the difference includes the null hypothesis value, the difference is not significant at the 5% level.
For example, in a study of respiratory disease in schoolchildren, children were followed at ages 5 and 14. We looked at the proportions of children with bronchitis in infancy and with no such history who were reported to have respiratory symptoms in later life (Holland et al. , 1978). We had 273 children with a history of bronchitis before age 5 years, 26 of whom were reported to have day or night cough at age 14. We had 1046 children with no bronchitis before age 5 years, 44 of whom were reported to have day or night cough at age 14. We shall test the null hypothesis that
difference. As a result, 95% confidence intervals and 5% significance tests sometimes give different answers near the cut-off point.
J. M. Bland 14 August 2006
References
Bland M. (2000) An Introduction to Medical Statistics, 3rd. Edition Oxford University Press.
Sadler LC, Davison T, McCowan LM. (2000) A randomised controlled trial and meta-analysis of active management of labour. British Journal of Obstetrics and Gynaecology 107 , 909-15.
