Principles of Biostatistics for Medicine

Confidence IntervalsObjectives:Students should know how to calculate a standard error, given a sample mean, standard deviation, and sample sizeStudents should know what a confidence interval is, and its purposeStudents should be able to construct and interpret 90%, 95%, and 99% confidence intervals, and know how they compare to each otherStudents should know how changes in variability and/or changes in sample size affect the width of a confidence interval

The first learning objective for this module is that students should know how to calculate a standard error, given a sample mean, standard deviation, and sample size. Second, students should know what a confidence interval is, and its purpose. Third, students should be able to construct and interpret 90%, 95%, and 99% confidence intervals, and know how they compare to each other. Finally, students should know how changes in variability and/or changes in sample size affect the width of a confidence interval.

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Confidence IntervalsHow precisely does the sample statistic estimate the population parameter?

To illustrate the calculation and interpretation of confidence intervals well use the HR data from our previous sample of 84 adults:

The sample mean HR was 74.0 bpmThe sample standard deviation was 7.5 bpm

Lets walk through an example demonstrating how the sample statistic can be used to estimate the population parameter. Well use our same heart rate data from the 84 adults. If you remember from previous examples, the sample mean heart rate for this group was 74 beats per minute and the standard deviation for the sample was 7.5 beats per minute.

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Calculating Confidence IntervalsWhat are the 90, 95, and 99% confidence intervals for our sample estimate of the true population heart rate?

90% CI: 74.0 1.645(0.8) = 74.0 1.3 = 72.7, 75.395% CI: 74.0 1.960(0.8) = 74.0 1.6 = 72.4, 75.699% CI: 74.0 2.575(0.8) = 74.0 2.1 = 71.9, 76.1

We want to calculate the 90, 95, and 99% confidence intervals for our sample estimate of the true population heart rate. First, we need to use our data to determine the standard error. We do this by dividing the sample standard deviation by the square root of the sample size. So, 7.5 divided by the square root of 84 is 0.8. To determine our 90% confidence interval, we perform the calculation 74 +/- 1.645 times 0.8. Performing this calculation gives us a confidence interval of 72.7 through 75.3. Lets do the same for a 95% confidence interval. In this case, our calculation is 74 +/- 1.960 times 0.8, which gives us an interval of 72.4 to 75.6. The calculation for a 99% confidence interval is 74 +/- 2.575 times 0.8, which gives us an interval of 71.9 to 76.1. The general formula for all confidence intervals is: confidence interval equals the point estimate +/- the confidence level multiplier times the standard error.

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Confidence Interval Example90% CI: 74.0 1.645(0.8) = 74.0 1.3 = 72.7, 75.3We can be 90% confident that the true population heart rateis between 72.7 and 75.3 bpm95% CI: 74.0 1.960(0.8) = 74.0 1.6 = 72.4, 75.6We can be 95% confident that the true population heart rateis between 72.4 and 75.6 bpm99% CI: 74.0 2.575(0.8) = 74.0 2.1 = 71.9, 76.1We can be 90% confident that the true population heart rateis between 71.9 and 76.1 bpmAs we increase the level of confidence, the interval widens because the larger the range between the lower and upper bounds, the more confident we can be that the interval contains the true mean.

So we can be 90% confident that the true population heart rate is somewhere in the interval of 72.7 and 75.3 beats per minute. We can be 95% confident that our true population heart rate is between 72.4 and 75.6 beats per minute. And, we can be 99% confident that the true population heart rate is somewhere in the interval of 71.9 to 76.1 beats per minute. Youll notice that as we increase our level of confidence, the interval widens because the larger the range between the lower and upper bounds, the more confident we can be that the interval contains the true population mean heart rate.

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Factors Affecting Width of a Confidence IntervalFactors that determine the size of a standard error and therefore the width of a confidence interval:

The amount of variability in the sample:The greater the variability (for sample means, the larger the standard deviation), the wider the confidence interval.

As sd increases, SE increases:When the variability among subjects in a sample is large, the variability among the means of repeated samples will also be large

You should be aware that there are some factors that will influence the size of the standard error and therefore the width of the confidence interval. First, the more variability that exists in the sample observations, the larger the standard deviation and therefore, the wider the confidence interval. The relationship between standard deviation and standard error dictates that as standard deviation increases, standard error increases as well. When the variability among subjects in a sample is large, the variability among the means of repeated samples will also be large.

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Factors Affecting Width of a Confidence IntervalFactors that determine the size of a standard error and therefore the width of a confidence interval:

The sample size:The greater the sample size, the more narrow the confidence interval.

As sample size increases, SE decreases: As more individuals in the population are sampled, the estimate of the population parameter will become more precise

A second factor that greatly influences the size of the standard error and the width of the confidence interval is the sample size. The greater the sample size, the smaller the standard error and the narrower the confidence interval will be. Increasing sample size leads to more precision in the population parameter.

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Sample Size and Confidence IntervalsFor the HR example:In our sample of 84 adults, we can be 95% confident that the that the true population heart rate is between 72.4 and 75.6 bpmIf we increase our sample size to 500 adults, we can be 95% confident that the true population heart rate is between 73.34 and 74.657 bpm

95% CI: 74.0 1.960(0.335) = 74.0 0.657 = 73.34, 74.657

For our heart rate data set, a sample size of 84 adults produces a 95% confidence interval between 72.4 and 75.6 beats per minute. If we increase the sample size to 500 adults, we can be 95% confident that the true population heart rate is between 73.34 and 74.657 beats per minute. The larger sample size decreases the width of the 95% confidence interval, and the estimation of the true heart rate in the population becomes more precise. 7

Confidence vs PrecisionThe level of confidence reflects the uncertainty/variability inherent in sampling. More variability in the sample means that the standard deviation will be larger and therefore, the interval will be wider. This is not the same as increasing or decreasing your level of confidence.More variability in the sample means less accuracy that the sample statistic will accurately reflect the true population parameter (less precision)The wider the confidence interval, the lower the precision.Larger sample size = lower standard error = narrower CI (less variability)

Understanding the relationship between confidence and accuracy can be very tricky. Remember, he level of confidence reflects the uncertainty or variability inherent in sampling. Since we usually only take one sample from a population, we want that sample to reflect the true value of the population parameter, as much as possible. When there is a lot of variability in the sample, the standard deviation for the sample is greater and therefore, the interval is wider. Increasing sample size will produce a narrower interval at the same level of confidence. This should not be confused with increasing or decreasing your level of confidence, say from a 90% to 95% confidence level, or from a 99% to 95% confidence.

When there is more variability in the sample, it means there is less accuracy that the sample statistic will accurately reflect the true population parameter. In other words, we have less precision.

Increasing the sample size lowers the standard error and produces a narrower, or less variable, confidence interval.8

Confidence Intervals in Medical ResearchConfidence intervals (usually 95%) around sample means are commonly reported in published medical research.Other sample statistics that are commonly reported with confidence intervals include:difference between 2 meansproportionsdifferences between 2 proportionscorrelationsrelative risksodds ratios

Published medial research usually reports 95% confidence intervals in the statistical results. Other sample statistics that are commonly reported with confidence intervals include difference between 2 means, proportions, differences between 2 proportions, correlations, relative risks, and odds ratios.

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