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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