Methods for Two Categorical Variables – Confidence Intervals
Let’s again consider the example regarding congenital heart defects and karyotype.
Karyotype
Down syndrome
Patau syndrome
Total
Congenital Heart Defect
Present
Absent
Total
24
36
60
20
5
25
44
41
85
Questions:
1. Find the proportion of patients with Down syndrome that have a congenital heart defect.
2. Find the proportion of patients with Patau syndrome that have a congenital heart defect.
Note that we could compare these proportions using the relative risk.
Instead of using the relative risk to quantify the difference between two karyotypes, we could consider
the difference in proportions (sometimes called the risk difference).
3. Find the difference in the proportions.
4. Interpret this quantity.
When discussing the results of this study, we want to make conclusions that allow us to compare the
risk of congenital heart defect in the ____________________ of all persons with Down syndrome versus
all persons with Patau syndrome. The difference we just calculated describes the size of the difference
in the proportions obtained in the _______________. To generalize these results to the population, we
must calculate a confidence interval for the risk difference.
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A general formula for a confidence interval is shown below.
Basic Form of Most Confidence Intervals
Estimate ± Quantile(Standard Deviation of Estimate)
Margin of Error
Example: To find the confidence interval of interest in the karyotype example, we must first calculate
the following quantities:


An estimate of the difference between the two proportions:
o
pˆ Down =
o
pˆ Patau =
o
pˆ Down  pˆ Patau =
An approximate quantile:
o
These quantiles will again come from the standard normal distribution.
Confidence Level
z Quantile

90%
95%
99%
The standard deviation of the estimate (this is known as the standard error). The standard
deviation for the difference between proportions can be computed as follows:
pˆ Down 1 - pˆ Down  pˆ Patau 1 - pˆ Patau 
+
=
n Down
n Patau
Finally, find the 95% confidence interval for the difference in proportions.
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Note that we can also find this confidence interval in JMP; however, JMP is using a different method
than the one describe above, so the results may differ slightly. Click on the red drop-down arrow next to
Contingency Analysis… and choose Two Sample Test for Proportions.
Guidelines for Interpreting a Confidence Interval for a Difference in Proportions:
1. Check whether ________ falls in the interval or not. If it does, it is _________________
(but not proof!) that the two proportions are equal.
2. If all values in a 95% confidence interval for p1 – p2 are positive, then you can infer that
p1 ____ p2. The interval tells you how much bigger (with 05% certainty) p1 is. Similarly,
if all values in a confidence interval for p1 – p2 are negative, then you can infer that
p1_____ p2. The interval tells you how much bigger p2 is.
Questions:
5. Interpret the confidence interval for this example.
6. Does this confidence interval agree with the results of the hypothesis test? Explain.
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Example: In a study of obesity, a random sample of 150 males was taken and 21 were found to be
obese, while 48 of the 200 females were found to obese. The data have been summarized in the
contingency table below.
Obese?
Gender
Male
Female
Total
No
129
152
281
Yes
21
48
69
Total
150
200
350
Research Question – Is there evidence that the proportion of males who are obese less than the
proportion of females who are obese?
Questions:
7. Find the proportion of females who are obese.
8. Find the proportion of males who are obese.
9. Find the difference in the sample proportions and interpret this quantity.
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10. Find the 95% confidence interval for a difference in proportions. To do so, calculate the
following:

An estimate of the difference between the two proportions:
o
pˆ males =
o
pˆ females =
o
pˆ males  pˆ females =

An approximate quantile:

The standard deviation of the estimate (this is known as the standard error). The
standard deviation for the difference between proportions can be computed as
follows:
pˆ males 1 - pˆ males  pˆ females 1 - pˆ females 
+
=
nmales
nfemales
Finally, find the 95% confidence interval for a difference in proportions.
11. Interpret the confidence interval found above.
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Note that we could have also carried out Fisher’s Exact Test to test the following hypotheses:
H0: The proportion of males who are obese is greater than or equal to the proportion of females
who are obese.
Ha: The proportion of males who are obese is less than the proportion of females who are
obese.
H0: pfemales ≤ pmales
Ha: pfemales > pmales
p-value = ______
Conclusion:
Note that the confidence interval allows us to go one step further with this conclusion; that is, the
confidence interval gives us an idea of how much larger the proportion of metro lakes with recent
information is.
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Example: A study was done to compare the side effects of taking a new drug. Two hundred individuals
were chosen for the study and half of them were given the drug and the other half was given a placebo.
Of the people given the drug, 15 of them experienced nausea, while only 5 of the people taking the
placebo experienced nausea.
Questions:
12. Create the contingency table for the scenario.
Nausea
No Nausea
Total
Drug
Placebo
Total
13. Using JMP, find the relative risk of having nausea for those individuals taking the drug compared
to those taking the placebo.
14. Interpret the relative risk found in Question 13.
15. Construct the 90% confidence interval for the difference in proportions.
16. Interpret the 90% confidence interval found in Question 15.
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Example: To study the difference in drug therapy adherence among subjects with depression who
received usual care and those who received care in a collaborative care model was the goal of a study
conducted by Finley et al. (Pharmacology 2003). The collaborative care model emphasized the role of
clinical pharmacists in providing drug therapy management and treatment follow-up. Of the 50 subjects
receiving usual care, 24 adhered to the prescribed drug regimen, while 50 out of the 75 subjects in the
collaborative care model adhered to the drug regimen.
Questions:
17. Construct a 99% confidence interval for the difference in proportions.
18. Interpret the 99% confidence interval found in Question 17.
19. Looking at the 99% confidence interval constructed in Question 17, can it be concluded that
there is a difference in the two care types in regards to subjects sticking to the drug regimen?
Explain.
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