Section 12 Part 1
Confidence Intervals and
Hypothesis Tests of Proportions
from Two groups
Section 12 Part 1 overview
• Often researchers are interested in estimating the difference
between two independent groups.
• When the variable of interest is continuous, appropriate
inference methods are
– Confidence Intervals of the difference between two means
– Hypothesis tests (t-tests) of equal means
– Nonparametric when sample size is small and normality assumption is
not met (not covered in PH6414)
• When the variable of interest is categorical (binary),
appropriate inference methods are
– Confidence Intervals of the difference in proportions (Section 12 Part
1)
– Hypothesis tests (z-tests) of equal proportions (Section 12 Part 1)
– Chi-square tests of Independence (Section 13)
PubH 6414 Section 12 Part 1
2
Section 12 Part 1 Outline
• Two sample Z-test of proportions to test the null
hypothesis of equal proportions
• Confidence Interval for the difference of two
proportions
• Both inference methods require that the normal
approximation to the binomial distribution are
met for both groups
– n*π > 5 and n*(1-π) ≥ 5 (for CI)
– n*π > 5 and n*(1-π) ≥ 5 (for Hypothesis test)
PubH 6414 Section 12 Part 1
3
Two sample z-test of proportions: Notation
• We have 2 distinct (independent) populations of some binary
variable.
• The population proportions (π) are the true proportions in
each population, the sample proportions (p) are estimates of
the population proportions.
Population Population Sample
proportion
size
Number in
sample with
success
1
π1
n1
x1
2
π2
n2
x2
PubH 6414 Section 12 Part 1
Sample
proportion
x1
p1 =
n1
x
p2 = 2
n2
4
Steps in Hypothesis Testing
Comparing Two Proportions
1. State the null hypothesis H0 and the
alternative hypothesis Ha.
2. Calculate the value of the test statistic on
which the test will be based.
3. Find the p-value for the observed data.
4. State a conclusion.
PubH 6414 Section 12 Part 1
5
Hypothesis Testing Steps:
Comparing two Proportions
• Step 1: State your hypotheses
– H0: π1-π2= π0
– Ha: π1-π2≠ π0 or
Ha: π1-π2 < π0 or Ha: π1-π2 > π0
PubH 6414 Section 12 Part 1
6
Hypothesis Testing Steps:
Comparing two Proportions
Step 2 : Calculate your test statistic (z statistic is appropriate
provided you have a SRS that meet this condition
n1 p1 ≥ 5 and n1 (1 − p1 ) ≥ 5 and n2 p2 ≥ 5 and n2 (1 − p2 ) ≥ 5
We are testing under the null hypothesis that π1=π2, so we create
and then,
X1 + X 2
p=
n1 + n2
z=
( p1 − p2 )
1
1
p (1 − p )( + )
n 1 n2
PubH 6414 Section 12 Part 1
7
Hypothesis Testing Steps: Comparing Two
Proportions
• Step 3: Calculate the p-value
One or two-sided.
• Step 4: Make your conclusion.
If p-value is small, you have evidence against the null.
If p-value is large, you do not have evidence against the null.
PubH 6414 Section 12 Part 1
8
Birth-Weight and Respiratory Tract Infections
• A study was designed to investigate whether low birth weight
(LBW: wt < 2500 gm) have the same proportion of lower respiratory
tract infection (LRTI) in the first 2 years as normal birth weight
infants (NBW).
• A random sample of 1000 infants from several urban clinics were
enrolled in the study.
• Infants were identified as LBW or NBW and followed for 2 years
PubH 6414 Section 12 Part 1
9
Birth-Weight and Respiratory Tract Infections
Scientific Question of Interest: Is there a difference in the proportion
of infants with any LRTI in the first 2 years between LBW and NBW
infants?
What are the null and alternative hypotheses to test this scientific
question?
PubH 6414 Section 12 Part 1
10
Birth-Weight and Respiratory Tract Infections
Use α = 0.05 as the significance level
What are the critical values are from the standard normal
distribution?
PubH 6414 Section 12 Part 1
11
Birth-Weight and Respiratory Tract Infections
1000 infants were enrolled in the study
• 64 of the infants were LBW, 936 were NBW
• After 2 years of follow-up – 15 of the LBW infants had at least one LRTI
– 112 of the NBW infants had at least one LRTI
Group
LBW
NBW
Sample
Size
Number
with LRTI
n1= 64
x1= 15
Proportion
with LRTI
p1= 0.234
n2= 936
x2= 112
p2= 0.120
PubH 6414 Section 12 Part 1
12
Birth-Weight and Respiratory Tract Infections
• Is the normal approximation is valid for both groups?
• Calculate the overall proportion (p) of LRTI:
x1 + x2
p=
=
n1 + n2
PubH 6414 Section 12 Part 1
= 0.127
13
Birth-Weight and Respiratory Tract Infections
Calculate the z-statistic: the difference between the
sample proportions divided by the SE of the difference.
Use the overall p to calculate the SE (diff).
z=
( p1 − p2 )
=
1
1
p(1 − p )( + )
n 1 n2
= 2.67
PubH 6414 Section 12 Part 1
14
Birth-Weight and Respiratory Tract Infections
What is the p-value of the is hypothesis test?
> 2*pnorm(-2.67, 0, 1)
[1] 0.007585125
PubH 6414 Section 12 Part 1
15
Birth-Weight and Respiratory Tract Infections
In this study 23% of low birth weight infants had at least one LRTI in
the first two years of life and 12% of normal birth weight infants
had at least one LRTI in the first two years of life. A significantly
different proportion of low birth weight infants had at least one
LRTI by age 2 than normal birth weight infants (p = 0.008).
PubH 6414 Section 12 Part 1
16
CI for the Difference in Population
Proportions
• Confidence interval for the difference in
population proportions:
( p1 − p2 ) ± Z * SE ( p1 − p2 )
where :
SE ( p1 − p2 ) =
p1 (1 − p1 ) p2 (1 − p2 )
+
n1
n2
n1 p1 ≥ 5 and n1 (1 − p1 ) ≥ 5 and n2 p2 ≥ 5 and n2 (1 − p2 ) ≥ 5
PubH 6414 Section 12 Part 1
17
SE (diff) and coefficient for CI of difference
between proportions
• Your book is incorrect in saying the SE for the
confidence interval is
SE ( p1 − p2 ) =
1 1
p (1 − p ) + 
 n1 n2 
• To do this you must assume (π1=π2).
PubH 6414 Section 12 Part 1
18
Binge Drinking and Gender
• Binge drinking on college campuses is a public
health concern.
• A survey of 17, 096 students revealed the
following results.
Group
N
X
p
Men
7180
1630
0.227
Women
9916
1684
0.170
PubH 6414 Section 12 Part 1
19
Binge Drinking and Gender
• Is the sample large enough to construct the CI?
• Construct the 95% CI for the difference
( p1 − p2 ) ± Z *
p1 (1 − p1 ) p2 (1 − p2 )
+
n1
n2
(0.227 − 0.170) ± 1.96
0.227 (0.773)
7180
+
0.170(0.830)
9916
0.057 ± 0.012
(0.045,0.069)
PubH 6414 Section 12 Part 1
20
Binge Drinking and Gender
The data suggest that men are more likely to be
frequent binge drinkers with an estimated risk
difference (men – women) of 5.7% and 95%
confidence interval from 4.5% to 6.9%.
PubH 6414 Section 12 Part 1
21
Confidence Interval and z-test
• Typically, if the 95% confidence interval does
not contain 0, the two-tailed z-test will result
in the decision to reject the null hypothesis of
equal proportions.
• However; the standard error is different for
the confidence interval and the hypothesis
test, just as with the one sample proportion.
PubH 6414 Section 12 Part 1
22