Statistic in Health Care Management: Assignment Week 3
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chapter6.pdf
Chapter 6
Confidence Interval Estimates
Learning Objectives
• Define point estimate, standard error,
confidence level and margin of error
• Compare and contrast standard error and
margin of error
• Compute and interpret confidence intervals for
means and proportions
• Differentiate independent and matched or
paired samples
Learning Objectives
• Compute confidence intervals for the
difference in means and proportions in
independent samples and for the mean
difference in paired samples
• Identify the appropriate confidence interval
formula based on type of outcome variable and
number of samples
Statistical Inference
• There are two broad areas of statistical inference, estimation and hypothesis testing.
• Estimation, the population parameter is unknown, and sample statistics are used to generate estimates of the unknown parameter.
Statistical Inference
• Hypothesis testing, an explicit statement or hypothesis is generated about the population parameter. Sample statistics are analyzed and determined to either support or reject the hypothesis about the parameter.
• In both estimation and hypothesis testing, it is assumed that the sample drawn from the population is a random sample.
Estimation
• Process of determining likely values for
unknown population parameter
• Point estimate is best single-valued estimate
for parameter
• Confidence interval is range of values for
parameter:
point estimate + margin of error
Estimation
A point estimate for a population parameter is the
"best" single number estimate of that parameter.
A confidence interval estimate is a range of values for
the population parameter with a level of confidence
attached (e.g., 95% confidence that the range or
interval contains the parameter).
Confidence Interval Estimates
point estimate + margin of error
point estimate + Z SE (point estimate)
where Z = value from standard normal distribution for desired confidence level and SE (point estimate) = standard error of the point estimate
Confidence Intervals for m
• Continuous outcome
• 1 Sample
n > 30 (Find Z in Table 1B)
n < 30 (Find t in Table 2,
df=n-1)
n
s ZX
n
s tX
Table 2. Critical Values of the t
Distribution
Table entries represent values from t distribution with upper tail area equal to a.
Confidence Level 80% 90% 95% 98% 99%
Two Sided Test a .20 .10 .05 .02 .01
One Sided Test a .10 .05 .025 .01 .005
df
1 3.078 6.314 12.71 31.82 63.66
2 1.886 2.920 4.303 6.965 9.925
3 1.638 2.353 3.182 4.541 5.841
4 1.533 2.132 2.776 3.747 4.604
5 1.476 2.015 2.571 3.365 4.032
6 1.440 1.943 2.447 3.143 3.707
7 1.415 1.895 2.365 2.998 3.499
8 1.397 1.860 2.306 2.896 3.355
9 1.383 1.833 2.262 2.821 3.250
10 1.372 1.812 2.228 2.764 3.169
Example 6.1.
Confidence Interval for m
In the Framingham Offspring Study (n=3534), the mean
systolic blood pressure (SBP) was 127.3 with a standard
deviation of 19.0. Generate a 95% confidence interval for the
true mean SBP.
n
s ZX
3534
19.0 96.1127.3
127.3 + 0.63
(126.7, 127.9)
Example 6.2.
Confidence Interval for m
In a subset of n=10 participants attending the Framingham
Offspring Study, the mean SBP was 121.2 with a standard
deviation of 11.1. Generate a 95% confidence interval for the
true mean SBP.
n
s tX
10
11.1 262.2121.2 121.2 + 7.94
(113.3, 129.1)
df=n-1=9, t=2.262
New Scenario
• Outcome is dichotomous (p=population proportion)
– Result of surgery (success, failure)
– Cancer remission (yes/no)
• One study sample
• Data
– On each participant, measure outcome (yes/no)
– n, x=# positive responses,
n
x p̂
Confidence Intervals for p
• Dichotomous outcome
• 1 Sample
(Find Z in Table 1B)
5)]p̂n(1,p̂min[n n
)p̂-(1p̂ Zp̂
Example 6.3.
Confidence Interval for p
In the Framingham Offspring Study (n=3532), 1219 patients
were on antihypertensive medications. Generate a 95%
confidence interval for the true proportion on antihypertensive
medication.
n
)p̂-(1p̂ Zp̂ 0.345
3532
1219 p̂
3532
0.345)-0.345(1 96.10.345
0.345 + 0.016
(0.329, 0.361)
New Scenario
• Outcome is continuous
– SBP, Weight, cholesterol
• Two independent study samples
• Data
– On each participant, identify group and measure
outcome
– )s(ors,X,n),s(ors,X,n 2
2
2221
2
111
Two Independent Samples
RCT: Set of Subjects Who Meet
Study Eligibility Criteria
Randomize
Treatment 1 Treatment 2
Mean Trt 1 Mean Trt 2
Two Independent Samples
Cohort Study - Set of Subjects Who
Meet Study Inclusion Criteria
Group 1 Group 2
Mean Group 1 Mean Group 2
Confidence Intervals for (m1m2)
• Continuous outcome
• 2 Independent Samples
n1>30
and n2>30 (Find Z in
Table 1B)
n1<30
or n2<30 (Find t in
Table 2,
df=n1+n2-2)
21
21 n
1
n
1 ZSp)X - X(
21
21 n
1
n
1 tSp)X - X(
Pooled Estimate of Common Standard
Deviation, Sp
• Previous formulas assume equal variances
(s1 2=s2
2)
• If 0.5 < s1 2/s2
2 < 2, assumption is reasonable
2nn
1)s(n1)s(n Sp
21
2
22
2
11
Example 6.5.
Confidence Interval for (m1m2)
Using data collected in the Framingham Offspring
Study, generate a 95% confidence interval for the
difference in mean SBP between men and women.
n Mean Std Dev
MEN 1623 128.2 17.5
WOMEN 1911 126.5 20.1
Assess Equality of Variances
• Ratio of sample variances: 17.52/20.12 = 0.76
2nn
1)s(n1)s(n Sp
21
2
22
2
11
0.1912.359 219111623
1)20.1(19111)17.5(1623 Sp
22
Confidence Intervals for (m1m2)
1911
1
1623
1 (19.0) 1.96 126.5) - (128.2
21
21 n
1
n
1 ZSp)X - X(
1.7 + 1.26
(0.44, 2.96)
New Scenario
• Outcome is continuous
– SBP, Weight, cholesterol
• Two matched study samples
• Data
– On each participant, measure outcome under each
experimental condition
– Compute differences (D=X1-X2)
– dd s,Xn,
Two Dependent/Matched Samples
Subject ID Measure 1 Measure 2
1 55 70
2 42 60
.
.
Measures taken serially in time or under different
experimental conditions
Crossover Trial
Treatment Treatment
Eligible R
Participants
Placebo Placebo
Each participant measured on Treatment and placebo
Confidence Intervals for md
• Continuous outcome
• 2 Matched/Paired Samples
n > 30 (Find Z in Table 1B)
n < 30 (Find t in Table 2,
df=n-1)
n
s ZX d
d
n
s tX d
d
Example 6.8.
Confidence Interval for md
In a crossover trial to evaluate a new
medication for depressive symptoms, patients’
depressive symptoms were measured after
taking new drug and after taking placebo.
Depressive symptoms were measured on a
scale of 0-100 with higher scores indicative of
more symptoms.
Example 6.8.
Confidence Interval for md
Construct a 95% confidence interval for the
mean difference in depressive symptoms
between drug and placebo.
The mean difference in the sample (n=100) is -
12.7 with a standard deviation of 8.9.
Example 6.8.
Confidence Interval for md
n
s ZX d
d
100
8.9 96.112.7-
-12.7 + 1.74
(-14.1, -10.7)
New Scenario
• Outcome is dichotomous
– Result of surgery (success, failure)
– Cancer remission (yes/no)
• Two independent study samples
• Data
– On each participant, identify group and measure
outcome (yes/no)
– 2211
p̂,n,p̂,n
Confidence Intervals for (p1p2)
• Dichotomous outcome
• 2 Independent Samples
(Find Z in Table 1B)
5)]p̂(1n,p̂n),p̂(1n,p̂min[n 22221111
2
22
1
11 21
n
)p̂(1p̂
n
)p̂-(1p̂ Z)p̂-p̂(
Example 6.10.
Confidence Interval for (p1-p2)
A clinical trial compares a new pain reliever to
that considered standard care in patients
undergoing joint replacement surgery. The
outcome of interest is reduction in pain by 3+
scale points. Construct a 95% confidence
interval for the difference in proportions of
patients reporting a reduction between
treatments.
Example 6.10.
Confidence Interval for (p1-p2)
Reduction of 3+ Points
Treatment n Number Proportion
New 50 23 0.46
Standard 50 11 0.22
Example 6.10.
Confidence Interval for (p1-p2)
2
22
1
11 21
n
)p̂(1p̂
n
)p̂-(1p̂ Z)p̂-p̂(
50
)22.00.22(1
50
0.46)-0.46(1 96.10.22)-(0.46
0.24 + 0.18
(0.06, 0.42)
Confidence Intervals for Relative Risk (RR)
• Dichotomous outcome
• 2 Independent Samples
exp(lower limit), exp(upper limit)
(Find Z in Table 1B)
2
222
1
111
n
)/xx-(n
n
)/xx-(n ZR)R̂ln(
Example 6.12.
Confidence Interval for RR
Reduction of 3+ Points
Treatment n Number Proportion
New 50 23 0.46
Standard 50 11 0.22
Construct a 95% CI for the relative risk.
Example 6.12.
Confidence Interval for RR
2.09 0.22
0.46
p̂
p̂ RR̂
2
1
50
39/11
50
27/23 1.96ln(2.09)
0.737 + 0.602 exp(0.135), exp(1.339)
(0.135, 1.339) (1.14, 3.82)
Confidence Intervals for Odds Ratio (OR)
• Dichotomous outcome
• 2 Independent Samples
exp(lower limit), exp(upper limit)
(Find Z in Table 1B)
)x(n
1
n
1
)x(n
1
x
1 ZR)Ôln(
222111
Example 6.14.
Confidence Interval for OR
Reduction of 3+ Points
Treatment n Number Proportion
New 50 23 0.46
Standard 50 11 0.22
Construct a 95% CI for the odds ratio.
Example 6.14.
Confidence Interval for OR
3.02 11/39
23/27
)x-/(nx
)x-/(nx RÔ
222
111
39
1
11
1
27
1
23
1 1.96ln(3.02)
1.105 + 0.870 exp(0.235), exp(1.975)
(0.235, 1.975) (1.26, 7.21)
