1Apr 18, 2023

Chapter 17Chapter 17Comparing Two ProportionsComparing Two Proportions

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In Chapter 17:

17.1 Data

[17.2 Risk Difference]

[17.3 Hypothesis Test]

17.4 Risk Ratio

[17.5 Systematic Sources of Error]

[17.6 Power and Sample Size]

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Data conditions• Binary response variables (“success/failure”)• Binary explanatory variable • Notation:

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Sample Proportions

1

11ˆ

n

ap

2

22ˆ

n

ap

Incidence proportion, exposed group:

Incidence proportion, non-exposed group:

Incidence proportion ≡ average risk

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Example: WHI Estrogen Trial

Random Assignment

Group 1 n1 = 8506

Group 2 n2 = 8102

Estrogen Treatment

Placebo

Compare risks of index disease

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2-by-2 TableDisease+ Disease− Total

Exposure + a1 b1 n1

Exposure − a2 b2 n2

Total m1 m2 N

1

11ˆ

n

ap

2

22ˆ

n

ap

Risk, exposed Risk, non-exposed

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WHI DataD+ D− Total

E+ 751 7755 8506

E- 623 7479 8102

Total 1374 15234 16608

8506

751ˆ1 p

07689.0

08829.0

8102

623ˆ 2 p

Compare these risks

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§17.4 Proportion Ratio (Relative Risk)

• Compare incidences from the two groups in form of a RATIO

• Quantifies effect of the exposure in relative terms

2

1

p

pRR

Relative Risk Parameter

Relative Risk Estimator (“RR hat”)

2

1

ˆ

ˆˆp

pRR

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Example: RR (WHI Data)+ − Total

Estrogen + 751 7755 8506

Estrogen − 623 7479 8102

2

1

ˆ

ˆˆp

pRR

08829.08506

751ˆ1 p 07689.0

8102

623ˆ 2 p

07689.0

08829.0 15.11483.1

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Interpretation• When p1 = p2, RR = 1 indicating “no

association”– RR > 1 positive association– RR < 1 negative association

• The RR indicates how much the exposure multiplies the risk over the baseline risk of the non-exposed group– RR of 1.15 suggests risk in exposed group is “1.15

times” that of non-exposed group

• Baseline RR is 1!– Thus, an RR of 1.15 is 0.15 (15%) above the baseline

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Confidence Interval for the RR

ln ≡ natural log, base e

RRSEzRR

eˆln

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ˆln

2211

1111ˆln

wherenanaRR

SE

To derive information about the precision of the estimate, calculate a (1– α)100% CI for the RR with this formula:

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90% CI for RR, WHID+ D− Total

E+ 751 7755 8506

E− 623 7479 8102

1.645 ,confidence 90%For

051920.0

1382.0)1483.1ln(ˆln

81021

6231

85061

7511

ˆln

z

SE

RR

RR

)25.1 ,05.1(

2236.0,0528.00854.01382.0)051920.0)(645.1(1382.0

eee