This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this
material constitutes acceptance of that license and the conditions of use of materials on this site.
Copyright 2008, The Johns Hopkins University and Brian Caffo. All rights reserved. Use of these materials
permitted only in accordance with license rights granted. Materials provided “AS IS”; no representations or
warranties provided. User assumes all responsibility for use, and all liability related thereto, and must independently
review all materials for accuracy and efficacy. May contain materials owned by others. User is responsible for
obtaining permissions for use from third parties as needed.
Lecture 20
Brian Caffo
Table of
contents
Outline
Recap
The delta
method
Derivation of
the delta
method
Lecture 20
Brian Caffo
Department of Biostatistics
Johns Hopkins Bloomberg School of Public Health
Johns Hopkins University
November 6, 2007
Lecture 20
Table of contents
Brian Caffo
Table of
contents
Outline
Recap
1 Table of contents
The delta
method
Derivation of
the delta
method
2 Outline
3 Recap
4 The delta method
5 Derivation of the delta method
Lecture 20
Brian Caffo
Table of
contents
Outline
Recap
The delta
method
Derivation of
the delta
method
1
Review two sample binomial results
2
Delta method
Lecture 20
Two sample binomials results
Brian Caffo
Table of
contents
Outline
Recall X ∼ Bin(n1 , p1 ) and Y ∼ Bin(n2 , p2 ). Also this
information is often arranged in a 2×2 table:
The delta
method
Derivation of
the delta
method
n12 = n1 − x
n22 = n2 − y
n11 = x
n21 = y
Recap
ˆ = p̂1 − p̂2
• RD
ˆ ˆ =
SE
RD
q
p̂1 (1−p̂1 )
n1
+
p̂2 (1−p̂2 )
n2
ˆ = p̂1
• RR
p̂2
ˆ =
• OR
q
(1−p̂1 )
(1−p̂2 )
p̂1 n1 + p̂2 n1
p̂1 /(1−p̂1 )
n11 n22
p̂2 /(1−p̂2 ) = n12 n21
ˆ
SE
ˆ =
log RR
ˆ
SE
ˆ =
log OR
q
1
n11
+
1
n12
+
CI = Estimate ± Z1−α/2 SEEst
1
n21
+
1
n22
n1
n2
Lecture 20
Standard errors
Brian Caffo
Table of
contents
Outline
Recap
The delta
method
• delta method can be used to obtain large sample
standard errors
• Formally, the delta methods states that if
Derivation of
the delta
method
θ̂ − θ
→ N(0, 1)
ˆ
SE
θ̂
then
f (θ̂) − f (θ)
→ N(0, 1)
ˆ
f 0 (θ̂)SE
θ̂
• Asymptotic mean of f (θ̂) is f (θ)
• Asymptotic standard error of f (θ̂) can be estimated with
ˆ
f 0 (θ̂)SE
θ̂
Lecture 20
Example
Brian Caffo
Table of
contents
Outline
Recap
The delta
method
Derivation of
the delta
method
• θ = p1
• θ̂ = p̂1
q
ˆ = p̂1 (1−p̂1 )
• SE
n1
θ̂
• f (x) = log(x)
• f 0 (x) = 1/x
• θ̂−θ
ˆ → N(0, 1) by the CLT
SE θ̂
ˆ log p̂ = f 0 (θ̂)SE
ˆ
• Then SE
1
θ̂
1
=
p̂1
• And
s
p̂1 (1 − p̂1 )
=
n1
s
(1 − p̂1 )
p̂1 n1
log p̂1 − log p1
q
→ N(0, 1)
(1−p̂1 )
p̂1 n1
Lecture 20
Putting it all together
Brian Caffo
Table of
contents
Outline
Recap
• Asymptotic standard error
The delta
method
Var(log R̂R) = Var{log(p̂1 /p̂2 )}
Derivation of
the delta
method
= Var(log p̂1 ) + Var(log p̂2 )
(1 − p̂1 ) (1 − p̂2 )
+
≈
p̂1 n1
p̂2 n2
• The last line following from the delta method
• The approximation requires large sample sizes
• The delta method can be used similarly for the log odds
ratio
Lecture 20
Motivation for the delta method
Brian Caffo
Table of
contents
Outline
Recap
• If θ̂ is close to θ then
The delta
method
f (θ̂) − f (θ)
Derivation of
the delta
method
θ̂ − θ
≈ f 0 (θ̂)
• So
f (θ̂) − f (θ)
f 0 (θ̂)
≈ θ̂ − θ
• Therefore
θ̂ − θ
f (θ̂) − f (θ)
≈
0
ˆ
ˆ
f (θ̂)SE θ̂
SE
θ̂