(50) ? =[
Var(β0) Cov(β0, β1) Cov(β1, log(σε))̂
Cov(β0, β1) Var(β1) Cov(β1, log(σε))]̂
Cov(β0, log(σε)) ̂ Cov(β1, log(σε)) ̂ Var(log(σε))̂
(51) ? =
[
∂(μ???) ̂
∂(β0)
∂(σ???)̂
∂(β0)
∂(μ???) ̂
∂(β1)
∂(σ???)̂
∂(β1)
∂(μ???) ̂
∂ (log(σε)) ̂
∂(σ???)̂
∂(log(σε))̂
]
=
[
−
1
β1
0
−
μ??? ̂
β1
−
σ??? ̂
β1
0
σε
β1 ]
=−
1
β1
[μ???
1 0
̂ σ???]̂
0 −σε
Thus, the conversion matrix is:
(52) ???𝒅 =[
Var(μ???) ̂ Cov(μ???, ̂ σ???)̂
Cov(μ???, ̂ σ???) ̂ Var(σ???)̂
]
where
Var(μ???) ̂ =
1
β1 2
(Var[β0] +
2μ???Cov[β0, ̂ β1] +μ???2Var[β1])̂
Var(σ???) ̂ =
1
β1 2 (σ???Cov[β0, ̂ β1] +μ???σ???Var[β1] ̂̂ −
σεCov[β0, log(σε)] − μ???σεCov[β1, ̂ log(σε)])
Cov(μ???, ̂ σ???) ̂ =
1
β1 2
(σ???2Var[β1] ̂ +
σε2Var[log(σε)] − 2σεσ???Cov[β1, ̂ log (σε)])
Appendix B2: POD Using SAS PROC REG
For the simple linear model, μ??? ̂ =− (β0 − 𝑦???)⁄β1 and σ??? ̂ =σε⁄β1. However, SAS [22, 25] returns Var(σε2) instead
of Var(σε). The derivatives with respect to β0 and β1 are the same, but new derivatives were calculated using a change of
variables with 𝜂̂ =σε2.
The two new derivatives are:
(53)
∂(μ???)̂
∂(𝜂̂)
=0
∂(σ???) ̂
∂(𝜂̂)
=
∂(σ???) ̂
∂(σε2)
=
∂ (σε
β1
)
∂(σε2)
=
∂
(
√σε2
β1
)
∂(σε2)
=
∂(√𝜂̂/β1)
∂(𝜂̂)
=
1
2√𝜂̂β1
=
1
2 √σε2 β1
=
1
2σεβ1

(54)
∂μ??? ̂
∂β0
=−
1
β1
∂σ???̂
β0
=0
∂μ??? ̂
∂β1
=−
ydec − β0
β1 2
=−
μ???̂
β1
∂σ??? ̂
∂β1
=
−σε
β1 2
=−
σ???̂
β1
∂μ???̂
∂σε2
=0
∂σ??? ̂
∂σε2
=
1
2σεβ1
The SAS linear model (PROC REG) returns the variance-covariance matrix in Equation 55, so the matching derivatives
matrix is in Equation 56, and the transition matrix is in Equation 57.
(55) ? =
[
Var(β0) Cov(β0, β1) Cov(β1, σε)2
Cov(β0, β1) Var(β1) Cov(β1, σε)2
Cov(β0, σε) 2 Cov(β1, σε) 2 Var(σε) 2 ]
(56) ? =
[
∂(μ???) ̂
∂(β0)
∂(σ???)̂
∂(β0)
∂(μ???) ̂
∂(β1)
∂(σ???)̂
∂(β1)
∂(μ???) ̂
∂(σε2)
∂(σ???)̂
∂(σε2) ]
=
[
−
1
β1
0
−
μ??? ̂
β1
−
σε
β1 2
0
1
2σεβ1]
=−
1
β1
[
1 0
μ??? ̂ σ???̂
0 −
1
2σε]
Thus, the conversion matrix is:
(57) ???𝒅 =[
Var(μ???) ̂ Cov(μ???, ̂ σ???)̂
Cov(μ???, ̂ σ???) ̂ Var(σ???)̂
]
where
Var(μ???) ̂ =
1
β1 2 (Var[β0] +2μ???Cov[β0, ̂ β1] +μ???2Var[β1])̂
Var(σ???) ̂ =
1
β2
1
(σ???Cov[β0, ̂ β1] +μ???σ???Var[β1] ̂̂ −
1
2σε Cov[β0, σε2] −
μ??? ̂
2σε Cov[β1, σε2])
Cov(μ???, ̂ σ???) ̂ =
1
β1 2 (σ???2Var[β1] ̂ +
1
4σε2 Var[σε2] −
1
β1 Cov[β1, σε2])