2Cov(?1, ?2) − 2Cov(?1, ?3) − 2Cov(?2, ?3) −
Var(?1) − Var(?2) − Var(?3) − 2Cov(?1, ?2) −
2Cov(?1, ?3) − 2Cov(?2, ?3)) =Cov(?1, ?1) +
Cov(?1, ?2) +Cov(?1, ?3) +Cov(?2, ?1) +
Cov(?2, ?2) +Cov(?2, ?3) +Cov(?3, ?1) +
Cov(?3, ?2) +Cov(?3, ?3))
Appendix B: Cautions About Software
The same parameters must be used in the variance-covariance matrix, ? and the derivative matrix, ?, but the outputs
from statistical software tools may differ. For example, the Survival package within R software returns the parameters
{β0, β1, log(σε)}, but SAS software’s PROC REG returns parameters {β0, β1, σε}. 2 Although the parameter values can be
transformed, such as σε 2 =exp(log(σε))2, the variance of these parameters is not easily transformed (e.g., Var[log(σε)] =
?).Therefore, if ? is in terms of the parameters {β0, β1, log(σε)}, then the derivatives in ? must also be with respect to
{β0, β1, log(σε)}. The ?, ?, and ?{???} matrices for R and SAS are provided in this appendix.
Appendix B1: POD Using R’s Survival Package
For the simple linear model, μ??? ̂ =(β0 − 𝑦???)⁄β1 and σ??? ̂ =σε⁄β1. However, the Survival package returns
Var(log(σε)) 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 τ =log(σε).
The two new derivatives are:
(48)
𝜕(μ???)̂
∂(τ)
=0
∂(σ???) ̂
∂(τ)
=
∂(σ???) ̂
∂(log (σε))
=
∂ (σε
β1
)
∂(log (σε))
=
∂ (exp(log(σε))
β1
)
∂(log (σε))
=
∂ (exp(τ)
β1
)
∂(τ)
=
exp(τ)
β1
=
exp(log(σε))
β1
=σε/β1
(49)
∂μ??? ̂
∂β0
=−
1
β1
∂σ???̂
β0
=0
∂μ??? ̂
𝜕β1
=−
𝑦??? − β0
β2
1
=−
μ???̂
β1
∂σ??? ̂
∂β1
=−σε
β1 2
=−
σ???̂
β1
∂μ???̂
∂ log(σε)
=0
∂σ??? ̂
∂ log(σε)
=
log(σε)
β1
The Survival package within R (survreg function) [24] returns the variance-covariance matrix in Equation 50, so the
matching derivatives matrix is in Equation 51, and the transition matrix is in Equation 52.

(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