Table 1 Previously published, alternative, and new formulations of the prediction error variance for a random effect u with ${\sigma }_g^2$, the assumptions pertinent to each formulation, the information used in each formulation, and the asymptotic sampling variances of each formulation

¹PEV_GC1 = ${\sigma }_{\text{g}}^{\text{2}}$ - Var($\widehat{\text{u}}$)

Cov(u, $\widehat{\text{u}}$) = Var($\widehat{\text{u}}$)

Var(u) = ${\sigma }_{\text{g}}^{\text{2}}$

$\widehat{\text{u}}$

2r⁴ ${\sigma }_{\text{g}}^4$/n

²PEV_GC2 = Var(u - $\widehat{\text{u}}$)

¹¹Cov(u, $\widehat{\text{u}}$) ≠/= Var($\widehat{\text{u}}$)

Var(u) = ${\sigma }_{\text{g}}^{\text{2}}$

u - $\widehat{\text{u}}$

2(1-r²)² ${\sigma }_{\text{g}}^4$/n

³ ${\text{PEV}}_{\text{GC3}}=\frac{\left[\frac{{\text{PEV}}_{\text{GC1}}}{{\text{Var(PEV}}_{\text{GC1}}\text{)}}\right]+\left[\frac{{\text{PEV}}_{\text{GC2}}}{{\text{Var(PEV}}_{\text{GC2}}\text{)}}\right]}{\frac{\text{1}}{{\text{Var(PEV}}_{\text{GC1}}\text{)}}+\frac{\text{1}}{{\text{Var(PEV}}_{\text{GC2}}\text{)}}}$

Cov(u - $\widehat{\text{u}}$, $\widehat{\text{u}}$) = 0

Var(u) = ${\sigma }_{\text{g}}^{\text{2}}$

$\widehat{\text{u}}$, u - $\widehat{\text{u}}$

{[2r⁴(1-r²)²]/[(1-r²)² + r⁴]}${\sigma }_{\text{g}}^4$/n

⁴PEV_FL = ${\sigma }_{\text{g}}^{\text{2}}$ - Cov(u, $\widehat{\text{u}}$)

Cov(u, $\widehat{\text{u}}$) = Var($\widehat{\text{u}}$)

Var(u) = ${\sigma }_{\text{g}}^{\text{2}}$

Cov(u, $\widehat{\text{u}}$)

r²(1+r²)${\sigma }_g^2$/n

⁵PEV_AF1 = ${\sigma }_{\text{g}}^{\text{2}}$ - [Var($\widehat{\text{u}}$)/Var(u)] ${\sigma }_{\text{g}}^{\text{2}}$

Cov(u, $\widehat{\text{u}}$) = Var($\widehat{\text{u}}$)

Var(u) ≠ ${\sigma }_{\text{g}}^{\text{2}}$

$\widehat{\text{u}}$, u

4r⁴(1-r²)${\sigma }_{\text{g}}^4$/n

⁶PEV_AF2 = [Var(u - $\widehat{\text{u}}$)/Var(u)] ${\sigma }_{\text{g}}^{\text{2}}$

¹¹Cov(u, $\widehat{\text{u}}$) ≠/= Var($\widehat{\text{u}}$)

Var(u) ≠ ${\sigma }_{\text{g}}^{\text{2}}$

u - $\widehat{\text{u}}$, u

4r²(1-r²)² ${\sigma }_{\text{g}}^4$/n

⁷ ${\text{PEV}}_{\text{AF3}}=\frac{\left[\frac{{\text{PEV}}_{\text{AF1}}}{{\text{Var(PEV}}_{\text{AF1}}\text{)}}\right]+\left[\frac{{\text{PEV}}_{\text{AF2}}}{{\text{Var(PEV}}_{\text{AF2}}\text{)}}\right]}{\frac{\text{1}}{{\text{Var(PEV}}_{\text{AF1}}\text{)}}+\frac{\text{1}}{{\text{Var(PEV}}_{\text{AF2}}\text{)}}}$

Cov(u - $\widehat{\text{u}}$, $\widehat{\text{u}}$) = 0

Var(u) ≠ ${\sigma }_{\text{g}}^{\text{2}}$

$\widehat{\text{u}}$, u - $\widehat{\text{u}}$, u

4r⁴ (1 - r²)² ${\sigma }_{\text{g}}^4$/n

⁸PEV_AF4 = ${\sigma }_{\text{g}}^{\text{2}}$ - [Cov(u, $\widehat{\text{u}}$)/Var(u)] ${\sigma }_{\text{g}}^{\text{2}}$

Cov(u, $\widehat{\text{u}}$) = Var($\widehat{\text{u}}$)

Var(u) ≠ ${\sigma }_{\text{g}}^{\text{2}}$

Cov(u, $\widehat{\text{u}}$), u

r²(1-r²)${\sigma }_g^2$/n

⁹PEV_NF1 = [1 - Cov(u, $\widehat{\text{u}}$)²/(Var(u) × Var($\widehat{\text{u}}$))]${\sigma }_{\text{g}}^{\text{2}}$

4r²(1-r²)² ${\sigma }_g^2$/n

¹⁰PEV_NF2 = {Var(u - $\widehat{\text{u}}$)/[Var($\widehat{\text{u}}$) + Var(u - $\widehat{\text{u}}$]}${\sigma }_{\text{g}}^{\text{2}}$

Cov(u - $\widehat{\text{u}}$, $\widehat{\text{u}}$) = 0

$\widehat{\text{u}}$ and u - $\widehat{\text{u}}$

4r⁴(1-r²)² ${\sigma }_{\text{g}}^4$/n