Variance Components

Variance Components — Nested (ANOVA)

Decomposes the total variation of a nested design via expected mean squares.

Statusvalidated

Version1.0.0

Minitab equivalentStat > ANOVA > Fully Nested ANOVA

Description

In the nested model the levels of each factor exist only inside one level of the preceding factor — part 1 from cavity 1 has nothing in common with part 1 from cavity 2. The terms are therefore the prefixes of the factor list: A, B(A), C(A,B), … The sequential (type I) sums of squares and their degrees of freedom give the mean squares; solving Henderson I's coefficient matrix of expected mean squares yields the variance components. Negative estimates are clamped to 0 after the solve — a reporting rule, not part of the computation.

Formulas

M S_{i} = \frac{S S _{i}}{d f _{i}}

Mean square from sequential sum of squares and degrees of freedom

EMS

E [M S_{i}] = σ^{2} + j \sum c_{ij} σ_{j}^{2}

Expected mean square as a linear combination of the components

σ²

\overset{σ}{^} = C^{- 1} M S

Henderson I: solving the EMS system

Anteil

p_{i} = 100 \cdot \frac{σ _{i}^{2}}{\sum _{j} σ _{j}^{2}}

Share of one component in the total variance

Assumptions

All factors are random effects
Effects are independent and normally distributed
The factor order matches the nesting

Limitations

On an unbalanced design the components depend on the method — prefer REML then
Negative estimates are clamped to 0, losing information in the process
Two to four factors

References

Henderson, C. R. (1953), Estimation of Variance and Covariance Components, Biometrics 9(2)
Searle, Casella & McCulloch (1992), Variance Components, Wiley
NIST/SEMATECH e-Handbook of Statistical Methods, Section 2.5