Estimation of
Variance Components  Estimating the Variation of Random Factors
The ANOVA method provides an integrative approach to estimating variance
components, because ANOVA techniques can be used to estimate the variance
of random factors, to estimate the components of variance in the dependent
variable attributable to the random factors, and to test whether the variance
components differ significantly from zero. The ANOVA method for estimating
the variance of the random factors begins by constructing the Sums of
squares and cross products (SSCP) matrix for the independent variables.
The sums of squares and cross products for the random effects are then
residualized on the fixed effects, leaving the random effects independent
of the fixed effects, as required in the mixed model (see, for example,
Searle, Casella, & McCulloch, 1992). The residualized Sums of squares
and cross products for each random factor are then divided by their degrees
of freedom to produce the coefficients in the Expected mean squares matrix.
Nonzero offdiagonal coefficients for the random effects in this matrix
indicate confounding, which must be taken into account when estimating
the population variance for each factor. For the Wheat.sta
data, treating both Variety and
Plot as random
effects, the coefficients in the Expected mean squares matrix show
that the two factors are at least somewhat confounded. The Expected mean
squares spreadsheet is shown below.
Expected Mean Squares (wheat.sta) 

Mean Squares Type: 1 
Source 
Effect
(F/R) 
VARIETY 
PLOT 
Error 
{1}VARIETY 
Random 
3.179487 
1.000000 
1.000000 
{2}PLOT 
Random 

1.000000 
1.000000 
Error 



1.000000 
The coefficients in the Expected mean squares matrix are used to estimate
the population variation of the random effects by equating their variances
to their expected mean squares. For example, the estimated population
variance for Variety using Type Sums of squares would be 3.179487 times
the Mean square for Variety plus
1 times the Mean square for Plot
plus 1 times the Mean square for Error.
The ANOVA method provides an integrative approach to estimating variance
components, but it is not without problems (i.e., ANOVA estimates of variance
components are generally biased, and can be negative, even though variances,
by definition, must be either zero or positive). An alternative to ANOVA
estimation is provided by maximum likelihood estimation. Maximum likelihood
methods for estimating variance components are based on quadratic forms,
and typically, but not always, require iteration to find a solution. Perhaps
the simplest form of maximum likelihood estimation is MIVQUE(0) estimation.
MIVQUE(0) produces Minimum Variance Quadratic Unbiased Estimators (i.e.,
MIVQUE). In MIVQUE(0) estimation, there is no weighting of the random
effects (thus the 0 [zero] after MIVQUE), so an iterative solution for
estimating variance components is not required. MIVQUE(0) estimation begins
by constructing the Quadratic sums of squares (SSQ) matrix. The elements
for the random effects in the SSQ matrix can most simply be described
as the sums of squares of the sums of squares and cross products for each
random effect in the model (after residualization on the fixed effects).
The elements of this matrix provide coefficients, similar to the elements
of the Expected Mean Squares matrix, which are used to estimate the covariances
among the random factors and the dependent variable. The SSQ matrix for
the Wheat.sta data is shown below.
Note that the nonzero offdiagonal element for Variety
and Plot again shows that the
two random factors are at least somewhat confounded.
MIVQUE(0) Variance Component Estimation (wheat.sta) 

SSQ Matrix 
Source 
VARIETY 
PLOT 
Error 
DAMAGE 
{1}VARIETY 
31.90533 
9.53846 
9.53846 
2.418964 
{2}PLOT 
9.53846 
12.00000 
12.00000 
1.318077 
Error 
9.53846 
12.00000 
12.00000 
1.318077 
Restricted Maximum Likelihood (REML) and
Maximum Likelihood (ML) variance component estimation methods are closely
related to MIVQUE(0). In fact, in the program, REML and ML use MIVQUE(0)
estimates as start values for an iterative solution for the variance components,
so the elements of the SSQ matrix serve as initial estimates of the covariances
among the random factors and the dependent variable for both REML and
ML.