Variance components

Variance components are estimates of a part of the total variability accounted for by a specified source of variability.

Random factors are factors where a number of levels are randomly sampled from the population, and the intention is to make inferences about the population. For example, a study might examine the precision of a measurement procedure in different laboratories. In this case, there are 2 variance components: variation within an individual laboratory and the variation among all laboratories. When performing the study it is impractical to study all laboratories, so instead a random sample of laboratories are used. More complex studies may examine the precision within a single run, within a single laboratory, and across laboratories.

Most precision studies use a nested (or hierarchical) model where each level of a nested factor is unique amongst each level of the outer factor. The basis for estimating the variance components is the nested analysis of variance (ANOVA). Estimates of the variance components are extracted from the ANOVA by equating the mean squares to the expected mean squares. If the variance is negative, usually due to a small sample size, it is set to zero. Variance components are combined by summing them to estimate the precision under different conditions of measurements.

The variance components can be expressed as a variance, standard deviation (SD), or coefficient of variation (CV). A point estimate is a single value that is the best estimate of the true unknown parameter; a confidence interval is a range of values and indicates the uncertainty of the estimate. A larger estimate reflects less precision.

There are numerous methods for constructing confidence intervals for variance components:
Estimator Description
Exact Used to form intervals on the inner-most nested variance components. Based on the F-distribution.
Satterthwaite Used to form intervals on the sum of the variances. Based on the F-distribution, as above, but uses modified degrees of freedom. Works well when the factors have equal or a large number of levels, though when the differences between them are large it can produce unacceptably liberal confidence intervals.
Modified Large Sample (MLS) Used to form intervals on the sum of variances or individual components. A modification of the large sample normal theory approach to constructing confidence intervals. Provides good coverage close to the nominal level in a wide range of cases.