  Gene expression is stochastic and displays variation ("noise") both within
and between cells. Intracellular (intrinsic) variance can be distinguished from
extracellular (extrinsic) variance by applying the law of total variance to
data from two-reporter assays that probe expression of identical gene pairs in
single-cells. We examine established formulas for the estimation of intrinsic
and extrinsic noise and provide interpretations of them in terms of a
hierarchical model. This allows us to derive corrections that minimize the mean
squared error, an objective that may be important when sample sizes are small.
The statistical framework also highlights the need for quantile normalization,
and provides justification for the use of the sample correlation between the
two reporter expression levels to estimate the percent contribution of
extrinsic noise to the total noise. Finally, we provide a geometric
interpretation of these results that clarifies the current interpretation.
