The fixed curve was modeled by second and third order polynomial regressions using 12 matrix structures of the

random variance and covariance matrix (G), maintaining the residual effects matrix (R) always equal to the VC.

The multiplicative

random variance coefficients of the other parameters are set at 0.05.

As in any regression analysis, including independent variables in a model is intended to reduce

random variance. A fraction of inside level variance in multilevel regression or a fraction of total variance in other regression models is explained by the independent variables.

Leonard (1975) considered a one-way random classification in which each level of the classification had a random mean and a

random variance that were modeled with exchangeable random variables, i.e.: [y.sub.ij] | [[theta].sub.i], [[sigma].sup.2.sub.i] ~ N([[theta].sub.i], [[sigma].sup.2.sub.i]), where [[theta].sub.i] | [[sigma].sup.2.sub.i] ~ N(0, [[sigma].sup.2.sub.i]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], with [[alpha].sub.i] | [[sigma].sup.2.sub.[alpha]] ~ N(0, [[sigma].sup.2.sub.[alpha]]).

By recognizing the resonance effects from ground vias, one can let the tool apply a random variance to the via pattern to avoid having all via-formed cavities resonating at the same frequency.

FIGURE 2 shows a contour stitching and a surface via pepper with random variance applied.

To produce the combined curve, we regressed these estimates against indicator variables for each level, using inverse variance weighting and allowing for a random variance component to capture heterogeneity in the association across cities.

where [d.sub.j] are dummy variables for the j exposure levels, [V.sub.ij] is the estimated variance in city i at level j, and [delta] is the estimated random variance component.

To combine results across cities, we used inverse variance weighted averages including a

random variance component to incorporate heterogeneity.

We extended their method to incorporate

random variance components.

These estimated the overall effect as a weighted average, with weights equal to the inverse of the sum of the square of the standard error plus a

random variance component.

The observed variances of facets are composed of three elements: common variances, specific variances, and

random variances (Nunnally, 1978).