I was trying to figure out satterthwaite's degrees of freedom for the combination of variance component to calculate confidence interval.

I can get the confidence interval for individual variances using lme4 package in R. I am interested in calculating confidence interval for combination of variance components. In order to calculate CI for variance combinations, degrees of freedom needed to be calculated based on one of the approximation method like Satterthwaite or Welch. I am seeking help to get the generalized formula to calculate degrees of freedom.

I would highly appriciate your help on this.

Thanks in advance, DJ