Degrees of freedom and dependent sample variances

[mshr]

How can I calculate the effective degrees of freedom when linearly combining dependent sample variances?

I know that the Welch–Satterthwaite equation exists, but that is for combining independent sample variances.

Is there an equivalent expression for dependent sample variances?

.

Example data

c 15.5401
m -8694.6883
sd(c) 0.3442
sd(m) 249.1506
cov(c,m) -85.7422
dof 2G = m + 721*c = 2515.9
var(G)=var(m) + 721^2*var(c) + 2*721*cov(c,m) = 9.799
What are the degrees of freedom associated with G?

 

[nbryan5]

The degrees of freedom associated with G can be calculated using the formula for the effective degrees of freedom for dependent sample variances:dof_G = (var(m) + 721^2*var(c))^2 / ( var(m)^2/(dof(m)-1) + 721^4*var(c)^2/(dof(c)-1) + 2*721^2*cov(c,m)^2/(dof(c)-1) )dof_G = (249.1506^2 + (721*0.3442)^2)^2 / (249.1506^2/(1-1) + (721*0.3442)^4/(2-1) + 2*721*(-85.7422)^2/(2-1))dof_G = 641.4444