Analytic Calculation of Covariance between Cosmological Parameters from
  Correlated Data Sets, with an Application to SPTpol
  Consistency checks of cosmological data sets are an important tool because
they may suggest systematic errors or the type of modifications to $\Lambda$CDM
necessary to resolve current tensions. In this work, we derive an analytic
method for calculating the level of correlations between model parameters from
two correlated cosmological data sets, which complements more computationally
expensive simulations. This method is an extension of the Fisher analysis that
assumes a Gaussian likelihood and a known data covariance matrix. We apply this
method to the SPTpol temperature and polarization CMB spectra (TE and EE). We
find weak correlations between $\Lambda$CDM parameters with a 9$\%$ correlation
between the TE-only and EE-only constraints on $H_0$ and a 25$\%$ and 32$\%$
correlation for log($A_s$) and $n_s$ respectively. Despite the negative
correlations between the TE and EE power spectra, the correlations in the
parameters are positive. The TE-EE parameter differences are consistent with
zero, with a PTE of 0.53, in contrast to the PTE of 0.017 reported by SPTpol
for the consistency of the TE and EE power spectra with $\Lambda$CDM. Using
simulations we find that the results of these two tests are independent and
that this difference can arise simply from statistical fluctuations. Ignoring
correlations in the TT-TE and TE-EE comparisons biases the $\chi^2$ low,
artificially making parameters look more consistent. Therefore, we conclude
that these correlations need to be accounted for when performing internal
consistency checks of the TT vs TE vs EE power spectra for future CMB analyses.
