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Consistency checks of cosmological data sets are an important tool becausethey may suggest systematic errors or the type of modifications to $\Lambda$CDMnecessary to resolve current tensions. In this work, we derive an analyticmethod for calculating the level of correlations between model parameters fromtwo correlated cosmological data sets, which complements more computationallyexpensive simulations. This method is an extension of the Fisher analysis thatassumes a Gaussian likelihood and a known data covariance matrix. We apply thismethod to the SPTpol temperature and polarization CMB spectra (TE and EE). Wefind weak correlations between $\Lambda$CDM parameters with a 9$\%$ correlationbetween 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 negativecorrelations between the TE and EE power spectra, the correlations in theparameters are positive. The TE-EE parameter differences are consistent withzero, with a PTE of 0.53, in contrast to the PTE of 0.017 reported by SPTpolfor the consistency of the TE and EE power spectra with $\Lambda$CDM. Usingsimulations we find that the results of these two tests are independent andthat this difference can arise simply from statistical fluctuations. Ignoringcorrelations in the TT-TE and TE-EE comparisons biases the $\chi^2$ low,artificially making parameters look more consistent. Therefore, we concludethat these correlations need to be accounted for when performing internalconsistency checks of the TT vs TE vs EE power spectra for future CMB analyses.

Analytic Calculation of Covariance between Cosmological Parameters from Correlated Data Sets, with an Application to SPTpol