Several correlations among parameters derived from modeling the high-energy properties of gamma-ray bursts (GRBs) have been reported. We show that well known examples of these have common features indicative of strong contamination by selection effects. We focus here on the impact of detector threshold truncation on the spectral peak versus isotropic equivalent energy release (E _pk-E _iso) relation, extended to a large sample of 218 Swift and 56 HETE-2 GRBs with and without measured redshift. The existence of faint Swift events missing from pre-Swift surveys calls into question inferences based on pre-Swift surveys which must be subject to complicated incompleteness effects. We demonstrate a generalized method for treating data truncation in correlation analyses and apply this method to Swift and pre-Swift data. Also, we show that the E _pk-E _γ ("Ghirlanda") correlation is effectively independent of the GRB redshifts, which suggests that its existence has little to do with intrinsic physics. We suggest that a physically based correlation, manifest observationally, must show significantly reduced scatter in the rest frame relative to the observer frame and must not persist if the assumed redshifts are scattered. As with the E _pk-E _γ correlation, we find that the pre-Swift, bright GRB E _pk-E _iso correlation of Amati does not rigorously satisfy these conditions.