  Several correlations among parameters derived from modelling the high-energy
properties of 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 ($E_{\rm pk}$-$E_{\rm
iso}$) 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 $E_{\rm pk}$-$E_{\gamma}$ ("Ghirlanda") correlation is effectively
independent of the GRB redshifts, which suggests 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 $E_{\rm pk}$-$E_{\gamma}$ correlation, we
find that the pre-Swift, bright GRB $E_{\rm pk}$-$E_{\rm iso}$ correlation of
\citet{amati06} does not rigorously satisfy these conditions.
