  We show how both smaller and more reliable p-values can be computed in
Bell-type experiments by using statistical deviations from no-signalling
equalities to reduce statistical noise in the estimation of Bell's S or
Eberhard's J. Further improvement is obtained by using Wilks' likelihood ratio
test based on the four tetranomially distributed vectors of counts of the four
different outcome combinations, one 4-vector for each of the four setting
combinations. The methodology is illustrated by application to the
loophole-free Bell experiments of 2015 and 2016 performed in Delft and Munich,
at NIST, and in Vienna respectively; and also to the earlier Innsbruck
experiment of Weihs et al. (1998) and the recent Munich experiment of Zhang et
al. (2022), which investigates use of a loophole-free Bell experiment as part
of a protocol for Device Independent Quantum Key Distribution, DIQKD.
