An incomplete variant of Wilson's congruence

This article examines the nontrivial solutions of the congruence $$(p-1)\cdots (p-r)\equiv -1(\mathrm{m}\mathrm{o}\mathrm{d}p)\mathrm{.}$$ We discuss heuristics for the proportion of primes $p$ that have exactly $N$ solutions to this congruence. We supply numerical evidence in favour of these conjectures, and discuss the algorithms used in our calculations.