“…Then, write Rearranging Equation 5, we obtain Because S denotes the sum of the entries in a ( k × k ) correlation matrix, it can be shown that Rearranging Equation 7, we find that Next, 4 after substituting Equation 6 into Equation 8, we have Thus, to generate a data set with k items that yields a known value of α ∈ (0, 1), a researcher can use Equation 9 to quickly determine r¯ij. With this quantity in hand, they can then use theory and methods described in Waller (2016) to generate all correlation matrices that produce the desired α. Note that for fixed c , where c ∈ (0, 1), if R is a k × k correlation matrix ( k ≥ 2) then the set double-struckS=falsefalse{R falsefalse| αfalse(Rfalse)=cfalsefalse} is infinite (where α( R ) denotes the α computed on set member R ).…”