“…Such a split into random subsamples was first proposed as part of a graphical inspection of the invariance of 1-PL Rasch model item parameters (Hambleton & Murray, 1983), and was soon employed for the LRT (e.g., Maier & Philipp, 1985, 1986Maier, Philipp, Buller, & Schiegel, 1987). There are numerous examples of using a random split (e.g., Devy, Lehert, Varlan, Genty, & Edan, 2015;Gnambs & Batinic, 2011;Kliem et al, 2015;Koller & Alexandrowicz, 2010;Rusch, Mair, Lowry, & Treiblmaier, 2013). However, little is known about Type I error and power of an LRT with a random split.…”