After model comparison with 'anova()' command, variation in level 3 turned out to be not significant which means that 3 level approach can be inappropriate for the data.

In this case, do I have to adopt a level 2 mixed approach? The problem is that i guess some variables are seemingly higher level variables (e.g., institution),which was proven to be insignificant, while other variables are relatively lower level variables (e.g., female ratio). In my guess, integrating different characteristics into a mixed model can be problematic. Additionally, since more than one effectsize were reported in a study, some variable like 'outcome type' can be a covariate explaining within study variation.

Succinctly, is it acceptable to put different characteristics variables in a equation in 2 level approach, where 3 level test was insignificant?