This paper shows both analytically and empirically that, ignoring nonlinear term (in this article, quadratic effects) in mixed modeling will lead to significant, but spurious random slope variance or cross-level interaction. Moreover, the power (fake) of the significant test is nontrivial at all.
Some thoughts on this article with possible future directions are:
Analytically, the parameter estimates of both fixed effects and random effects will be biased. This has been derived for the first case in which the fitted model includes a random slope term. For the second case in which the fitted model includes a cross-level interaction term, the paper does not derive the bias for parameter estimates except for the spurious interaction coefficient.
Ignoring higher-order effects other than quadratic effects can be expected to lead to biased estimates and spurious coefficients. Ignoring cross-level interaction or random slope might also lead to spurious quadratic effect. There are thousands of possibilities and it is extremely time consuming to explore every one of them. Moreover, even if you explore all possibilities, it is simply impossible to decide the true (or best, or correct) model.
One commonly suggested way is to use data diagnostics. Residuals can be plotted against the predicted values to judge for model misspecification. It is simple for practitioners but has little to do if which term of the model is omitted is unknown. Because one still has to try every potential effect (interaction, quadratic, etc. ) and to test their significance.
Another way is to use a general method to evaluate the influence of misspecification. First derivative of the estimating function with respect of the target parameter can give information of whether it is positively biased (larger than 0), negatively biased (smaller than 0), or unbiased (equal to zero). This method does not require derivation of parameter estimates and is easier to apply than the method in this article. There are several parameters in the table that are not biased due to misspecification. It can also be detected by the derivative method.