Why You Should Always Include a Random Slope for the Lower-Level Variable Involved in a Cross-Level Interaction

Mixed effects multilevel models are often used to investigate cross-level interactions, a specific type of context effect that may be understood as an upper-level variable moderating the association between a lower-level predictor and the outcome. We argue that multilevel models involving cross-level interactions should always include random slopes on the lower-level components of those interactions. Failure to do so will usually result in severely anti-conservative statistical inference. Monte Carlo simulations and illustrative empirical analyses highlight the practical relevance of the issue. Using European Social Survey data, we examine a total 30 cross-level interactions. Introducing a random slope term on the lower-level variable involved in a cross-level interaction, reduces the absolute t-ratio by 31% or more in three quarters of cases, with an average reduction of 42%. Many practitioners seem to be unaware of these issues. Roughly half of the cross-level interaction estimates published in the European Sociological Review between 2011 and 2016 are based on models that omit the crucial random slope term. Detailed analysis of the associated test statistics suggests that many of the estimates would not meet conventional standards of statistical significance if estimated using the correct specification. This raises the question how much robust evidence of cross-level interactions sociology has actually produced over the past decades.