{'fraction_non_alphanumeric': 0.03217391304347826, 'fraction_numerical': 0.011304347826086957, 'mean_word_length': 5.430167597765363, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Double Decomposition of Level-1 Variables in Multilevel Models: An Analysis of the Flynn Effect in the NSLY Data

ABSTRACT This paper introduces an extension of cluster mean centering (also called group mean centering) for multilevel models, which we call “double decomposition (DD).” This centering method separates between-level variance, as in cluster mean centering, but also decomposes within-level variance of the same variable. This process retains the benefits of cluster mean centering but allows for context variables derived from lower level variables, other than the cluster mean, to be incorporated into the model. A brief simulation study is presented, demonstrating the potential advantage (or even necessity) for DD in certain circumstances. Several applications to multilevel analysis are discussed. Finally, an empirical demonstration examining the Flynn effect (Flynn, 1987), our motivating example, is presented. The use of DD in the analysis provides a novel method to narrow the field of plausible causal hypotheses regarding the Flynn effect, in line with suggestions by a number of researchers (Mingroni, 2014; Rodgers, 2015).