Are the absent always wrong? Dealing with zero values in urban scaling

Both theoretical and empirical studies have shown the ability of scaling laws to reveal processes of emergence in urban systems. Nevertheless, a controversy about the robustness of results obtained with these models on empirical cases remains, regarding for instance the definition of the 'city' considered or the way the estimations are performed. Another source of bias is highlighted in this contribution, with respect to the non-ubiquitous character of some urban attributes (i.e. their partial absence from several cities of the system). The problem with the zero count for cities where these attributes are absent is that the technical necessities of usual estimation procedures make the analysis ignore them altogether even when they represent some valid information. This could seriously impact the results. A precise exploration of the effects of this arbitrary filtering is conducted here, and several solutions are proposed to overcome this limitation. In a case study about foreign investment towards French cities, we show that some erroneous conclusions about a hierarchical diffusion could be drawn when adopting the classical ordinary least squares approach. The framework we suggest specifies how it is possible to avoid misinterpretations deriving from the exclusion of zero values by using methods of analysis which deal with zero values specifically. The conclusion of a diffusion of foreign investment in the French urban system is then rejected.