Indeed there was was a study published in 2007 by Ian Walker "Bicycle helmet wearing is associated with closer overtaking by drivers". Olivier and Walter (2013 – ‘OW’) concluded their re-analysis of Walker’s (2007 – ‘W7’) data with the strong statement “helmet wearing is associated with a small difference in passing distance and is not associated with close passing”


The original study considered around 2400 passes on a single rider, riding near Bath in the UK. This study has been used extensively to underpin the statement that drivers are more likely to pass, more closely, riders with helmets.


It seems to me that the original paper was a very small sample, based on single rider in a very limited geographical area. There is also some dispute regarding the statistical validity of the original research.