Co-author of the paper checking in here, to clarify...
It's more that there is a very strong, robust statistical association between exposure to fine particular matter (to which emissions of NOx can ultimately contribute, by way of particulate nitrate) and premature mortality. This is based on regressions (not necessarily linear, but you're on the right track) of health data and ambient pollutant concentrations, when other co-founders are accounted for.
This association was first noticed in some famous studies (aka the "6 cities studies") in the US in the 90's and has been repeated 100's of times since then. Essentially, exposure to 10 ug/m3 of fine particulate matter over the course of a year can increase your chances of death owing to heart disease and lung cancer by several percent. This information was used in the current paper being discussed, but not something that we contributed to here (i.e., we used results from the literature). What we did do, however, was a lot of super-computer simulations to estimate how much particulate nitrate (and ozone) are formed from the diesel NOx emissions, and where they are formed, in order to calculate (given previously identified concentration-response relationships) impacts on human health.