Linear controls are not enough to account for multiplicative confound effects on air rage

Regarding the recent publication in PNAS of an article on predictors of air rage (1), I am concerned that the main regression analyses did not control sufficiently for the two highly correlated factors of number of seats and length of flight.

The dependent variable analyzed, raw incidence of air rage per flight, is naturally confounded by the number of passengers, modeled in this analysis by number of seats, times the length of the flight, modeled by miles. The authors controlled for both main effects, but not for the product (i.e., interaction), of these two factors. Effectively, they controlled for number of seats, and additionally for the fraction of variance in flight time not accounted for by the high correlation with number of seats. This method does not control enough for the mere opportunity for person-level events during a flight, which depends on the full product of those two factors.

All things being equal, this analysis would overestimate the positive impact of factors positively correlated with seats/flight length (e.g., presence of first class on a flight in model 1) and underestimate the positive impact of factors negatively correlated with seats/flight length (e.g., the front boarding factor in model 2).

It would be also desirable to control for quadratic effects of each term on top of the interaction (i.e., the length² × seats and seats² × length terms.) These products would control for differences in incidents due to incremental effects of time and flight size per person-hour. For example, the length² × seats factor models whether there is more air rage on longer flights, apart from the mere fact that they have more passengers and more time for a random event to emerge. This factor would represent possible effects of fatigue or environmental frustration on longer flights, effects that might be expected and would also confound the effects of the focal variables.

My suggestion is for the authors to repeat all their basic and robust logistic regression models, but to calculate and include the following additional predictor effects simultaneously with or before the predictor variables of interest (i.e., presence of first class, front boarding):

• Seats × flight length interaction

• Seats² × flight length interaction

• Seats × flight length² interaction

Additionally, it would be useful to know the zero-order effects of the variables of interest upon incidents, controlling only for seats, flight length, and their interaction. Zero-order effects would help determine whether the overall effect of other factors is to suppress or facilitate the focal effects.

I have every expectation that these authors will be interested in making sure that the analyses in their articles are correct and will cooperate in providing any analyses of their dataset required in their reply.