Questionable association between front boarding and air rage

DeCelles and Norton (1) conclude that physical inequality (the presence of a first-class cabin) on airplanes is associated with a greater number of air rage incidents in economy class, and that situational inequality (boarding from the front rather than the middle of the airplane) is associated with a greater number of air rage incidents in both economy class and first class. Their study has many flaws that invalidate their conclusions, but we focus on just one, their failure to recognize a statistical artifact in their analyses.

Decelles and Norton’s (1) table S2 shows that the correlation between front boarding and economy class incidents equals −0.035 (odds ratio 0.1954), and the correlation between front boarding and first class incidents equals −0.019 (odds ratio 0.1498); that is, the association between front boarding and air rage is moderately sized and negative. In table 2 of ref. 1, however, the odds ratios predicting economy class incidents and first class incidents from front boarding and several covariates equal 2.1754 and 11.8594, respectively, indicating that the association between front boarding and air rage is moderately sized and positive.

What has happened here? The surprising reversal of the direction of the association between front boarding and air rage is an example of a statistical phenomenon known as “negative suppression” (2) in psychology and as “qualitative confounding” (3) in epidemiology. Consider a least-squares regression analysis with a criterion Y and two correlated predictors X₁ and X₂ (all variables standardized). Score X₁ and X₂ so that the correlations between Y and X₁ and between Y and X₂ are nonnegative, and so that the correlation between Y and X₁ exceeds the correlation between Y and X₂. Ordinarily, when X₂ is added to the regression containing only X₁, the partial regression coefficient for both X₁ and X₂ has the same sign as, and a magnitude less than, its corresponding simple regression coefficient.

However, if the correlation between X₁ and X₂ is greater than the ratio of the correlation between Y and X₂ and the correlation between Y and X₁ (rX1X2>rYX2/rYX1), then negative suppression occurs. The partial regression coefficient for X₂ reverses sign; the partial regression coefficient for X₁ keeps the same sign but increases in magnitude. For example, if rYX1 equals 0.44, rYX2 equals 0.10, and rX1X2 equals 0.60, then rYX1.X2 equals 0.59 and rYX2.X1 equals −0.26.

DeCelles and Norton (1) used logistic rather than least-squares regression (producing odds ratios rather than standardized regression coefficients), and used multiple predictors rather than just two, but something akin to negative suppression must have occurred. Without the data, however, it is not possible to determine which variables were responsible for the association reversal and exactly how that reversal occurred.

So, does front boarding have a negative association with air rage, or a positive association, as DeCelles and Norton (1) claim? It is impossible to know, given the information provided. Suffice it to say that, generally, suppression effects are considered statistical artifacts unless there is a strong theoretical explanation for their occurrence (4). No such explanation is provided by DeCelles and Norton (1).