I'm sorry, I am trying to explain something that is very clear in my head, and I'm pretty sure that I'm right, but I haven't had bio-stats training specifically so I do not know the language precisely. My background is in physics, computer science, and epistemology.The functions I am referring to are estimators of cognitive indicators (like backwards digit span, say, that they use in the paper), as a function of working hours.Take a look at page 21 for some plots. For each of the cognitive indicators, the estimator is a downwards parabola as a function of working hours. What I am saying is that this is an artifact of the analysis. The shape could be far different -- in fact it could be a bad case of curve fitting. Additionally -- why not just directly plot the data as a scatter plot or a binned average of cognitive indicators for bins between, say, 20 - 25 hours, 25 - 30 hours, etc? Then at least we could see if the parabolas are close to the data...

 There's nothing wrong with the inclusion of a quadratic term in a linear model if the variable is significant, which is clearly is according to Table IV.You can't just plot a single predictor against the sample outcome and expect the plot to be particularly revealing in multiple regression. Plus, this isn't even multiple regression; this is a two-stage least squares multiple regression. The working hour (WH) variables are instruments, not predictors. See page 6.Instrumental variables exist specifically to deal with the case of a possible bidirectional causal association between predictor and response.
 I'm afraid I don't think you are understanding my point, but apparently it is a difficult one to make.I'm unfortunately too busy to make it clearer, so I will just leave you with a koan.Why not include a third order term in the regression? What about an n-th order term? What assumptions do we "bake into" the results of a statistical regression as an effect of including, or not, any function on the original data?The statistical significance of the quadratic term is actually dependent upon the presence of any more complex or higher order terms in the regression, just as the coefficients and the statistics of the linear term will depend on the presence of the second order term in the analysis.I'm not saying you should never include a quadratic term in a regression, I'm saying we should understand what the regression is doing when it is fitting a model.
 Why not include insignificant terms in the regression? Maybe because they're insignificant?I understand what the regression is doing. The authors understand. You do not, though.You've already admitting not to having a background in stats, yet you keep throwing around words like "significance" and "model fitting" without having the faintest clue what they mean mathematically. I'm sorry, but I can't fit several semesters of undergrad-level stats in these comment boxes.