This is a paper on the work that the article cited: https://www.melbourneinstitute.com/downloads/working_paper_s...There is a critical mathematical mistake with their use of statistical analysis which basically, as an artifact, bakes in this "conclusion" into the "results".It comes about because they manually add "Working hours-squared/100" into the factor analysis, and then, low and behold, there is a parabola correlated with the data. It's a downward facing one, because an upward facing one, or no correlation will not match as well. But the model is not reality.There are numerous other conceptual problems with the study, like "what is work?" and a lack of real data on the nature of work beyond what is correlated with socioeconomic indicators, like University education, however, it is remarkable that even at the level of statistic analysis, the work is critically flawed.I am also disappointed that nobody else on hacker news seems to have pointed this out.Hopefully, this can serve as yet another example, of how to interpret and understand data the new, be it hacker news or otherwise, presents us with. Statistical data analysis is both hard, and often times misleading.

 I skimmed the paper and couldn't find this "critical mistake". The nonlinear model you describe is common and appropriate to situations like this.`````` y ~ x + x^2 `````` A mistake would be`````` y ~ y^2 `````` And even then, it would be appropriate if we're modeling an autoregressive process.`` y_{t} ~ y_{t-1}``
 I'll try to state in this language what I think is the conceptual flaw.The stated conclusion is "there is an optimum number of work hours, and it is less than 40."However, their method of analysis is this:"We fit a non-linear model that is a quadratic model cognitive ability y ~ ax + b x^2 for x work hours / week, and we found a statistical optimum around 25 hours"The problem with this is that you're trying to find the best fit of a parabola to the data. And you have tons of samples where the work hours are very few / none (unemployed). Because there is a fairly strong correlation with unemployment and cognitive indicators, the parabola is already being "forced down" near hours worked = 0.Now in this parabola model of estimated cognitive indicators vs work hours, either you are going to get a minima -- and it goes to infinity at working hours -> infinity (of course in real life it cannot really do this, because we only have so many hours in the week, but the statistical model will suggest it) -- or, you are going to get a maxima, which is what actually happens.It could well be in the data that the indicators are that there is roughly flat, or even increasing response of cognitive indicators to working hours when the number of working hours is beyond a nominal value, but that the unemployed population has somewhat lower indicators.In this case the model will automatically become a downward curving, parabola with a maxima, suggesting decline with increasing work hours -- even though this is not what the data directly suggests.This maxima, the fact that there even is a "work hour optimum" that is a smooth, quadratic curve, is a mirage -- the model is not the data.A remaining question is why the optimum is less than 40 hours. It is relatively easy to construct a statistical case in which it is a curve fitting artifact, despite that there is no direct data even at at the suggested optimum.One could in principle check to see if this is the case. The data may be available.For now, there's few graphs on page 20. It really doesn't seem to me that there is a significant distinction between the part time and full time groups -- in fact, the biggest difference is that more women who have a high reading score are not unemployed. Men who have a higher symbols score are more likely to be full-time employed instead of part-time, slightly -- but the converse is true for men with higher reading scores. The difference is not very distinct.
 > Because there is a fairly strong correlation with unemployment and cognitive indicatorsYou're arguing that there's an endogeneity effect -- that poor cognition causes less working? That's a common problem. The authors discuss their use of an instrumental variable technique to avoid this issue.> parabola is already being "forced down" near hours worked = 0Not sure what you mean. Typically a model like this includes a constant to allow for a non-zero dependent variable when all the explanatory variables are zero. To do otherwise in this case would be absurd. The idea that the average non-working person has zero cognitive function...> smooth, quadratic curve, is a mirageEver heard of a Taylor polynomial?
 You'll learn a lot more if you ask, instead of "what could be wrong about what this person is saying", you ask "what could be right about it?"
 That's exactly what I'm asking for: a clearer explanation.I don't believe the paper's conclusion, but I don't understand your criticism of it. If you're saying the estimated curve is inappropriate, a better argument would be that they should include more terms of the work-hours Taylor expansion to get a better fit. Or perhaps there are confounding variables left out of the model.
 Um, what? The model is right there on page 5 and there's nothing wrong with it. The predictors include working hours, (working hours)^2, and others. The outcome is score on a cognitive assessment.What's your agenda in trying to discredit this study with FUD, I wonder? Hopefully this can serve as yet another example of how to ignore commenters that sound like they know what they're talking about but actually don't.
 Look, I am 100% for the conclusion that we should not as a default case have people working too much! Even a 40 hour work week, for work that is statistically usual today, is in my opinion inhumane.That philosophical belief however does not win out in the context of this particular analysis.My agenda is to support sensible discourse on the question of how our brains adapt and how we live, using rigorous thought. Maybe I'm not seeing the problem with my reasoning but it seems pretty clear to me.The problem is that you automatically get an upward or downward parabola, plotting cognitive scores versus working hours, if you have a non-zero coefficient in working hours^2It's very unlikely that you'll get an upward parabola, especially given the strong anticorrelation between unemployment and cognitive scores that they use.In science, you have a judgment call for which predictors you plug into the statistical analysis. Choosing any particular function, be it working hours^2, sin(working hours/100), or even, say, ballmerpeak(alcohol content) will effect the statistical results in factor analysis or anything else.Since we chose only linear and quadratic functions to get coefficients for in our statistical analysis, the functions are going to be a parabola -- either up or down.
 The coefficient for the squared component could easily be zero. This is called "not statistically significant". There is no guarantee for a parabolic shape.
 The measure of possible data sets where, by standard statistical analysis, the coefficient for the squared component is zero, is tiny. There's no guarantee for a parabola, just a very high degree of certainty.Your other point, that the parabola could be "not statistically significant," is true.But given a strong degree of significant correlation between unemployment and the cognitive indicators, even if the dependence is totally flat for the cognitive indicators between 5 hours worked and 100 hours worked, you will still get a parabola by this method of statistical analysis.Do not forget, this is model fitting.
 > is zeroSigh. No coefficient is ever exactly zero, just very close [0]. I didn't think I needed to explain that when writing, "could be zero."> you will still get a parabolaIf the squared parameter is not statistically significant, the author will likely drop it from the model. In that case, we would not see a parabolic model and the paper wouldn't exist. The authors would have moved on to a different topic, or found a different dataset.If the coefficient is so small that it is indistinguishable from zero (not significant), then we ignore the associated variable entirely. To do otherwise would require us to discuss an infinity of possible variables as if they mattered to the model.> correlation between unemployment and cognitive indicatorsIf you're arguing that the author should have dropped all observations of unemployed persons from the dataset, that's completely separate and has nothing to do with parabolas.[0] "ever" loosely defined.
 "The functions are going to be a parabola." What functions? A plot of which two variables? I have no idea what you're talking about and I'm a grad student in biostats.
 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.