More reflections on a study (EDITED)

Okay, before I get to the meat of this post, this quote by Andrew Gelman is dynamite:

. . . I’m generally suspicious of arguments in which the rebound is bigger than the main effect.
 

How many "counter-intuitive" studies would survive this kind of skepticism.  Not that a rebound effect can't be larger, but like many unlikely things it requires a higher level of proof.

The context is an education study which suggests that the more parents pay for education the lower the grades of the student will be.   The authors apparently tried to control for a lot of possible confounders (like SAT scores) but the whole process ends up looking like "what not to do in regression analysis". 

There is an intermediate variable (problem), a restriction of range problem (extrapolating parental support out to values that exceed annual income), and an issue with differential drop-out that does not seem to be addressed.  All of these points are present in Andrew's nice write up. 

What I want to focus on is the sharp counter-factual.  I am not always a fan of counter-factual reasoning, but I think that it would provide a ton of clarity in this case.  The real claim is that if you decreased exposure X (parental support) then you would increase outcome Y (GPA).  The direct causal model would suggest that the fastest way to improve student grades would be to make your contributions zero.  But, the last time I looked, Pell grants require a non-zero parental contribution in most cases (it is a little hard to tell precisely what the thresholds are but they definitely are not zero for most students).  So clearly this is a floor on parental contributions (and if it was the only source of contributions the effect would become the richer the parents the worse the grades of the student). 

So maybe, to have a realistic counter-factual, the exposure should be dollars of support above the minimum expected contribution? 

So, really what we have to be talking about the the effect of a marginal dollar separate from the (non-linear) scale of what the parents are required to pay.  But, even there, the direction is unclear.  Imagine that your not especially inspired child gets admission to Stanford but they are struggling with the material.  Do you insist, on principle, that they get a job or do you pay more so that they have a better chance to be a "C average" Stanford graduate (which is much better than a Stanford drop-out). So the causal direction is actually unclear. 

But if the idea is that giving more resources to students decreases performance then there are a lot of experiments we could try.  For example, we could decrease wages (for everyone including upper managment) and see if performance goes up.  Or we could randomize students to improved levels of support.  Better yet, we could look at experiments that have already been done:

 

We examine the impacts of a private need-based college financial aid program distributing grants at random among first-year Pell Grant recipients at thirteen public Wisconsin universities. The Wisconsin Scholars Grant of $3,500 per year required full-time attendance. Estimates based on four cohorts of students suggest that offering the grant increased completion of a full-time credit load and rates of re-enrollment for a second year of college. An increase of $1,000 in total financial aid received during a student’s first year of college was associated with a 2.8 to 4.1 percentage point increase in rates of enrollment for the second year.
 

 So not only is the main effect in the opposite direction (at least in terms of retention) but it has precisely the impact on a GPA analysis that Andrew expects: students are more likely to leave with lower levels of support.  Do we think that leaving school is completely independent of performance (that there is no GPA difference between the drop-outs and those who persist)?  Or is parental support different, in some magic way, than government grant support?  People are more careful stewards of government money than they are of money from their close community (and think about what this would mean for charity versus government welfare programs, if true)?

I agree that the current form of this study is impossible to interpret. 

[EDIT: Talking with Mark, it is clear that I was unclear on one point above.  The experiments show money from a specific source (i.e. government funding) go in a specific direction but don't at all address whether money from parents has a similar causal effect (Mark is promising to talk about this in a post himself).  The issues of selection, intermediate variables, and experimental evidence from other sources are all important, but without re-analyzing the data it is impossible to prove the directionality of the bias.  As an epidemiologist I am trained to speculate on bias direction/strength but I recognize that is all I am doing.  ]