monkey's uncle

Priceless. Steve Pinker wrote a spectacular review of Malcolm Gladwell’s latest book, What the Dog Saw and Other Adventures, in the New York Times today. I regularly read and enjoy Gladwell’s essays in the New Yorker, but I find his style sometimes problematic, verging on anti-intellectual, and I’m thrilled to see a scientist of Pinker’s stature calling him out.

Pinker coins a term for the problem with Gladwell’s latest book and his work more generally.  Pinker’s term: “The Igon Value Problem” is a clever play on the Eigenvalue Problem in mathematics.  You see, Gladwell apparently quotes someone referring to an “igon value.” This is clearly a concept he never dealt with himself even though it is a ubiquitous tool in the statistics and decision science about which Gladwell is frequently so critical.  According to Pinker, the Igon Value Problem occurs “when a writer’s education on a topic consists in interviewing an expert,” leading him or her to offering “generalizations that are banal, obtuse or flat wrong.”  In other words, the Igon Value Problem is one of dilettantism.  Now, this is clearly a constant concern for any science writer, who has the unenviable task of rendering extremely complex and frequently quite technical information down to something that is simultaneously accurate, understandable, and interesting. However, when the bread and butter of one’s work involves criticizing scientific orthodoxy, it seems like one needs to be extremely vigilant to get the scientific orthodoxy right.

Pinker raises the extremely important point that the decisions we make using the formal tools of decision science (and cognate fields) represent solutions to the inevitable trade-offs between information and cost.  This cost can take the form of financial cost, time spent on the problem, or computational resources, to name a few. Pinker writes:

Improving the ability of your detection technology to discriminate signals from noise is always a good thing, because it lowers the chance you’ll mistake a target for a distractor or vice versa. But given the technology you have, there is an optimal threshold for a decision, which depends on the relative costs of missing a target and issuing a false alarm. By failing to identify this trade-off, Gladwell bamboozles his readers with pseudoparadoxes about the limitations of pictures and the downside of precise information.

Pinker is particularly critical of an analogy Gladwell draws in one of his essays between predicting the success of future teachers and future professional quarterbacks.  Both are difficult decision tasks fraught with uncertainty.  Predicting whether an individual will be a quality teacher based on his or her performance on standardized tests or the presence or absence of teaching credentials is an imperfect process just as predicting the success of a quarterback in the N.F.L. based on his performance at the collegiate level.  Gladwell argues that anyone with a college degree should be allowed to teach and that the determination of the qualification for the job beyond the college degree should only be made after they have taught. This solution, he argues, is better than the standard practice of  credentialing, evaluating, and “going back and looking for better predictors.” You know, science? Pinker doesn’t hold back in his evaluation of this logic:

But this “solution” misses the whole point of assessment, which is not clairvoyance but cost-effectiveness. To hire teachers indiscriminately and judge them on the job is an example of “going back and looking for better predictors”: the first year of a career is being used to predict the remainder. It’s simply the predictor that’s most expensive (in dollars and poorly taught students) along the accuracy- cost trade-off. Nor does the absurdity of this solution for professional athletics (should every college quarterback play in the N.F.L.?) give Gladwell doubts about his misleading analogy between hiring teachers (where the goal is to weed out the bottom 15 percent) and drafting quarterbacks (where the goal is to discover the sliver of a percentage point at the top).

This evaluation is spot-on. As a bit of an aside, the discussion of predicting the quality of prospective quarterbacks also reminds me of one of the great masterpieces of statistical science and the approach described by this paper certainly has a bearing on the types of predictive problems of which Gladwell ruminates.  In a 1975 paper, Brad Efron and Carl Morris present a method for predicting 18 major league baseball players’ 1970 season batting average based on their first 45 at-bats. The naïve method for predicting (no doubt, the approach Gladwell’s straw “we” would take) is simply to use the average after the first 45 at-bats. Turns out, there is a better way to solve the problem, in the sense that you can make more precise predictions (though hardly clairvoyant).  The method turns on what a Bayesian would call “exchangeability.”  Basically, the idea is that being a major league baseball player buys you a certain base prediction for the batting average.  So if we combine the averages across the 18 players and with each individual’s average in a weighted manner, we can make a prediction that has less variation in it.  A player’s average after a small number of at-bats is a reflection of his abilities but also lots of forces that are out of his control — i.e., are due to chance.  Thus, the uncertainty we have in a player’s batting based on this small record is partly due to the inherent variability in his performance but also due to sampling error.  By pooling across players, we combine strength and remove some of this sampling error, allowing us to make more precise predictions. This approach is lucidly discussed in great detail in my colleague Simon Jackman’s new book, draft chapters of which we used when we taught our course on Bayesian statistical methods for the social sciences.

Teacher training and credentialing can be thought of as strategies for ensuring exchangability in teachers, aiding the prediction of teacher performance.  I am not an expert, but it seems like we have a long way to go before we can make good predictions about who will become an effective teacher and who will not.  This doesn’t mean that we should stop trying.

Janet Maslin, in her review of What the Dog Saw, waxes about Gladwell’s scientific approach to his essays. She writes that the dispassionate tone of his essays “tames visceral events by approaching them scientifically.” I fear that this sentiment, like the statements made in so many Gladwell works, reflects the great gulf between most educated Americans and the realities of scientific practice (we won’t even talk about the gulf between less educated Americans and science).  Science is actually a passionate, messy endeavor and sometimes we really do get better by going back and finding better predictors.

It’s that time of the year again, it seems, when I have lots of students writing proposals to submit to NSF to fund their graduate education or dissertation research.  This always sets me to thinking about the practice of science and how one goes about being a successful scientist. I’ve written about “productive stupidity” before, and I still think that is very important. Before I had a blog, I composed a series of notes on how to write a successful NSF Doctoral Dissertation Improvement Grant when I saw the same mistakes over and over again sitting on the Cultural Anthropology panel.

This year, I’ve find myself thinking a lot about what Craig Loehle dubbed “the Medawar Zone.” This is an nod to the great British scientist, Sir Peter Medawar, whose book, The Art of the Soluble: Creativity and Originality in Science, argued that best kind of scientific problems are those that can be solved.  In his classic (1990) paper Loehle argues that “there is a general parabolic relationship between the difficulty of a problem and its likely payoff.” Re-reading this paper got me to thinking.

In Loehle’s figure 1, he defines the Medawar Zone.  I have reproduced a sketch of the Medawar Zone here.

Now, what occurred to me on this most recent reading of this paper is that for a net payoff curve to look like this, the benefits with increased difficulty of the problem are almost certainly concave.  That is, they show diminishing marginal returns to increased difficulty.  Hard to say what the cost curve with difficulty would be – linear? convex? Either way, there is an intermediate maximum (akin to Gadgil and Bossert’s analysis of intermediate levels of reproductive effort) and the best plan is to pick a problem of intermediate difficulty because that is where the scientific benefits, net of the costs, are maximized.

Suppose that a dissertation is a risky endeavor.  This is not hard for me to suppose since I know many people from grad school days who had at least one failed dissertation project.  Sometimes this led to choosing another, typically less ambitious project.  Sometimes it led to an exit from grad school, sans Ph.D.  Stanford (like Harvard now, but not when I was a student) funds its Ph.D. students for effectively the entirety of their Ph.D.  This is a great thing for students because nothing interferes with your ability to think and be intellectually productive than worrying about how you’re going to pay rent.  The downside of this generous funding is that students do not have much time to come up with an interesting dissertation project, write grants, go to the field, collect data, and write up before their funding runs out. So, writing a dissertation is risky.  There is always a chance that if you pick too hard a problem, you might not finish in time and your funding will run out. Well, it just so happens that the combination of a concave utility function and a risk of failure is pretty much the definition of a risk-averse decision-maker.

Say there is an average degree of difficulty in a field.  A student can choose to work on a topic that is more challenging than the average but there is the very real chance that such a project will fail and in order for the student to finish the Ph.D., she will have to quickly complete work on a problem that is easier than the average.  Because the payoff curve with difficulty is concave, it means that the amount you lose relative to the mean if you fail is much greater than the amount you gain relative to the mean if you succeed.  That is, your downside cost is much greater than your upside benefit.

In the figure, note that d1>>d2.  Here, I have labeled the ordinate as w, which is the population genetics convention for fitness (i.e., the payoff).  The bar-x is the mean difficulty, while x2 and x1 are the high and low difficulty projects respectively.

The way that economists typically think about risk-aversion is that a risk-averse agent is one who is willing to pay a premium for certainty.  This certainty premium is depicted in the dotted line stretching back horizontally from the vertical dashed line at x=xbar to the utility curve.  The certain payoff the agent is willing to accept vs. the uncertain mean is where this dotted line hits the utility curve. Being at this point on the utility curve (where you have paid the certainty premium) probably puts you at the lower end of the Medawar Zone envelope, but hopefully, you’re still in it.

I think that this very standard analysis actually provides the graduate student with pretty good advice. Pick a project you can do and maybe be a bit conservative.  The Ph.D. isn’t a career – it’s a launching point for a career. The best dissertation, after all, is a done dissertation.  While I think this is sensible advice for just about anyone working on a Ph.D., the thought of science progressing in such a conservative manner frankly gives me chills.  Talk about a recipe for normal science!  It seems what we need, institutionally, is a period in which conservatism is not the best option. This may just be the post-doc period.  For me, my time at the University of Washington (CSSS and CSDE) was a period when I had unmitigated freedom to explore methods relevant to what I was hired to do.  I learned more in two years than in – I’d rather not say how many – years of graduate school. The very prestigious post-doctoral programs such as the Miller Fellowships at Berkeley or the Society of Fellows at Harvard or Michigan seem like they are specifically designed to provide the environment where the concavity of the difficulty-payoff curve is reversed (favoring gambles on more difficult projects).

There is, unfortunately, a folklore that has diffused to me through graduate student networks that says that anthropologists need to get a faculty position straight out of their Ph.D. or they will never succeed professionally.  This is just the sort of received wisdom that makes my skin crawl and, I fear, is far too common in our field.  If our hurried-through Ph.D.s can’t take the time to take risks, when can we ever expect them to do great work and solve truly difficult problems?

As we enter flu season, I think that the importance of hand-washing can not be overstated for maintaining health. Here is a (somewhat ugly) flyer published by CDC:

You’d think CDC could hire some graphic designers!

Two things that I think many people don’t appreciate: (1) washing hands with hot soapy water is better than using alcohol-based sanitizer and (2) you need to wash your hands for a long time to get best results.  A funny story on NPR last spring provides some ideas for timing your hand-washing.  I’m afraid I can’t help but sing Bohemian Rhapsody (starting at time stamp 3:06 for 20 secs) while washing my hands ever since hearing this story…

Perhaps it’s just me being a bit groggy from jet-lag, but I just read one of the most bizarre things I think I have ever seen in the New York Times.  There is a generally very interesting article by Sarah Kershaw on so-called “cougars,” older women who have sexual relationships with younger men. It was the first I had ever heard the term – shows what I know. As the article concludes, Kershaw makes the following statement:

The paradox, of course, is that the older-woman relationship makes perfect sense when it comes to life expectancy, with women outliving men by an average of five years. But with men’s fertility far outlasting women’s, biology makes the case for the older-man scenario, and recent research has even suggested that older men having children with younger women is a key to the survival of the human species.

Say what?! Survival of the species??

It’s a pretty strange statement that strangely lacks attribution, particularly given how well referenced all the other scholarly work discussed in the article is.  I wonder if it isn’t a vague allusion to the work of my colleague Shripad Tuljapurkar who has shown that systematic differences in mean age of childbearing would mitigate the so-called “wall of death” predicted by W.D. Hamilton’s famous paper on the evolution of senescence.

My last post, which I had to cut short, discussed the recent paper by Scheffer et al. (2009) on the early warning signs of impending catastrophe. This paper encapsulates a number of things that I think are very important and relate to some current research (and teaching interests). Scheffer and colleagues show the consequences on time series of state observations when a dynamical system characterized by a fold bifurcation is forced across its attractor where parts of this attractor are stable and others are unstable.  In my last post, I described the fold catastrophe model as an attractor that looks like an “sideways N.” I just wanted to briefly unpack that statement.  First, an attractor is kind of like an equilibrium.  It’s a set of points to which a dynamical system evolves.  When the system is perturbed, it tends to return to an attractor.  Attractors can be fixed points or cycles or extremely complex shapes, depending upon the particulars of the system.

The fold catastrophe model posits an attractor that looks like this figure, which I have more or less re-created from Scheffer et al. (2009), Box 1.

The solid parts of the curve are stable — when the system state is perturbed when in the vicinity of this part of the attractor, it tends to return, as indicated by the grey arrows pointing back to the attractor.  The dashed part of the attractor is unstable — perturbations in this neighborhood tend to move away from the attractor.  This graphical representation of the system makes it pretty easy to see how a small perturbation could dramatically change the system if the current combination of conditions and system state place the system on the attractor near the neighborhood where the attractor changes from stable to unstable.  The figure illustrates one such scenario.  The conditions/system state start at point F1. A small forcing perturbs the system off this point across the bifurcation.  Further forcing now moves the system way off the current state to some new, far away, stable state.  We go from a very high value of the system state to a very low value with only a very small change in conditions.  Indeed, in this figure, the conditions remain constant from point F1 to the new value indicated by the white point — just a brief perturbation was sufficient to cause the drastic change.  I guess this is part of the definition of a catastrophe.

The real question in my mind, and one that others have asked, is how relevant is the fold catastrophe model for real systems?  This is something I’m going to have to think about. One thing that is certain is that this is a pedagogically very useful approach as it makes you think… and worry.

There is an extremely cool paper in this week’s Nature by Scheffer and colleagues. I’m too busy right now to write much about it, but I wanted to mention it, even if only briefly.  The thing that I find so remarkable about this paper is that it’s really not the sort of thing that I usually like.  The paper essentially argues that there are certain generic features of many systems as they move toward catastrophic change.  The paper discusses epileptic seizures, asthma attacks, market collapses, abrupt shifts in oceanic circulation and climate, and ecological catastrophes such as sudden shifts in rangelands, or crashes of fish or wildlife populations. At first, it sounds like the vaguely mystical ideas about transcendent complexity, financial physics, etc.  But really, there are a number of very sensible observations about dynamical systems and a convincing argument that these features will be commonly seen in real complex systems.

The basic idea is that there are a number of harbingers of catastrophic change in time series of certain complex systems.  The model the authors use is the fold catastrophe model, where there is an attractor that folds back on itself like a sideways “N”.  As one gets close to a catastrophic bifurcation, a very straightforward analysis shows that the rate of return to the attractor decreases (I have some notes that describe the stability of the equilibria of simple population models here. The tools discussed in Scheffer et al. (2009) are really just generalizations of these methods).  As the authors note, one rarely has the luxury of measuring rates of return to equilibria in real systems but, fortunately, there are relatively easily measured consequences of this slow-down of rates of return to the attractor. They show in what I think is an especially lucid manner how the correlations between consecutive observations in a time series will increase as one approaches one of these catastrophic bifurcation points. This increased correlation has the effect of increasing the variance.

So, two ways to diagnose an impending catastrophe in a system that is characterized by the fold bifurcation model are: (1) an increase in variance of the observations in the series and (2) an increase in the lag-1 autocorrelation.  A third feature of impending catastrophes does not have quite as intuitive an explanation (at least for me), but is also relatively straightforward.  Dynamical systems approaching a catastrophic bifurcation will exhibit increased skewness to the fluctuations as well as flickering.  The skewness means that the distribution of period-to-period fluctuations will become increasingly asymmetric.  This has to do with the shape of the underlying attractor and how the values of the system are forced across it. Flickering means that the values will bounce back and forth between two different regimes (say, high and low) rapidly for a period before the catastrophe.  This happens when the system is being forced with sufficient strength that it is bounced between two basins of attraction before getting sucked into a new one for good (or at least a long time).

In summary, there are four generic indicators of impending catastrophe in the fold bifurcation model:

1. Increased variance in the series
2. Increased autocorrelation
3. Increased skewness in the distribution of fluctuations
4. Flickering between two states

There are all sorts of worrisome implications in these types of models for climate change, production systems, disease ecology, and the dynamics of endangered species.  What I hope is that by really getting a handle on these generic systems, we will develop tools that will help us identify catastrophes soon enough that we might actually be able to do something about some of them.  The real challenge, of course, is developing tools that give us the political will to tackle serious problems subject to structural uncertainty. I won’t hold my breath…

On 29-31 October, we will be holding our next installment of the Stanford Workshops in Formal Demography and Biodemography, the result of an ongoing grant from NICHD to Shripad Tuljapurkar and myself.  This time around, we will venture onto the bleeding edge of biodemography.  Specific topics that we will cover include:

• The use of genomic information on population samples
• How demographers and biologists use longitudinal data
• The use of quantitative genetic approaches to study demographic questions
• How demographers and biologists model life histories

Information on the workshop, including information on how to apply for the workshop and a tentative schedule, can be found on the IRiSS website. We’ve got an incredible line-up of international scholars in demography, ecology, evolutionary biology, and genetics coming to give research presentations.

The workshop is intended for advanced graduate students (particularly students associated with NICHD-supported Population Centers), post-docs, and junior faculty who want to learn about the synergies between ecology, evolutionary biology, and demography. Get your applications in soon — these things fill up fast!

One common frustration that I have heard expressed about R is that there is no automatic way to plot error bars (whiskers really) on bar plots.  I just encountered this issue revising a paper for submission and figured I'd share my code.  The following simple function will plot reasonable error bars on a bar plot.

Now let's use it.  First, I'll create 5 means drawn from a Gaussian random variable with unit mean and variance.  I want to point out another mild annoyance with the way that R handles bar plots, and how to fix it.  By default, barplot() suppresses the X-axis.  Not sure why.  If you want the axis to show up with the same line style as the Y-axis, include the argument axis.lty=1, as below. By creating an object to hold your bar plot, you capture the midpoints of the bars along the abscissa that can later be used to plot the error bars.

Now let's say we want to create the very common plot in reporting the results of scientific experiments: adjacent bars representing the treatment and the control with 95% confidence intervals on the estimates of the means.  The trick here is to create a 2 x n matrix of your bar values, where each row holds the values to be compared (e.g., treatment vs. control, male vs. female, etc.). Let's look at our same Gaussian means but now compare them to a Gaussian r.v. with mean 1.1 and unit variance.

Clearly, a sample size of 100 is too small to show that the means are significantly different. The effect size is very small for the variability in these r.v.'s.  Try 10000.

That works. Maybe I'll show some code for doing power calculations next time...

It's been a while since I last posted about swine flu.  Alas, it is still with us. The most recent data from CDC show that swine flu is still with us and that we should steel ourselves for a heckuva flu season this autumn and winter.  The curve peaks around the middle of June, but this is well past a typical flu season.  The influenza virus apparently does not survive well when the absolute humidity rises as temperatures rise and the air can hold more moisture.  When the weather gets cold again in the northern hemisphere and the absolute humidity drops, the virus will better survive outside of its infected host and transmission will increase.

Here is the epidemic curve as it currently stands:

CDC reported confirmed influenza cases for 2008-2009

It reassuringly appears to be tailing off, but in reality, it is just experiencing a summer lull (remember, also, that there is quite a bit of under-reporting at this point).  It should start to pick up in October or so when the bars representing the incident cases will almost certainly dwarf the current ones.  We're working on a number of flu-related projects, including the very precise measurement of within-school contact networks (recently funded by NSF!) as well as a project on perceptions of vaccination and (we hope) the measurement of vaccine opinion clustering.  My collaborator on this project, Marcel Salathé, has a terrific paper with Sebastian Bonhoeffer at ETH on the impact of opinion clustering on infectious disease eradication through vaccination. Their work shows that the standard estimates of necessary vaccination coverage required to protect the population through herd immunity are overly optimistic if people who share anti-vaccination beliefs, and therefore do not vaccinate themselves or their families, cluster in a population.  I will try to update, but I fear it will prove to be a very busy Autumn for me...

I'm no longer on vacation which means that I have much less time to devote to blogging.  I just wanted to follow up on the last couple posts though before I jump back into the fray. I received some very stimulating comments from Edward Hugh and Aslak Berg, who are economists and contributers to the Demography Matters blog. They pointed to a recent blog post that Aslak wrote in response to my defense of the recent Nature paper by Myrskylä et al. Given how hysterical debate (ostensibly) over health care in the United States has been of late,  it is very refreshing to have a rational debate with intellectual give and take, arguments backed up by evidence, concern over truth, etc. You know, all those things that don't seem to matter in contemporary American political discourse? So, my thanks to my interlocutors.

My basic reply is that I don't disagree with much Ed and Aslak have said.  I nonetheless think that the Myrskylä et al. paper is of fundamental interest.  How can that be?  Well, I think that this turns on the question of causality. Does high HDI cause higher fertility? I think that this is unlikely in the strict sense.   We can use a handy graphical formalism called a directed acyclic graph (DAG) to illustrate causality (Judea Pearl, who pioneered the use of DAGs in causal analysis, has some very nice slides explaining both causal inference and the use of simple DAGs.  There is a whole group at Carnegie Mellon including Peter Spirtes, Richard Scheines, and Clark Glymour who work on the use of statistics and causal inference. Causal DAGs, as discussed in Pearl (1995), are a non-parametric generalization of path analysis and linear structural relations models first developed by Sewell Wright and familiar to geneticists, psychometricians, and econometricians).  The idea that HDI somehow causes fertility can be encapsulated in the following simple graph:

An arrow leads from HDI directly to fertility, indicating that HDI "causes" fertility. The thing is, I don't believe this at all in the strictest sense.  HDI is a composite measure that includes six quantities (life expectancy at birth, log-per capita GDP at PPP in $US, adult literacy, and primary, secondary and tertiary school enrollment fractions).  This alone leads me to think that the results described by Myrskylä et al. are really (interesting) correlations and not causal relations. I suspect that Myrskylä and colleagues also think this.  In the discussion, the authors speculate on what it is about very high HDI that allows fertility to increase from its lowest levels generally seen at intermediate-high HDI. Their leading hypothesis relates to social structures that allow women to simultaneously be part of the workforce and have children: "analyses on Europe show that nowadays a positive relationship is observed between fertility and indicators of innovation in family behaviour or female labour-force participation." They further suggest that the more conservative social mores of the rich East Asian countries may be why their fertility continues to plummet: "Failure to answer to the challenges of development with institutions that facilitate work–family balance and gender equality might explain the exceptional pattern for rich eastern Asian countries that continue to be characterized by a negative HDI–fertility relationship."  The causal graph here might look like this:

I've made the line between HDI and fertility dashed to indicate that the direct influence is reduced -- it's possible that its only influence is indirectly through childcare.  Now HDI causes changes in childcare structures and these are what have the major causal impact on fertility.  Really, I suspect it is more than that, of course.  One possibility is the existence of relatively high-fertility immigrants in many of these high-HDI countries. In the United States, the fertility of foreign- and native-born women (based on the most recent analysis of the Census Bureau's Current Population Survey) was 2.1 and 1.8 respectively.   So foreign-born women in the United States have (period) TFRs that are nearly 20% higher than native-born women.  Similar results apply to European countries.  Is it possible that it's not childcare arrangements but the fraction of foreign-born that is different between the high-HDI European and East Asian countries?  If that's true, what's going on with Canada? It's not difficult to construct a story relating HDI to immigration: as development continues to increase and the skills of a workforce (and wages demanded by it) increase there are two forces increasing further immigration.  First of all, the country becomes a more attractive destination.  Secondly, as the skills/wages of the native labor force increase, there is need to find people who are willing to do the less highly skilled and lower paid labor.  The existence of high fertility migrants is an example of unmeasured heterogeneity, which is the bugaboo of demography and causal inference.  In this case, I think the heterogeneity might really be the object of interest and not simply a nuisance for causal inference.

My guess is that there are multiple causes.  Something like this seems likely to me:

with a number of other causes almost certainly contributing (either directly or indirectly) as well.

What I think is so valuable about the paper by Myrskylä and colleagues is that it makes us ask what the causal stories might be. What these scholars have done is initiate a chain of abductive reasoning.  Charles Sanders Pierce first identified abduction as a form of logical inference. Describing abduction, he wrote, "The surprising fact, C, is observed; But if A were true, C would be a matter of course, Hence, there is reason to suspect that A is true" (Collected papers: 5.189). Abduction is basically the process through which new hypotheses are created. Myrskylä have just revealed surprising fact C, namely, that fertility appears to increase with very high HDI.  We are surprised because all the previous literature on the relationship between economic development and fertility showed that the two were negatively related. Our goal now is to elucidate what A (almost certainly a multi-factorial quantity) is.  I like this paper because I see it as starting a new and productive area of research not because it identifies the cause of increased fertility in low-fertility countries.

The problematic correlations that Aslak notes (i.e., that the countries that show J-shaped HDI-TFR curves longitudinally are culturally related) may actually aid us in our quest to uncover the causal mechanism(s) that explains the HDI-TFR relation (more unmeasured heterogeneity). This, of course, would be a miserable situation if we thought that HDI was strictly causal since then HDI and whatever this latent cultural variable would be almost completely confounded.  But their very relationship may aid us in identifying what the actual causal mechanism is.

I look forward to more work in this exciting and important area of demographic research.  Maybe one of these days I'll write more on causal directed acyclic graphs. It's a pretty cool approach to science and one that I think merits much more attention in the social sciences