This is a quick note on profiling your compiled code on the mac. It is important not to guess when figuring out where the bottlenecks in your code are, and for this reason, the R manual has several suggestions on how to profile compiled code running within R. All of the methods are platform dependent, with linux requiring command line tools (sprof, and oprofile). On the mac there are some convenient GUI tools built in to help you out.

Our test code is drawn from the ergm package, and fits a network model to a small graph. This model is fit using markov chain monte carlo, so we expect most of the time to be spent drawing MCMC samples.

library(ergm)
flomarriage = network(flo,directed=FALSE)
flomarriage %v% "wealth" = c(10,36,27,146,55,44,20,8,42,103,48,49,10,48,32,3)
for(i in 1:10) gest = ergm(flomarriage ~ kstar(1:2) + absdiff("wealth") + triangle)

The quickest way to get an idea of where your program is spending time is to open Activity Monitor.app and select the R process while the above code is running. Then click sample, which should result in something like:

In order to get down to the interesting code, you’ll have to wade though a bunch of recursive Rf_eval and do_begin calls used by R. Finally though we see the MCMCSample function in ergm.so which contains almost all of the samples. So it is as we expected, the MCMC sampling takes up all the time. Underneeth that we see calls to:

• ChangeStats: calculates the network statistics
• MH_TNT: calculates the metropolis proposal
• ToggleEdge: alters the network to add (or remove) an edge

But how much these contribute is a bit difficult to see here. Enter the Time Profiler in Instruments.app. You can launch this program from within Xcode.app under Xcode – Open Developer Tool – Instruments. Once Instruments is open, click Library and drag the Time Profiler into the program. Then, while the R code is running, select ‘Attach to Process’ – R and click record. You should get something like:
Like in the Activity Monitor, we have to drill down past all of the recursive Rf_eval calls, but once we do, we see that ChangeStats is taking 44.6% of the time, MH_TNT takes 30.9% and ToggleEdges takes 11.7%. We could further expand these out to identify possible targets for optimization within each of these functions.

So there you have it. Apple makes it really easy to profile your compiled code. One issue that I ran into is to make sure that your Xcode is up-to-date and that Instruments is version 4.0 or above. I experienced some crashes profiling R with older versions.

I’m usually quite a big fan of the content syndicated on R-Bloggers (as this post is), but I came across a post yesterday that was as statistically misguided as it was provocative. In this post, entitled “The Surprisingly Weak Case for Global Warming,” the author (Matt Asher) claims that the trend toward hotter average global temperatures over the last 130 years is not distinguishable from statistical noise. He goes on to conclude that “there is no reason to doubt our default explaination of GW2 (Global Warming) – that it is the result of random, undirected changes over time.”

These are very provocative claims which are at odds with the vast majority of the extensive literature on the subject. So this extraordinary claim should have a pretty compelling analysis behind it, right?…

Unfortunately that is not the case. All of the author’s conclusions are perfectly consistant with applying an unreasonable model, inappropriate to the data. This in turn leads him to rediscover regression to the mean. Note that I am not a climatologist (neither is he), so I have little relevant to say about global warming per se, rather this post will focus on how statistical methodologies should pay careful attention to whether the data generation process assumed is a reasonable one, and how model misspecification can lead to professional embarrassment.

His Analysis

First, let’s review his methodology. He looked at the global temperature data available from NASA. It looks like this:

He then assumed that the year to year changes are independent, and simulated from that model, which yielded:

Here the blue lines are temperature difference records simulated from his model, and the red is the actual record. From this he concludes that the climate record is rather typical, and consistant with random noise.

A bit of a fly in the ointment though is that he found that his independence assumption does not hold. In fact he finds a negative correlation between one years temperature anomaly and the next:

Any statistician worth his salt (and indeed several of the commenters noted) that this looks quite similar to what you would see if there were an unaccounted for trend leading to a regression to the mean.

Bad Model -> Bad Result

The problem with using an autoregressive model here is that it is not just last year’s temperatures which determine this year’s temperatures. Rather, it would seem to me as a non-expert, that temperatures from one year are not the driving force for temperatures for the next year (as an autoregressive model assumes). Rather there are underlying planetary constants (albedo and such) that give a baseline for what the temperature should be, and there is some random variation which cause some years to be a bit hotter, and some cooler.

Remember that first plot, the one with the cubic regression line. Let’s assume that data generation process is from that regression line, with the same variance of residuals. We can then simulate from the model to create an fictitious temperature record. The advantage of doing this is that we know the process that generated this data, and know that there exists a strong underlying trend over time.

If we fit a cubic regression model to the data, which is the correct model for our simulated data generation process, it shows a highly significant trend.

              Sum Sq  Df F value    Pr(>F)
poly(year, 3)  91700   3  303.35 < 2.2e-16 ***
Residuals      12797 127

We know that this p-value (essentially 0) is correct because the model is the same as the one generating the data, but if we apply Mr. Asher’s model to the data we get something very different.

His model finds a non-significant p-value of .49. We can also see the regression to the mean in his model with this simulated data.

So, despite the fact that, after you adjust for the trend line, our simulated data is generating independent draws from a normal distribution, we see a negative auto-correlation in Mr. Asher’s model due to model misspecification.

Final Thoughts

What we have shown is that the model proposed by Mr. Asher to “disprove” the theory of global warming is likely misspecified. It fails to to detect the highly significant trend that was present in our simulated data. Furthermore, if he is to call himself a statistician, he should have known exactly what was going on because regression to the mean is a fundamental 100 year old concept.

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The data/code to reproduce this analysis are available here.

The 2012 election is over and in the books. A few very close races remain to be officially decided, but for the most part everything has settled down over the last week. By all accounts it was a very good night for the Democrats, with wins in the presidency, senate and state houses. They also performed better than expected in the House of Representatives. If we take the results as they stand now, the Democrats will have 202 seats v.s. the Republican’s 233 seats, a pick-up of 9 relative to the 2010 election. This got me thinking about how the Republicans could have maintained 54% of the house seats during a Democratic wave election.

Since I’m a statistician, the obvious answer was to go get the data and find out. Over the last few days, I was able to obtain unofficial vote counts for each congressional race except 3. The results from two unopposed Democrats in Massachusetts’ 1st and 2nd district, and one unopposed Republican from Kansas’ 1st district will not be available until the final canvas is done in December. Not including these districts 55,709,796 votes were cast for Democrats, v.s. 55,805,487 for Republicans, a difference of less than 100,000 votes.

Both of the candidates from Massachusetts also ran unopposed in 2008, where they garnered 234,369 and 227,619 votes respectively, which seems like a good estimate for what we can expect this year. In 2008 Kansas’ 1st district had a total of 262,027 votes cast. We will assign all of them to the Republican candidate as an (over) estimate, as the old 2002 district lines are similar to their 2012 placements. Using these numbers we get estimated vote counts of 56,171,784 for the Democrats and 56,067,514 for the Republicans, leading to a 100,000 vote advantage for the Democrats. Either way, this election was a real squeaker in terms of popular vote, which in percentage terms is 50.046% to 49.954%. Until the full official results are released, I would classify this as “too close to call,” but find a Democrat popular vote victory likely.

Nate Silver recently looked at the differences in the number of votes cast in 2008 v.s. those already reported in 2012, and found that California has reported 3.4 million fewer votes this election than 2008, suggesting that millions of mail-in ballots remain waiting to be certified. Somewhat similarly, New York reported 1.5 million fewer votes, though this may be due (at least in part) to the effects of hurricane Sandy. These were the only two states with differences > 1 million, and they both lean heavily Democratic, which may well push the popular vote totals for the Democrats into “clear winner” territory.

In a certain sense, the winner of the popular vote is irrelevant, as the only thing that matters in terms of power is the number of seats held by a party. But if the Democrats win, then they can reasonably claim that they (even as the minority party) are the ones with the mandate to enact their agenda.

Even though the popular vote is razor close in the current unofficial results, the distribution of seats is certainly not. We shouldn’t expect there to be a perfect correspondence between the popular vote and seats, because who wins the seats depends on how the population is divided up into congressional districts. As a simplified example consider three districts, each with 100,000 voters. District 1 is heavily Democratic, with 100% of votes going to the democratic candidate, while districts 2 and 3 are swing districts where 51% of the vote goes to the Republicans. In this simplified example, the Democrats would win nearly 2/3 of the popular vote, but only 1/3 of the seats.

So, if we expect there to be a discrepancy between popular vote and seat totals, the question then becomes: Is this election unusual? To answer this, we can look at the official historical record going back to 1942 which gives us the following:

In almost every election in the last 70 years, the party with the popular vote victory also won the majority of seats. The only exception to this was 1996, where the Democrats won the popular vote, but failed to attain a majority in congress. If 2012 turns out to be a popular vote victory for the Democrats, then their seat deficit would be unheard of in living memory. On the other hand, there was a very close election in 1950, where the Republicans lost by a margin of just 0.04%, but ended up with only 199 seats to the Democrats 234.

Redistricting, Gerrymandering, oh my!

Whether or not the official vote tally goes to the Democrats, the 2012 House of Representatives electoral landscape is unusual. The seat totals clearly favor the Republicans, but why? Some have noted that the congressional district lines were redrawn following the 2010 Census, a process that in many states is controlled by the state legislature, and that many of these lines seem to be constructed to maximize the political advantage of one party or the other. I was initially skeptical of these claims. After all, the two years most comparable to 2012 are 1996 and 1950, neither of which are particularly near a redistricting period. As is often the case, an examination of the data revealed evidence that was counter to my expectations.

First, let us explore how one would draw lines that maximize party advantage. Remember in our simple three district example, the advantage was gained by bunching all of the Democratic voters into one district, and maintaining slim advantages in the other two. So, the rule is to spread out your own vote into many districts, and consolidate your opponents voters into just a few. This can be a difficult geometric “problem” to solve, but an easy way to think about one possible route toward a solution is to focus on the swing districts. If a district is very close, you will want to alter its location so that it has just enough more Republican votes to make it a safe bet for your party. Ideally you will want to take these Republican votes from a safe Democratic district, making it even more Democratic. What you can then expect when you look at the distribution of districts is a clump of very safe Democratic districts, a clump of safe, but not overwhelmingly safe, Republican districts, and very few toss-up districts. This is what we call a bimodal distribution, and the process of redistricting with a focus on political gain is known as gerrymandering. There is nothing particularly Republican about it. Both political parties engage in gerrymandering, the difference being whether or not they control the redistricting process, and to what degree are they willing to contort the shape of their districts.

The Republican party was in control of the redistricting of 174 districts, v.s. just 44 for the Democrats. Just to give a specific example, Ohio had a Republican controlled redistricting process, which though wildly successful for Republicans, lead to torturously shaped districts. Republicans won 52.5% of the vote in Ohio, but obtained a staggering 75% of the seats.

But that is just one state, what about the rest of the districts? If we plot the proportion of people in each district that go to the Democrats, broken down by who was in control of redistricting we see clear evidence of clumping into safe, but not overwhelmingly safe territory for the redistricting party.

The way to read this plot is that each point represents a district, with its position being the proportion of people in the district who voted for the Democratic candidate. The lines represent a smoothed estimate of the distribution, with wider parts indicating increased density. Among the districts where Republicans had control, we see a characteristic bimodal distribution, with many districts chosen to be relatively safe Republican strongholds, with a smaller group of heavily Democratic districts. The same trend may be present in the Democratically controlled districts, but there are too few of them to see the trend clearly. Alternatively, in districts where the decision making is split between parties, or delegated to an independent group, we see a more natural distribution of vote shares, with a good number of toss-up districts.

But can we prove that it swung the election?

To parse out the effect of redistricting on the election results in a non-descriptive sense, it is necessary to construct (a simple idealized) model of how redistricting affects the probability of winning a district. To do this, we assume that the probability of winning a district within a state is related to the partisanship of the state (i.e. we expect a higher proportion of Republican seats in Alabama as compared to New York), which we will measure by the share of the vote that Romney won in the state. Second, control of redistricting (either Democratic, Republican, or Independent/Split) yields an increase (or decrease) in the odds of a win. We slap these two together into a standard statistical model known as a logistic regression, which yields the following:

This table tells us two important things through its p-values. With p-values, the closer to 0, the more significant the relationship. First, that the partisanship of a district matters, because the proportion voting for Romney is very significantly related to the probability of congressional control. Secondly, control of redistricting matters. If the redistricting process were fair, we would see the magnitude of the trends that we see in this data only once every thousand years (10 years between redistricting / 0.010 ). This gives us a high degree of confidence that political concerns are the driving force in redistricting when politicians are put in charge of it (shocking, I know).

This effect is relatively large. Consider an idealized swing state, where Romney won exactly 50% of the vote. Our model estimates the proportion of the congressional delegation who are Democrats as:

Here the model estimates that districts from an independent/split source are roughly unbiased with an estimated proportion of seats of 48%. If the Republicans control the process then we would expect only 31% of the delegation to be Democrats. This is slightly above what we observed in Ohio, where only 25% of the delegation were Democrats, despite Romney’s percentage of the vote being nearly 50%. With the Democrats in control we see a smaller bias than the Republicans with an estimated 56% belonging to the Democratic party, but our confidence in this estimate (denoted by the red dotted lines) is small due to the small number of redistricting opportunities that they had.

A perfect world

What would have happened if the entire redistricting process was controlled by independent or bipartisan agreement? Well, our simple model can tell us something about that. By counterfactually assuming that all redistricting was done independently or by split legislature, it estimates that the proportion of Democrats in the current congress would be, on average, 52%. If Democrats got to pick all the lines it would be 59%, and under complete Republican control it would be just 37%.

 Conclusions

So what lessons can we take away from this analysis? Control of the redistricting process is a powerful tool for those who would use it as a tool. The model estimates that who controls the process could yield  17% point swings (56%-39%) in a typical swing state. This is consistent both with the results from the general election as a whole, and Ohio in particular. While I would love to see the implementation of an algorithmic solution to the problem of district formation (if only to hear the pundits trying to say minimum isoperimetric quotient), it appears that independent commissions, or even bipartisan agreement within a legislative body are sufficient to have fair lines. As of now, 6 states decide their districts based on an independent commission, and it is hard to think of an honest argument why every state should not adopt this model. Economics teaches us that incentives matter, and if you give politicians the incentive to bias districts in their favor, it is a safe bet that they will do so.

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Fellows Statistics provides experienced, professional statistical advice and analysis for the corporate and academic worlds.