Beyond compare

Comparisons are the lifeblood of graphing. When we plot two points, the eye immediately judges their relative positions, and a conclusion is drawn. Usiually, of course, we have more than two points, and more elaborate conclusions may be drawn about trends, reversals, and the like. And sometimes we want to compare two (or more) complete graphs. A reasonable desire, but one fraught with peril.

Here is a recent graph from the estimable New York Times economic columnist David Leonhardt. It compares the recent economic performance, represented by the Gross Domestic Product, of three nations: US, Germany, and the UK. The point of the article is that austerity, as practiced in Germany, has caused that nation’s recovery to lag behind the US, which has pursued a more stimulus-driven approach. The primary evidence given is the rightmost points in the graph: at the end of 2010  Germany appears not to have recovered to the same point as the US.

To allow for easier comparisons, we re-create this graph, using data obtained from http://stats.oecd.org.

Notice that the graphs are compared by indexing them to a single point: the first quarter of 2008. This might seem reasonable; after all, that was the peak, or near peak, for all three economies, and could be taken as the starting point for our recent travel through the valley of (economic) death. And the graphs have to be indexed, or somehow converted to a common unit, since the absolute size of the economies are very different.

But there is a serious problem with picking one point on the graph for indexing. Note that the germany economy had a spike just in that quarter, while the US economy showed a dip. Using that point for indexing favors the US in subsequent comparisons.

To make this point more concrete, suppose we chose the previous quarter (the last quarter of 2007), for indexing? That is also reasonable, since it marked the point at which US growth stalled. Here is the resulting graph.

Woah! All of a sudden, Germany is right up there with the US in terms of its recovery. Both nations are back to 100% of the value in the third quarter of 2007. Not also that this matching of US and german economies seems more reasonable than the previous, since the two curves, rather than a single point, are better matched.

But this leads us to note that an altogether more reasonable way of matching several curves, that avoids the transient blips of one nations fortunes, is to match the entire curves as well as possible. This amounts to matching their average values. We have done this in our next graph. Each curve is now plotted as a percentage of the average over the interval.

Again, we see that the US and Germany are effectively tied in their recovery. In this graph, even the UK doesn’t look quite as bad. The main difference between the US and Germany, viewed in this way, is in the magnitude of their deviations: Germany has swung more wildly from boom to bust.

Even this equitable way of comparing curves has a problem: it depends on the interval over which we average. It would (probably) make no sense to allow economic patterns from a century ago to influence a comparison of recent trajectory of US, Germany, and the UK. So the average should be local, restricted to values in the neighborhood of the relevant data. But how local? Alas, there is no fixed answer. It depends. And that means that it is variable that can be manipulated to make a point.

In summary, the student of graphs should always be vigilant for comparisons of several curves that require “indexing.”

References:

Why Budget Cuts Don’t Bring Prosperity By DAVID LEONHARDT

New York Times. Published: February 22, 2011

http://www.nytimes.com/2011/02/23/business/economy/23leonhardt.html?_r=1&scp=2&sq=leonhardt&st=cse

My data from Organisation for Economic Co-operation and Development (OECD): “VPVOBARSA: Millions of US dollars, volume estimates, fixed PPPs, OECD reference year, annual levels, seasonally adjusted.” from http://stats.oecd.org/

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