# Mathematicians Create Objective Quality of Life Index

Here’s a thorny problem: to develop an objective way to rank countries according to the quality of life they offer their citizens.

There are various ways of approaching this problem. For example, the Economist Intelligence Unit compiles its quality of life index using surveys, a useful technique but one that is hard to show is objective. Another widely quoted index, the Life Quality Index is based on life expectancy at birth and the gross domestic product per person but is only able to rank countries by applying a correction factor for each country that some critics say is open to bias.

Is there another way? Andrei Zinovyev at the Institut Curie in Paris and Alexander Gorban at the University of Leicester in the UK think so, using a mathematical technique developed in the mid-90s that can cut through this kind of problem .

They chose several widely-measured and well-studied indices on which to base their index: GDP per capita, life expectancy at birth, infant mortality rate and the incidence of tuberculosis. This data from 2005 is available for 162 countries.

Zinovyev and Gorban then plot this data in four-dimensional space. To create a ranking, the important question is whether there is a linear function that reduces this four-dimensional dataset to a one-dimensional set. Unsurprisingly, the answer turns out to be no. “Any linear mapping will inevitably give strong distortions in one or other region of data space,” they say. That’s what makes this problem tricky.

However, in the mid-90s a group of mathematicians devised a technique for reducing the dimensionality of complex data sets. This technique is essentially equivalent to connecting various data points together with springs and allowing the system to relax; hence it’s name: elastic mapping. The trick is to find an arrangement of springs that “flattens” the data set, or in other words, reduces its dimensionality.

And that’s basically what Zinovyev and Gorban have done, creating what they call the Nonlinear Quality of Life Index in the process.

Here are the top and bottom 5 from 2005:

1. Luxembourg

2. USA

3. Norway

4. Ireland

5. Iceland

.

.

.

158. Zambia

159. Mozambique

160. Zimbabwe

161. Kenya

162. Swaziland

No real surprises there, although there are some interesting features of the list. For example Equatorial Guinea is ranked at 140 although its GDP per capita is more than Saudi Arabia’s ranked at 37. That’s because of Equatorial Guinea’s appalling health statistics: 123 infant mortalities per 10,000 inhabitants, for example, compared to 21 in Saudi Arabia.

For similar reasons, Russia is ranked 71st despite having a GDP per capita that is significantly higher than other countries with a similar ranking.

Every list throws ups anomalies like this. The important point about this one is that it is done objectively and transparently.

That’s important because these kinds of indices are widely used by economists and politicians as a measure of economic and social development and so used to determine spending polices and legislation.

Objectivity is hard to come by when making these kinds of decisions. If the people who matter would agree to use it, this index could help.

Ref: arxiv.org/abs/1008.4063: Nonlinear Quality of Life Index