Title: Step 5: Improving the model
Course: UGQuants (SSPS) > Section: Multiple Regression

Step 5:   Improving the model

While we have managed to find some evidence of a relationship between national wealth and female life expectancy, it is pitched at a rather general level.  The likelihood is that you have probably wondered whether there wouldn’t be differences depending on which part of the world we are talking about that go beyond differences in wealth.  If we have other variables that measure some of these additional factors, we could explore this association further through extending our model and thereby improving the information we produce.

The regression line allows us to differentiate countries according to whether their female life expectancy 'performance' is better or worse than we should expect it to be, given their level of GDP/head.  Thus, a poor state might still have relatively low life expectancy compared to richer countries, but do very well indeed for a state with its level of resources.  Those countries above the regression line are doing well, while those below it are doing relatively badly.  If we look at our scatterplot again, there seems to be a number of countries with low life expectancies where GDP per head is much higher than we might expect it to be, given the low level of female life expectancy.  These countries pull the regression line down and reduce the amount of variation accounted for by our X variable.  Is there something particular about these countries?  If we label these outlying data points by the name of the country, it might give us a clue.

There does appear to be some relationship with geographical location.   Now we might wish to link the data points with different colours according to the continent they are in and calculate separate regression lines for each.

It turns out that almost all of these countries are located in Southern Africa.  Comparing our R-squared values, our very simple model works extremely well for South and East Asia and the Pacific, very well for Europe, Central Asia and North America and acceptably well for Latin America, the Caribbean, the Middle East and North Africa, but rather poorly for Sub-Saharan Africa.  What might we conclude from this?  One possibility is that there is another factor at work that is specific to Sub-Saharan Africa.  It might make sense to produce two models: one for Southern Africa and another for the rest of the world.

Another possibility is that there is another factor at work in countries with low GDP/head or low female life expectancy.  If you examine either of the previous diagrams, it looks as if the data points cluster rather more tightly around the regression line on the right hand side and upper half of the scatterplot; i.e. in higher GDP and higher female life expectancy countries.  Thus, GDP per head might be a more powerful explanation of differences in female life expectancy for countries with higher levels of female life expectancy.  In fact, if we were to investigate this further we would find that for countries with female life expectancy from around 60 upwards our model works extremely well (the R-squared is 0.70).  We could look for ways to improve our model for these countries, but the contribution of any other variable would probably be fairly small.

Thus one technique we can use in refining a model is to decide whether one model is indeed appropriate for all our data.  Here we have briefly considered two possibilities: that we build different models for different areas of the world, or models for different ranges of female life expectancy or GDP/head.  Yet another approach is to search for further variables that might improve our model.  Let us do this, concentrating on countries with low life expectancy for women; i.e. those countries with female life expectancy at birth below 60 years.

For countries with female life expectancy below age 60 our model is hopeless, and that is why the results for Africa, where most of these low life expectancy countries are located, are rather poor.  It explains nothing at all.  Thus, rather than having one model for all countries, we could decide to have two models: one for countries with low female life expectancy and one for countries with high female life expectancy.  This seems a reasonable thing to do, given that our very first scatterplot suggested a very different relationship between GDP and female life expectancy for different values of life expectancy.  Let us define high life expectancy countries as those where women can expect to live over 60 years at birth.  We have a simple and powerful model for these countries that explains almost three quarters of all the variation in female life expectancy using the log of GDP.

Now, if we look at countries with low life expectancies, we can see that GDP/head is a poor predictor.  However, we now face a further challenge.  The vast majority of countries with such low life expectancy (30 out of 32 or 93.75% of the countries for which we have data) are in Sub-Saharan Africa.  We cannot know from our data whether there is something specific about Southern Africa that makes it different, or if it is a general issue about GDP/head being less relevant in countries with very low life expectancy.  This problem of two different variables coinciding is called collinearity.  Sometimes we can find ways round this problem, or sometimes (as here) it may simply be an aspect of social reality.  In this case the fact is that there are very few countries outside sub-Saharan Africa with low female life expectancy.

However, using our general knowledge, we might reflect on what might be distinct about the African experience.  One possibility, for example, is that the virulence of the HIV pandemic in that continent, combined with the tremendous difficulties in securing affordable supplies of the relevant drugs (whose patents are jealously protected by the drug companies who own them), mean that the determinants of life expectancy are different in that continent than in others.  It might also be that the data, on either GDP or female life expectancy, are less reliable in Africa than elsewhere.  Many African countries are plagued by corruption, secrecy and a general lack of democratic control, making the manipulation of 'official' data easier.  Corrupt leaders may be tempted to massage such sensitive statistics in order to claim greater legitimacy for their policies.  However it is hardly the case that 'massaging' data for public relations purposes is exclusive to Africa.  Let us therefore explore the HIV issue, proceeding on the assumption that our data are reliable (in fact, due to poor data availability, we often have to fall back on slightly older data than we would like).

Let us look to see if there are other variables that correlate better with life expectancy in these countries.  The table below shows some correlations with female life expectancy for Sub-Saharan countries. GDP/head has virtually no effect.  It looks as if a major factor is infant mortality.  Mean life expectancy is depressed because so many children die from easily preventable diseases before their first or fifth birthday.  (You might like to check this correlation for high life expectancy countries: it will be much weaker, as in such countries less of the variation in mortality will be attributable to much smaller differences in a very low level of infant mortality). Healthcare provision also appears to be important.  The fewer hospital beds there are per 1000 inhabitants of a country, the lower is life expectancy.  However, we need to be careful; we only have information on healthcare provision for 34 of our 48 countries.  It might be dangerous to assume that what was going on in the other 14 countries was the same.  We might be on safer ground when confining our analysis to these 34 countries. Or we could use the variable on state health expenditure. This is the sort of compromise lack of suitable data often forces on the analyst.  The situation is even worse when we consider data on economic equality.  Only 13 of our countries have supplied this data.

Look at the correlation coefficients in the table carefully and note anything unusual or improbable.

Independent variables | Life expectancy at birth, female (years) (Sub-Saharan region)
GDP per capita | 0.381
N | 45
Prevalence of HIV (% of people aged 15-49) | -.452
N | 45
Mortality rate, under age 5 (per 1,000) | -.653
N | 47
Income share held by highest 10%* | -.430
N | 13
Income share held by lowest 10%* | .308
N | 13
Health expenditure, total (% of GDP) | -.306
N | 45
Hospital beds (per 1,000 people)** | .335
N | 32

Source: WDI 2011/*WDI 2008-2010 (most recent available value)/**WDI 2008-2011 (most recent available value).

Did you notice that when we look at the state health expenditure variable closely, we can see something rather counter-intuitive?  It appears that the more poor countries spend on health, the lower is life expectancy!  While some extreme libertarians might want to argue that this shows that state intervention can only ever worsen things, there is a more plausible explanation: impoverished countries ravaged by the AIDS pandemic have been forced to devote increasing amounts of their scarce resources to palliating its effects.  In other words, the chief direction of cause and effect may lie in the opposite direction to that which we are hypothesising in our model.  We might imagine that without such intervention the situation would be even worse.  Indeed, the strongest correlation (apart from that with infant mortality) is that between life expectancy and prevalence of HIV.

Let us check the form of the relationship between life expectancy and HIV prevalence using a scatterplot as before.

Once again it looks as if we might improve things by 'straightening' our line through transforming our variables.  We will take the natural log of HIV prevalence (lnHIV) to achieve this.

Our data points are now more evenly distributed instead of fanning out on one side and our correlation coefficient is higher, but a new issue has arisen.  Our regression line is being pulled away from most of our coordinates by a few outlying data points much further from the regression line than all the others.