Title: Outliers
Course: UGQuants (SSPS) > Section: Multiple Regression

Outliers

In the previous diagram we can see that there are at least two, possibly three, countries that have very high life expectancy, despite a noticeable prevalence of HIV.  They might be what we call an outlier; that is to say they are much further from the regression line than our other countries.  We can use SPSS to check which case this is (by looking directly at the data point, or by asking SPSS to label all the cases in the scatterplot, which usually makes it difficult to read).  It turns out that our outlier candidates are Cape Verde and Mauritius with 1% of the population between 15 and 49 carrying HIV and a life expectancy about ten years higher than we might expect, as well as Namibia, with a rate of 13.4% of HIV in the relevant population and also a female life expectancy about 10 years higher than our model would suggest.

Again we face a decision.  Since we are using the vertical squared distances from the regression line to make our line fit, such outliers can have a big impact on our results.  We could undoubtedly improve our model by excluding these countries.  However, we cannot just go around throwing out pieces of evidence that do not fit our theory without a very good reason.  Either we need to be able to suggest what the special reasons are that could explain why Cape Verde, Mauritius and/or Namibia fit our model so poorly, or we need to accept that this case illustrates something that is missing in our model that we have yet to account for.

What reasons could we find to justify the exclusion of these cases? Mauritius and Cape Verde are both island states. Perhaps there is something special about their circumstances because of that. Cape Verde is also actually on about the same level as the southern-most areas of the Sahara, so we might challenge if it is a typical Sub-Saharan country at all.

Namibia, on the other hand, certainly fits that description. However, remember when we first spoke about the reliability of data from states like these? If we cross-check the information provided by the WDI with figures from the CIA World Factbook 2012, we find that they see female life expectancy at birth in Namibia at an average value of 51.69 years in contrast to the 62.88 years in our dataset. This value would fit our data much better.

If we re-do the scatterplot with those three countries excluded we obtain a much tighter model, with an R-squared value of 38.1%. We can explain nearly 40% of the variation in female life expectancy across Sub-Saharan countries now. (Ideally, we would present our model by pointing out that it applies only to Sub-Saran countries on the main continent and by including Namibia with a corrected value for female life expectancy.)

We identified these countries as outliers merely by looking at the scatterplot.  However, how do we decide exactly how far a case needs to be from the regression line before we need to consider its impact on the model as a whole?  The answer to this question lies in the analysis of residuals (i.e. the distances between our observed data and the datapoints predicted by our regression model), which we will consider a little more closely later on in this module.