Title: Step 6: Working with more than one independent variable
Course: UGQuants (SSPS) > Section: Multiple Regression

Step 6:    Working with more than one independent variable

So far we have achieved four things in developing our model.

1 | We've discovered that the determinants of female life expectancy seem to change as it gets higher.
2 | For relatively high female life expectancies GDP per capita seems to do a pretty good job in accounting for differences.
3 | For lower life expectancies, mostly in African countries, HIV seems to be a major factor.
4 | We've identified some outliers that affect our model (Mauritius and Cape Verde), which we can reasonably safely remove as they appear to be special cases.

Where do we go next?  This is where multiple regression comes into its own, as we can look at what happens when we add different variables to our model, controlling for other variables.  Let us concentrate on low female life expectancy countries.  We can run a number of different models at once.  We know that HIV prevalence has an effect, so we will want to include it in our model.  Next we could test if, taking account of HIV, GDP/capita acts in a similar way to its effect in higher life expectancy countries. To do this we first enter our lnHIV variable, and then, as a second model, we enter the lngdp variable to see if it now has any effect, having controlled for the effect of HIV.  We do this by entering the two variables as successive blocks in our model, as shown in the following screen shots of the regression dialog box.  Enter the lnHIV variable, click on 'next' and enter the lngdp variable. SPSS will then report the results from each of these two models, the first with only lnHIV entered and the second with both lnHIV and lngdp entered.

When we run a multiple regression model, SPSS reports the multiple correlation coefficient R and the coefficient of determination for each model.  This shows that ln(HIV) on its own ‘explains’ about 40% of female life expectancy, but with ln(GDP) as well it increases to about 45% explanation (remember: look at the adjusted R Square).

SPSS also shows the two kinds of coefficients as before.

The ’unstandardized coefficients’ tell us what the change in the value of our dependent variable will be for a one unit change in the value of each independent variable, controlling for all the others.  We can use these unstandardised coefficients (known as B coefficients) to calculate estimates of what the value of our dependent variable would be for given values of our independent variables.  Since these coefficients are expressed in the units of the independent variables, their size depends on these units.  If we want to compare the impact each of the different independent variables on the dependent, controlling for all the other independent variables, we need to standardise.  T

his is what the Beta coefficients are. They estimate how much of the variation of the dependent variable is accounted for by each independent variable, controlling for all the others, because they tell us the correlation between our variable of interest and the dependent variable, controlling for all the other variables in the model and expressed in units of standard deviation.  Thus in model 2, controlling for the effect of HIV prevalence, a change of one standard deviation unit in log GDP is associated with an increase of 0.28 standard deviations in female life expectancy.

Once we control for HIV, GDP has the effect we might expect it to have, with a beta coefficient of 0.28.  When we looked at the simple correlaton of LE_female and the lngdp for these countries there was virtually no relationship.  R-squared was 0.00!  The effect of GDP was obscured by the fact that some relatively more developed African countries nevertheless face dreadful problems from the HIV epidemic.  HIV provides a lot more explanation, and by looking at the relative size of the standardised coefficients we can see that it has about three times the effect that is offered by GDP.

We could go on to improve our model further, testing other variables, using the same powerful capacity of multiple regression to tell us the impact of each variable in the model, controlling for all the others.  The regression coefficients are now partial regression coefficients as they give results controlling for the other variables in the model.  Finally, the Beta coefficients are standardized (in units of standard deviation) so that we can compare the contribution of different variables irrespective of the unit of measurement that they happen to be in.

We can also see that once we control for GDP, the role of HIV actually increases, while we get the relationship we might expect, controlling for HIV, between GDP/capita and life expectancy.  We can now look again at the variable that gave a curious result in its negative correlation with life expectancy: medical expenditure.  When we put it alone into a regression equation with LE_female we find that its beta coefficient is -.303.  This is what we should expect since, in a regression equation with a single independent, the size of the beta coefficient is equal to the Pearson correlation coefficient for the two variables.  What happens, though, when we enter this variable along with that for ln(GDP) and ln(HIV)?  To look at the effect we can get SPSS to enter variables into the regression equation in stages, or 'blocks'.

SPSS then gives us the results separately for each successive new variable.  In this case we can compare the results of two models, one with only the GDP and HIV variables, and the second with the medical expenditure variable added in.  As you can see from the output below, once we control for GDP and the prevalence of HIV, medical expenditure has less independent effect.  Its beta coefficient is reduced to less than two thirds of its previous size.