Title: Step 4: Obtaining the values for the regression equation
Course: UGQuants (SSPS) > Section: Multiple Regression

Step 4:   Obtaining the values for the regression equation

We can get SPSS to calculate the values for our formula:

Female life expectancy in years = a + b * Natural log of (GDP per capita in US$)

We do this using the Regression... Linear command from the Analyze menu.  In the dialog box LE_female will be our dependent variable and we will put lngdp in the independent variables box.  Make sure that 'enter' appears in the method selection box, as illustrated in the screen shot below.

The other features of the dialog box can be ignored for the present.  Compare your own output with the output shown below.

SPSS Linear regression output

Variables Entered/Removed

Model | Variables Entered | Variables Removed | Method
1 | LNGDP | . | Enter

a All requested variables entered.
b  Dependent Variable: LE_female

In multiple regression we are often interested in the effect of entering or removing variables from our model and the first section of the output records this.  Our model is a very simple one with only one independent variable, which appears in the ‘Variables Entered’ column.  Later, when we input subsequent independent variables, SPSS will give us results for successive models with different selections of variables.  Depending on how you have set the preferences in your copy of SPSS, you may find small differences: for example SPSS may show variable labels rather than variable names (you can choose your preferred output settings by going to the Edit menu and clicking on ‘Options’).

Model Summary

Model | R | R Square | Adjusted R Square | Std. Error of the Estimate
1 | .796 | .634 | .631 | 6.482673

a  Predictors: (Constant), LNGDP

The next section of output summarises the results of our model, which SPSS has given the number 1.  In some circumstances we might test several models at once, which SPSS would number consecutively.  It records the correlation between lngdp and LE_female (R) and the coefficient of determination for the model (R-squared).  You should use the adjusted R-squared value, which takes account of the number of independent variables in the model and the number of cases.  As the number of independent variables increases and the number of cases gets smaller, the estimate for R-Squared is reduced.  As a very rough rule of thumb, you should avoid using models that have fewer than 10 cases for each independent variable in your model.

In this model we can see that our single independent variable has accounted for close to two thirds (0.63) of the variation in our dependent variable.  Remember that this can be calculated by comparing the sum of squared squares (SS) for our model or regression (the sum of the squares of the difference between the mean value of Y and those estimated by our regression equation for each observation) and the corresponding sum of squared errors obtained with no independent variable (the sum of the squares of the difference between our observed values for Y and the mean of Y).  The more variation there is to explain in our Y variable, the larger this second amount will be.  To the extent that our X variable accounts for some of this variation, the regression sum of squares will be greater.

It also gives us the Standard Error of the Estimate of the regression coefficient for our independent variable, analogous to the standard error of the mean when we were looking at sampling variation.

The third section of the output gives the values for the sum of squares (SS).

The total sum of squares is the sum of the squares of the deviation of each observed value of the dependent (Y) variable from its mean. This tells us how much variation there is in the dependent variable that we are trying to account for.  It is the same calculation that we used when calculating the variance of a variable except that here we do not divide by the number of observations.

The regression sum of squares is the sum of the squares of the difference between the mean of Y and the value predicted for Y for each observation.  This tells us how much of the variation in Y is accounted for by our regression model.

The residual sum of squares is the sum of the squared residuals for each observation (the difference for each observation between the observed value of Y and that predicted by the model).  This tells how much of the variation in Y our model does not account for.

Dividing the regression sum of squares by the total sum of squares gives us the proportion of the variation in Y accounted for by our regression model, or R-squared.  If you do this for the results above you will find that 11992.978/18927.111 = 0.634

.

This section of the output also reports the significance (written here as ‘Sig.’ but usually reported as a p value) based on the F statistic. This tells us the probability (if our data come from a random sample - which in our case here they do not) that in the population, none of the variables in our model has an effect: that is the probability of a type I error if we reject a null hypothesis that our model offers no improvement in our capacity to predict the values for LE_female over just using the mean for Y.

The last section is the most important:

It tells us that our constant, intercept or value of (a) is 29.768 years and (b), the regression coefficient or the slope of our line, is 5.355. Putting these in our formula we get:

Female life expectancy in years = 29.768 + (5.355*lngdp)

This might seem to be a complex formula, but in the world of computers it poses few problems.  It means that our best estimate of female life expectancy (Y-hat) starts from just under 30 years when there is (theoretically) no GDP (i.e. x = 0) and increases by 5.36 years for each unit increase in the log of GDP.  Thus, if we wished to interpret our equation for specific values, we could calculate the following expected values of female life expectancy:

                If GDP = US$10,000, female life expectancy would be estimated to be 29.77 + (5.36 * ln(10000)) = 79.14 years

                If GDP = US$5,000, female life expectancy would be estimated to be 29.77 + (5.36 * ln(5000)) = 75.42 years

                If GDP = US$1,000, female life expectancy would be estimated to be 29.77 + (5.36  * ln(1000)) = 66.80 years

The non-linear effect of changes in GDP on female life expectancy are revealed, despite using an equation of a straight line, such that an increase from US$1,000 to US$5,000 increases life expectancy by roughly 8 ½ years, while an increase from US$5,000 to US$10,000 increases life expectancy by only 3 ¾ years.  We could also use this approach to examine whether the relationship between GDP and female life expectancy had changed over time, using data from other years.

The last section of the output also shows beta or standardized coefficients for each independent variable.  These coefficients measure the amount of change in our dependent variable associated with change in our independent variable in units of standard deviation (analogous to Z scores).  For a model with only one independent variable, the beta coefficient will be the same as r, the correlation coefficient.  (You may be able to see why: both measure the standardised amount of variation in Y for a unit change in X,).  Standardised coefficients are extremely useful in multiple linear regression as they allow us to compare directly the impact on our dependent variable of each separate independent variable, even though they may be measured in different units, controlling for all the other independent variables in the model.