Nonlinearity and Outliers

We return to a fully specified model, including popOOO and tpm. We retain compact, however, as a better measure of urban form than popden.

To establish linearity between the dependent and independent variables and outliers, we first investigate the relationship between the dependent variable and the independent variables using scatterplots. Go to “Graphs” —* “Legacy Dialogs" —* “Scatter/Dot" (Figure 12.21). We will begin with lane miles per 1000 population as an independent variable and VMT per capita as a dependent variable (Figure 12.22).

Figure 12.21 Simple Scatterplot Window

Figure 12.22 Scatterplot Between Lane Miles per 1000 Population and VMT Per Capita

The relationship looks roughly linear, though the slope may be leveling off a bit at high Im values. There may also be slight nonlinearity between compactness and VMT per capita. Nonlinearity can bias linear regression coefficients.

The scatterplot between population in 1000s and VMT per capita (Figure 12.23) shows that there may be outliers with extremely large values, particularly New York and Los Angeles. Outliers can bias regression coefficients. The plot also suggests the possibility of something called heteroscedasticity. The same is true of the relationship between transit passenger miles per capita and VMT per capita. Put simply, heteroscedasticity (also spelled heteroskedasticity) refers to the circumstance in which the variability of a variable is unequal across the range of values of a second variable that predicts it. We will come back to these issues later.

We now introduce a common test for outliers, using something called leverage values to identify them. Click on the Save button in the Linear Regression window, and check “Leverage values." Run a regression using Im, popOOO, tpm,fuel, incOOO, and compact as independent variables, and of course vmt as the dependent variable. You now have a new final column in your dataset labeled LEV_1. By convention, leverage values of more than approximately 0.20 signify' outliers. By this convention, Los Angeles and New York are outliers. We could delete these two cases from the dataset,

Figure 12.23 Scatterplot Between Population in 1000s and VMT Per Capita

but with only 157 cases, we hate to lose two. Moreover, these two UZAs are unique and important cases from a planning standpoint. Fortunately, there is another way to handle outliers that allows them to be retained in this example.

Log-transformation

Incorrect functional form refers to the relationship between dependent and independent variables being modeled incorrectly (e.g., linear form when it is non-linear). In order to handle outliers and nonlinearity-—both violating the assumptions of OLS regression, we may need to transform the data. The most common method of transforming data is to take the natural logarithms of one or more variables. A very' useful feature of linear regression is that the model results remain valid if one or even both sides of the equation change form through logarithm, polynomial expansion, etc.

The use of logarithms often gives the dependent variable a more normal distribution, while making relationships among our variables more nearly linear, and reducing the influence of outliers (such as New York and Los Angeles).

Please note that in the dataset, all variables are already presented both in linear and logged forms. Thus, the following steps for log-transformation are optional.

To create a new log-transformed variable, click the “Transform” menu and select “Compute Variable” (Figure 12.24).

In the Compute Variable window (Figure 12.25), type the name of your new variable in the "Target Variable” field, for example: Invmtcap. Then, in the “Numeric Expression" field, type the equation for the new variable, in this case ln(vmt). When you click "OK,” SPSS will add the new variable’s column with the transformed values of the corresponding variable. Repeat these steps for any variable you wish to compute. Note that this step is optional as your data already have a log-transformed version for most variables.

By taking the natural logarithms of VMT per capita and population in 1000s, and creating a scatterplot as just described, we find a relatively linear relationship between the variables. Note that this relationship is not perfectly linear and is still heteroscedastic—in fact, there may still be concerns about inefficient estimates—but it is nonetheless an improvement over the previous scatterplot.

Also, the scatterplot (Figure 12.26) shows that outliers become less obvious.

When performing logarithmic transformations, researchers can take the natural log of the left-hand (dependent variable) side of the equation, the right-hand (independent variable) side of the equation, or both. Logging just one side produces a semi-log model (log linear or linear log), while logging both sides produces a doublelog (log-log) model. This latter option offers a major advantage, because the coefficient estimates of a double-log regression model can be interpreted as elasticities.

Figure 12.24 Finding “Compute Variable” Option

Figure 12.25 Log-Transformation of VMT Per Capita Variable

Elasticity, or effect size, describes the proportional change in the dependent variable given a 1 percent shift in the independent variable. An estimated coefficient of —2, for instance, would indicate that every 1 percent increase in the independent variable corresponds to a 2 percent decrease in the dependent variable.

We estimated a double-log model using our sample dataset, Invmt as our dependent variable, and Inlm, InpopOOO, Intmp, Infuel, InincOOO, and Incompact as independent variables. Doing so, we obtained the results shown in Figure 12.27.

Note that the model as a whole has remained significant, and that the coefficient of population size approaches significance at the 0.10 level. Knowing that the coefficients represent elasticities, we can conclude that a 10 percent increase in lane miles per 1000 population is associated with a 2.93 percent increase in VMT per capita. Likewise, a 10 percent increase in compactness index is associated with a 2.06 percent decrease in VMT per capita, respectively. Note that New York and Los Angeles are no longer outliers, according to the leverage values. Overall, we conclude that the double-log transformation has improved the performance of our model and has produced more interesting results. However, the R-squared (and overall explanatory' power) of the model is still lower than we would ideally like it to be.

Figure 12.26 Scatterplot Between Log-Transformed Population Size and Log-Transformed VMT Per Capita