In the last section, we were concerned with finding the most appropriate regression model that would best fit a set of non-linear (i.e. non-proportional) data through a process of ”straightening out” the data by transforming one or more of its variables. In this section, we will be concerned with how certain changes in the independent variable of such a non-proportional model bring about certain changes in its dependent variable by interpreting the model’s parameters in a way that is reminiscent of the way we study the slope parameter of a proportional model. Specifically, we will look at two ways to measure change for both the response and the explanatory variable: total change and percent change.
Total change is usually a level dependent quantity for non-proportional models. This means that we get very different amounts of total change at different levels of X values, even for the same total change in X. However, the idea of percent change incorporates this level dependency in its very definition. In fact, we have four basic combinations of the ways of measuring change. By examining these different combinations, we can develop a way of interpreting the parameters of regression models that we produce, for linear and many nonlinear models:
Total change in response variable vs. total change in explanatory variable
Total change in response variable vs. percent change in explanatory variable
Percent change in response variable vs. total change in explanatory variable
Percent change in response variable vs. percent change in explanatory variable
However, it is not always easy to appreciate, and hence interpret, the parameters in the form in which they appear in the regression equations, as they appear in the first part of this chapter. This situation becomes apparent as we look at the chart of various models on page ??. It is not obvious, for example, why a model whose response variable has been logged and whose explanatory variable has not been logged is called an exponential model; likewise, it is not obvious why a model whose response variable as well as its explanatory variable has been logged is called a power model. Using the rules of exponents and logarithms, we shall rework each of these two regression models so that their coefficients become readily identifiable as the parameters in an exponential and a power function, respectively. From here, we will be able to readily interpret the effects of change in logarithmic, exponential, and power models in terms of their parameters in such a way that accounts for their level.
For example, we will find that the parameters in a logarithmic model are more easily interpreted if we look at the total change in the response variable contrasted to a 1% change in the explanatory variable. Exponential models on the other hand, are more easily interpreted by considering the percent change in the response variable contrasted to the total change in the explanatory variable. Interpreting the parameters in power functions is most easily done by examining the percent change in the response variable compared to a 1% change in the explanatory variable. For all of these models, the total or percent change in the response variable will depend directly on the values of the parameters in the model. Other non-proportional models, such as the quadratic or square root models, are not so easy to interpret in terms of their parameters and must await further developments in a later chapter.