Page 146 - Statistics II for Dummies

130 Part II: Using Different Types of Regression to Make Predictions

As with any regression model, you can’t estimate the value of y for x-values
outside the range of where data was collected. If you try to do this, you commit
a no-no called extrapolation. It refers to trying to make predictions beyond
where your data allows you to. You can’t be sure that the model you fit to
your data actually continues ad infinitum for any old value of x. In the quiz-
score example (see Figure 7-6), you really can’t estimate quiz scores for study
times higher than six hours using this model because the data doesn’t show
anyone studying more than six hours. The model likely levels off after six
hours to a score of ten, indicating that studying more than six hours is overkill.
(You didn’t hear that from me though!)

Going Up? Going Down? Go Exponential!

Exponential models work well in situations where a y variable either
increases or decreases exponentially over time. That means the y variable
either starts out slow and then increases at a faster and faster rate or starts
out high and decreases at a faster and faster rate.

Many processes in the real world behave like an exponential model: for exam-
ple, the change in population size over time, average household incomes
over time, the length of time a product lasts, or the level of patience one has
as the number of statistics homework problems goes up.

In this section, you familiarize yourself with the exponential regression model
and see how to use it to fit data that either rise or fall at an exponential rate.
You also discover how to build and assess exponential regression models
in order to make accurate predictions for a response variable y, using an
explanatory variable x.

Recollecting exponential models

x
Exponential models have the form y = αβ . These models involve a constant,
β, raised to higher and higher powers of x multiplied by a constant, α. The
constant β represents the amount of curvature in the model. The constant α
is a multiplier in front of the model that shows where the model crosses the
y-axis (because when x = 0, y = α * 1).

An exponential model generally looks like the upper part of a hyperbola
(remember those from advanced algebra?). A hyperbola is a curve that crosses
the y-axis at a point and curves downward toward zero or starts at some point
and curves upward to infinity (see Figure 7-11 for examples). If β is greater
than 1 in an exponential model, the graph curves upward toward infinity.
If β is less than 1, the graph curves downward toward zero. All exponential
models stay above the x-axis.

12_466469-ch07.indd 130 7/24/09 9:39:10 AM