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Chapter 7: When Data Throws You a Curve: Using Nonlinear Regression
The techniques and criteria you use to do this are the same as those I dis-
cuss in a previous section “Assessing the fit of a polynomial model.”
The math magic from step three works courtesy of the definition of loga-
b ^h
y
b =
a. Suppose you have the equation
y +
rithm, which says log a =
log 10 y = 2 + 3x. Now if you take ten to the power of each side, you get
3
x
2 +
()
y
10
. By the definition of logarithm, the tens cancel out on the left
10
=
log 10
2+3x
side and you get y = 10
. This model is exponential because x is in the expo-
nent. You can take step two up another notch to include the general form of
the straight line model y = b 0 + b 1 x. Using the definition of logarithm on this
line, you get log 10 _i
y.
b x
10
b 0 +
b x +
y =
b 0 +
=
1
1
If these steps seem dubious to you, stick with me. By looking at the example
in the next section, you can see each step firsthand and that will help a great
deal. In the end, actually finding predictions by using an exponential model is
a lot easier to do than it is to explain.
Spreading secrets at an exponential rate 145
Often, the best way to figure something out is to see it in action. By using
the secret-spreading quiz example from Figure 7-1, you can work through the
series of steps from the preceding section to find the best-fitting exponential
model and use it to make predictions.
Checking the scatterplot
Your goal in step one is to make a scatterplot of the secret-spreading data
and determine whether the data resembles the curved function of an expo-
nential model. Figure 7-1 shows the data for the spread of a number of people
knowing the secret, as a function of the number of days. You can see that the
number of people starts out small, but then as more and more people tell
more and more people, the number grows quickly until the secret isn’t a
secret anymore. This is a good situation for an exponential model, due to the
amount of upward curvature in this graph.
Letting Minitab do its thing to log(y)
In step two, you let Minitab find the best-fitting line to the log(y) data
(see the section “Searching for the best exponential model” to find out how
to do this in Minitab). The output for the analysis of the secret-spreading
data is in Figure 7-9. You can see in Figure 7-9 that the best-fitting line is
log(y) = –0.19 + 0.28 x, where y is the number of people knowing the secret
*
and x is the number of days.