Lottery: A tax on the statistically-challenged.

Chapter 4

Sec 4.1

Linear regression using the LSRL is not the only model for describing
data. Some data just are not best described linearly. Non-linear
relationships between two quantitative variables can sometimes be* changed*
into linear relationships by* transforming *one or both variables.
Removal of outliers from data may cause a drop in the correlation so that linear
no longer does a satisfactory job of describing the data. The bulk of the
data may not be linear at all.

Transforming can be thought of as re-expressing the data. We may want to transform either the explanatory variable x, or the response variable y in a scatter plot, or maybe even both. We will call the transformed variable "t" when talking about the transforming in general.

Transforming data amounts to changing the scale. Linear transformations as discussed in Chapter 1 change units by addition or multiplication. Recall, adding a constant amount to each observation does NOT change the spread but does add that constant to the center and quartiles. Multiplying by a constant multiplies both the measure of center and the measure of spread by the same constant. BUT linear transformations of this type CANNOT STRAIGHTEN OUT A CURVED RELATIONSHIP BETWEEN TWO VARIABLES.

Some non-linear functions that have been studied in Algebra include quadratic, logarithmic, reciprocal (negative power), square root, and other power functions.

Monotonic Transformations:

A monotonic function f(t) moves in ONE direction as its argument t increases.

A monotonic increasing function preserves the ORDER of data. If a>b, then f(a)>f(b).

A monotonic decreasing function reverses the ORDER of data. If a>b, then f(a)<f(b).

The graph of a linear function is a straight line. The graph of a monotonic increasing function is increasing EVERYWHERE. A monotonic decreasing function has a graph that is decreasing EVERYWHERE. A function can be monotonic over some range of t without being everywhere monotonic. For example, the square (quadratic, parabolic) function t

The increasing monotonic functions are linear, quadratic t

2) next choose a power or logarithmic transformation that

Other things to remember:

1) Power transformations t

2) Power transformations tp for powers p less than 1 (and logarithmic for p = 0) are concave down. They pull in the right tail and push out the left tail. Effect strengthens as p decreases.

There are a variety of transformations available to us depending
on the appearance of the data. Using a "trial and error" or "try it and
see" approach isn't very satisfactory or efficient. It is much more
logical to begin with a theory or mathematical model that we expect to describe
a relationship The transformation needed to make the relationship
linear is then a consequence of the model. One of the most common models
is EXPONENTIAL growth.

A variable grows linearly over time if it ADDS a fixed increment in each equal
time period. Exponential growth occurs when a variable is MULTIPLIED by a
fixed number in each time period. (Exponential growth increases by a fixed*
percentage *of the previous total). We have studied exponential
growth before when dealing with interest problems, bacterial growth, or epidemic
situations.

What does exponential growth look like??? Can we trust our eyes to recognize it??? How can we be sure that we have a correct model???

When exponential growth is suspected from viewing the data, the first step is to calculate the ratios of consecutive terms to see that they are approximately the same. Exponential form is y = ab

log (a) + x log (b). When we plot log (y) against x, we should observe a straight line for the transformed data. Apply least-squares regression to the transformed data and check the correlation (r) value and the coefficient of determination (r

We will do Ex. 4.6 page 209 by hand and using technology.

POWER LAW MODELS

When you order pizza you order by its diameter, like 10 inches...BUT when you eat pizza you eat its AREA or A = Πr

When we are dealing with things of the same general form, whether circles or fish or people, we expect area to go up with the

Biologists have found that MANY characteristics of living things are described quite closely by power laws. There are more mice than elephants, and more flies than mice, the abundance of species follows a power law with number of eggs a bird lays, and so on. Sometimes the powers can be predicted from geometry, but sometimes they are mysterious. Why, for example, does the rate at which animals use energy go up as the 3/4 power of their body weight?? This particular relationship is called "Kleiber's law and works from bacteria to whales. Therefore, power laws are a good place to start in simplifying relationships for

Exponential growth models become linear when we apply the logarithm transformation to just the response variable y.

Power law model is y = a x^{p}

Take the log of both sides to get log (y) = log (a) + p log (x)

Taking the log of BOTH variables straightens the scatter plot of y against
x.

The power p becomes the slope of the line that links log y to log x.

Follow the steps above to verify that the regression is a good fit using r and r^{2}.

Great, specific directions for accomplishing this appear on page 219 in the
technology toolbox.

IF taking logs of both variables makes a scatter plot linear, a power law is a
reasonable model for the original data. We can even estimate what power
the law involves by regressing log y on log x and using the slope of the
regression line as a good estimate of the power. Remember the slope is
only an estimate of the p in an underlying power model. The greater the
scatter of the points in the scatter plot about the fitted line, the smaller the
confidence that this estimate is accurate. See E. 4.8 page 215 for an
example