Power modelsRelationships between categorical

variables

Exponential and Power Transformations

for achieving linearity

Exponential data benefits from taking the logarithm of the response variable.

Power models may benefit from taking the logarithm of both variables.

Power law models

y = axp

When do we see power models?

Area

Volume

Abundance of species

Power Law Models

The theory behind logarithms makes it that taking the logarithm of both variables in a power model yields a linear relationship between log x and log y.

Power regression models are appropriate when a variable is proportional to another variable to a power.

y = axp

Power Law Models

Notice the power p in the power law becomes the slope of the straight line that links logx to logy.

y = axp

yieldslogy= loga + plogx

We can even roughly estimate what power p the law involves by finding the LSRL of logy on logx and using

the slope of the line as an estimate of the power.

Body and brain weight of 96 species of mammals

You might remember this example from last time.

When we plot the logarithm of brain weight against the logarithm of body weight for all 96

species we get a fairly linear form. This suggests that a power law governs this relationship.

log ( ) 1.01 0.72log ( )brain size body weight

Prediction from a power model

Bigfoot is estimated to weigh about 280 pounds or 127 kilograms. Use the model to predict Bigfoot's

brain weight.

Inverse Transformation for a Power Model

Use an inverse transformation to find the model that fits the original data.

log ( ) 1.01 0.72log ( )brain size body weight

ˆ 10a by x

What's a planet anyway?

Predict Xena's period of revolution from the data if it is 9.5 billion miles from the sun.

Graphs: Scatterplot, Linearized plot, Residual plot

Numerical summaries: r and r2

Model:Give an equation for our linear model with a statement of how well it fits the data

Interpretation: Is our model sufficient for making predictions?Predict Xena's period of revolution

(an astronomical unit is 93 million miles)

What's a planet anyway?

Relationships between categorical variables

College Students

● Two way table ● Row variable● Column variable

College Students

● Marginal distributions

● Round off error

College Students

● percents

● Calculate the marginal distribution of age group in percents.

College Students

● Each marginal distribution from a two-way table is a distribution for a single categorical variable.

● Construct a bar graph that displays the distribution of age for college students.

College Students

● To describe relationships among or compare categorical variables, calculate appropriate percents.

College Students

● Conditional distributions● Compare the percents of women in each age group

by examining the conditional distributions.● Find the conditional distribution of gender, given that

a student is18 to 24 years old.

Computer Outputof a two-way table

Read the last paragraph on page 297