Power models Relationships between categorical variables
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Power modelsRelationships between categorical
variables
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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.
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Power law models
y = axp
When do we see power models?
Area
Volume
Abundance of species
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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
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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.
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Body and brain weight of 96 species of mammals
You might remember this example from last time.
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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
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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.
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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
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What's a planet anyway?
Predict Xena's period of revolution from the data if it is 9.5 billion miles from the sun.
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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?
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Relationships between categorical variables
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College Students
● Two way table ● Row variable● Column variable
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College Students
● Marginal distributions
● Round off error
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College Students
● percents
● Calculate the marginal distribution of age group in percents.
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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.
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College Students
● To describe relationships among or compare categorical variables, calculate appropriate percents.
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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.
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Computer Outputof a two-way table
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Read the last paragraph on page 297