Stats 155 Class Notes 2012-10-10

Review

\( R^2 \) between zero and 1.

• Use delta with nested models to decide if a model is better than junk.
• SAT versus GPA, \( R^2 \approx 0.16 \). Based on thousands of student records.
• Little \( r \) and the meaning of \( r=0.40 \)

Question about grades: How would you decide if the SAT is deficient or whether it is as good as it can possibly be and grades are deficient?

Survey Project

1. Hand out project description They are going to be creating a Google Form to collect the data.
2. Ask for suggestions about project areas
3. Have students form groups around common interests.

Partial versus total

Suppose I have a used car. I'm going away for a year and thinking of selling it. On the other hand, it would be nice to have a car available when I get back. How much will it cost me to delay selling the car for a year?

Consider several models of used car prices fitted to data on used Hondas:

cars = fetchData("used-hondas.csv")

## Retrieving from http://www.mosaic-web.org/go/datasets/used-hondas.csv

mod1 = lm(Price ~ Age, data = cars)
mod2 = lm(Price ~ Mileage, data = cars)
mod3 = lm(Price ~ Mileage + Age, data = cars)
mod4 = lm(Price ~ Mileage * Age, data = cars)

Each of the first three is nested in the 4th. You can play around with the various models this way:

fetchData("mLM.R")
mLM(Price ~ Age * Mileage, data = cars)

Include and exclude terms to try to answer this question:

Which is the right model to use to inform my car-selling decision?

Tempting to use model 1, since Age is the only variable that I'm interested in.

xyplot(fitted(mod1) + Price ~ Age, data = cars)

It's a bit hard to see the model. Let's try another way of plotting it.

f1 = makeFun(mod1)
plotPoints(Price ~ Age, data = cars)
plotFun(f1(Age) ~ Age, add = TRUE)

How much the price goes down with a year depends on how old the car is, but you can get the rate from the derivative of the function. Let's evaluate that derivative for an 8-year old car with 50,000 miles:

f1 = makeFun(mod1)
df1 = D(f1(Age) ~ Age)
df1(Age = 8)

##     1 
## -1559 

Or, since I'm really thinking about a 1-year difference:

f1(Age = c(9, 8), Mileage = 50000)

##    1    2 
## 6465 8023 

Take the difference.
QUESTION: How come I get the same answer for the finite-difference and the derivative?

But let's consider mod4

f4 = makeFun(mod4)
plotFun(f4(Age = a, Mileage = m) ~ a & m, a.lim = c(0, 10), m.lim = c(0, 1e+05), 
    levels = 1000 * (1:20), npts = 200)
plotPoints(Mileage ~ Age, data = cars, add = TRUE, pch = 20, col = "red")

Examine the change as age goes up by one year. Should I hold mileage constant or should I let the mileage change with age in the typical way?

Here's the same question another way: Do I want to compare cars with different mileages and different ages, or do I want to compare cares with different ages and the same mileage.

Calculating the partial derivative or partial change:

• Take a finite-difference approach

f4(Age = c(8, 9), Mileage = 50000)

##     1     2 
## 12815 12238 

• Use partial derivatives

df4da = D(f4(Age = Age, Mileage = Mileage) ~ Age)
df4da(Age = 8, Mileage = 50000)

##    1 
## -577 

• Do it analytically, using the coefficients to construct the model polynomial.

Review of Partial Derivatives

Relate to the two-variable polynomial: \( f(x,y) = a_0 + a_1 x + a_2 y + a3 x y + ... \)

Relate to the three-variable polynomial.

• Calculation from model function by evaluating at nearby points

Real-world examples

• Insurance example: Should we pay people not to be on the College's health plan?
• NGOs and development aid: Speaker Christie Hansen talked about how donors for health funding in Kenya ended up funding other activities as the government moved money away from health toward agriculture, etc.
• Merit aid. Will it reduce net tuition or increase it. Depends on whether we substantially change the yield.

Activity

Total-vs-partial In-class activity

Car data: work with the two-dimensional graphs