5/27/2015

Review

In Chapter 7, we saw:

  • Scatterplots can show us relationships between two numeric variables.
  • Correlation describes the strength and direction of linear relationships between two numeric variables.
  • We call the \(X\) variable the explanatory variable, because it explains the \(Y\) variable.
  • We call the \(Y\) variable the response because it responds to changes in \(X\).

Where do we go from here?

  • If correlation tells us how well the points fit around a line, what line do they fit around?
  • How can we use this line to describe the relationship further?
  • Can we use it to make predictions?

Fuel Efficiency vs. Weight

Fuel Efficiency vs. Weight

What can we see?

  • \(r = -0.87\)
  • There is a fairly strong negative linear relationship between the weight of a car and its fuel efficiency
  • As we increase the weight of a car, the fuel efficiency tends to decrease.

What else might we want to do?

  • Describe the general trend with a line
  • Make a prediction of what the fuel efficiency of a car weighing 4500 lbs is.

The Line of Best Fit

So how do we find the line the best describes the general trend?

  • The most commonly used method is called the least squares regression (LSR).
  • The least squares regression line is the line that comes closest to all of the points simultaneously.
  • At any value of the \(X\) variable, the line is our prediction for the mean or expected value of the \(Y\) variable.
  • This lets us analyze values of \(Y\) given that we know what \(X\) is.

Fuel Efficiency vs. Weight

The Linear Model

Defining Least Squares Regression:

  • LSR is a linear model
  • A mathematical model is just a formula that describes something in the real world (e.g., Area = Base \(\times\) Height)
  • A statistical model does the same thing, but it accounts for variability or uncertainty (e.g., The Normal Model)
  • LSR calculates the best possible line to describe the overall trend between our variables from our data

The Least Squares Line

Recall from algebra that we can describe a line using the formula:

  • \(y = mx + b\)

In statistics, we use the same method with different, but we label things differently:

  • \(\hat{y} = b_0 + b_1 x\)

What do the terms mean?

  • \(b_1\) \((m)\) is the slope
  • \(b_0\) \((b)\) is the y-intercept
  • \(\hat{y}\) is our prediction for the mean of \(Y\) when \(X = x\)

Algebra Review: Lines

Consider the line: \(\hat{y} = 1 + 2x\)

\(x\) \(\hat{y} = 1 + 2x\) \(\hat{y}\)
\(0\) \(\hat{y} = 1 + 2(0)\) \(1\)
\(1\) \(\hat{y} = 1 + 2(1)\) \(3\)
\(2\) \(\hat{y} = 1 + 2(2)\) \(5\)

What do the coefficients tell us?

  • \(b_0 = 1\) tells us the value of \(\hat{y}\) when \(X = 0\)
  • \(b_1 = 2\) tells us that every time \(X\) goes up by one unit, \(\hat{y}\) increases by \(2\)

Fuel Efficiency vs. Weight

For our car data set, the regression line is:

\[ \widehat{mpg} = 37.2851 -5.3445wt\]

What does this tell us? Keep in mind that weight is in thousands of pounds.

  • For every added 1000 pounds of weight, fuel efficiency drops by \(5.3445\) miles per gallon
  • If a car weighs 0 lbs, it's predicted efficiency is \(37.285\) mpg

This brings up an important note:

  • A car can't weigh 0 lbs.
  • In statistics, the y-intercept is often not interpretable, we just use it to draw the line and make predictions.

Fuel Efficiency vs. Weight

\[\widehat{mpg} = 37.2851 -5.3445wt\]

If a car weighed \(4500\) lbs, what would we expect its efficiency to be?

  • \(\widehat{mpg} = 37.28511 -5.3445(4.5)\)
  • \(\widehat{mpg} = 37.285 - 24.05\)
  • \(\widehat{mpg} = 13.23\)

How do we interpret this?

  • We predict that the average car weighing \(4500\) lbs gets \(13.23\) mpg

Hitting Points

An important note:

  • The Least Squares Regression line is not gauranteed to hit any one of our observed points
  • In fact, it's entirely possible that the line will not hit any of our observed points
  • There is only one place that we know the line will pass through: \((\bar{x}, \bar{y})\)

Hitting Points

Residuals

We said the the Least Squares Regression line doesn't hit every point in our scatterplot, so how can we tell how well it does?

  • For each point \((x, y)\), we predict the value \((x, \hat{y})\)
  • For every observation we have, we can see how far off we were by finding the residual
  • For a given \(x\), the residual is: \(y - \hat{y}\)
  • This is the vertical distance between the line and the true value of \(y\).
  • If our estimate was too high, the residual will be negative
  • If our estimate was too low, the residual will be positive

Residuals: Example

Let's pick on car in our data set, the Camaro Z28. This car gets \(13.3\) mpg and weighs \(3840\) lbs.

  • \(\widehat{mpg} = 37.2851 -5.34455wt\)
  • \(\widehat{mpg} = 37.2851 -5.3445(3.84)\)
  • \(\widehat{mpg} = 37.2851 - 20.5228\)
  • \(\widehat{mpg} = 16.76\)

So how'd we do?

  • \(mpg - \widehat{mpg} = 13.3 - 16.76 = -3.46\)
  • We overestimated by 3.46 mpg

Visualizing Residuals

Residuals: Example

Let's pick another car, the Fiat 128. It's efficiency is \(32.4\) mpg and it weights \(2200\) lbs.

  • \(\widehat{mpg} = 37.2851 -5.34455wt\)
  • \(\widehat{mpg} = 37.2851 -5.3445(2.2)\)
  • \(\widehat{mpg} = 37.2851 - 11.76\)
  • \(\widehat{mpg} = 25.53\)

So how'd we do?

  • \(mpg - \widehat{mpg} = 32.4 - 25.53 = 6.87\)
  • The model underestimated by 6.87 mpg

Visualizing Residuals

Residuals in Reverse

Imagine you bought a car that weighs \(2780\) pounds, and I told you the residual was \(-1.03\). What was the car's fuel efficiency?

  • \(mpg - \widehat{mpg} = -1.03\)
  • \(\widehat{mpg} = 37.2851 -5.34455wt\)
  • \(\widehat{mpg} = 37.2851 -5.3445(2.78)\)
  • \(\widehat{mpg} = 37.2851 - 14.86\)
  • \(\widehat{mpg} = 22.43\)
  • \(mpg - 22.43 = -1.03\)
  • \(mpg = -1.03 + 22.43 = 21.4\)

The Least Squares Line

We've seen how to check the prediction for a single value, but how do we know we did the best we could?

  • The LSR line comes the closest to all of the points simultaneously
  • It does this by finding the line which has the smallest residuals overall
  • Since negative residuals are just as important as positive ones, we square them to force them to be positive
  • Because of this, we focus on the squared residuals

We call our line the Least Squares Regression line because it:

  • Minimizes the sum of the squared residuals
  • This means that it minimizes the sum of the squared vertical distances between the points and the line.

Relationship to Correlation

Recall:

  • Correlation is the strength of the linear relationship between \(X\) and \(Y\).
  • \(-1 \le r \le 1\)
  • The direction of the relationship is indicated by the sign of \(r\)

Because of this,

  • The sign of \(r\) will always match the sign of \(b_1\).

Measure of Fit: \(R^2\)

In order to evaluate how good the model is, we need a measurement of how well it fits. For this, we use the \(R^2\) statistic.

  • \(R^2\) is the fraction of the variability in the response \((Y)\) variable exlained by the \(X\) variable.
  • Do changes in \(X\) explain changes in \(Y\)?

So what is \(R^2\)?

  • If we only have one \(X\) variable, \(R^2 = r^2\)
  • \(-1 \le r \le 1 \quad \to \quad 0 \le R^2 \le 1\)
  • If \(R^2 = 1\), knowing \(X\) lets us perfectly predict \(Y\)
  • If \(R^2 = 0\), \(X\) tells us nothing about \(Y\)

Fuel Efficiency vs. Weight

What is the \(R^2\) of our model that predicts fuel efficiency from weight?

  • \(r = -0.87\), there is a strong negative correlation
  • \(R^2 = (-0.87)^2\)
  • \(R^2 = 0.76\)
  • So the weight of cars explains 76% of the variability in their fuel efficiency
  • This makes sense, obviously other properties (number of cylinders, transmission, design, etc.) will play a role in a car's fuel efficiency

Units in Regression

In Correlation:

  • The correlation coefficient has no units
  • Changes in units (e.g., lbs \(\to\) kg) had no effect on \(r\)

In Regression:

  • The slope is "rise over run", or "change in \(Y\) over change in \(X\)"
  • The slope is measured in units of \(Y\) over units of \(X\)
  • Changing units can significantly change our line

Fuel Efficiency vs. Weight: Changes in Units

So far, we've been measuring the weight in 1000s of pounds

  • \(b_1 = -5.3445\)
  • This represents how much the fuel efficiency in mpg changes when we increase weight by 1000 pounds

What if we represented the weight directly in pounds?

  • The same relationship needs to exist, so changing the weight by 1000 pounds still needs to move mpg down by 5.3445
  • In order for this to be true, the slope needs to be divided by 1000
  • \(b_1 = -5.3445/1000 = -0.0053445\)
  • So changing increasing the weight by a single pound decreases mpg by 0.0053445.

Regression: Outliers

How do outliers affect regression?

If they're above or below the line (extreme in \(Y\)):

  • They can affect the slope drastically
  • They can decrease \(R^2\)

If they fall along the line, but are extreme in \(X\)

  • They can inflate \(R^2\) and make us think the relationship is stronger than it really is

Fuel Efficiency vs. Weight

Fuel Efficiency vs. Weight

Horsepower vs. Engine Displacement

Horsepower vs. Engine Displacement