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AP Statistics
Chapter 8
Linear Regression
Objectives:
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Linear model
Predicted value
Residuals
Least squares
Regression to the mean
Regression line
Line of best fit
Slope intercept
se
R2
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Fat Versus Protein: An Example
• The following is a scatterplot of total fat
versus protein for 30 items on the Burger King
menu:
The Linear Model
• The correlation in this example is 0.83. It says
“There seems to be a linear association
between these two variables,” but it doesn’t
tell what that association is.
• We can say more about the linear relationship
between two quantitative variables with a
model.
• A model simplifies reality to help us
understand underlying patterns and
relationships.
The Linear Model
• The linear model is just an equation of a
straight line through the data.
– The points in the scatterplot don’t all line up, but
a straight line can summarize the general pattern
with only a couple of parameters.
– The linear model can help us understand how the
values are associated.
The Linear Model
• Unlike correlation, the linear model requires
that there be an explanatory variable and a
response variable.
• Linear Model
1. A line that describes how a response variable y
changes as an explanatory variable x changes.
2. Used to predict the value of y for a given value of
x.
3. Linear model of the form:
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The Linear Model
• The model won’t be perfect, regardless of the
line we draw.
• Some points will be above the line and some
will be below.
• The estimate made from a model is the
predicted value (denoted as ŷ ).
The Linear Model
• The predicted value:
y
• Putting a hat on y is standard statistics
notation to indicate that something has been
predicted by a model. Whenever you see a hat
over a variable name or symbol, you can
assume it is the predicted version of that
variable or symbol.
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Example: Linear Model
Observed
Values
Predicted
Values
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The Linear Model
• The linear model will not pass exactly through all the
points, but should be as close as possible.
• A good linear model makes the vertical distances
between the observed points and the predicted
points (the error) as small as possible.
• This “error” doesn’t mean it’s a mistake. Statisticians
often refer to variability not explained by the model
as error.
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The Linear Model
Predicted value ŷ (y-hat)
Observed value y
Error = observed – predicted (y – ŷ)
Residuals
• This “error”, the difference between the
observed value and its associated predicted
value is called the residual.
• To find the residuals, we always subtract the
predicted value from the observed one:
residual  observed  predicted  y  ŷ
Residuals
• Symbol for residual is: e
• Why e for residual?
• Because r is already taken. No, the e stands for
“error.”
• So,
e yy
Residuals
• A negative residual means
the predicted value’s too big
(an overestimate).
• A positive residual means
the predicted value’s too
small (an underestimate).
• In the figure, the estimated
fat of the BK Broiler chicken
sandwich is 36 g, while the
true value of fat is 25 g, so
the residual is –11 g of fat.
“Best Fit” Means Least Squares
• Some residuals are positive, others are negative, and,
on average, they cancel each other out.
• So, we can’t assess how well the line fits by adding up
all the residuals.
• Similar to what we did with deviations, we square the
residuals and add the squares.
• The smaller the sum, the better the fit.
• The line of best fit is the line for which the sum of the
squared residuals is smallest, the least squares line.
Least – Squares Regression Line (LSRL)
• The LSRL is the line that minimizes the sum of
the squared residuals between the observed
and predicted y values (y – ŷ).
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Correlation and the Line
• What we know about correlation from chapter 7 can lead us
to the equation of the linear model.
• Start with a scatterplot of standardized values.
Original scatterplot - fat
versus protein for 30 items on
the Burger King menu.
Standardized scatterplot –
zy (standardized fat) vs zx
(standardized protein).
Correlation and the Line
• What line would you choose to model
the relationship of standardized values?
• Start at the center of the line. If an item
has average protein x , should it have
average fat y ?
• Yes, so the line should pass through the
point  x, y  . This is the first property of
the line we are looking for, it must
always pass through the point x, y  .
• In the plot of z-scores, the point  x, y  is
the origin and then the line passes
through the origin (0, 0).
Correlation and the Line
• The equation for a line that
passes through the origin is
y = mx.
• So the equation on our
standardized plot will be
z y  mz x .
• We use z y to indicate that
the point on the line is the
predicted value z y, not the
actual value zy.
Correlation and the Line
• Many lines with different slopes
pass through the origin. Which
one best fits our data? That is,
which slope determines the line
that minimizes the sum of the
squared residuals?
• It turns out that the best choice
for slope is the correlation
coefficient r.
• So, the equation of the linear
model is
z y  rz x .
Correlation and the Line
•
z y  rz x .
• What does this tell us?
• Moving one standard deviation
away from the mean in x moves
us r standard deviations away
from the mean in y.
• Put generally, moving any
number of standard deviations
away from the mean in x moves
us r times that number of
standard deviations away from
the mean in y.
How Big Can Predicted Values Get?
• z y  rz x .
• r cannot be bigger than 1 (in absolute value),
so each predicted y tends to be closer to its
mean (in standard deviations) than its
corresponding x was.
• This property of the linear model is called
regression to the mean; the line is called the
regression line.
The Term Regression
• Sir Francis Galton related the heights of sons to the heights of
their fathers with a regression line. The slope of the line was
less than one.
• That is, sons of tall fathers were tall, but not as much above
the mean height as their fathers had been above their mean.
Sons of short fathers were short, but generally not as far from
their mean as their fathers.
• Galton interpreted the slope correctly as indicating a
“regression” toward the mean height. And regression stuck as
a description of the method he used to find the line.
The Regression Line in Real Units
• Remember from Algebra that a straight line can be written as:
y  mx  b
• In Statistics we use a slightly different notation:
ŷ  b0  b1 x
• We write ŷ to emphasize that the points that satisfy this
equation are just our predicted values, not the actual data
values.
• This model says that our predictions from our model follow a
straight line.
• If the model is a good one, the data values will scatter closely
around it.
The Regression Line in Real Units
• We write b1 and b0 for the slope and intercept
of the line.
• b1 is the slope, which tells us how rapidly ŷ
changes with respect to x.
• b0 is the y-intercept, which tells where the line
crosses (intercepts) the y-axis.
The Regression Line in Real Units
• In our model, we have a slope (b1):
– The slope is built from the correlation and the
standard deviations:
b1  r
sy
sx
– Our slope is always in units of y per unit of x.
The Regression Line in Real Units
• In our model, we also have an intercept (b0).
– The intercept is built from the means and the
slope:
b0  y  b1 x
– Our intercept is always in units of y.
Example: Fat Versus Protein
• The regression line for the
Burger King data fits the
data well:
– The equation is
y  6.8  0.97 x
The predicted fat content for a BK Broiler chicken sandwich (with 30 g of
protein) is 6.8 + 0.97(30) = 35.9 grams of fat.
Calculate Regression Line by Hand
• First calculate the following for the data;
1. The means
2. The standard deviations
3. The correlation r
• Then the LSRL is
1. Slope
2. y- - intercept
Calculate Regression Line on TI-83/84
• Enter the data into lists: explanatory variable
L1 and response L2
• STAT/CALC/LinReg(a+bx)/L1,L2,VARS/YVARS/FUNCTION/Y1
• Your display on the screen shows
LinReg(a+bx)L1,L2,Y1.
– This creates the LSRL and stores it as function Y1.
– The LSRL will now overlay your scatterplot.
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Graphing the LSRL by Hand
• The equation of the LSRL makes prediction
easy. Just substitute an x-value into the
equation and calculate the corresponding yvalue.
• Use the equation to calculate two points on
the line. One at each end of the line (ie. low xvalue and high x-value).
• Plot the two points and draw a line.
Example: Calculate the LSRL by hand and on
the calculator (r = -.64).
LSRL by Hand
• Calculate the slope
– From 2-VAR Stats
–
• Calculate the y-intercept
– From 2-VAR Stats
–
LSRL by Hand - Continued
• Then the LRSL is;
–
• Or in the context of the problem
–
By Calculator
• STAT/CALC/LinReg(a+bx)/L1,L2,VARS/YVARS/FUNCTION/Y1
– y=a+bx
– a=109.8738406
– b=-1.126988915
– r2=.4099712614
– r=-.6402899823
•
• or
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Your Turn:
• Calculate the linear model by hand using
r=.894.
Your Turn:
• Calculate and graph
the linear model
using the Ti-83/84.
Year
Powerboat Reg.
(1000s)
Manatees
Killed
1977
447
13
1978
460
21
1979
481
24
1980
498
16
1981
513
24
1982
512
20
1983
526
15
1984
559
34
1985
585
33
1986
614
33
1987
645
39
1988
675
43
1989
711
50
1990
719
47 37
Facts About LSRL
• The distinction between explanatory and
response variables is essential in regression.
LSR uses the distances of the data points
from the line in only the y direction. If the 2
variables are reversed, you get a different
LSRL.
• The LSRL always passes through the point
.
More Facts on LSRL
• There is a close connection between
correlation and the slope of the LSRL.
A change of one standard deviation in x
corresponds to a change of r standard
deviations in y.
Residuals Revisited
• The linear model assumes that the
relationship between the two variables is a
perfect straight line. The residuals are the part
of the data that hasn’t been modeled.
Data = Model + Residual
or (equivalently)
Residual = Data – Model
e  y  ŷ
Or, in symbols,
Residuals Revisited
• Residuals help us to see whether the model
makes sense.
• When a regression model is appropriate,
nothing interesting should be left behind.
• After we fit a regression model, we usually
plot the residuals in the hope of
finding…nothing.
Residuals Revisited
• The residuals for the BK menu regression look appropriately
boring:
• The sum of the residuals is always equal to zero.
Finding Residuals
• Use the LSRL to find predicted values for each
observed value and calculate the
corresponding residual.
• Example:
– Data – (0,1) (1,6) (2,8) (3,13) (4,13)
– LSRL ŷ = 2 + 3.1x
– e  y  ŷ
Calculated Residuals
Residual Plot
• Is a scatterplot of the residuals against the
explanatory variable (x).
• The residuals are plotted on the vertical axis.
• The explanatory variable (x) is plotted on the
horizontal axis.
• The residual plot helps us assess the fit of a
regression line.
Residual Plot
• Whenever you calculate a LSRL on the TI-83/84, the calculator
automatically calculates the residuals for that particular LSRL
and stores them in a list named RESIDS.
• To create a residual plot on the calculator, make sure you
calculate the LSRL first.
Another Example - Data
Scatterplot
Residual Plot
What to Look for on the Residual Plot
• Random points, no pattern – data fits the linear
model.
• Curved pattern – the relationship is not linear.
• Increasing (or decreasing) spread about the zero
line as x increases – Prediction of y will be less
(more) accurate for larger values of x.
Random Points, No Pattern
Curved Pattern
Increasing Spread
Residual Plot
• Individual points with large residuals – these
points are outliers in the vertical (y) direction.
Outlier in the
y direction
Residual Plot
• Individual points that are extreme in the x
direction – these points are outliers in the
horiztonal (x) direction, such points may not
have large residuals, but they can be very
important.
Outlier in the
x direction
Residual Plot
• No regression analysis is complete without a
display of the residuals to check that the
model is reasonable.
• Because the residuals are the “left over” after
the model describes the relationship, they
often reveal subtleties that were not clear
from a plot of the original data.
Importance of Checking the Residual
Plot
• The scatterplot of the data seems to indicate
that a linear model will be a good fit.
Next, the LSRL
• The LSRL ŷ = 8.8 +7x yieds an r = .9992 and r2
= .9984.
Now the Residual Plot
• The residual plot, however, displays a distinctly curved pattern, indicating
that a nonlinear model will be a better fit for the data.
• The pattern you see in the residual hints at a model you may need to fit
your data.
Moral
• ALWAYS LOOK AT A RESIDUAL PLOT OF YOUR
DATA!
• Any function is linear if plotted over a small
enough interval.
• A residual plot will help you to see patterns in
the data that may not be apparent in the
original graph.
The Residual Standard Deviation
• If the residual plot shows no interesting
pattern, we can look at how large the
residuals are. After all, we are trying to make
them as small as possible.
• Since the mean of the residuals is always zero,
it makes sense to look at how they vary or
their standard deviation.
The Residual Standard Deviation
• The standard deviation of the residuals, se, measures how
much the points spread around the regression line.
• For se to make sense, the residuals should all share the same
variation or spread.
• Check to make sure the residual plot has about the same
amount of scatter throughout. Check the Equal Variance
Assumption with the Does the Plot Thicken? Condition.
• We can check the Equal Variance Assumption in the original
scatterplot or in the residual plot.
Equal Variance Assumption
• Use either the original scatterplot or the residual plot.
• Scatterplot: Variation about the regression line is about the same.
• Residual plot: Variation about the zero residual line is about the same.
• Therefore, both plots show the residuals meet the equal Variance
Assumption.
The Residual Standard Deviation
• We estimate the SD of the residuals using:
se 
2
e

n2
• We don’t need to subtract the mean because the
mean of the residuals e  0.
• We divide by n-2 rather than n-1. We used n-1 for s
when we estimated the mean (used x for µ). Now we
are estimating both slope and the y-intercept, so we
use n-2. We subtract one more for each parameter
we estimate.
The Residual Standard Deviation
• Then it’s a good to make a histogram of the
residuals.
• It should look unimodal and roughly
symmetric.
• Then we can apply the 68-95-99.7 Rule to see
how well the regression model describes the
data.
2
R —The Variation Accounted For
• The variation in the residuals is the key to assessing how well
the model fits.
• Compare the variation of
the response variable with
the variation of the residuals.
• In the BK menu items example,
total fat has a standard deviation
of 16.4 grams. The standard deviation
of the residuals is 9.2 grams.
2
R —The Variation Accounted For
• If the correlation were 1.0 and
the model predicted the fat
values perfectly, the residuals
would all be zero and have no
variation.
• As it is, the correlation is
0.83—not perfection.
• However, we do see that the
model residuals had less
variation than total fat alone.
• We can determine how much
of the variation is accounted
for by the model and how
much is left in the residuals.
2
R —The Variation Accounted For
• The squared correlation, r2, gives the fraction
of the data’s variance accounted for by the
model.
• Thus, 1 – r2 is the fraction of the original
variance left in the residuals.
• For the BK model, r2 = 0.832 = 0.69, so 31% of
the variability in total fat has been left in the
residuals.
2
R —The Variation Accounted For
• All regression analyses include this statistic, although
by tradition, it is written R2 (pronounced “Rsquared”). An R2 of 0 means that none of the
variance in the data is in the model; all of it is still in
the residuals.
• When interpreting a regression model you need to
Tell what R2 means.
– In the BK example, according to the model 69% of the
variation in total fat is accounted for by variation in the
protein content.
2
R —The Variation Accounted For
• The R2 is the Coefficient of Determination, indicates
how well the model (the LSRL) fits the data. R2 values
are between 0 and 1. The closer R2 is to 1, the better
the regression line explains the response. R2 gives
the fraction of the variability of y that is explained by
the least squares linear regression on x.
• Example: R2=.73 means 73% of the total variation in
y is explained by the linear model.
2
How Big Should R Be?
• R2 is always between 0% and 100%. What
makes a “good” R2 value depends on the kind
of data you are analyzing and on what you
want to do with it.
• The standard deviation of the residuals can
give us more information about the usefulness
of the regression by telling us how much
scatter there is around the line.
2
Reporting R
• Along with the slope and intercept for a
regression, you should always report R2 so
that readers can judge for themselves how
successful the regression is at fitting the data.
• Statistics is about variation, and R2 measures
the success of the regression model in terms
of the fraction of the variation of y accounted
for by the regression.
More on Outliers
• Outlying points can strongly influence regression.
• A point can be an outlier because its x-value is
extraordinary, because its y-value is extraordinary, or
because it deviates from the overall pattern of the
data.
• A point has leverage and is call an influential point if
its removal causes the slope of the regression line to
change dramatically.
More on Outliers
• Although the indicated point lies outside the overall pattern
of the data set, its removal has little effect on the regression
line.
• It would not be considered an influential point.
More on Outliers
• The point in the lower right-hand corner of the graph is an
outlier in the x-direction.
• Its removal causes the slope of the regression line to change
dramatically.
• This point has leverage and is an influential point.
Influential Point - Summary
As seen in the prior examples, points that are
outliers in the x-direction on a scatterplot are
often influential on the calculation of the LSRL
or linear model.
Regression: Assumptions and Conditions
• Quantitative Variables Condition:
– Regression can only be done on two quantitative
variables (and not two categorical variables), so
make sure to check this condition.
• Straight Enough Condition:
– The linear model assumes that the relationship
between the variables is linear.
– A scatterplot will let you check that the
assumption is reasonable.
Regression: Assumptions and Conditions
• If the scatterplot is not straight enough, stop
here.
– You can’t use a linear model for any two variables,
even if they are related.
– They must have a linear association or the model
won’t mean a thing.
• Some nonlinear relationships can be saved by
re-expressing the data to make the scatterplot
more linear.
Regression: Assumptions and Conditions
• It’s a good idea to check linearity again after
computing the regression when we can
examine the residuals.
• Does the Plot Thicken? Condition:
– Look at the residual plot -- for the standard
deviation of the residuals to summarize the
scatter, the residuals should share the same
spread. Check for changing spread in the residual
scatterplot.
Regression: Assumptions and Conditions
• Outlier Condition:
– Watch out for outliers.
– Outlying points can dramatically change a
regression model.
– Outliers can even change the sign of the slope,
misleading us about the underlying relationship
between the variables.
• If the data seem to clump or cluster in the
scatterplot, that could be a sign of trouble
worth looking into further.
Reality Check:
Is the Regression Reasonable?
• Statistics don’t come out of nowhere. They are
based on data.
– The results of a statistical analysis should reinforce
your common sense, not fly in its face.
– If the results are surprising, then either you’ve
learned something new about the world or your
analysis is wrong.
• When you perform a regression, think about
the coefficients and ask yourself whether they
make sense.
What Can Go Wrong?
• Don’t fit a straight line to a nonlinear relationship.
• Beware extraordinary points (y-values that stand off
from the linear pattern or extreme x-values).
• Don’t extrapolate beyond the data—the linear model
may no longer hold outside of the range of the data.
• Don’t infer that x causes y just because there is a
good linear model for their relationship—association
is not causation.
• Don’t choose a model based on R2 alone.
What have we learned?
• When the relationship between two
quantitative variables is fairly straight, a linear
model can help summarize that relationship.
– The regression line doesn’t pass through all the
points, but it is the best compromise in the sense
that it has the smallest sum of squared residuals.
What have we learned?
• The correlation tells us several things about the
regression:
– The slope of the line is based on the correlation,
adjusted for the units of x and y.
– For each SD in x that we are away from the x mean, we
expect to be r SDs in y away from the y mean.
– Since r is always between –1 and +1, each predicted y
is fewer SDs away from its mean than the
corresponding x was (regression to the mean).
– R2 gives us the fraction of the response accounted for
by the regression model.
What have we learned?
• The residuals also reveal how well the model
works.
– If a plot of the residuals against predicted values
shows a pattern, we should re-examine the data
to see why.
– The standard deviation of the residuals quantifies
the amount of scatter around the line.
What have we learned?
• The linear model makes no sense unless the
Linear Relationship Assumption is satisfied.
• Also, we need to check the Straight Enough
Condition and Outlier Condition with a
scatterplot.
• For the standard deviation of the residuals, we
must make the Equal Variance Assumption. We
check it by looking at both the original scatterplot
and the residual plot for Does the Plot Thicken?
Condition.
Put it Altogether Example
REVIEW
Example: Creating and Using a Least Squares
Regression Line
A student wanted to see how good she was at
predicting distances measured in meters. She
randomly selected 7 distances to estimate. After
recording her estimated distances, she used a range
finder to find the actual distances. Next is a
scatterplot of the Actual Distances vs. Estimated
Distances and the corresponding residual plot.
Make a Picture
Check Conditions for Regression
• Quantitative Variables Condition: Actual Distance
and Estimated Distance are quantitative.
• Straight Enough Condition: The data follow a straight
line pattern.
• Outlier Condition: There are no outliers in the data.
• Does the Plot Thicken? Condition: The residual plot
shows no obvious patterns.
• The scatterplot follows a linear pattern with no
apparent outliers. The residual plot shows no
discernable pattern, so a linear model is appropriate.
Numerical Analysis
• Use your calculator to find the equation of the LSRL,
if given the data; however, you need to be able to
read various computer outputs to be successful on
the AP Statistics exam.
There will be things on the
printout that you might not
be familiar with, such as
the F-ratio. Focus on
finding the information you
need to write the equation
of the LSRL (ie. Slope & yintercept).
Typical Questions on the Regression
• State the equation of the LSRL. Define any
variables used.
Question
• Interpret the slope and the y-intercept of the
LSRL.
Question
• State and interpret the correlation
coefficient.
Question
• State and interpret the coefficient of
determination.
Question
• Predict the actual distance from an object
when the estimated distance is 4 meters.
Question
• For an estimated distance of 4 meters, the
actual distance from the object was 4.2
meters. What is the residual for this data
value?