Transcript Document

Chapter 5
Regression
BPS - 3rd Ed.
Chapter 5
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Objectives of Regression
 To
describe the change in Y per unit
X
 To predict the average level of Y at a
given level of X
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“Returning Birds” Example
Plot data first to see if
relation can be described
by straight line
(important!)
Illustrative data from
Exercise 4.4
Y = adult birds joining
colony
X = percent of birds
returning, prior year
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If data can be described by
straight line
…
describe relationship with equation
Y = (intercept) + (slope)(X)
 May also be written:
Y = (slope)(X) + (intercept)
Intercept  where line crosses Y axis
Slope  “angle” of line
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Linear Regression

Algebraic line  every point falls on line:
exact y = intercept + (slope)(X)
line  scatter cloud suggests a
linear trend:
 Statistical
“predicted y” = intercept + (slope)(X)
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Regression Equation
ŷ = a + bx, where
– ŷ (“y-hat”) is the predicted value of Y
– a is the intercept
The TI calculators
reverse a & b!
– b is the slope
– x is a value for X
 Determine
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a & b for “best fitting line”
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What Line Fits Best?
If we try to draw the
line by eye, different
people will draw
different lines
We need a method
to draw the “best
line”
This method is
called “least
squares”
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The “least squares” regression line
Each point has:
Residual
= observed y – predicted y
= distance of point from prediction line
The least squares line
minimizes the sum of
the square residuals
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Calculating Least Squares
Regression Coefficients
 Formula
(next slide)
 Technology
– TI-30XIIS
– Two variable Applet
– Other
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Formulas
b
= slope coefficient
a = intercept coefficient
sy
br
sx
a  y  bx
where sx and sy are the standard deviations of
the two variables, and r is their correlation
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Technology: Calculator
BEWARE!
TI calculators label the slope and
intercept backwards!
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Regression Line
 For
the “bird data”:
a = 31.9343
b = 0.3040
 The linear regression equation is:
ŷ = 31.9343  0.3040x
The slope (-0.3040) represents the
average change in Y per unit X
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Use of Regression for Prediction
Suppose an individual colony has 60% returning (x =
60). What is the predicted number of new birds for
this colony?
Answer:
ŷ = a + bx = 31.9343  (0.3040)(60) = 13.69
Interpretation: the regression model predicts
13.69 new birds (ŷ) for a colony with x = 60.
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Prediction via Regression Line
Number of new birds and Percent returning
When X = 60,
the regression
model predicts
Y = 13.69
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Case Study
Per Capita Gross Domestic Product
and Average Life Expectancy for
Countries in Western Europe
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Regression Calculation
Case Study
Country
Austria
Belgium
Finland
France
Germany
Ireland
Italy
Netherlands
Switzerland
United Kingdom
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Per Capita GDP (x)
21.4
23.2
20.0
22.7
20.8
18.6
21.5
22.0
23.8
21.2
Chapter 5
Life Expectancy (y)
77.48
77.53
77.32
78.63
77.17
76.39
78.51
78.15
78.99
77.37
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Life Expectancy and GDP
(Europe)
Case Study (Life Expectancy)
Life expectancy (yrs)
79
78
77
76
18
19
20
21
22
23
24
Per Capital GDP
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Regression Calculation
by Hand (Life Expectancy Study)
Calculations:
x  21.52
s x  1.532
y  77.754
sy  0.795
r  0.809
 0.795 
br
 (0.809)
  0.420
sx
 1.532 
a  y  bx  77.754 - (0.420)(21 .52)  68.716
sy
ŷ = 68.716 + 0.420x
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BPS/3e Two Variable Applet
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Applet: Data Entry
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Applet: Calculations
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Applet: Scatterplot
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Applet: least squares line
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Interpretation
Life Expectancy Case Study
ŷ = 68.716 + (0.420)X
 Slope: For each increase in GDP 
0.420 years increase in life expectancy
 Prediction example: What is the life
expectancy in a country with a GDP of
20.0?
ANSWER:
ŷ = 68.716 + (0.420)(20.0) = 77.12
 Model:
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Coefficient of Determination (R2)
(Fact 4 on p. 111)
 “Coefficient
of determination, (R2)
Quantifies the fraction of the Y “mathematically
explained” by X
Examples:
 r=1:
R2=1:
 r=.7:
R2=.49: regression line explains almost half
(49%) of the variation in Y
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regression line explains all (100%) of
the variation in Y
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We are NOT going to cover the analysis of
residual plots (pp. 113-116)
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Outliers and Influential Points
 An
outlier is an observation that lies far
from the regression line
 Outliers
in the y direction have large
residuals
 Outliers
in the x direction are influential
– removal of influential point would markedly
change the regression and correlation values
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Outliers:
Case Study
Gesell Adaptive Score and Age at First Word
After removing
child 18
r2 = 11%
From all the data
r2 = 41%
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Cautions
About Correlation and Regression
 Describe
 Are
only linear relationships
influenced by outliers
 Cannot
be used to predict beyond the
range of X (do not extrapolate)
 Beware
of lurking variables (variables other
than X and Y)
– Association does not always equal causation!
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Do not extrapolate (Sarah’s height)
 Sarah’s
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height (cm)
height is
plotted against her
age
 Can you predict her
height at age 42
months?
 Can you predict her
height at age 30
years (360 months)?
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90
85
80
30 35 40 45 50 55 60 65
age (months)
30
Do not extrapolate (Sarah’s height)
 Regression
ŷ = 71.95 + .383(42) = 88
(Reasonable)

At age 360 months:
210
190
height (cm)
equation:
ŷ = 71.95 + .383(X)
 At age 42 months:
170
150
130
110
ŷ = 71.95 + .383(360) =
209.8
(That’s over 17 feet
tall!)
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70
30
90 150 210 270 330 390
age (months)
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Caution: Correlation does not always
mean causation
Even very strong correlations
may not correspond to a causal
relationship between x and y
(Beware of the lurking variable!)
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