Stat 401 B – Lecture 6
Simple Linear Regression
„Question
„Is
annual carbon dioxide
concentration related to
annual global temperature?
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Simple Linear Regression
„Response variable, Y.
„ Annual
(o
global temperature
C).
„Explanatory (predictor)
variable, x.
„ Annual atmospheric CO2
concentration.
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Regression model
Y = μ y| x + ε
•Y
represents a value of the response
variable.
•μ y| x represents the population mean
response for a given value of the
explanatory variable, x.
•ε represents the random error
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Stat 401 B – Lecture 6
Linear Model
Y = μ y| x + ε = β 0 + β1 x + ε
β0
The Y-intercept parameter.
β1
The slope parameter.
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Conditions
„The relationship is linear.
„The random error term, ε , is
„ Independent
„ Identically
distributed
„ Normally distributed with
standard deviation, σ .
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15.0
Temp
14.5
14.0
13.5
300
350
CO2
400
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Stat 401 B – Lecture 6
Describe the plot.
„Direction – positive/negative.
„Form – linear/non-linear.
„Strength.
„Unusual points?
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Method of Least Squares
„Find estimates of
β 0 and β1
such that the sum of squared
vertical deviations from the
estimated straight line is the
smallest possible.
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Least Squares Estimates
βˆ1 = ∑
(x − x )( y − y )
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∑ (x − x )
βˆ0 = y − βˆ1 x
yˆ = βˆ0 + βˆ1 x
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Stat 401 B – Lecture 6
Bivariate Fit of Temp By CO2
15.0
Temp
14.5
14.0
13.5
300
350
400
CO2
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Linear Fit
Linear Fit
yˆ = βˆ0 + βˆ1 x
„Predicted Temp = 9.8815 +
0.012584*CO2
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Interpretation
„Estimated Y-intercept.
„This
does not have an
interpretation within the
context of the problem.
Having no CO2 in the
atmosphere is not
reasonable given the data.
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Stat 401 B – Lecture 6
Interpretation
„Estimated slope.
„For
each additional 1 ppmv
of CO2, the annual global
temperature goes up
0.012584 o C, on average.
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Bivariate Fit of Temp By CO2
15.0
Temp
14.5
14.0
13.5
300
350
400
CO2
Linear Fit
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How Strong?
„The strength of a linear
relationship can be
measured by R2, the
coefficient of determination.
„RSquare in JMP output.
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Stat 401 B – Lecture 6
How Strong?
R2 =
SS Model
SSTotal
R2 =
0.80145
= 0.806
0.99450
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Interpretation
„80.6% of the variation in the
global temperature can be
explained by the linear
relationship with carbon
dioxide concentration.
„19.4% is unexplained.
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Interpretation
„There is a fairly strong
positive linear relationship
between carbon dioxide
concentration and global
temperature.
„Cause and effect?
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Stat 401 B – Lecture 6
Cause and Effect?
„There is a strong positive
linear relationship between
the number of 2nd graders in
communities and the number
of crimes committed in those
communities.
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Connection to Correlation
„If you square the correlation
coefficient, r, relating carbon
dioxide to global temperature
you get R2, the coefficient of
determination.
r = ± R 2 = + 0.806 = +0.898
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Connection to Correlation
⎛ sy ⎞
⎟⎟
⎝ sx ⎠
βˆ1 = r ⎜⎜
s y is the standard deviation of the y values
s x is the standard deviation of the x values
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