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
global temperature
(o C).
Explanatory (predictor)
variable, x.
 Annual atmospheric CO2
concentration.
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Regression model
Y   y| x  
•Y
represents a value of the response
variable.
• y| x represents the population mean
response for a given value of the
explanatory variable, x.
• represents the random error
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Linear Model
 y| x  0  1 x
0
The Y-intercept parameter.
1
The slope parameter.
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Conditions
The relationship is linear.
The random error term,
 , is
 Independent
 Identically
distributed
 Normally distributed with
standard deviation,  .
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 y| x  0  1 x
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15.0
Temp
14.5
14.0
13.5
300
350
CO2
400
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Describe the plot.
Direction – positive/negative.
Form – linear/non-linear.
Strength.
Unusual points?
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CO2 and Temperature.
There is a fairly strong, positive
linear relationship between
CO2 and temperature.
Larger (smaller) values of CO2
are associated with larger
(smaller) values of
temperature.
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Method of Least Squares
Find estimates of
 0 and 1
such that the sum of squared
vertical deviations from the
estimated straight line is the
smallest possible.
10
residual
residual
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Least Squares Estimates
Slope:
Intercept:
Line of
Best fit:
ˆ1
x  x  y  y 


 x  x 
2
ˆ
ˆ
 0  y  1 x
ˆy  ˆ  ˆ x
0
1
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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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Linear Fit
yˆ  ˆ0  ˆ1 x
Predicted Temperature =
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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Interpretation
Estimated slope.
For
each additional 1 ppmv
of CO2, the annual global
temperature goes up
o
0.012584 C, on average.
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Bivariate Fit of Temp By CO2
15.0
Temp
14.5
14.1755
14.0
341.2305
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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How Strong?
SS Model
R 
SSTotal
2
0.80145
R 
 0.806
0.99450
2
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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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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   0.806  0.898
2
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Connection to Correlation
s


y
ˆ1  r  
 sx 
s y is the standard deviation of the y values
s x is the standard deviation of the x values
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