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Linear Regression/Correlation
• Quantitative Explanatory and Response
Variables
• Goal: Test whether the level of the response
variable is associated with (depends on) the
level of the explanatory variable
• Goal: Measure the strength of the
association between the two variables
• Goal: Use the level of the explanatory to
predict the level of the response variable
Linear Relationships
• Notation:
– Y: Response (dependent, outcome) variable
– X: Explanatory (independent, predictor) variable
• Linear Function (Straight-Line Relation):
Y = a + b X (Plot Y on vertical axis, X horizontal)
– Slope (b): The amount Y changes when X increases by 1
 b > 0  Line slopes upward (Positive Relation)
 b = 0  Line is flat (No linear Relation)
 b < 0  Line slopes downward (Negative Relation)
– Y-intercept (a): Y level when X=0
Example: Service Pricing
• Internet History Resources (New South
Wales Family History Document Service)
• Membership fee: $20A
• 20¢ ($0.20A) per image viewed
• Y = Total cost of service
• X = Number of images viewed
 a = Cost when no images viewed
 b = Incremental Cost per image viewed
• Y = a + b X = 20+0.20X
Example: Service Pricing
Total Cost vs Images Viewed
www.ihr.com.au

60

cost = 20.00 + 0.20 * im ages

R-Square = 1.00

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50
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cost
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0
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100
images
150
200
Linear Regression
Probabilistic Models
• In practice, the relationship between Y and X is
not “perfect”. Other sources of variation exist.
We decompose Y into 2 components:
– Systematic Relationship with X: a + b X
– Random Error: e
• Random respones can be written as the sum of
the systematic (also thought of as the mean) and
random components: Y = a + b X + e
• The (conditional on X) mean response is:
E(Y) = a + b X
Least Squares Estimation
• Problem: a, b are unknown parameters, and must
be estimated and tested based on sample data.
• Procedure:
–
–
–
–
^
Sample n individuals, observing X and Y on each one
Plot the pairs Y (vertical axis) versus X (horizontal)
Choose the line that “best fits” the data.
Criteria: Choose line that minimizes sum of squared
vertical distances from observed data points to line.
Least Squares Prediction Equation:
Y = a  bX
( X  X )(Y  Y )

b=
(X  X )
2
a = Y bX
Example - Pharmacodynamics of LSD
• Response (Y) - Math score (mean among 5 volunteers)
• Predictor (X) - LSD tissue concentration (mean of 5 volunteers)
• Raw Data and scatterplot of Score vs LSD concentration:
80
70
60
LSD Conc (x)
1.17
2.97
3.26
4.69
5.83
6.00
6.41
50
40
SCORE
Score (y)
78.93
58.20
67.47
37.47
45.65
32.92
29.97
30
20
1
2
LSD_CONC
Source: Wagner, et al (1968)
3
4
5
6
7
Example - Pharmacodynamics of LSD
Score (y)
78.93
58.20
67.47
37.47
45.65
32.92
29.97
350.61
LSD Conc (x)
1.17
2.97
3.26
4.69
5.83
6.00
6.41
30.33
x-xbar
-3.163
-1.363
-1.073
0.357
1.497
1.667
2.077
-0.001
y-ybar
28.843
8.113
17.383
-12.617
-4.437
-17.167
-20.117
0.001
Sxx
10.004569
1.857769
1.151329
0.127449
2.241009
2.778889
4.313929
22.474943
Sxy
-91.230409
-11.058019
-18.651959
-4.504269
-6.642189
-28.617389
-41.783009
-202.487243
Syy
831.918649
65.820769
302.168689
159.188689
19.686969
294.705889
404.693689
2078.183343
(Column totals given in bottom row of table)
350.61
30.33
Y=
= 50.087
X=
= 4.333
7
7
 202.4872
b=
= 9.01
a = Y  b X = 50.09  (9.01)( 4.33) = 89.10
22.4749
^
Y = 89.10  9.01X
SPSS Output and Plot of Equation
a
c
i
d
a
i
i
c
c
B
e
i
M
t
E
g
1
(
4
8
6
0
L
9
3
7
4
2
a
D
Math Score vs LSD Concentration (SPSS)
80.00

Linear Regression
70.00

60.00
score

50.00

40.00



30.00
1.00
2.00
score = 89.12 + -9.01 * lsd_conc
R-Square4.00
= 0.88 5.00
3.00
6.00
lsd_conc
Example - Retail Sales
• U.S. SMSA’s
• Y = Per Capita Retail Sales
• X = Females per 100 Males
Per Capita Retail Sales vs Females per 100 Males

Linear Regression
40.00
i
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a
c
pcsale s
30.00
d
a
f
f
i
i
c
c
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S
B
e
i
E
M
t
g
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1
(
1
3
6
0
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20.00
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F
3
8
1
9
0
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pcsales =-9.85

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* f100m
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 0.08
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R-Square
=
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10.00
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0.00
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50.00
75.00

100.00
f100m
125.00
a
D
^
Y = 9.851  0.163 X
Residuals
• Residuals (aka Errors): Difference between
observed values and predicted values: e = Y  Y^
• Error sum of squares:
^
SSE =  (Y  Y )
2
• Estimate of (conditional) standard deviation of Y:
^
SSE
=
=
n2
^
2
(
Y

Y
)

n2
Linear Regression Model
•
•
•
•
Data: Y = a  b X + e
Mean: E(Y) = a  b X
Conditional Standard Deviation: 
Error terms (e) are assumed to be independent and
normally distributed
Parameter
Estimator
b
( X  X )(Y  Y )

b=
(X  X )
a
a  bX
a = Y bX
a  bX
2
^

^
 =
2
(
Y

Y
)

n2
Example - Pharmacodynamics of LSD
^
Y = 89.10  9.01X
Y
X
78.93
58.20
67.47
37.47
45.65
32.92
29.97
1.17
2.97
3.26
4.69
5.83
6.00
6.41
Yhat
e=Y-Yhat
78.5583
0.3717
62.3403
-4.1403
59.7274
7.7426
46.8431
-9.3731
36.5717
9.0783
35.04
-2.12
31.3459
-1.3759
e^2
0.138161
17.14208
59.94785
87.855
82.41553
4.4944
1.893101
253.8861
253.8861
SSE = 253.8861   =
= 7.13
72
^
Correlation Coefficient
• Slope of the regression describes the direction
of association (if any) between the explanatory
(X) and response (Y). Problems:
– The magnitude of the slope depends on the units of
the variables
– The slope is unbounded, doesn’t measure strength of
association
– Some situations arise where interest is in association
between variables, but no clear definition of X and Y
• Population Correlation Coefficient: r
• Sample Correlation Coefficient: r
Correlation Coefficient
• Pearson Correlation: Measure of strength of
linear association:
– Does not delineate between explanatory and
response variables
– Is invariant to linear transformations of Y and X
– Is bounded between -1 and 1 (higher values in
absolute value imply stronger relation)
– Same sign (positive/negative) as slope
r=
 ( X  X )(Y  Y )
 ( X  X )  (Y  Y )
2
2
 sX
= 
 sY

b

Example - Pharmacodynamics of LSD
• Using formulas for standard deviation from
beginning of course: sX = 1.935 and sY = 18.611
• From previous calculations: b = -9.01
 1.935 
r =
(9.01) = 0.94
 18.611 
This represents a strong negative association between math
scores and LSD tissue concentration
Coefficient of Determination
• Measure of the variation in Y that is “explained”
by X
– Step 1: Ignoring X, measure the total variation in Y
(around its mean):
2
TSS =  (Y  Y )
– Step 2: Fit regression relating Y to X and measure the
unexplained variation in Y (around its predicted
^
values):
SSE =  (Y  Y ) 2
– Step 3: Take the difference (variation in Y “explained”
by X), and divide by total:
TSS  SSE
2
r =
TSS
Example - Pharmacodynamics of LSD
TSS =  (Y  Y ) 2 = 2078.183
^
SSE =  (Y  Y ) 2 =253.89
r2 =
2078.183  253.89
= 0.88 = (0.94) 2
2078.183
TSS
80.00

Mean
score

Linear Regression
70.00
70.00


60.00
score
80.00
SSE

60.00

50.00
50.00

Mean = 50.09

40.00
40.00



30.00
1.00


30.00
1.00
2.00
3.00
4.00
lsd_conc
5.00
6.00

score = 89.12 + -9.01 * lsd_conc
2.00 R-Square
3.00 = 0.88
4.00
5.00
6.00
lsd_conc
Inference Concerning the Slope (b)
• Parameter: Slope in the population model (b)
• Estimator: Least squares estimate: b
^
^
^
• Estimated standard error:


b =
(X  X )
2
=
sX n 1
• Methods of making inference regarding population:
– Hypothesis tests (2-sided or 1-sided)
– Confidence Intervals
Significance Test for b
• 2-Sided Test
– H0: b = 0
– HA: b  0
T .S . : tobs =
b
^
b
P  val : 2 P(t | tobs |)
• 1-sided Test
– H0: b = 0
– HA+: b > 0 or
– HA-: b < 0
T .S . : tobs =
b
^
b
P  val  : P(t  tobs ) P  val  : P(t  tobs )
(1-a)100% Confidence Interval for b
^
^
b  ta / 2,n  2  b  b  ta / 2,n  2

(X  X )
2
• Conclude positive association if entire interval above 0
• Conclude negative association if entire interval below 0
• Cannot conclude an association if interval contains 0
• Conclusion based on interval is same as 2-sided hypothesis test
Example - Pharmacodynamics of LSD
^
2
(
X

X
)
= 22.475

n = 7 b = 9.01  = 50.72 = 7.12
7.12
b =
= 1.50
22.475
^
• Testing H0: b = 0 vs HA: b  0
T .S . : tobs
 9.01
=
= 6.01
1.50
P = 2 P(t | 6.01 |)  0
• 95% Confidence Interval for b :
 9.01  2.571(1.50)   9.01  3.86  (12.87,5.15)
t.025,5
Analysis of Variance in Regression
• Goal: Partition the total variation in y into
variation “explained” by x and random variation
^
^
( yi  y ) = ( yi  y i )  ( y i  y )
2
^
^
 ( y  y) =  ( y  y )   ( y  y)
2
i
i
i
2
i
• These three sums of squares and degrees of freedom are:
•Total (TSS)
dfTotal = n-1
• Error (SSE)
dfError = n-2
• Model (SSR)
dfModel = 1
Analysis of Variance in Regression
Source of
Variation
Model
Error
Total
Sum of
Squares
SSR
SSE
TSS
Degrees of
Freedom
1
n-2
n-1
Mean
Square
MSR = SSR/1
MSE = SSE/(n-2)
F
F = MSR/MSE
• Analysis of Variance - F-test
• H0: b = 0
HA: b  0
MSR
T .S . : Fobs =
MSE
P  val : P( F  Fobs )
F represents the F-distribution with 1 numerator and n-2
denominator degrees of freedom
Example - Pharmacodynamics of LSD
• Total Sum of squares:
TSS =  ( yi  y) 2 = 2078.183
dfTotal = 7  1 = 6
• Error Sum of squares:
^
SSE =  ( yi  y i ) 2 = 253.890
df Error = 7  2 = 5
• Model Sum of Squares:
^
SSR =  ( y i  y) 2 = 2078.183  253.890 = 1824.293
df Model = 1
Example - Pharmacodynamics of LSD
Source of
Variation
Model
Error
Total
Sum of
Squares
1824.293
253.890
2078.183
Degrees of
Freedom
1
5
6
Mean
Square
1824.293
50.778
•Analysis of Variance - F-test
• H0: b = 0
HA: b  0
MSR
T .S . : Fobs =
= 35.93
MSE
P  val : P( F  35.93) = .002
(See next slide)
F
35.93
Example - SPSS Output
b
O
m
d
F
S
M
i
a
g
f
a
1
R
2
1
2
8
2
R
1
5
6
T
3
6
a
P
b
D
Significance Test for Pearson
Correlation
• Test identical (mathematically) to t-test for b,
but more appropriate when no clear
explanatory and response variable
• H0: r = 0 Ha: r  0 (Can do 1-sided test)
r
• Test Statistic:
t =
obs
• P-value: 2P(t|tobs|)
(1  r 2 ) /( n  2)
Model Assumptions & Problems
• Linearity: Many relations are not perfectly linear,
but can be well approximated by straight line over
a range of X values
• Extrapolation: While we can check validity of
straight line relation within observed X levels, we
cannot assume relationship continues outside this
range
• Influential Observations: Some data points
(particularly ones with extreme X levels) can exert
a large influence on the predicted equation.
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