Transcript slides

Introduction to Linear and
Logistic Regression
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Basic Ideas
Linear Transformation
Finding the Regression Line
Minimize sum of the quadratic residuals
Curve Fitting
Logistic Regression
Odds and Probability
Basic Ideas
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Jargon
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IV = X = Predictor (pl. predictors)
DV = Y = Criterion (pl. criteria)
Regression of Y on X
Linear Model = relations between IV and DV
represented by straight line.
Y    X   (population values)
i

i
i
A score on Y has 2 parts – (1) linear function of
X and (2) error.
Basic Ideas (2)
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Yi  a  bX i  ei
Sample value:
Intercept – place where X=0
Slope – change in Y if X changes 1 unit
If error is removed, we have a predicted value
for each person at X (the line): Y   a  bX
Suppose on average houses are worth about 50.00
Euro a square meter. Then the equation relating price
to size would be Y’=0+50X. The predicted price for
a 2000 square meter house would be 250,000 Euro
Linear Transformation
1 to 1 mapping of variables via line Y   a  bX
 Permissible operations are addition and
multiplication (interval data)

4
0
3
5
3
0
2
5
Y
2
=
0
1
5
1
0
h
a
1
Y
=
n
5
Y
1
C
in
g
Y
Y
C
=
5
3
0
2
0
+
0
1
5
2 Y
+
0
hg
X=
2
Y
+
a
2
5
X
=
Y
X
nt
gh
+
5
=
2
+
5
1
0
0
0
0
2
4
6
X
Add a constant
8
1
0
0
2
4
6
8
X
Multiply by a constant
Linear Transformation (2)
Centigrade to Fahrenheit
240
 Note 1 to 1 map
200
160
 Intercept?
120
 Slope?
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Degrees F
Y   a  bX
212 degrees F, 100 degrees C
80
40
32 degrees F, 0 degrees C
0
0
30
60
90
120
Degrees C
Intercept is 32. When X (Cent) is 0, Y (Fahr) is 32.
Slope is 1.8. When Cent goes from 0 to 100 (run), Fahr goes
from 32 to 212 (rise), and 212-32 = 180. Then 180/100 =1.8 is
rise over run is the slope. Y = 32+1.8X. F=32+1.8C.
Standard Deviation and
Variance
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Square root of the variance, which is the sum of
squared distances between each value and the
mean divided by population size (finite
population)
1 N

  xi  x
N i1


Example
• 1,2,15 Mean=6
1 6
2
•

 (2  6)  (15  6) 2
 40.66
3
2
=6.37

2
Correlation Analysis
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Correlation coefficient (also called Pearson’s product moment
coefficient)
rXY
x


i
 x y i  y 
(n 1) X  Y
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If rX,Y > 0, X and Y are positively correlated (X’s values increase as
Y’s). The higher, the stronger correlation.
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rX,Y = 0: independent;

rX,Y < 0: negatively correlated
ig
Regression of Weight on Height
X
61
105
62
120
63
120
65
160
65
120
68
145
69
175
70
160
72
185
75
210
N=10
N=10
mean=67
mean=150
=4.57
 = 33.99
R
R
ee
gg
rr
e
Wt
2
2
4
0
1
0
Y   a  bX
Y
=
-
1
8
0
1
5
0
R
R
W
Ht
1
2
9
0
6
0
6
6
u
0
6
0
6
2
H
6
4
7
6
e
7
8
7
0
ig
Correlation (r) = .94.
Regression equation: Y’=-316.86+6.97X
Predicted Values & Residuals
Y   a  bX
•Y’ is called the
predicted value
•Y-Y’ the residual
(RS)
•The residual is the
error
•Mean of Y’ and Y
is the same
•Variance of Y is
equal to the
variance Y’ + RS
Numbers for linear part and error.
N
Ht
Wt
Y'
RS
1
61
105
108.19
-3.19
2
62
120
115.16
4.84
3
63
120
122.13
-2.13
4
65
160
136.06
23.94
5
65
120
136.06
-16.06
6
68
145
156.97
-11.97
7
69
175
163.94
11.06
8
70
160
170.91
-10.91
9
72
185
184.84
0.16
10
75
210
205.75
4.25
mean
67
150
150.00
0.00

4.57
33.99
31.85
11.89
V
20.89
1155.56
1014.37
141.32
Finding the Regression Line
Need to know the correlation, standard deviation and means
of X and Y
Y
b  rXY
To find the intercept, use:
X
a  Y  bX

Suppose rXY = .50, X = .5, meanX = 10, Y = 2, meanY = 5.
Slope
Intercept
2
b  .5  2
.5
a  5  2(10)  15
Y '  15  2 X
Line of Least Squares
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Assume linear relations is reasonable, so the 2
variables can be represented by a line. Where should
the line go?
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Place the line so errors (residuals) are small
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The line we calculate has a sum of errors = 0
It has a sum of squared errors that are as small as possible;
the line provides the smallest sum of squared errors or least
squares

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Minimize sum of the quadratic
residuals
n
SRS min   (RS) 2
i1
RSi  a  bXi Yi
n
SRS min   (a  bX i  Yi ) 2
i1
• Derivation equal 0
n
  (a  bX i  Yi ) 2
i1
a
n
0
  (a  bX i  Yi ) 2
i1
b
0
n
  (a  bX i  Yi ) 2
i1
a
0
n
n
i1
i1
2 (a  bX i  Yi )    (a  bX i  Yi )

a
0
n

2n  (a  bX i  Yi )  0
i1
n
n
n
 a   bX  Y
i

i1
i1
i
i1
n
n
i1
i1
a  n  b X i  Yi
n
  (a  bX i  Yi ) 2
i1
b
0
n
n
i1
i1
2 (a  bX i  Yi )    (a  bX i  Yi )

b
n
n
i1
i1
0
2 (a  bX i  Yi ) X i  0

n
n
n
i1
i1
i1
2
a
X

bX
 i  i   X iYi

n
n
n
i1
i1
i1
a X i  b X i2   X iYi

The coefficients a and b are found by
solving the following system of linear
equations

 n

n

 X i
i1

 n

 X i a  Yi 
i1
   i1 
n
n

b 
2 
 X i   X iYi

i1

i1
n
Curve Fitting
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Linear Regression
Y  a  bX
Exponential Curve
Y  ae bX

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Logarithmic Curve
a0
Y  a  bln( x)


Power Curve

Y  aX b
a0

The coefficients a and b are found by
solving the following system of linear
equations

 n

n
 ˆ
 X i
i1

 n

ˆ
ˆ
X
Y


 i aˆ  i 
i1
   i1 
n
n
b  ˆ ˆ 
2 
ˆ
 X i   X iYi

i1

i1
n


with

Linear Regression
aˆ : a
Xˆ i : X i

Exponential Curve
aˆ : ln( a)
Xˆ i : X i
Yˆi : Yi
Logarithmic Curve
aˆ : a
Xˆ i : ln( X i )
 Power Curve
aˆ : ln( a)
Xˆ i : ln( X i )
Yˆi : ln(Y)i

Yˆi : Yi
Yˆi : ln(Yi )
Multiple Linear Regression
Ti  a  bXi  cYi

The coefficients a, b and c are found by
solving the following system of linear
equations

 n

 n
 X i
i1n
 Y
i

i1
n
X
i
i1
n
X
2
i
i1
n
X Y
i i
i1

 n

Yi    Ti 
i1
a  i1 
n
   n

 X iYib  X iTi
i1
i1


c
  n
n
Yi2  YiTi 

i1

i1
n
Polynomial Regression
Yi  a  bX i  cX i2

The coefficients a, b and c are found by
solving the following system of linear
equations

 n

 n
 X i
i1
n
 X 2
 i

i1
n
X
i
i1
n
X
2
i
i1
n
X
i1

 n

 X   Yi 
i1
a  i1 
n
n

  
3 
 X i b  X iYi 
i1
i1


c
  n
n
 X i4   X i2Yi

i1

i1
n
3
i
2
i
Logistic Regression


Variable is binary (a categorical variable that
has two values such as "yes" and "no") rather
than continuous
binary DV (Y) either 0 or 1


For example, we might code a successfully kicked
field goal as 1 and a missed field goal as 0
or we might code yes as 1 and no as 0 or admitted
as 1 and rejected as 0 or Cherry Garcia flavor ice
cream as 1 and all other flavors as zero.

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If we code like this, then the mean of the distribution
is equal to the proportion of 1s in the distribution.
For example if there are 100 people in the
distribution and 30 of them are coded 1, then the
mean of the distribution is .30, which is the
proportion of 1s
The mean of a binary distribution so coded is
denoted as P, the proportion of 1s
The proportion of zeros is (1-P), which is sometimes
denoted as Q
The variance of such a distribution is PQ, and the
standard deviation is Sqrt(PQ)


Suppose we want to predict whether someone
is male or female (DV, M=1, F=0) using height
in inches (IV)
We could plot the relations between the two
variables as we customarily do in regression.
The plot might look something like this
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

None of the observations (data points) fall on the
regression line
They are all zero or one

Predicted values (DV=Y)correspond to
probabilities
If linear regression is used, the predicted
values will become greater than one and less
than zero if one moves far enough on the Xaxis
 Such values are theoretically inadmissible

a bX
e
1
P : Y 

a bX
1 e
1 e(a bX )
Linear vs. Logistic regression
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Odds and Probability
P
odds 
1 P
 P 
log( odds)  log it(P)  ln 

1 P 
log it(P)  a  bX
P
 e a  bX
1 P
a  bX
e
P
1 e a  bX
Linear regression!
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






Basic Ideas
Linear Transformation
Finding the Regression Line
Minimize sum of the quadratic residuals
Curve Fitting
Logistic Regression
Odds and Probability