Introduction to Linear and Logistic Regression

Basic Ideas Linear Transformation Finding the Regression Line Minimize sum of the quadratic residuals Curve Fitting Logistic Regression Odds and Probability

Basic Ideas Jargon

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.

A score on Y has 2 parts – (1) linear function of X and (2) error.

Y Xi i i= + +α β ε (population values)

Basic Ideas (2) 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 ei i i= + +

′= +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 Permissible operations are addition and

multiplication (interval data)

1086420X

40

35

30

25

20

15

10

5

0

Y

Changing the Y Intercept

Y=5+2XY=10+2XY=15+2X

Add a constant

1086420X

30

20

10

0

Y

Changing the Slope

Y=5+.5XY=5+X

Y=5+2X

Multiply by a constant

′= +Y a bX

Linear Transformation (2) Centigrade to Fahrenheit Note 1 to 1 map Intercept? Slope?

1209060300Degrees C

240

200

160

120

80

40

0D

eg

ree

s F

32 degrees F, 0 degrees C

212 degrees F, 100 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.

′= +Y a bX

Standard Deviation and Variance Square root of the variance, which is the sum of

squared distances between each value and the mean divided by population size (finite population)

Example• 1,2,15 Mean=6

• =6.37

€

1− 6( )2

+ (2 − 6)2 + (15 − 6)2

3= 40.66

€

=1

N∗ x i − x( )

2

i=1

N

∑

Correlation Analysis

Correlation coefficient (also called Pearson’s product moment coefficient)

If rX,Y > 0, X and Y are positively correlated (X’s values increase as

Y’s). The higher, the stronger correlation.

rX,Y = 0: independent; rX,Y < 0: negatively correlated€

rXY =x i − x ( )∑ y i − y ( )

(n −1)σ Xσ Y

Regression of Weight on HeightHt Wt

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

767472706866646260Height in Inches

240

210

180

150

120

90

60

Weight in Lbs

Regression of Weight on Height

Regression of Weight on Height

Regression of Weight on Height

Rise

Run

Y= -316.86+6.97X

Correlation (r) = .94.

Regression equation: Y’=-316.86+6.97X

′= +Y a bX

Predicted Values & Residuals

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

Numbers for linear part and error.

•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

′= +Y a bX

Finding the Regression Line

Need to know the correlation, standard deviation and means of X and Y

€

b = rXY

σ Y

σ X

To find the intercept, use: XbYa −=

Suppose rXY = .50, X = .5, meanX = 10, Y = 2, meanY = 5.

25.

25. ==b

15)10(25 −=−=aXY 215' +−=Slope

Intercept

Line of Least Squares Assume linear relations is reasonable, so the 2

variables can be represented by a line. Where should the line go?

Place the line so errors (residuals) are small 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

Minimize sum of the quadratic residuals

• Derivation equal 0

€

SRSmin = (RS)2

i=1

n

∑

€

RSi = a + bX i −Yi

€

SRSmin = (a + bX i

i=1

n

∑ −Yi)2

€

∂ (a + bX i −Yi)2

i=1

n

∑∂a

= 0

€

∂ (a + bX i −Yi)2

i=1

n

∑∂b

= 0

€

∂ (a + bX i −Yi)2

i=1

n

∑∂a

= 0

€

2 (a + bX i −Yi)i=1

n

∑ ⋅∂ (a + bX i −Yi)i=1

n

∑∂a

= 0

€

2n (a + bX i

i=1

n

∑ −Yi) = 0

€

ai=1

n

∑ + bX i

i=1

n

∑ = Yi

i=1

n

∑

€

a ⋅n + b X i

i=1

n

∑ = Yi

i=1

n

∑

€

∂ (a + bX i −Yi)2

i=1

n

∑∂b

= 0

€

2 (a + bX i −Yi)i=1

n

∑ ⋅∂ (a + bX i −Yi)i=1

n

∑∂b

= 0

€

2 (a + bX i

i=1

n

∑ −Yi) X i

i=1

n

∑ = 0

€

ai=1

n

∑ X i + bX i2

i=1

n

∑ = X iYi

i=1

n

∑

€

a X i

i=1

n

∑ + b X i2

i=1

n

∑ = X iYi

i=1

n

∑

The coefficients a and b are found by solving the following system of linear equations

€

n X i

i=1

n

∑

X i

i=1

n

∑ X i2

i=1

n

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

a

b

⎡

⎣ ⎢

⎤

⎦ ⎥=

Yi

i=1

n

∑

X iYi

i=1

n

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

Curve Fitting

Linear Regression

Exponential Curve

Logarithmic Curve

Power Curve

€

Y = a + bX

€

Y = aebX a > 0

€

Y = aX b a > 0€

Y = a + b ln(x)

The coefficients a and b are found by solving the following system of linear equations

€

n ˆ X ii=1

n

∑

ˆ X ii=1

n

∑ ˆ X i2

i=1

n

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

ˆ a

b

⎡

⎣ ⎢

⎤

⎦ ⎥=

ˆ Y ii=1

n

∑

ˆ X i ˆ Y ii=1

n

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

with

Linear Regression

Exponential Curve

Logarithmic Curve

Power Curve

€

ˆ a := a ˆ X i := X iˆ Y i := Yi

€

ˆ a := ln(a) ˆ X i := X iˆ Y i := ln(Y )i

€

ˆ a := a ˆ X i := ln(X i) ˆ Y i := Yi

€

ˆ a := ln(a) ˆ X i := ln(X i) ˆ Y i := ln(Yi)

Multiple Linear Regression

€

Ti = a + bX i + cYi

€

n X i

i=1

n

∑ Yi

i=1

n

∑

X i

i=1

n

∑ X i2

i=1

n

∑ X iYi

i=1

n

∑

Yi

i=1

n

∑ X iYi

i=1

n

∑ Yi2

i=1

n

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

a

b

c

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

Ti

i=1

n

∑

X iTi

i=1

n

∑

Yi

i=1

n

∑ Ti

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

The coefficients a, b and c are found by solving the following system of linear equations

Polynomial Regression

€

Yi = a + bX i + cX i2

€

n X i

i=1

n

∑ X i2

i=1

n

∑

X i

i=1

n

∑ X i2

i=1

n

∑ X i3

i=1

n

∑

X i2

i=1

n

∑ X i3

i=1

n

∑ X i4

i=1

n

∑

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

a

b

c

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

Yi

i=1

n

∑

X iYi

i=1

n

∑

X i2

i=1

n

∑ Yi

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

The coefficients a, b and c are found by solving the following system of linear equations

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.

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

None of the observations (data points) fall on the regression line

They are all zero or one

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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 X-axis

Such values are theoretically inadmissible

€

P := Y =ea +bX

1+ ea +bX=

1

1+ e−(a +bX )

Linear vs. Logistic regression

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Odds and Probability

€

odds =P

1− P

log(odds) = log it(P) = lnP

1− P

⎛

⎝ ⎜

⎞

⎠ ⎟

€

logit(P) = a + bX

P

1− P= ea +bX

P =ea +bX

1+ ea +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