Regression Analysis

Intro to OLS Linear Regression

Regression AnalysisDefined as the analysis of the statistical relationship among variablesIn it’s simplest form there are only two variables:Dependent or response variable (labeled

as Y) Independent or predictor variable (labeled

as X)

Statistical Relationships- A warning

Be aware that as with correlation and other measures of statistical association, a relationship does not guarantee or even imply a causality between the variablesAlso be aware of the difference between a mathematical or functional relationship based upon theory and a statistical relationship based upon data and its imperfect fit to a mathematical model

Simple Linear RegressionThe basic function for linear regression is Y=f(X) but the equation typically takes the following form:Y=α+βX+ε α - Alpha – an intercept component to the model that

represents the models value for Y when X=0 β - Beta – a coefficient that loosely denotes the nature of

the relationship between Y and X and more specifically denotes the slope of the linear equation that specifies the model

ε - Epsilon – a term that represents the errors associated with the model

XY

XY

Examplei in this case is a “counter” representing the ith observation in the data set

ii XY

observation (i) number of ice cream cones sold (Y) cost of icecream cones (X)1 84 $2.502 89 $3.003 92 $3.254 96 $2.255 98 $1.756 102 $2.757 113 $2.008 114 $1.509 122 $1.25

10 127 $1.00

Ice Cream Demand

80

90

100

110

120

130

$0.75$1.00

$1.25$1.50

$1.75$2.00

$2.25$2.50

$2.75$3.00

$3.25$3.50

Ice Cream Cone Cost

Ice

Crea

m C

one'

s So

ldAccompanying Scatterplot

Accompanying Scatterplot with Regression Equation

Ice Cream Demand

y = -16.094x + 137.81R2 = 0.7078

80

90

100

110

120

130

$0.75$1.00

$1.25$1.50

$1.75$2.00

$2.25$2.50

$2.75$3.00

$3.25$3.50

Ice Cream Cone Cost

Ice

Crea

m C

one'

s So

ld

What does the additional info mean?

α - Alpha – 138 conesβ - Beta – -16 cones/$1 increase in costε - Epsilon – still present and evidenced by the

fact that the model does not fit the data perfectly

R2 - a new term, the Coefficient of Determination - a value of 0.71 is pretty good considering that the value is scaled between 0 and 1 with 1 being a model with a perfect agreement with the data

Coefficient of Determination

In this simple example R2 is indeed the square of RRecall that R is often the symbol for the Pearson Product Moment Correlation (PPMC) which is a parametric measure of association between two variablesR (X,Y) = -0.84 in this case –0.84^2=0.71

A Digression into HistoryAdrien Legendre- the original author of “the method of least squares”, published in 1805

The guy that got the credit-Carl-Fredrick- the “giant” of early statisticsAKA – Gauss – published the theory of least squares in 1821

Back on Topic – a recap of PPMC or r

From last semester:The PPMC coefficient is essentially the

sum of the products of the z-scores for each variable divided by the degrees of freedom

Its computation can take on a number of forms depending on your resources

What it looks like in equation form:

The sample covariance is the upper center equation without the sample standard deviations in the denominatorCovariance measures how two variables covary and it is this measure that serves as the numerator in Pearson’s r

1nzz

r yx

yxssnyyxx

r)1(

))((

1)( 2

nxx

sx

22 )()(

))((

yyxx

yyxxr

nyynxx

nyxxyr

/)(/)(

/)()(2222

Computationally EasierMathematically Simplified

Take home messageCorrelation is a measure of association between two variablesCovariance is a measure of how the two variables vary with respect to one anotherBoth of these are parametrically based statistical measures – note that PPMC is based upon z-scoresZ-scores are based upon the normal or Gaussian distribution - thus these measures as well as linear regression based upon the method of least squares is predicated upon the assumption of normality and other parametric assumptions

OLS definedOLS stands for Ordinary Least SquaresThis is a method of estimation that is used in linear regressionIts defining and nominal criteria is that it minimizes the errors associated with predicting values for YIt uses a least squares criterion because a simple “least” criterion would allow positive and negative deviations from the model to cancel each other out (using the same logic that is used for computations of variance and a host of other statistical measures)

2

1

)(min i

n

ii YY

The math behind OLSRecall that the linear regression equation for a single independent variable takes this form: Y=α+βX+ε

2

1

)(min

n

iii XY

(i) (Y) (X)1 84 $2.502 89 $3.003 92 $3.254 96 $2.255 98 $1.756 102 $2.757 113 $2.008 114 $1.509 122 $1.25

10 127 $1.00

Since Y and X are known for all I and the error term is immutable, minimizing the model errors is really based upon our choice of alpha and beta

2

1

)(min

n

iii XY This is this under the condition that S is the

total sum of squared deviations from i =1 to n for all Y and X for an alpha and beta

2

1

)(),(

n

iii XYS

The correct alpha and beta to minimize S can be found by taking the partial derivative for alpha and beta by setting each of them equal to zero for the other, yielding

0)(1

n

iii XY for alpha, and 0)(

1

n

iiii XYX for beta

which can be further simplified to

n

i

n

iii

n

iii XXYX

1 1

2

1

for alpha and

for beta

n

ii

n

ii XnY

11

Refer to page 436 for the the text’s more detailed description of the computations for solving for alpha and beta

n

i

n

iii

n

iii XXYX

1 1

2

1

n

ii

n

ii XnY

11

Given these, we can easily solve for the more simple alpha via algebra

n

ii

n

ii XnY

11

is

n

X

n

Yn

ii

n

ii

11

and since X(bar) is the sum of all X(I) from 1 to n diveded by n and the same can be said for Y(bar) we are left with XYSince the mean of both X and

Y can be obtained from the data, we can calculate the intercept or alpha very simply if we know the slope or beta

Once we have a simple equation for alpha, we can plug it into the equation for beta and then solve for the slope of the regression equation

n

i

n

iii

n

iii XXYX

1 1

2

1

n

X

n

Yn

ii

n

ii

11

n

i

n

iii

n

ii

n

iin

iii XX

n

X

n

YYX

1 1

211

1

n

ii

n

ii

n

i

n

iiin

iii X

n

X

n

XYYX

1

2

2

11 1

1

Multiply by n and you get

isolate beta and we have

n

i

n

iii

n

i

n

iii

n

iii XXnXYYXn

1

2

1

2

1 11

n

i

n

iii

n

i

n

iii

n

iii

XXn

XYYXn

1

2

1

2

1 11

XYAlpha or the regression intercept

Beta or the regression slope

n

i

n

iii

n

i

n

iii

n

iii

XXn

XYYXn

1

2

1

2

1 11

Given this info, let’sHead over to the lab and get some hand’s on practice using the small and relatively simple ice cream sale’s data setWe will cover the math behind the coefficient of determination on Thursday and introduce regression with multiple independent variables