Purpose of Regression Analysis

• Regression analysis is used primarily to model causality and provide prediction– Predicts the value of a dependent (response)

variable based on the value of at least one independent (explanatory) variable

– Explains the effect of the independent variables on the dependent variable

Types of Regression Models

Positive Linear Relationship

Negative Linear Relationship

Relationship NOT Linear

No Relationship

Simple Linear Regression Model

• Relationship between variables is described by a linear function

• The change of one variable causes the change in the other variable

• A dependency of one variable on the other

PopulationRegressionLine (conditional mean)

Population Linear Regression

Population regression line is a straight line that describes the dependence of the average value (conditional mean)average value (conditional mean) of one variable on the other

Population Y intercept

Population SlopeCoefficient

Random Error

Dependent (Response) Variable

Independent (Explanatory) Variable

ii iY X

YX

Population Linear Regression

(continued)

ii iY X

= Random Error

Y

X

(Observed Value of Y) =

Observed Value of Y

YX iX

i

(Conditional Mean)

Sample regression line provides an estimateestimate of the population regression line as well as a predicted value of Y

Sample Linear Regression

Sample Y Intercept

SampleSlopeCoefficient

Residual0 1i iib bY X e

0 1Y b b X Sample Regression Line (Fitted Regression Line, Predicted Value)

Sample Linear Regression

• and are obtained by finding the values of and that minimizes the sum of the squared residuals

• provides an estimateestimate of • provides and estimateestimate of

0b 1b 0b1b

0b

1b

(continued)

22

1 1

ˆn n

i i ii i

Y Y e

Sample Linear Regression

(continued)

Y

XObserved Value

YX iX

i

ii iY X

0 1i iY b b X

ie

0 1i iib bY X e 1b

0b

Interpretation of the Slope and the Intercept

• is the average value of Y when

the value of X is zero.

• measures the change in the

average value of Y as a result of a one-unit

change in X.

| 0E Y X

1

|E Y X

X

• is the estimatedestimated average

value of Y when the value of X is zero.

• is the estimatedestimated change in

the average value of Y as a result of a one-unit

change in X.

(continued)

ˆ | 0b E Y X

1

ˆ |E Y Xb

X

Interpretation of the Slope and the Intercept

Simple Linear Regression: Example

You want to examine the linear dependency of the annual sales of produce stores on their size in square footage. Sample data for seven stores were obtained. Find the equation of the straight line that fits the data best.

Annual Store Square Sales

Feet ($1000)

1 1,726 3,681

2 1,542 3,395

3 2,816 6,653

4 5,555 9,543

5 1,292 3,318

6 2,208 5,563

7 1,313 3,760

Scatter Diagram: Example

0

2000

4000

6000

8000

10000

12000

0 1000 2000 3000 4000 5000 6000

Square Feet

An

nu

al

Sa

les

($00

0)

Excel Output

Equation for the Sample Regression Line: Example

0 1ˆ

1636.415 1.487i i

i

Y b b X

X

From Excel Printout:

CoefficientsIntercept 1636.414726X Variable 1 1.486633657

Excel Output

Regression Statistics

Multiple R 0.970557

R Square 0.941981

Adjusted R Square 0.930378

Standard Error 611.7515

Observations 7

ANOVA

  df SS MS FSignificance

F

Regression 1 30380456 30380456 81.17909 0.000281

Residual 5 1871200 374239.9

Total 6 32251656      

 Coefficient

sStandard

Error t Stat P-value Lower 95% Upper 95%Intercept 1636.415 451.4953 3.624433 0.015149 475.8109 2797.019X Variable 1 1.486634 0.164999 9.009944 0.000281 1.06249 1.910777

Graph of the Sample Regression Line: Example

0

2000

4000

6000

8000

10000

12000

0 1000 2000 3000 4000 5000 6000

Square Feet

An

nu

al

Sa

les

($00

0)

Y i = 1636.415 +1.487X i

Interpretation of Results: Example

The slope of 1.487 means that for each increase of one unit in X, we predict the average of Y to increase by an estimated 1.487 units.

The model estimates that for each increase of one square foot in the size of the store, the expected annual sales are predicted to increase by $1487.

ˆ 1636.415 1.487i iY X

How Good is the regression?

• R2

• Confidence Intervals• Residual Plots• Analysis of Variance• Hypothesis (t) tests

Measure of Variation: The Sum of Squares

SST = SSR + SSE

Total Sample

Variability

= Explained Variability

+ Unexplained Variability

Measure of Variation: The Sum of Squares

• SST = total sum of squares – Measures the variation of the Yi values around their

mean Y

• SSR = regression sum of squares – Explained variation attributable to the relationship

between X and Y

• SSE = error sum of squares – Variation attributable to factors other than the

relationship between X and Y

(continued)

Measure of Variation: The Sum of Squares

(continued)

Xi

Y

X

Y

SST = (Yi - Y)2

SSE =(Yi - Yi )2

SSR = (Yi - Y)2

_

_

_

The Coefficient of Determination

•

• Measures the proportion of variation in Y that is explained by the independent variable X in the regression model

2 Regression Sum of Squares

Total Sum of Squares

SSRr

SST

Coefficients of Determination (r 2) and Correlation (r)

r2 = 1, r2 = 1,

r2 = .8, r2 = 0,Y

Yi = b0 + b1Xi

X

^

YYi = b0 + b1Xi

X

^Y

Yi = b0 + b1Xi

X

^

Y

Yi = b0 + b1Xi

X

^

r = +1 r = -1

r = +0.9 r = 0

Linear Regression Assumptions

1. Linearity

2. Normality– Y values are normally distributed for each X– Probability distribution of error is normal

2. Homoscedasticity (Constant Variance)

3. Independence of Errors

Residual Analysis

• Purposes– Examine linearity – Evaluate violations of assumptions

• Graphical Analysis of Residuals– Plot residuals vs. Xi , Yi and time

Residual Analysis for Linearity

Not Linear Linear

X

e eX

Y

X

Y

X

Residual Analysis for Homoscedasticity

Heteroscedasticity Homoscedasticity

SR

X

SR

X

Y

X X

Y

• Y values are normally distributed around the regression line.

• For each X value, the “spread” or variance around the regression line is the same.

Variation of Errors around the Regression Line

X1

X2

X

Y

f(e)

Sample Regression Line

Residual Plot

0 1000 2000 3000 4000 5000 6000

Square Feet

Residual Analysis:Excel Output for Produce Stores Example

Excel Output

Observation Predicted Y Residuals1 4202.344417 -521.34441732 3928.803824 -533.80382453 5822.775103 830.22489714 9894.664688 -351.66468825 3557.14541 -239.14541036 4918.90184 644.09816037 3588.364717 171.6352829

Residual Analysis for Independence

Not Independent Independente e

TimeTime

Residual is plotted against time to detect any autocorrelation

No Particular PatternCyclical Pattern

Graphical Approach

Inference about the Slope: t Test

• t test for a population slope– Is there a linear dependency of Y on X ?

• Null and alternative hypotheses– H0: 1 = 0 (no linear dependency)– H1: 1 0 (linear dependency)

• Test statistic–

–

1

1

1 1

2

1

where

( )

YXb n

bi

i

b St S

SX X

. . 2d f n

Example: Produce Store

Data for Seven Stores: Estimated Regression Equation:

The slope of this model is 1.487.

Is square footage of the store affecting its annual sales?

Annual Store Square Sales

Feet ($000)

1 1,726 3,681

2 1,542 3,395

3 2,816 6,653

4 5,555 9,543

5 1,292 3,318

6 2,208 5,563

7 1,313 3,760

Yi = 1636.415 +1.487Xi

Inferences about the Slope: t Test Example

H0: 1 = 0

H1: 1 0

.05

df 7 - 2 = 5

Critical Value(s):

Test Statistic:

Decision:

Conclusion:There is evidence that square footage affects annual sales.

t0 2.5706-2.5706

.025

Reject Reject

.025

From Excel Printout

Reject H0

Coefficients Standard Error t Stat P-valueIntercept 1636.4147 451.4953 3.6244 0.01515Footage 1.4866 0.1650 9.0099 0.00028

1b 1bS t

0 1 1 2 2i i i k ki iY b b X b X b X e

Population Y-intercept

Population slopes Random Error

The Multiple Regression Model

Relationship between 1 dependent & 2 or more independent variables is a linear

function

Dependent (Response) variable for sample

Independent (Explanatory) variables for sample model

1 2i i i k ki iY X X X

Residual

Population Multiple Regression Model

Bivariate model

X2

Y

X1Y|X = 0 + 1X 1i + 2X 2i

0

Y i = 0 + 1X 1i + 2X 2i + i

ResponsePlane

(X 1i,X 2i)

(O bserved Y )

i

X2

Y

X1Y|X = 0 + 1X 1i + 2X 2i

0

Y i = 0 + 1X 1i + 2X 2i + i

ResponsePlane

(X 1i,X 2i)

(O bserved Y )

i

Sample Multiple Regression Model

X2

Y

X1

b0

Y i = b0 + b1X 1 i + b2X 2 i + e i

ResponsePlane

(X 1i, X 2i)

(O bserved Y)

^

e i

Y i = b0 + b1X 1 i + b2X 2 i

X2

Y

X1

b0

Y i = b0 + b1X 1 i + b2X 2 i + e i

ResponsePlane

(X 1i, X 2i)

(O bserved Y)

^

e i

Y i = b0 + b1X 1 i + b2X 2 i

Bivariate model

Sample Regression PlaneSample Regression Plane

Simple and Multiple Regression Compared

• Coefficients in a simple regression pick up the impact of that variable plus the impacts of other variables that are correlated with it and the dependent variable.

• Coefficients in a multiple regression net out the impacts of other variables in the equation.

Simple and Multiple Regression Compared:Example

• Two simple regressions:– –

• Multiple regression:–

0 1

0 1

Oil Temp

Oil Insulation

0 1 2Oil Temp Insulation

Multiple Linear Regression Equation

Too complicated

by hand! Ouch!

Interpretation of Estimated Coefficients

• Slope (bi)

– Estimated that the average value of Y changes by bi for each 1 unit increase in Xi holding all other variables constant (ceteris paribus)

– Example: if b1 = -2, then fuel oil usage (Y) is expected to decrease by an estimated 2 gallons for each 1 degree increase in temperature (X1) given the inches of insulation (X2)

• Y-intercept (b0)

– The estimated average value of Y when all Xi = 0

Multiple Regression Model: Example

Oil (Gal) Temp Insulation275.30 40 3363.80 27 3164.30 40 10

40.80 73 694.30 64 6

230.90 34 6366.70 9 6300.60 8 10237.80 23 10121.40 63 3

31.40 65 10203.50 41 6441.10 21 3323.00 38 3

52.50 58 10

(0F)

Develop a model for estimating heating oil used for a single family home in the month of January based on average temperature and amount of insulation in inches.

1 2ˆ 562.151 5.437 20.012i i iY X X

Sample Multiple Regression Equation: Example

CoefficientsIntercept 562.1510092X Variable 1 -5.436580588X Variable 2 -20.01232067

Excel Output

For each degree increase in temperature, the estimated average amount of heating oil used is decreased by 5.437 gallons, holding insulation constant.

For each increase in one inch of insulation, the estimated average use of heating oil is decreased by 20.012 gallons, holding temperature constant.

0 1 1 2 2i i i k kiY b b X b X b X

Confidence Interval Estimate for the Slope

Provide the 95% confidence interval for the population slope 1 (the effect of temperature on oil consumption).

11 1n p bb t S

Coefficients Lower 95% Upper 95%Intercept 562.151009 516.1930837 608.108935X Variable 1 -5.4365806 -6.169132673 -4.7040285X Variable 2 -20.012321 -25.11620102 -14.90844

-6.169 1 -4.704

The estimated average consumption of oil is reduced by between 4.7 gallons to 6.17 gallons per each increase of 10 F.

Coefficient of Multiple Determination

• Proportion of total variation in Y explained by all X variables taken together

–

• Never decreases when a new X variable is added to model– Disadvantage when comparing models

212

Explained Variation

Total VariationY k

SSRr

SST

Adjusted Coefficient of Multiple Determination

• Proportion of variation in Y explained by all X variables adjusted for the number of X variables used–

– Penalize excessive use of independent variables– Smaller than– Useful in comparing among models

2 212

11 1

1adj Y k

nr r

n k

212Y kr

Coefficient of Multiple Determination

Regression StatisticsMultiple R 0.982654757R Square 0.965610371Adjusted R Square 0.959878766Standard Error 26.01378323Observations 15

Excel Output

SST

SSRr ,Y 2

12

Adjusted r2

reflects the number of explanatory variables and sample size

is smaller than r2

Interpretation of Coefficient of Multiple Determination

•

– 96.56% of the total variation in heating oil can be explained by different temperature and amount of insulation

•

– 95.99% of the total fluctuation in heating oil can be explained by different temperature and amount of insulation after adjusting for the number of explanatory variables and sample size

2,12 .9656Y

SSRr

SST

2adj .9599r

Using The Model to Make Predictions

Predict the amount of heating oil used for a home if the average temperature is 300 and the insulation is six inches.

The predicted heating oil used is 278.97 gallons

1 2

ˆ 562.151 5.437 20.012

562.151 5.437 30 20.012 6

278.969

i i iY X X

Testing for Overall Significance

• Shows if there is a linear relationship between all of the X variables together and Y

• Use F test statistic• Hypotheses:

– H0: …k = 0 (no linear relationship)

– H1: at least one i ( at least one independentvariable affects Y )

• The null hypothesis is a very strong statement• Almost always reject the null hypothesis

Test for Significance:Individual Variables

• Shows if there is a linear relationship between the variable Xi and Y

• Use t test statistic• Hypotheses:

– H0: i 0 (no linear relationship)

– H1: i 0 (linear relationship between Xi and Y)

Residual Plots

• Residuals vs.– May need to transform variable

• Residuals vs.– May need to transform variable

• Residuals vs. time– May have autocorrelation

Y

1X

2X1X

2X

Residual Plots: Example

Insulation Residual Plot

0 2 4 6 8 10 12

No Discernable Pattern

Temperature Residual Plot

-60

-40

-20

0

20

40

60

0 20 40 60 80

Re

sid

ua

ls

Maybe some non-linear relationship

The Quadratic Regression Model

• Relationship between one response variable and two or more explanatory variables is a quadratic polynomial function

• Useful when scatter diagram indicates non-linear relationship

• Quadratic model :–

• The second explanatory variable is the square of the first variable

20 1 1 2 1i i i iY X X

Quadratic Regression Model

(continued)

Quadratic models may be considered when scatter diagram takes on the following shapes:

X1

Y

X1X1

YYY

2 > 0 2 > 0 2 < 0 2 < 0

2 = the coefficient of the quadratic term

X1

Dummy Variable Models

• Categorical explanatory variable (dummy variable) with two or more levels:

• Yes or no, on or off, male or female, • Coded as 0 or 1• Only intercepts are different• Assumes equal slopes across categories• The number of dummy variables needed is

(number of levels - 1)• Regression model has same form:

0 1 1 2 2i i i k ki iY X X X

Dummy-Variable Models (with 2 Levels)

Given:

Y = Assessed Value of House

X1 = Square footage of House

X2 = Desirability of Neighborhood =

Desirable (X2 = 1)

Undesirable (X2 = 0)

0 1 1 2 0 2 1 1ˆ (1) ( )i i iY b b X b b b b X

0 if undesirable 1 if desirable

0 1 1 2 0 1 1ˆ (0)i i iY b b X b b b X

Same slopes

0 1 1 2 2i i iY b b X b X

Undesirable

Desirable Location

Dummy-Variable Models (with 2 Levels)

(continued)

X1 (Square footage)

Y (Assessed Value)

b0 + b2

b0

Same slopes

Intercepts different

1b

Interpretation of the Dummy Variable Coefficient (with 2 Levels)

Example:

0 1 1 2 2 1 2ˆ 20 5 6i i i i iY b b X b X X X

1X : GPA2X

0 Female

1 Male

Y: Annual salary of college graduate in thousand $

On average, male college graduates are making an estimated six thousand dollars more than female college graduates with the same GPA.

:

Dummy-Variable Models (with 3 Levels)

1

2 3

Given:

Assessed Value of the House (1000 $)

Square Footage of the House

Style of the House = Split-level, Ranch, Condo

(3 Levels; Need 2 Dummy Variables)

1 if Split-level 1

0 if not

Y

X

X X

0 1 1 2 2 3 3

if Ranch

0 if not

iY b b X b X b X

Interpretation of the Dummy Variable Coefficients (with 3 Levels)

With the same footage, a Split-level will have an estimated average assessed value of 18.84 thousand dollars more than a Condo.With the same footage, a Ranch will have an estimated average assessed value of 23.53 thousand dollars more than a Condo.

1 2 3

2

1

3

1

1

Given the Estimated Model:

ˆ 20.43 0.045 18.84 23.53

For Split-level 1 :

ˆ 20.43 0.045 18.84

For Ranch 1 :

ˆ 20.43 0.045 23.53

For Condo:

ˆ 20.43 0.045

i i i i

i i

i i

i i

Y X X X

X

Y X

X

Y X

Y X

Dummy Variables

• Predict Weekly Sales in a Grocery Store• Possible independent variables:

– Price– Grocery Chain

• Data Set:– Grocery.xls

• Interaction Effect?

Interaction Regression Model

• Hypothesizes interaction between pairs of X variables– Response to one X variable varies at different levels

of another X variable

• Contains two-way cross product terms –

• Can be combined with other models – E.G., Dummy variable model

0 1 1 2 2 3 1 2i i i i i iY X X X X

Effect of Interaction

• Given:–

• Without interaction term, effect of X1 on Y is measured by 1

• With interaction term, effect of X1 on Y is measured by 1 + 3 X2

• Effect changes as X2 increases

0 1 1 2 2 3 1 2i i i i i iY X X X X

Y = 1 + 2X1 + 3(1) + 4X1(1) = 4 + 6X1

Y = 1 + 2X1 + 3(0) + 4X1(0) = 1 + 2X1

Interaction Example

Effect (slope) of X1 on Y does depend on X2 value

X1

44

88

1212

0000 110.50.5 1.51.5

YY = 1 + 2X1 + 3X2 + 4X1X2

Interaction Regression Model Worksheet

Case, i Yi X1i X2i X1i X2i

1 1 1 3 3

2 4 8 5 40

3 1 3 2 6

4 3 5 6 30

: : : : :

Case, i Yi X1i X2i X1i X2i

1 1 1 3 3

2 4 8 5 40

3 1 3 2 6

4 3 5 6 30

: : : : :

Multiply X1 by X2 to get X1X2. Run regression with Y, X1, X2 , X1X2

• Hypothesize interaction between pairs of independent variables

• Contains 2-way product terms

Evaluating Presence of Interaction

0 1 1 2 2 3 1 2i i i i i iY X X X X

Using Transformations

• Requires data transformation• Either or both independent and dependent

variables may be transformed• Can be based on theory, logic or scatter

diagrams

Inherently Linear Models

• Non-linear models that can be expressed in linear form– Can be estimated by least squares in linear form

• Require data transformation

Transformed Multiplicative Model (Log-Log)

1 20 1 2Original: i i i iY X X

0 1 1 2 2Transformed: ln ln ln ln lni i i iY X X

Y

X1

Y

X1

1 1

10 1 11 0

1 1

1 1

Similarly for X2

Square Root Transformation

Y

X1

0 1 1 2 2i i i iY X X

1 > 0

1 < 0

Similarly for X2

Transforms one of above model to one that appears linear. Often used to overcome heteroscedasticity.

Linear-Logarithmic Transformation

Y

X1

0 1 1 2 2ln( ) ln( )i i i iY X X

1 > 0

1 < 0

Similarly for X2

Transformed from an original multiplicative model

Exponential Transformation(Log-Linear)

Y

X1

0 1 1 2 2i iX Xi iY e Original Model

1 > 0

1 < 0

Transformed Into: 0 1 1 2 2 1ln lni i iY X X

Model Building / Model Selection

• Find “the best” set of explanatory variables among all the ones given.

• “Best subset” regression (only linear models)– Requires a lot of computation (2N regressions)

• “Stepwise regression”• “Common Sense” methodology

– Run regression with all variables– Throw out variables not statistically significant– “Adjust” model by including some excluded variables, one at a

time

• Tradeoff: Parsimony vs. Fit

Association ≠ Causation !

Regression Limitations

• R2 measures the association between independent and dependent variables

Association ≠ Causation !• Be careful about doing predictions that involve

extrapolation• Inclusion / Exclusion of independent

variables is subject to a type I / type II error

Multi-collinearity

• What?– When one independent variable is highly correlated (“collinear”)

with one or more other independent variables– Examples:

• square feet and square meters as independent variables to predict house price (1 sq ft is roughly 0.09 sq meters)

• “total rooms” and bedrooms plus bathrooms for a house

• How to detect?– Run a regression with the “not-so-independent” independent

variable (in the examples above: square feet and total rooms) as a function of all other remaining independent variables, e.g.:

• X1 = β0 + β2 X2 + …+ βk Xk

– If R2 of the above regression is > 0.8, then one suspects multi-collinearity to be present

Multi-collinearity

• What effect?– Coefficient estimates are unreliable– Can still be used for predicting values for Y– If possible, delete the “not-so-independent”

independent variable

• When to check?– When one suspects that two variables measure the

same thing, or when the two variables are highly correlated

– When one suspects that one independent variable is a (linear) function of the other independent variables

(continued)