Linear Regression as a Fundamental Descriptive Tool

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© 2019 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or distribution without the prior written consent of McGraw-Hill Education

Learning Objectives

Construct a regression line for a dichotomous treatment

Construct a regression line for a multi-level treatment

Explain both intuitively and formerly the formulas generating a regression line for a single treatment

Distinguish the use of sample moment equations from estimation via least squares

Distinguish regression equations for single and multiple treatments

Describe a dataset with multiple treatments using multiple regression

Explain the difference between linear regression and a regression line

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Scatterplot of Price and Sales

How do we summarize the relationship between these two variables?

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The Regression Line for a Dichotomous Treatment

Dichotomous treatment

Two treatment statuses—treated and untreated

Regression analysis

The process of using a function to describe the relationship among variables

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The Regression Line for a Dichotomous Treatment : An Intuitive Approach

Draw a line through these data that will best describe the relationship between Price and Treatment

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The Regression Line for a Dichotomous Treatment: An Intuitive Approach

In general, the formula for a line is: Y = f(X) = b + mX,

where b is the intercept and m is the slope of the line

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Line Describing the Relationship Between Profits and Treatment

What is the equation for the line shown here?

Profits = 208.33 – 20 × Treatment

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Line Describing the Relationship Between Profits and Price

Knowing the two point on the Profits/Price line, solve for slope and intercept

Profits = 248.33 – 40 × Price

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The Regression Line for a Dichotomous Treatment

Whenever there is a dichotomous treatment, a line can be built describing the relationship between the treatment and outcome by using the means for each treatment status called the regression line for a dichotomous treatment

Set f(0) = and f(1)

The equation for the line is:

Outcome = +

(- ) × Treatment

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The Regression Line for a Dichotomous Treatment: A Formal Approach

Observed outcomes in terms of two points on a line

Profiti = f(1.00) + ei if Pricei = 1.00

Profiti = f(1.50) + ei if Pricei = 1.50

i delineates between different observations, (i

ei is the residual for the observation i.

The residual is the difference between the observed outcome and the corresponding point on the regression line for a given observation

ei = Yi – f(xi)

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Scatterplot of Residuals for Price of $1.00 when f(1.00) = $220

FIRST RESIDUAL IS 20. THIS MEANS THE ACTUAL PROFIT WE OBSERVE (240) IS 20 HIGHER THAN WHAT WE OBSERVE (220).

SECOND RESIDUAL IS -20. THIS MEANS THE ACTUAL PROFIT WE OBSERVE (200) IS 20 HIGHER THAN WHAT WE OBSERVE (220).

THIRD RESIDUAL IS -35. THIS MEANS THE ACTUAL PROFIT WE OBSERVE (240) IS 20 HIGHER THAN WHAT WE OBSERVE (185).

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The Regression Line for a Dichotomous Treatment: A Formal Approach

Residuals for price of $1.00 when f(1.00) = $220

The average residual is [20 + (-20) + (-35)]/3 = -11.67

A choice for f(1.00) is best if it tends to neither overshoot nor undershoot the observed outcomes. That means, a choice for f(1.00) is best if the corresponding residuals are on average, zero.

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The Regression Line for a Dichotomous Treatment: A Formal Approach

For the residuals to average zero means:

THE RESIDUALS TO AVERAGE ZERO, BEST CHOICE FOR f(1.00):

Similarly, when price is $1.50, the best choice is the average of profits when the price is $1.50 = (205 + 170 + 190)/3 = 188.33

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The Regression Line for a Multi-Level Treatment: An Intuitive Approach

Multi-level treatment is a treatment that can be administered in more than one quantity

HERE, PRICES ARE, $1.00, $1.50, $2.00. PRICE OF $1.00 IS UNTREATED AND A $0.50 PRICE INCREASE IS THE TREATMENT.

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The Regression Line for a Multi-Level Treatment: An Intuitive Approach

The approach we used for the dichotomous treatment generally does not work for a multi-level treatment

The problem is that when three or more points are plotted on a graph, it is generally the case that they might not fall on the same line

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The Regression Line for a Multi-Level Treatment: An Intuitive Approach

Line attempting to connect average profits to the following price levels:

f(1.00) = 208.33

f(1.50) = 188.33

f(2.00) = 160

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The Regression Line for a Multi-Level Treatment: An Intuitive Approach

Using the average outcome to plot the points for each treatment level generally will result in not being able to connect three points on a single line when there more than two treatment levels

f(1.00) = b + m × 1.00 208.33 = b + m × 1.00

f(1.50) = b + m × 1.50 188.33 = b + m × 1.50

f(2.00) = b + m × 2.00 160 = b + m × 2.00

We cannot solve for m and b as there are three equations to solve but only two unknowns

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The Regression Line for a Multi-Level Treatment: An Intuitive Approach

Rather than plot an “ideal” point for each treatment level and then solve for the corresponding slope and intercept, try to directly solve for the slope and intercept of the line believed to best describe the describes the data

It should not generally overshoot or undershoot the data

Its tendency to over or undershoot the data across specific price levels should not depend on the price level

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Two Candidate Lines for Describing Profits and Price Data

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The Regression Line for a Multi-Level Treatment: A Formal Approach

For our example, we have three levels and nine points. Expressing them in terms of intercept and slope:

Profiti = b + m × 1.00 + ei, if Pricei = 1.00

Profiti = b + m × 1.50 + ei, if Pricei = 1.50

Profiti = b + m × 2.00 + ei, if Pricei = 2.00

Here i takes on the values one through nine, since there are nine points. Residuals, ei, are the difference between the observed profit and the corresponding point on the line for a given observation.

Ei = Profiti – b – m × Pricei

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The Regression Line for a Multi-Level Treatment: A Formal Approach

Applying the same approach used for a dichotomous treatment, solve for the “best” line by finding a slope and intercept that makes the residuals average zero for each price point.

THIS AGAIN GIVES US THREE EQUATIONS AND TWO UNKNOWNS.

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The Regression Line for a Multi-Level Treatment: A Formal Approach

Alternative way of defining what makes a line the best to describe the data. Criteria includes:

It should not generally overshoot or undershoot the data

Its tendency to over or undershoot the data across specific price levels should not depend on the price level

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The Regression Line for a Multi-Level Treatment: A Formal Approach

Translating these criteria in terms of residuals:

The residuals for all data points average to zero

The size of the residuals is not correlated with the treatment level

Expressing these two criteria in equation form:

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The Regression Line for a Multi-Level Treatment: A Formal Approach

The first equation ensures that the residual average zero across all observations, and the second equation ensures that the size of the residuals is not related to Price level

Solving these two equations yields:

m = -48.33

b = 258.06

The line that best fits the data, where “best” implies residuals that average zero and are not correlated with the treatment:

Profit = 258.06 – 48.33 × Price

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The Regression Line for a Multi-Level Treatment: A Formal Approach

Simple regression line

The slope is the sample covariance of the treatment and outcome divided by the sample variance of the treatment

The intercept is the mean value of the outcome minus the slope times the mean value of the treatment

Y = b + mX

Solving for m and b yields the following formulas for the slope and intercept of a simple regression line:

m =

b = – m

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The Regression Line for a Multi-Level Treatment: A Formal Approach

Applying these generalized formulas for our dichotomous price/profit example:

USING THE FORMULAS FOR VARIANCE AND COVRIANCE:

sCov (Profit, Price) = -3,

sVar (Price) = 0.075,

= 1.25 and = 198.33

Plugging these into our formulas,

m = -3/0.075 = -40, and

b = 198.33 – (-40)1.25= 248.33.

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Sample Moments and Least Squares

Sample moment

The mean of a function of a random variable(s) for a given sample

For example, for a sample size 20 that contains information on salaries, is a sample moment, where Salaryi is the random variable and the function is defined as f(a) = a3

Ordinary least squares

The process of solving for the slope and intercept that minimize the sum of the squared residuals

Minb,m =Yi – b – mXi)2

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Sample Moments and Least Square

Objective function

A function ultimately wished to be maximized or minimized

For ordinary least squares, the objective function is the sum of squared residuals ()

Least absolute deviations (LAD)

Use the sum of the absolute value of the residuals as the objective function and solve for the slope and intercept that minimize it

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Ordinary Least Square vs Least Absolute Deviation for Describing a Dataset

LINE A IS CLOSER TO THE OUTLIER, SO IT IS COMING FROM OLS AND LINE B IS COMING FROM LAD.

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Regression for Multiple Treatments

CHOLESTEROL LEVEL AND DRUG DOSES FOR 15 INDIVIDUALS.

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Regression for Multiple Treatments

Single vs. Multiple Treatments

Cholesterol = 235.17 – 0.997 × Drug A

Cholesterol = 205.83 – 0.107 × Drug B

Cholesterol outcome as follows:

Cholesteroli = b + m1DrugAi + m2DrugBi + ei

Expressing the OLS criteria in equation form:

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Regression Output in Excel for Cholesterol Regressed on Drug A and Drug B

HERE WE HAVE THE VALUES FOR:

b = 256.20,

m1 = -1.259, AND

m2 = -0.514.

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Regression Plane for Cholesterol Regressed on Drug A and Drug B

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Multiple regression

Solving for a function that best describes the data the implies the use of OLS (or equivalently, the sample moment equations)

Single regression the process that produces the simple regression line for a single treatment

Multiple Regression

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Multiple Regression

For a sample size of N with K treatments, the associated equations are:

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What Makes Regression Linear?

Linear regression is the process of fitting a function that is linear in its parameters to a given dataset

Y = b + m1X1 + m2X2 + … + mKXK

Here {b, m1, …, mK} are the parameters for this function

The use of linear regression does not at all imply construction of a line to fit the data

Linear regression is linear in the parameters but not necessarily the treatment(s)

It allows for an unlimited number of possible “shapes” for the relationship between the outcome and any particular treatment

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