Scatterplots regression
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Scatterplots & Regression. Week 3 Lecture MG461 Dr. Meredith Rolfe. Key Goals of the Week. What is regression? When is regression used? Formal statement of linear model equation Identify components of linear model Interpret regression results:

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Scatterplots regression

Scatterplots & Regression

Week 3 Lecture

MG461

Dr. Meredith Rolfe


Key goals of the week

Key Goals of the Week

  • What is regression?

  • When is regression used?

  • Formal statement of linear model equation

  • Identify components of linear model

  • Interpret regression results:

    • decomposition of variance and goodness of fit

    • estimated regression coefficients

    • significance tests for coefficients

MG461, Week 3 Seminar


Who has studied regression or linear models before

Who has studied regression or linear models before?

Taken before

Not taken before


Which group are you in

Which group are you in?

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Group 7

Group 8

Which group are you in?


Regression is a set of statistical tools to model the conditional expectation

Regression is a set of statistical tools to model the conditional expectation…

of one variable on another variable.

of one variable on one or more other variables.


Linear model background

Linear Model Background

MG461, Week 3 Seminar


Recap theoretical system

Recap: Theoretical System

Y

X

MG461, Week 3 Seminar


What is regression

What is regression?

Regression is the study of relationships between variables

It provides a framework for testing models of relationships between variables

Regression techniques are used to assess the extent to which the outcome variable of interest, Y, changes dependent on changes in the independent variable(s), X

MG461, Week 3 Seminar


Conditional dependence correct answer vs prior exposure

Conditional Dependence: Correct Answer vs. Prior Exposure


When to use regression

When to use Regression

  • We want to know whether the outcome, y, varies depending on x

    • We can use regression to study correlation (not causation) or make predictions

  • Continuous variables (but many exceptions)

MG461, Week 3 Seminar


Questions we might care about

Questions we might care about

Does an change in X lead to an change in Y

Do higher paid employees contribute more to organizational success?

Do large companies earn more? Do they have lower tax rates?

MG461, Week 3 Seminar


How to answer the questions

How to answer the questions?

Observational (Field) Study

Experimental Study

Random assignment to different contribution levels or levels of pay (?)

? Random assignment to country and/or tax rate, quasi-experiment?

  • Collect data on income and a measure of contributions to the organization

  • Collect data on corporate profits in various regions (which companies, which regions)

MG461, Week 3 Seminar


When to use regression1

When to use Regression

  • We want to know whether the outcome, y, varies depending on x

  • Continuous variables (but many exceptions)

  • Observational data (mostly)

MG461, Week 3 Seminar


Example 1 pay and performance

Example 1: Pay and Performance

Pay

Performance

Runs

Yearly Salary

Y

X

MG461, Week 3 Seminar


Scatterplot salaries vs runs

Scatterplot: Salaries vs. Runs

Y

X

MG461, Week 3 Seminar


Scatterplot salaries vs runs1

Scatterplot: Salaries vs. Runs

Δy

Δx

MG461, Week 3 Seminar


What is the equation for a line

What is the equation for a line?

What is the equation for a line?

y=ax2

y=ax

y=ax+b

y=x+b


Equation of a regression line

Equation of a (Regression) Line

Population Parameters

Intercept

Slope

But… x and y are random variables, we need an equation that accounts for noise and signal

MG461, Week 3 Seminar


Simple linear model

SimpleLinear Model

Observation or data point, i, goes from 1…n

Error

Intercept

Coefficient

(Slope)

Dependent

Variable

Independent

Variable

MG461, Week 3 Seminar


The relationship must be linear

The relationship must be LINEAR

  • The linear model assumes a LINEAR relationship

  • You can get results even if the relationship is not linear

  • LOOK at the data!

  • Check for linearity

MG461, Week 3 Seminar


When to use regression2

When to use Regression

We want to know whether the outcome, y, varies depending on x

Continuous variables (but many exceptions)

Observational data (mostly)

The relationship between x and y is linear

MG461, Week 3 Seminar


Understanding the key points

Understanding the key points

What is regression?

When is regression used?

Formal statement of linear model equation

MG461, Week 3 Seminar


Understand what regression is

Understand what regression is…

Strongly Agree

Agree

Disagree

Strongly Disagree

Mean =

Median =


Understand when to use regression

Understand when to use regression

Strongly Agree

Agree

Disagree

Strongly Disagree

Mean =

Median =


Know how to make formal statement of linear model

Know how to make formal statement of linear model

Strongly Agree

Agree

Disagree

Strongly Disagree

Mean =

Median =


Discussion

Discussion

What is regression?

When is regression used?

Formal statement of linear model equation

MG461, Week 3 Seminar

26


Model estimation

Model Estimation

MG461, Week 3 Seminar


Which model parameter do we not need to estimate

Which model parameter do we NOT need to estimate?

β0

β1

xi

σ2


Goal estimate the relationship between x and y

Goal: Estimate the Relationship between X and Y

("beta hat one")

("beta hat zero")

  • We can also estimate the error variance σ2 as

Estimate the population parameters

β0 and β1

MG461, Week 3 Seminar


What would good estimates do

What would “good” estimates do?

Minimize explained variance

Minimize distance to outliers

Minimize unexplained variance


Finding the best line unexplained variance

Finding the Best Line: Unexplained Variance

ei

MG461, Week 3 Seminar


Ordinary least squares ols criteria

Ordinary Least Squares (OLS) Criteria

("beta hat one")

("beta hat zero")

Minimize “noise” (unexplained variance) defined as residual sum of squares (RSS)

MG461, Week 3 Seminar


Ols estimates of beta hat

OLS Estimates of Beta-hat

MG461, Week 3 Seminar


Mean of x and y

Mean of x and y

MG461, Week 3 Seminar


Variance of x

Variance of x

MG461, Week 3 Seminar


Variance of y

Variance of y

MG461, Week 3 Seminar


Covariance of x and y

Covariance of x and y

MG461, Week 3 Seminar


Ols estimates of beta hat1

OLS Estimates of Beta-hat

Note the similarity between ß1 and the slope of a line: change in y over change in x (rise over run)

MG461, Week 3 Seminar


Why squared residuals

Why squared residuals?

Y

X

Geometric intuition: X and Y are vectors, find the shortest line between them:

MG461, Week 3 Seminar


Decomposing the variance

Decomposing the Variance

  • As in Anova, we now have:

    • Explained Variation

    • Unexplained Variation

    • Total Variation

  • This decomposition of variance provides one way to think about how well the estimated model fits the data

MG461, Week 3 Seminar


Total squared residuals syy

Total Squared Residuals (SYY)

MG461, Week 3 Seminar


Explained vs unexplained squared residuals

Explained vs. Unexplained Squared Residuals

MG461, Week 3 Seminar


R 2 and goodness of fit

R2 and goodness of fit

MG461, Week 3 Seminar

RSS (residual sum of squares) =

unexplained variation

TSS (total sum of squares) =

SYY, total variation of dependent variable y

ESS (explained sum of squares) =

explained variation (TSS-RSS)


Examples of high and low r 2

Examples of high and low R2

Graph 1

Graph 2

MG461, Week 3 Seminar


Which graph had a high r 2

Which graph had a high R2?

Graph 1

Graph 2


Recall that r 2 ess tss or 1 rss tss what values can r 2 take on

Recall that R2= ESS/TSS or 1-(RSS/TSS). What values can R2 take on?

Can be any number

Any number between -1 and 1

Any number between 0 and 1

Any number between 1 and 100


Examples of high and low r 21

Examples of high and low R2

R2=0.29

R2=0.87

MG461, Week 3 Seminar


Interpretation of hats

Interpretation of ß-hats

ß-hat0 : intercept, value of yi when xi is 0

ß-hat1: average or expected change in yi for every 1 unit change in xi

MG461, Week 3 Seminar


Visualization of coefficients

Visualization of Coefficients

Δy=β1

Δx=1

β0

MG461, Week 3 Seminar


Ols estimates pay for runs

OLS estimates: Pay for Runs

Salaryi = Beta-hat0 + Beta-hat1 * Runsi + errori

MG461, Week 3 Seminar


Ols estimates of regression line

OLS estimates of Regression Line

Salary = -34 + 27.47*Runs

MG461, Week 3 Seminar


Interpretation of hats1

Interpretation of ß-hats

y (Salary) = -34 + 27.47x (Runs)

ß-hat0 : players with no runs don’t get paid (not really – come back to this next week!)

ß-hat1: Each additional run translates into $27,470in salary per year

OR: a difference of almost $1 million/year between a player with an average (median) and an above average (80%) number of runs

MG461, Week 3 Seminar


Significance of results

Significance of Results

Model Significance

Coefficient Significance

H0: ß1=0, there is no relationship (covariation) between x and y

HA: ß1≠0, there is a relationship (covariation) between x and y

Application: a single estimated coefficient

Test: t-test

**assumes errors (ei) are normally distributed

  • H0: None of the 1 (or more) independent variables covary with the dependent variable

  • HA: At least one of the independent variables covaries with d.v.

  • Application: compare two fitted models

  • Test: Anova/F-Test

  • **assumes errors (ei) are normally distributed

MG461, Week 3 Seminar


Ols estimates pay for runs1

OLS estimates: Pay for Runs

Assuming normality, we can derive estimated standard errors for the coefficients

MG461, Week 3 Seminar


Ols estimates pay for runs2

OLS estimates: Pay for Runs

And using these, calculate a

t-statistic and test for whether or not the coefficients are equal to zero

MG461, Week 3 Seminar


Ols estimates pay for runs3

OLS estimates: Pay for Runs

And finally, the probability of being wrong (Type 1) if we reject H0

MG461, Week 3 Seminar


Plotting confidence intervals

Plotting Confidence Intervals

MG461, Week 3 Seminar


Agree or disagree the lecture was clear and easy to follow

Agree or Disagree, “The lecture was clear and easy to follow”

Strongly Agree

Agree

Somewhat Agree

Neutral

Somewhat Disagree

Disagree

Strongly Disagree


Next time

Next time..

Multiple independent variable

OLS assumptions

What to do when OLS assumptions are violated

MG461, Week 3 Seminar


Team scores

Team Scores


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