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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

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

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

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

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26

model estimation
Model Estimation

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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

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

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

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