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# Scatterplots & Regression - PowerPoint PPT Presentation

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

Week 3 Lecture

MG461

Dr. Meredith Rolfe

• 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?

Taken before

Not taken before

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…

of one variable on another variable.

of one variable on one or more other variables.

Linear Model Background conditional expectation…

MG461, Week 3 Seminar

Recap: Theoretical System conditional expectation…

Y

X

MG461, Week 3 Seminar

What is regression? conditional expectation…

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: conditional expectation…Correct Answer vs. Prior Exposure

When to use Regression conditional expectation…

• 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 conditional expectation…

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?

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How to answer the questions? conditional expectation…

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 Regression conditional expectation…

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

• Continuous variables (but many exceptions)

• Observational data (mostly)

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Example 1: Pay and Performance conditional expectation…

Pay

Performance

Runs

Yearly Salary

Y

X

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Scatterplot: conditional expectation…Salaries vs. Runs

Y

X

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Scatterplot: conditional expectation…Salaries vs. Runs

Δy

Δx

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What is the equation for a line? conditional expectation… What is the equation for a line?

y=ax2

y=ax

y=ax+b

y=x+b

Equation of a (Regression) Line conditional expectation…

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 conditional expectation…Linear Model

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

Error

Intercept

Coefficient

(Slope)

Dependent

Variable

Independent

Variable

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The relationship must be LINEAR conditional expectation…

• 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 Regression conditional expectation…

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 conditional expectation…

What is regression?

When is regression used?

Formal statement of linear model equation

MG461, Week 3 Seminar

Understand what regression is… conditional expectation…

Strongly Agree

Agree

Disagree

Strongly Disagree

Mean =

Median =

Understand when to use regression conditional expectation…

Strongly Agree

Agree

Disagree

Strongly Disagree

Mean =

Median =

Know how to make conditional expectation…formal statement of linear model

Strongly Agree

Agree

Disagree

Strongly Disagree

Mean =

Median =

Discussion conditional expectation…

What is regression?

When is regression used?

Formal statement of linear model equation

MG461, Week 3 Seminar

26

Model Estimation conditional expectation…

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Which model parameter conditional expectation…do we NOT need to estimate?

β0

β1

xi

σ2

Goal: Estimate the Relationship between X and Y conditional expectation…

("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? conditional expectation…

Minimize explained variance

Minimize distance to outliers

Minimize unexplained variance

Finding the Best Line: conditional expectation…Unexplained Variance

ei

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Ordinary Least Squares (OLS) Criteria conditional expectation…

("beta hat one")

("beta hat zero")

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

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OLS conditional expectation…Estimates of Beta-hat

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Mean of conditional expectation…x and y

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Variance of conditional expectation…x

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Variance of conditional expectation…y

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Covariance of conditional expectation…x and y

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OLS conditional expectation…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)

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Why squared residuals? conditional expectation…

Y

X

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

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Decomposing the Variance conditional expectation…

• 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

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Total conditional expectation…Squared Residuals (SYY)

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Explained vs. Unexplained conditional expectation…Squared Residuals

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R conditional expectation…2 and goodness of fit

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RSS (residual sum of squares) =

unexplained variation

TSS (total sum of squares) =

SYY, total variation of dependent variable y

ESS (explained sum of squares) =

Examples of high and low conditional expectation…R2

Graph 1

Graph 2

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Which graph had a high conditional expectation…R2?

Graph 1

Graph 2

Recall that R conditional expectation…2= 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 conditional expectation…2

R2=0.29

R2=0.87

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Interpretation of conditional expectation…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

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Visualization of Coefficients conditional expectation…

Δy=β1

Δx=1

β0

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OLS estimates: Pay for Runs conditional expectation…

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

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OLS estimates of Regression Line conditional expectation…

Salary = -34 + 27.47*Runs

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Interpretation of conditional expectation…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 conditional expectation…

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 Runs conditional expectation…

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

MG461, Week 3 Seminar

OLS estimates: Pay for Runs conditional expectation…

And using these, calculate a

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

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OLS estimates: Pay for Runs conditional expectation…

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

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Plotting Confidence Intervals conditional expectation…

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

Strongly Agree

Agree

Somewhat Agree

Neutral

Somewhat Disagree

Disagree

Strongly Disagree

Next time.. conditional expectation…

Multiple independent variable

OLS assumptions

What to do when OLS assumptions are violated

MG461, Week 3 Seminar

Team Scores conditional expectation…