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Live Lecture 6.1

Correlation, Linear Regression

Summer 2020

DATA 8

Spring 2020

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Announcements

  • Final project info has been released: review the guidelines and datasets
  • Lab 10 due 07/28 at 11:59pm PST
  • HW 10 due 07/30 at 11:59pm PST

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Agenda

  1. Correlation
  2. Linear Regression
  3. The Regression Equation

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Guessing the Future

  • Based on incomplete information

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  • One way of making predictions:
    • To predict an outcome for an individual,
    • find others who are like that individual
    • and whose outcomes you know.
    • Use those outcomes as the basis of your prediction.

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The Correlation Coefficient r

  • Measures linear association
  • Based on standard units
  • -1 ≤ r ≤ 1
    • r = 1: scatter is perfect straight line sloping up
    • r = -1: scatter is perfect straight line sloping down
  • r = 0: No linear association; uncorrelated

r = 0

r = 0.2

r = 0.5

r = 0.8

r = 0.99

r = -0.5

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Definition of r

average of

product of

x in standard units

and

y in standard units

Correlation Coefficient (r) =

Measures how clustered the scatter is around a straight line

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Watch Out For ...

  • False conclusions of causation
  • Nonlinearity
  • Outliers
  • Ecological Correlations

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Discussion Question

For each pair, which one will have a higher value of r?

a)

c)

b)

d)

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Discussion question

True or False?

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  1. If x and y have a correlation of 1, then one must cause the other.

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  • If the correlation of x and y is close to 0, then knowing one will never help us predict the other.

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  • If x and y have a correlation of -0.8, then they have a negative association.

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Agenda

  • Correlation
  • Linear Regression
  • The Regression Equation

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Galton's Heights

  • Oval shaped

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  • Moderate positive correlation

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  • How can we predict child height from mid-parent height?

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Galton's Heights

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Galton's Heights

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Nearest Neighbor Regression

A method for prediction:

  • Group each x with similar (nearby) x values
  • Average the corresponding y values for each group

For each x value, the prediction is the average of the y values in its nearby group.

The graph of these predictions is the “graph of averages”.

If the association between x and y is linear, then points in the graph of averages tend to fall on a line.

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Agenda

  • Correlation
  • Linear Regression
  • The Regression Equation

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Regression Line Equation

In original units, the regression line has this equation:

Lines can be expressed by slope & intercept

estimated y in standard units

x in standard units

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Regression Line

Standard Units

(0, 0)

1

r

Original Units

(Average x,� Average y)

SD x

r * SD y

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Slope and Intercept

estimate of y = slope * x + intercept

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Discussion Question

Suppose we use linear regression to predict candy prices (in dollars) from sugar content (in grams). What are the units of each of the following?

  • r

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  • The slope

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  • The intercept

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Discussion Question

A course has a midterm (average 70; standard deviation 10)�and a final (average 79; standard deviation 12)

If the scatter diagram comparing midterm & final scores for students has an oval shape with correlation 0.75, then...

What do you expect the average final score would be for students who scored 90 on the midterm?

How about 60 on the midterm?

(Demo)

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