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Lecture 30

Linear Regression

DATA 8

Spring 2022

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Announcements

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Correlation Coefficient

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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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Care in Interpretation

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Watch Out For ...

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

(Demo)

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Discussion question

True or False?

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If the correlation of x and y is close to 0, then knowing one cannot help us predict the other.

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Chocolate and Nobel Prizes

https://www.biostat.jhsph.edu/courses/bio621/misc/Chocolate%20consumption%20cognitive%20function%20and%20nobel%20laurates%20(NEJM).pdf

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Prediction

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Predicting Heights

  • Oval shaped

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  • Moderate positive correlation

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  • How can we predict child height from the parents’ average height?

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Average of parents’ heights

Child’s (adult) height

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Approach to Prediction

Average of parents’ heights

Child’s (adult) height

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Predicted Heights

Average of parents’ heights

Child’s (adult) height

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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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Where is the prediction line?

r = 0.99

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Where is the prediction line?

r = 0.0

(Demo)

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Linear Regression

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Linear Regression

A statement about x and y pairs

  • Measured in standard units (su)
  • Describing the deviation of x from 0 (the average of x's)
  • And the deviation of the corresponding y from 0 (the average of y's)

On average, y deviates from 0 less than x deviates from 0

Not true for all points — a statement about averages

Regression line

correlation