Correlation

• Problems with regression
• Correlation
• Using correlation

Here's a Scenario

• Market research firm
• Studying preference for a new type of computer
• What is the relationship between people's degree of computer expertise and their preference for the new brand?
• Get a measure of their preference
• 0 - 100 scale
• Self-rating of expertise
• 1-9 scale

Expertise vs Preference

 
 
 
 
 

Correlation

• Correlation is a measure of mathematical association.
• There are man possible correlations we might measure.
• Typically, when we talk about correlation, we are referring to Pearson's correlation coefficient (called r)
• Correlation is defined as:

WOW, we can make sense of the equation!  It's like calculating a t.
 

What is going on?

• Each observation has its mean removed
• Each observation is divided by its standard deviation
• The resulting value has no units.
• What are the properties of the correlation
• Range
• -1:  Perfect negative relationship
•  1:  Perfect positive relationship
•  0:  No relationship at all.

A Positive Relationship (r=.54)

 
 
 
 
 
 

A Negative Relationship (r=-.62)

 
 
 
 
 
 

No Relationship (r=-.08)

 
 
 
 
 
 
 

Important Things to Know

• Correlation measures the linear association between the variables.
• No good for nonlinear associations
• The correlation coefficient is not affected b changes in the units of measurement of the variables
• A correlation of 1 or -1 indicates that the observed points all fall on a straight line.

What does this value signify?

• The square of the correlation is the proportion of variance in one variable that can be accounted for b the other.
• This value is abbreviated r²
• What does this mean?
• How much of the variability in a variable can be predicted from the regression line?
• How much comes from other factors?
• How far do the points lie from the regression line?

Proportion of Variance (r=.54)

 
 
 
 
 
 

Resistance

• Regression and correlation coefficients are not resistant
• The are strongly influenced b outliers
• Must graph our data to ensure that the effects our see are not due to outliers.

Resistance (r=1.0 vs r=.54)

 
 
 
  Demo on outliers in correlation/regression
 
 
 
 

Extrapolation

• A strong relationship suggests that you can predict the value of one variable from the value of another.
• Prediction is only valid within the range for which measurements were taken.
• Consider height and basketball ability
• True for correlation and regression

Interpreting Correlations

• A correlation coefficient is just a description of our data
• What it means depends on how data were collected.
• All the correlation implies is a numerical relationship
• No causality is implied.
• Imagine we found a positive correlation between atmospheric pollution levels and murder rates for 100 counties in the United States.
• Why would this be?
• The third variable problem

Summary

• Correlation measures linear association
• Units are stripped off the measurements
• Correlation ranges from -1 to +1
• Correlations must be interpreted in light of the way the data were collected.