Correlation

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Correlation
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Descriptive technique Describes the
relationship between two variables Variables are
observed or measured but rarely manipulated thus
we CANNOT infer causation
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You have a pair of observations -one for each
variable Eg height and weight Anxiety and
introversion You measure all people on both
variables N number of pairs
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• Characteristics of the relationship measured
• Direction is it positive or negative-
• Indicated by the sign of the correlation
• 2) The strength of the relationship
• Indicated by the number of the correlation
• /- 1 is a perfect correlation
• 0 indicates no linear relationship
• 3) The form is it linear?

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Mean heights of a group of children in Kalama, an
Egyptian village that is the site of a study of
nutrition in developing countries. The data were
obtained by measuring the heights of all 161
children in the village each month over several
years. Age in months Height in cm
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Scatterplot of age vs height
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• Correlation does not mean causation
• Correlation does not mean causation
• Correlation does not mean causation
• Correlation does not mean causation
• Correlation does not mean causation
• Correlation does not mean causation
• Correlation does not mean causation

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• Before Jonas Salk found the polio vaccine
  researchers looked for relationship with polio
  and anything any protection
• Found a correlation between soft-drink sales and
  polio. The more soft-drinks sold the more cases
  of polio
• Or factoid...
• The more churches per square mile the higher the
  crime.....

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So why use correlation if we cant infer
causation???

• 1) prediction if know the relationship then can
  predict the value of one variable if know the
  other.
• 2) validity if design a new test and you want
  to see if it tests the same thing as the old one.
  
• 3) reliability will the test change over time
  or over different observers
• 4) test a theory

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• r amount that X and Y vary together
• amount that X and Y vary separately
• r covariability of X and Y
• variability of X and Y separately

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r is unchanged if

• 1) interchange the two variables call height X
  and age Y or height Y and age X
• 2) Add a constant to all the values of one
  variable
• 3) Multiply all the values of one variable by a
  constant. (changes mean and sd but not the
  relative positions)

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age
Ht2
Ht X 2
ht

• 18.00 76.10 78.10 152.20 .
• 19.00 77.00 79.00 154.00 .
• 20.00 78.10 80.10 156.20 .
• 21.00 78.20 80.20 156.40 .
• 22.00 78.80 80.80 157.60 .
• 23.00 79.70 81.70 159.40 .
• 24.00 79.90 81.90 159.80 .
• 25.00 81.10 83.10 162.20 .
• 26.00 81.20 83.20 162.40 .
• 27.00 81.80 83.80 163.60 .
• 28.00 82.80 84.80 165.60 .
• 29.00 83.50 85.50 167.00 .

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2
X2
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Coefficient of Determination

• r is correlation
• r2 is coefficient of determination

Tells you how much of the variability is
explained by the correlation
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Interpretation

• No causation
• 2) Value of r is influenced by range
• 3) A zero correlation tells you that there is no
  linear relationship.
• 4) Outliers can have a huge impact

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