Correlation

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Correlation Chi-Square
2
Correlation Coefficient

• Descriptive statistic
• degree of relationship between 2 variables
• if we know value of 1 variable
• how well can we predict value of other
• r correlation coefficient
• between -1 and 1
• 0 no relationship

3
Scatter Diagrams

• Also called scatter plots
• 1 variable Y axis other X axis
• plot point at intersection of values
• look for trends
• e.g., height vs shoe size

4
Scatter Diagrams
84
78
Height
72
66
60
Shoe size
5
Slope value of r

• Determines sign
• positive or negative
• From lower left to upper right
• positive

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Slope value of r

• From upper left to lower right
• negative

7
Width value of r

• Magnitude of r
• draw imaginary ellipse around most points
• Narrow r near -1 or 1
• strong relationship between variables
• straight line perfect relationship (1 or -1)
• Wide r near 0
• weak relationship between variables

8
Width value of r
Strong negative relationship
Weak relationship
r near 0
r near -1
9
Measures of Correlation

• Several different measures
• depends on level of measurement
• Pearsons r
• interval/ratio
• Spearmans rs
• ordinal and interval/ratio
• Others for nominal and different combinations of
  levels of measurement

10
Factors that affect size of r

• Nonlinear relationships
• Pearsons r does not detect more complex
  relationships
• r near 0

Y
X
11
Factors that affect size of r

• Range restriction
• eliminate values from 1 or both variable
• r is reduced
• e.g. eliminate people under 72 inches

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Correlation and Causation

• Causation requires correlation, but...
• Correlation does not imply causation!
• Does not mean 1 variable causes changes in the
  other
• e.g. of household appliances negatively
  correlated with family size
• appliances as effective birth control?
• Changes may be caused by a third unknown variable
  

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Chi SquareA nonparametric hypothesis test
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Parametric vs. Nonparametric Tests

• Parametric hypothesis test
• about population parameter (m or s2)
• z, t, F tests
• interval/ratio data
• Nonparametric tests
• do not test a specific parameter
• nominal ordinal data
• frequency data

15
Chi-square (C2)

• Nonparametric tests
• same 4 steps as parametric tests
• Chi-square test for goodness of fit
• single variable
• Chi-square test for independence
• two variables
• Same formula for both
• degrees of freedom different

16
C2 Test for Goodness of Fit

• Test about proportions in distribution
• p proportion
• 2 different forms of H0
• No preference
• category proportions are equal
• No difference from comparison population
• e.g., student population
• 55 female and 45 male?
• H1 the proportions are different

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Null Hypotheses C2
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Sample Data C2

• Frequency
• Expected frequency (fe)
• fe pn
• Observed frequency (fo)
• S f n
• Degrees of freedomGoodness of fit
• C-1
• C number of cells (categories)
• C2cv from table B.6, page A-33

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Chi-square (C2)
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C2 Test for Independence

• 2 variables
• are they related or independent
• H0 also 2 forms
• no relationship between variables
• distribution of 1 variable for the categories of
  other
• Same formula as Goodness of Fit
• different df

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C2 Test for Independence

• Differences from Goodness of Fit
• df (R-1)(C-1)
• R rows
• C columns
• Expected frequency for each cell

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C2 Test for Independence
A
B
C
D
F
Sec 1
Sec 2
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