Correlation

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Correlation
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Overview

• To frame our discussion, consider

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Outline

• Bivariate Data
• Correlation
• Models

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Bivariate Data

• For bivariate data each observation consists of
  two values. Observations are provided as an
  ordered pair.

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Representation

• Graphically bivariate data is viewed through the
  use of scatter plots.

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Example

• On a recent project we collected data for each
  module in the software developed. The table
  contains the module size in KLOC, the number of
  control flow paths and the number of post-release
  faults.

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Scatter Plot
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Scatter Plot

• A scatter plot shows the relationship between two
  quantitative variables measured on the same
  individual.

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Interpreting the Scatter Plot

• Look at
• Form
• Do the data points form a shape?
• Direction
• Does the shape formed indicate a direction?
• Strength
• Is there little scatter about the shape?

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Associations

• Two variables can be either
• Positively associated
• Or
• Negatively associated

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Positively Associated

• Two variables are positively associated when
  above average values in one variable results in
  above average values in the other.

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Negatively Associated

• Two variables are negatively associated when
  above average values in one variable results in
  below average values in the other.

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Example Data

• Team 1 Team 2
• Effort LOC Effort LOC
• 16.7 6050 23.5 5030
• 22.6 8363 12.6 7353
• 32.2 13334 19.4 12314
• 3.9 5942 15.6 8942
• 17.3 3315 19.2 5315
• 67.7 38988 22.3 30988
• 10.1 38614 34.5 32614
• 19.3 12762 24.8 14762
• 10.6 13510 31.6 11510
• 59.5 26500 29.9 28500

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Example Scatter Plot
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Correlation

• The correlation measures the strength and
  direction of a linear relationship between two
  variables. Correlation is usually written as r.

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Assumptions

• The distribution of both variables is
  approximately normal.
• The relation between the variables is linear.
  The bounding figure for the scatter plot is an
  ellipse.
• Homoscedastic (homogeneity of variance)

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Example with Correlation
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Properties of Correlations

• It makes no difference which variable is x and
  which is y.
• Correlation requires both variables be
  quantitative.
• Correlation is always between -1 and 1.
• Correlation looks for linear relationships.

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Interpreting r

• lt.20 slight correlation almost no relationship
• .20 - .40 low correlation small relationship
• .40 - .70 moderate correlation substantial
  relationship
• .70 - .90 high correlation marked relationship
• gt.90 very high correlation solid relationship
• (Use the values as absolute value. Hence, r -.25
  is a small negative relationship.)

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Sample (r.91)
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Relationships

• We look for patterns in scatter plots.
• Positive
• Negative
• Modeling the relationship with a straight line.

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Visualizing Correlation
r1.0
r-0.54
r0.85
r0.42
http//www.psychstat.smsu.edu/introbook/sbk17.htm
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Correlation Coefficient

• Since visual inspection is difficult to rely on,
  we fall back on mathematical techniques to search
  for relationships.

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• The coefficient of correlation, r, measures the
  strength of the linear relationship that exists
  within a sample.

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Alternately
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Relation to Sum of Squares
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Example

• A real-estate developer is interested in
  determining the relationship between family
  income (in thousands of dollars) and the size of
  their homes (in hundreds of square feet). A
  random sample of 10 families results in the
  following data

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Example 1
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Example 1
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Properties

• -1.0rlt1.0
• The larger r, the stronger the linear
  relationship.
• r near zero indicates there is no linear
  relationship.
• The sign of r tells whether the relationship is
  positive (direct) or negative (inverse).

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Example 2