Correlations

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Correlations
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Outline

• What is a correlation?
• What is a scatterplot?
• What type of information is provided by a
  correlation coefficient
• Pearson correlation
• How is pearson calculated
• Hypothesis testing with pearson
• Correlation causation
• Factors affecting correlation coefficient
• Coefficient of determination
• Correlation in research articles
• Other types of correlation

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Distinguishing Characteristics of Correlation

• Correlational procedures involve one sample
  containing all pairs of X and Y scores
• Neither variable is called the IV or DV
• Use the individual pair of scores to create a
  scatterplot

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Correlation Coefficient

• Describes three characteristics of the
  relationship
• Direction
• Form
• Degree

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What Is A Large Correlation?

• Guidelines
• 0.00 to lt.30 low
• .30 to lt.50 moderate
• gt.50 high
• While 0 means no correlation at all, and 1.00
  represents a perfect correlation, we cannot say
  that .5 is half as strong as a correlation of 1.0

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Pearson Correlation

• Used to describe the linear relationship between
  two variables that are both interval or ratio
  variables
• The symbol for Pearsons correlation coefficient
  is r
• The underlying principle of r is that it compares
  how consistently each Y value is paired with each
  X value in a linear manner

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Calculating Pearson r
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Calculating Pearson r

• There are 3 main steps to r
• Calculate the Sum of Products (SP)
• Calculate the Sum of Squares for X (SSX) and the
  Sum of Squares for Y (SSY)
• Divide the Sum of Products by the combination of
  the Sum of Squares

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Pearson Correlation - Formula
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1) Sum of Products

• To determine the degree to which X Y covary
  (numerator)
• We want a score that shows all of the deviation X
  Y have in common
• Sum of Products (also known as the Sum of the
  Cross-products)
• This score reflects the shared variability
  between X Y
• The degree to which X Y deviate from the mean
  together

SP ?(X MX)(Y MY)
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Sums of Product Deviations

• Computational Formula

• n in this formula refers to the number of pairs
  of scores

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2) Sum of Squares X Y

• For the denominator, we need to take into account
  the degree to which X Y vary separately
• We want to find all the variability that X Y do
  not have in common
• We calculate sum of squares separately (SSX and
  SSY)
• Multiply them and take the square root

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2) Sum of Squares X Y

• The denominator

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Hypothesis testing with r

• Step 1) Set up your hypothesis
• Ho ? 0 There is no correlation in the
  population between the number of errors and the
  number of drinks
• H1 ? ? 0 There is a correlation in the
  population between the number of errors and the
  number of drinks

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Hypothesis testing with r

• Step 2) Find your critical r-score
• Alpha and degrees of freedom
• a .05, two-tailed
• Degrees of freedom n 2

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Hypothesis testing with r

• Step 3) Calculate your r-obtained
• Step 4) Compare the r-obtained to r-critical, and
  make a conclusion
• If r-obtained lt r-critical fail to reject Ho
• If r-obtained gt r-critical reject Ho

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Correlation and Causality

• A statistical relationship can exist even though
  one variable does not cause or influence the
  other
• Correlational research CANNOT be used to infer
  causal relationships between two variables

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Correlation and Causality

• When two variables are correlated, three possible
  directions of causality
• 1st variable causes 2nd
• 2nd variable causes 1st
• Some 3rd variable causes both the 1st and the 2nd
• There is inherent ambiguity in correlations

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Factors Affecting CorrelationWatch out for
outliers
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Factors Affecting Correlation Restriction of
Range
No relationship here
Strong relationship here
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Coefficient Of Determination

• The squared correlation (r2) measures the
  proportion of variability in the data that is
  explained by the relationship between X and Y
• Coefficient of Non-Determination (1-r2)
  percentage of variance not accounted for in Y

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Correlation in Research Articles
Coleman, Casali, Wampold (2001). Adolescent
strategies for coping with cultural diversity.
Journal of Counseling and Development, 79, 356-362
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Other Types of Correlation

• Spearmans Rank Correlation
• variable X is ordinal and variable Y is ordinal
• Point-biserial correlation
• variable X is nominal and variable Y is interval
• Phi-coefficient
• variable X is nominal and variable Y is also
  nominal
• Rank biserial
• variable X is nominal and variable Y is ordinal

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Example 2
Hours (X) Errors (Y)
0 19
1 6
2 2
4 1
4 4
5 0
3 3
5 5
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Create Scatterplot
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Step 1- Set Up Your Hypothesis

• Ho
• H1

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Step 2 - Find critical r-score

• Alpha and degrees of freedom
• a .05, two-tailed
• Degrees of freedom n 2

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Step 3 - Calculate r-obtained
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Step 4 - Compare R-obtained To R-critical, Make
A Conclusion
Step 4 Compute r2