topics: correlation the road map examining “bi-variate” relationships through pictures examining...
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Topics: Correlation
• The road map
• Examining “bi-variate” relationships through pictures
• Examining “bi-variate” relationships through numbers
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Correlational Research
• Exploration of relationships between variables for better understanding
• Exploration of relationships between variables as a means of predicting future behavior.
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Correlation: Bi-Variate Relationships
• A correlation describes a relationship between two variables
• Correlation tries to answer the following questions:– What is the relationship between variable X and variable Y?
– How are the scores on one measure associated with scores on another measure?
– To what extent do the high scores on one variable go with the high scores on the second variable?
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Types of Correlation Studies
• Measures of same individuals on two or more different variables
• Measures of different individuals on the “same” variable
• Measures of the same individuals on the “same” variable(s) measured at different times
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Representations of Relationships
• Tabular Representation: arrangement of scores in a joint distribution table
• Graphical Representation: a picture of the joint distribution
• Numerical Represenation: a number summarizing the relationship
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Scatter Plot: SAT/GPA(Overachievement Study)
SAT
1300120011001000900
GPA
4.0
3.5
3.0
2.5
2.0
1.5
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Creating a Scatter Plot
• Construct a joint distribution table
• Draw the axis of the graph
– Label the abscissa with name of units of the X variable
– Label the ordinate with the name of the units of the Y variable
• Plot one point for each subject representing their scores on each variable
• Draw a perimeter line (“fence”) around the full set of data points trying to get as tight a fit as possible.
• Examine the shape:– The “tilt”
– The “thickness”
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Reading the Nature of Relationship
• Tilt: The slope (or slant) of the scatter as represented by an imaginary line.
– Positive relationship: The estimated line goes from lower-left to upper right (high-high, low-low situation)
– Negative relationship: The estimated line goes from upper left to lower right (high-low, low-high situation)
– No relationship: The line is horizontal or vertical because the points have no slant
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Examples of Various Scatter Plots Demontrating Tilt
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Reading the Strength of Relationship
• Shape: the degree to which the points in the scatter plot cluster around the imaginary line that represents the slope.– Strong relationship: If oval is elongated and
thin.– Weak relationship: If oval is not much longer
than it is wide.– Moderate relationship: Somewhere in between.
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Examples of Various scatter plots Demontrating Shape (Strength)
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Numerical Representation: The Correlation Coefficient
• Correlation Coefficient = numerical summary of scatter plots. A measure of the strength of association between two variables.
• Correlation indicated by ‘r’ (lowercase)
• Correlation range: -1.00 0.00 +1.00
• Absolute magnitude: is the indicator of the strength of relationship. Closer to value of 1.00 (+ or -) the stronger the relationship; closer to 0 the weaker the relationship.
• Sign (+ or -): is the indication of the nature (direction,)tilt) of the relationship (positive,negative).
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Types of Correlation Coefficients
Scale of
Measurement
Interval, Ratio Ordinal Nominal Dichotomous
Artificial
Dichotomy
Interval,Ratio Pearson Product
Moment
Ordinal Spearman Rho
Kendall Tau
Nominal Cramer's V
Dichotomous,
Artificial
Dichotomy
Point Biserial
Biserial
Phi
Tetrachoric
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Influences on Correlation Coefficients
• Restriction of range
• Use of extreme groups
• Combining groups
• Outliers (extreme scores)
• Curvilinear relationships
• Sample size
• Reliability of measures
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Restriction of Range: Example
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Using Extreme Groups Example
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Combining Groups Example
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Outliers (Extreme Scores) Example
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Curvilinear Examples
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Coefficient of Determination
• Coefficient of Determination: the squared correlation coefficient
• The proportion of variability in Y that can be explained (accounted for) by knowing X
• Lies between 0 and +1.00
• r2 will always be lower than r
• Often converted to a percentage
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Coefficient of Determination:Graphical Display
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Some Warnings
• Correlation does not address issue of cause and effect: correlation ≠ causation
• Correlation is a way to establish independence of measures
• No rules about what is “strong”, “moderate”, “weak” relationship