Topics: Correlation

• The road map

• Examining “bi-variate” relationships through pictures

• Examining “bi-variate” relationships through numbers

Correlational Research

• Exploration of relationships between variables for better understanding

• Exploration of relationships between variables as a means of predicting future behavior.

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?

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

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

Scatter Plot: SAT/GPA(Overachievement Study)

SAT

1300120011001000900

GPA

4.0

3.5

3.0

2.5

2.0

1.5

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”

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

Examples of Various Scatter Plots Demontrating Tilt

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.

Examples of Various scatter plots Demontrating Shape (Strength)

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).

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

Influences on Correlation Coefficients

• Restriction of range

• Use of extreme groups

• Combining groups

• Outliers (extreme scores)

• Curvilinear relationships

• Sample size

• Reliability of measures

Restriction of Range: Example

Using Extreme Groups Example

Combining Groups Example

Outliers (Extreme Scores) Example

Curvilinear Examples

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

Coefficient of Determination:Graphical Display

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