Chapter 3: Descriptive Statistics: Bivariate Correlation
Bivariate Correlation: Relationship between two variables
Concerned with two things:
1. Is there a relationship?
2. How strong/weak is the relationship? (if there is)
Can be one of three types:
1. High-high, low-low
2. High-low, low-high
3. Little systematic tendency
Correlation Coefficient:
- A measure of the correlation between two variables
- Ranges from -1 to 1
- Toward -1 means high-low (inverse relationship; negative correlation)
- Toward +1 means high-high (direct; positive correlation)
- Depending on value can use words like weak, moderate, strong to describe
Correlation Matrix
- Table of correlation coefficients between different pairs of variables
Different kinds of correlation coefficients used (depending on type of variables)
Type of
correlation
coefficient used
Var 1
Var 2
Pearson’s ProductMoment (r)
Quantitative (interval
or ratio); represents
raw score
Quantitative but
represents rank
Quantitative but
represents rank
Quantitative (interval
or ratio);
represents raw score
Quantitative but
represents rank
Quantitative but
represents rank
Raw Scores
Raw Scores
Dichotomous (1 or 0)
Artificial Dichotomy
Nominal
Dichotomous (1 or 0)
Artificial Dichotomy
Dichotomous (1 or 0)
Artificial Dichotomy
Nominal
Spearman’s Rho
Kendall’s Tau
Point Biserial
Biserial Correlation
Phi
Tetrachoric
Cramer’s V
Remarks
Similar to Rho but
differs in treatment of
ties (Tau does a better
job)
If only two categories,
Cramers’ V identical
to Phi
Warnings about correlation:
1.
2.
3.
Correlation is not equal to Cause
Coefficient of determination (square of correlation coefficient)
a. Suggests how much variability in either variable is explained by the
other variable
Outliers
4.
5.
6.
Linearity
a. Needs to check whether the two variables have a linear relationship to
begin with
b. If relationship is curvilinear, Pearson’s correlation will underestimate
the strength of the relationship
Correlation and Independence
a. The two instruments used to measure the variables should (ideally) be
independent
b. Hence, the correlation between them should be low
Relationship Strength
a. Choice of words still subjective; check raw data if necessary and possible