Correlation
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(Linear) correlations describe straight-line relationships between two variables. Correlations can range between -1 (perfect negative) and +1 (perfect positive), with 0 indicating no straight-line relationship.
[edit] Introduction
When we ask questions such as "Is X related to Y?", "Does X predict Y?", and "Does X account for Y"?, we are interested in measuring and better understanding the relationship between two variables.
Correlation measures the extent to which:
- Variables covary
- One variable depends on another variable
- Values for one variable can be predicted from values of another variable
The correlation between variables X and Y can be denoted by rXY.
A variety of bivariate correlational statistics are available, the choice of which depends on the variables' level of measurement:
- Contingency table, Pearson's chi-square test, Phi/Cramer's V
- Spearman's rho, Kendall's tau-b
- Pearson product-moment correlation coefficient
Correlational analyses should be accompanied by appropriate graphs, such as:
[edit] The world is made of covariation
Responses to a single variable will vary (i.e., they will be distributed across a range).
Responses to two or more variables may covary, i.e., they may share some variation e.g., when the value of one variable is high, the other also tends to be relatively high (or low).
The world is made of covariation! Look around – look closely - everywhere we look, there are patterns of covariation, i.e., when two or more states tend to co-occur - e.g., higher rainfall tends to be associated with lusher plant growth.
From the distribution of stars to the behaviour of ants, we can observe predictable co-occurrence of phenomena (they tend to occur together) - e.g., when students study harder, they tend to perform better on assessment tasks.
[edit] Visual inspection of scatterplots is essential
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[edit] Correlation does not equal causation
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[edit] Range restriction
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[edit] Coefficient of determination
- When a correlation coefficient is squared, this gives the coefficient of determination which expresses the percentage of variance shared between the two variables.
- Lecture slide
- References
- Allen & Bennett, 2010, p. 173
- Howell, 2010, p. 344
- Coefficient of determination (Wikipedia)
[edit] Activities
Test yourself: This is a pre-quiz to see what you already know - Introductory quiz
Correlation guess: Correlation guess
Tutorial: Correlation (Tutorial)
[edit] See also
- Correlation (Lecture)
- Correlation (Wikipedia)
- Point-biserial correlation coefficient
[edit] External links
- 11 ways to look at the chi-squared coefficient for contingency tables
- 13 ways to look at the correlation coefficient
- Correlation (Annis, 2008)
- Correlation (Garson, 2008)
- Correlation (Plonsky, 2006)
- Correlation (Trochim, 2006)
- Correlation coefficient (Hopkins, 2000)
- Correlation coefficients (Calkins, 2005)
- Lecture - Correlation (Neill, 2010)
- New poll shows correlation is causation (Humour)
- Rank order correlation (Lowry, 2008)
- Tutorial - Correlation (Neill, 2010)
- Understanding correlation (Rummel, 1976)