Association Claims and R-squared

Association Claims

• Claims that two variables have a predictable relationship.
• Scatter plots show the strength and direction of the relationship.

Correlation Coefficient (R)

• Indicates the strength and direction of the relationship.

R-squared

• Also known as the proportion of variance accounted for (PVAF).

• Represents the percentage of variable $y$ that can be predicted by variable $x$.

• $r^2$ measures how well one variable can be predicted based on knowing the other.

  • Example: SAT score ($x$) and freshman GPA ($y$) with $r = 0.5$, then $r^2 = 0.25$, meaning SAT score accounts for 25% of freshman GPA.

• Remaining variance is explained by other factors.

Perfect Positive Correlation

• Example: Weight in pounds and weight in kilograms have a correlation of $1.0$.
• $r^2 = 1$, meaning weight in pounds accounts for 100% of the variance in weight in kilograms.

Correlation vs. Causation

• Association does not equal causation.

  • Example: Ice cream sales and drowning deaths correlate positively, but neither causes the other; both are related to the season/month.

• Coincidental correlations can occur.

  • Example: Number of people who drown by falling into a pool correlates with the number of films Nicolas Cage appeared in, but it is a coincidence.

• Association claims are not as strong as causal claims.