March 18, 2019

## Plan for Today:

### (1) Correlation

• correlation
• technical details
• random association
• $$p$$ values

## Correlation

#### correlation:

degree of association or relationship between the observed values taken by two variables ($$X$$ and $$Y$$)

• Many different ways of doing this (compare group means, regression) are all fundamentally about correlation.
• correlations have a direction:
• positive: implies that as $$X$$ increases, $$Y$$ increases
• negative: $$X$$ increases, $$Y$$ decreases
• correlations have strength (has nothing to do size of effect):
• strong: $$X$$ and $$Y$$ almost always move together
• weak: $$X$$ and $$Y$$ do not move together very much
• There is also a technical definition of correlation (later)

## Correlation

### What is it?

(Pearson) correlation: also has specific mathematical definition (you don't need to know it):

$r = \frac{\sum_{i}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_i^n(x_i - \bar{x})^2}\sqrt{\sum_i^n (y_i - \bar{y})^2}}$

This captures extent to which deviations from mean of $$X$$ move with deviations from mean of $$Y$$.

## Correlation

### What is it?

mathematically: correlation is the degree of linear association between $$X$$ and $$Y$$

• Takes values between $$-1$$ and $$1$$
• Values close to $$1$$ or $$-1$$ suggest high degree of linear association
• Values close to $$0$$ suggest low degree of linear association
• Value of correlation does not tell us how much $$Y$$ changes with $$X$$

## Correlation

### What is it?

negative correlation: (correlation $$< 0$$) values of $$X$$ and $$Y$$ move in opposite direction:

• higher values of $$X$$ appear with lower values of $$Y$$
• lower values of $$X$$ appear with higher values of $$Y$$

positive correlation: (correlation $$> 0$$) values of $$X$$ and $$Y$$ move in same direction:

• higher values of $$X$$ appear with higher values of $$Y$$
• lower values of $$X$$ appear with lower values of $$Y$$