1. Psychology 202a Advanced Psychological Statistics October 22, 2015

2. The plan for today Conditioning on a continuous variable Introducing correlation and regression

3. Continuous conditional distributions The scatterplot. Focusing on conditional center. Two natural questions: –How strong is the relationship? –What is the relationship? Correlation and regression.

4. How strong is the relationship? Pearson product-moment correlation coefficient Population:  (rho) Sample: r We will develop three ways to understand the correlation coefficient

5. First way to understand correlation A scale-free covariance Covariance:

6. Covariance (continued) Problem: magnitude depends on scale of X and Y Solution: remove the scale by standardizing Pearson’s r:

7. Problem with that way of understanding r Does not make it absolutely clear that the relationship must be linear in order for r to make sense as a measure of strength of association.

8. What is the relationship? Linear regression:

9. Estimation of regression parameters Slope estimate: Intercept estimate:

10. Regression as a Model Regression as a model for conditional mean What about all those other aspects of a distribution?

11. Estimating Regression Why are the estimates what they are? Definition: residual is an estimate of the error component of the model:

12. Estimating Regression The line that fits best is the one that minimizes the residuals. Once again, negative residuals balance positive residuals… …so we make the residuals positive by squaring them.

13. The Principle of Least Squares This criterion for best fit is known as the principle of least squares. You will also see it referred to as “ordinary least squares” … …or as “OLS” for short. See me if you are interested in why the OLS estimates are what they are.

14. Decomposing the sum of squares Recall that the model can be broken down into two components: –the part we do understand –the part we don’t understand The sum of squares can be broken down into corresponding components. These components have the same additive relationship as the model components.