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Psychology 202a Advanced Psychological Statistics October 22, 2015 The plan for today • Conditioning on a continuous variable • Introducing correlation and regression Continuous conditional distributions • The scatterplot. • Focusing on conditional center. • Two natural questions: – How strong is the relationship? – What is the relationship? • Correlation and regression. How strong is the relationship? • Pearson product-moment correlation coefficient • Population: r (rho) • Sample: r • We will develop three ways to understand the correlation coefficient First way to understand correlation • A scale-free covariance • Covariance: ( X M X )( Y MY ) Cov X ,Y . N 1 Covariance (continued) • Problem: magnitude depends on scale of X and Y • Solution: remove the scale by standardizing • Pearson’s r: r X ,Y Cov X ,Y s X sY . 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. What is the relationship? • Linear regression: Y 0 1X . Estimation of regression parameters • Slope estimate: ˆ1 X M X Y MY . X M X 2 • Intercept estimate: ˆ0 MY ˆ1M X . Regression as a Model • Regression as a model for conditional mean • What about all those other aspects of a distribution? Estimating Regression • Why are the estimates what they are? • Definition: residual is an estimate of the error component of the model: Y i 0 1X i i i Y i 0 1X i e i Y i ˆ0 ˆ1X i 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. 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. 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.