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Correlation and Regression

Correlation and Regression. Basic Concepts. An Example. We can hypothesize that the value of a house increases as its size increases. Said differently, size and house value “covary” or “co-relate.”

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Correlation and Regression

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  1. Correlation and Regression Basic Concepts

  2. An Example • We can hypothesize that the value of a house increases as its size increases. • Said differently, size and house value “covary” or “co-relate.” • Further, we can hypothesize that the relationship is a simple linear one, e.g., that as size increases, house value increases in a similar linear fashion. • Hence we can use the simple linear equation, • y = a + bx, to describe the relationship

  3. We Ask Two Questions… • Is there a relationship and how strong is it? • and • What is the relationship? • We answer the first with a new statistic, a “correlation” coefficient. • We answer the second with a linear regression model.

  4. Two Questions • We started with Correlation Monday. • We continue today with Regression.

  5. Terms • Independent and Dependent variables • Scatterplots • Correlation, correlation coefficient, r • Regression, regression coefficient, b • Regression, regression constant, a • Ordinary Least Squares (OLS) equation: y = a + bx + e

  6. Issues • Defining relationships • Nature of the relationship: for the moment, linear • Strength of the relationship (using r) • Direction of the relationship (using r and b) • Calculation of the relationship: y = a + bx + e

  7. Some useful websites • http://davidmlane.com/hyperstat/A60659.html • http://www.ruf.rice.edu/~lane/stat_sim/reg_by_eye/index.html

  8. Illustration • Case A. x= 2.5, y=2 • Case B. x=8, y = 7

  9. Linear Trend

  10. What if there are lots of data points?

  11. If there are more data points?How do we summarize the relationships in the data? ?

  12. Solution: Least Squares Regression, The Best Linear Fit

  13. Some Theory • Knowing nothing else, the best estimate of a variable is its mean.

  14. The Regression Model does better… • Deviation from y = yi – ymean

  15. A Regression equation… • Measures the nature of the relationship between x and y using a linear model • Measures the direction of the relationship • Accompanying statistics, for the time being, r, measures the strength of the relationship.

  16. Understanding the Improvement, measuring the deviations from the mean

  17. More Terms • Yi – the value of a particular case • Y mean – mean value of y • Y hat – y with a ^ above it soŷ • (Yi – Ymean) = total deviation from mean Y • (Yhat – Ymean) = explained deviation of Yi from Y mean • (Yi – Yhat) = unexplained deviation of Yi from Y mean

  18. Bivariate Regression • Relationships are modeled using the equation, y = a + bx + e • Translation: The values of an interval level dependent variable, y, can be “predicted” or “modeled” by adding a constant, a, to the product of a slope coefficient, b, times the values of the independent variable, x, and an error term, e.

  19. Estimating the Equation, y = a + bx + e • The regression equation is calculated by finding the equation that minimizes the sum of the squared deviations between the data points, the y’s, and the predicted y’s, also called y hat.

  20. Correlation Coefficient: r • A measure of the strength of a linear relationship between two interval variables, x and y • Ranges from – 1 to + 1 • The higher the value of r (e.g., the closer to -1 or + 1, the stronger the relationship between x and y

  21. Correlation Coefficient calculation • r = Covariance of x and y divided by the product of the standard deviation of x and the standard deviation of y • Covariance is the sum of the products of the deviations of the cases divided by N.

  22. Equations...

  23. Calculating a and b

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