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Regression and Correlation (and scatter plots)

Regression and Correlation (and scatter plots). Outline. Making a Scatter plot Calculating the Regression Line The Correlation Alternative Procedures Further Considerations of Regressions and Correlations. Estimating the speed of a car. Four Steps. Regression Lines. Different types

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Regression and Correlation (and scatter plots)

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  1. Regression and Correlation(and scatter plots)

  2. Outline • Making a Scatter plot • Calculating the Regression Line • The Correlation • Alternative Procedures • Further Considerations of Regressions and Correlations

  3. Estimating the speed of a car

  4. Four Steps

  5. Regression Lines • Different types • Here, a straight line that gets close to most of points. • One way to define “close to most points” is by finding the line that minimizes the sum of the squared vertical distances from line to the points. Minimizing ∑ei2. • Called ordinary least squares (OLS) regression.

  6. What is a straight line? • In mathematics: y = ax + b a is the slope b is the intercept • In statistics: yi = β0 + β1 xi + ei Subscripts show people have different values on variables Β0 is the intercept β1 is the slope ei are the residuals: the vertical distance between the line and the points.

  7. Scatter plot with regression line and residuals

  8. How to find the OLS line?

  9. Lots of calculations Estimate for β1 = 358.8/625.6 = 0.5735 Estimate for β0 = 14.4 - .5735(18.8) = 3.62.

  10. SStotal = SSmodel + SSresidual • SStotal is 956.4 • SSresidual is 750.6 • SSmodel = SStotal - SSresidual = 956.4 – 750.6 = 205.8 • R2= SSmodel / SStotal = 205.8/956.4 = 0.22 • R2 = the proportion of variation accounted for by the model. • Square root of R2, R (or r), also useful.

  11. Testing hypothesis R2 = 0 in the population • dfmodel = # variables in the model (here 1) • dferror = n - dfmodel – 1 (here 10 – 1 – 1 = 8) • MSSmodel = SSmodel /dfmodel (here 205.8/1 = 205.8) • MSSerror = SSerror /dferror (here 750.6/8 = 93.8) • F(dfmodel, dferror) = MSSmodel/MSSerror • F(1,8) = 205.8/93.8 = 2.19 • If computer doing the calculations, p = .18.

  12. Scatter plots with several values at same coordinate

  13. Pearson’s Correlation r Square root of R2, but keeping the sign. Ranges from -1 to 1, with negative associations having negative r values.

  14. Testing if r = 0 in the population Also worth calculating confidence intervals (see text)

  15. An Alternative Procedure: Spearman's rS • Rank the data, and then run Pearson’s • Some complications and variations if there are lots of ties. • Impact of univariate outliers (both those that increase the r and decrease it) r = .94 rS = .78 r = .76 rS = .74

  16. Further Considerations • What we have discussed has been for straight lines. Look at scatter plots to see if other techniques for curves necessary. • Correlation does not imply causation (but it suggests that somewhere in the network of hypotheses that includes these two variables that there are causal relationships).

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