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Correlation

Correlation. 10/31. Relationships Between Continuous Variables. Some studies measure multiple variables Any paired-sample experiment Training & testing performance; personality variables; neurological measures Continuous independent variables How are these variables related?

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Correlation

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  1. Correlation 10/31

  2. Relationships Between Continuous Variables • Some studies measure multiple variables • Any paired-sample experiment • Training & testing performance; personality variables; neurological measures • Continuous independent variables • How are these variables related? • Positive relationship: tend to be both large or both small • Negative relationship: when one is large, other tends to be small • Independent: value of one tells nothing about other

  3. Scatterplots • Graph of relationship between two variables, X and Y • One point per subject • Horizontal coordinate X • Vertical coordinate Y Weight = 181.7 Height = 67.7

  4. Correlation • Measure of how closely two variables are related • Population correlation: r (rho) • Sample correlation: r • Direction • r > 0: positive relationship; big X goes with big Y • r < 0: negative relationship; big X goes with small Y • Strength • ±1 means perfect relationship • Data lie exactly on a line • If you know X, you know Y • 0 means no relationship • Independent: Knowing X tells nothing about Y

  5. r = -1 r = -.75 r = -.5 r = -.25 r = 0 r = .25 r = .5 r = .75 r = 1

  6. Computing Correlation • Get z-scores for both samples • Multiply all pairs • Get average by dividing by n – 1 • Positive relationship • Positive zX tend to go with positive zY • Negative zX tend to go with negative zY • zXzY tends to be positive • Negative relationship • Positive zX tend to go with negative zY • Negative zX tend to go with positive zY • zXzY tends to be negative zY > 0 MY zY < 0 MX zX < 0 zX > 0 zY > 0 MY zY < 0 MX zX < 0 zX > 0

  7. Computing Correlation Y X

  8. Correlation and Linear Relationships • Correlation measures how well data fit on straight line • Assumes linear relationship between X and Y • Not useful for nonlinear relationships r = 0 Performance Arousal

  9. Predicting One Variable from Another • Knowing one measure gives information about others from same subject • Knowing a person’s weight tells about his height • Goal: Come up with a rule or function that uses X to compute best estimate of Y • Y (Y-hat) • Predicted value of Y • Function of X • Best prediction of Y based on X

  10. Linear Prediction • Simplest way to predict one variable from another • Straight line through data • Y is linear function of X X= 71

  11. How Good is the Prediction? • Sometimes data fall nearly on a perfect line • Strong relationship between variables • r near ±1 • Good prediction • Sometimes data are more scattered • Weak relationship • r near 0 • Can’t predict well Y Y Y X X X

  12. How Good is the Prediction? • Goal: Keep error close to zero • Minimize mean squared error:

  13. Correlation and Prediction • Best prediction line minimizes MSError • Closest to data; best “fit” • Correlation determines best prediction line • Slope = r when plotting z-scores: r = .75 zY slope = .75 zX

  14. r = -1 r = -.75 r = -.5 r = -.25 r = 0 r = .25 r = .5 r = .75 r = 1

  15. Explained Variance Without knowing X Knowing X Original Variance Explained Variance Reduction from knowing X Residual Variance

  16. Properties of Correlation • Measures relationship between two continuous variables • How well data are fit by a straight line • Sign of r shows direction of relationship • Magnitude of r shows strength of relationship • Strongest relationships have r = ±1; weak relationships have r ≈ 0 • Best prediction line minimizes error of prediction (MSError) • Correlation gives slope of line (when using z-scores): • r2 equals proportion of variance in one variable explained by other • Reduction from original variance (sY2) to residual variance (MSError)

  17. Review Find the correlation of r = .7. A C B D

  18. Review Calculate the correlation between X and Y. zX=[ -0.1 -1.4 -1.0 -0.4] zY=[ -0.4 -1.3 -1.0 -0.1] • -.94 • -.70 • -.02 • -.001

  19. Review The correlation between IQ and number of bicycles owned is r = .6. Predict the IQ of someone who owns 4 bikes (zbike = 2.5). Recall that µIQ = 100 and sIQ = 15. • 101.5 • 109.0 • 122.5 • 137.5

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