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Correlation 10/30

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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

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

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Correlation Measure of how closely two variables are related Population correlation: (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

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

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Computing Correlation 1.Get z-scores for both samples 2.Multiply all pairs 3.Get average by dividing by n 1 Positive relationship Positive z X tend to go with positive z Y Negative z X tend to go with negative z Y z X z Y tends to be positive Negative relationship Positive z X tend to go with negative z Y Negative z X tend to go with positive z Y z X z Y tends to be negative z X > 0z X < 0 MXMX MYMY z Y > 0 z Y < 0 MXMX z X > 0z X < 0 MYMY z Y > 0 z Y < 0

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Computing Correlation XYX M X Y M Y zXzX zYzY zX zYzX zY 5801.00.27.00 810331.16.80.93 34-2-3-.77-.80.62 913461.551.602.48 22-3-5-1.16-1.341.55 470-.39.00 45-2-.39-.53.21 M X = 5M Y = 7 = 5.80 s X = 2.6s Y = 3.7r =.97 X Y

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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 Arousal Performance r = 0

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Predicting One Variable from Another Knowing one measure gives information about others from same subject Knowing a persons 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

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

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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 Cant predict well X Y X Y X Y

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

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Correlation and Prediction Best prediction line minimizes MS Error Closest to data; best fit Correlation determines best prediction line Slope = r when plotting z-scores: zYzY zXzX r =.75 slope =.75

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

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Explained Variance Without knowing XKnowing X Original Variance Explained Variance Reduction from knowing X Residual Variance

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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 (MS Error ) Correlation gives slope of line (when using z-scores): r 2 equals proportion of variance in one variable explained by other Reduction from original variance (s Y 2 ) to residual variance (MS Error )

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

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Review Calculate the correlation between X and Y. z X = [-0.1-1.4-1.0-0.4] z Y = [-0.4-1.3-1.0-0.1] A.-.94 B.-.70 C.-.02 D.-.001

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Review The correlation between IQ and number of bicycles owned is r =.6. Predict the IQ of someone who owns 4 bikes (z bike = 2.5). Recall that IQ = 100 and IQ = 15. A.101.5 B.109.0 C.122.5 D.137.5