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9. Regression and correlation

9. Regression and correlation - 9 Linear Regression and...

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9. Linear Regression and Correlation Data: y: a quantitative response variable x: a quantitative explanatory variable (Chapter 8: Recall that both variables were categorical ; later chapters have multiple explanatory variables) For example (Wagner et al., Amer. J. Community Health , vol. 16, p. 189) y = mental health, measured with Hopkins Symptom List (presence or absence of 57 psychological symptoms) x = stress level (a measure of negative events weighted by the reported frequency and subject’s subjective estimate of impact of each event) We consider: Is there an association? (test of independence ) How strong is the association? (uses correlation ) How can we describe the nature of the relationship, e.g., by using x to predict y ? ( regression equation, residuals)
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Linear Relationships Linear Function (Straight-Line Relation): y = α + β x expresses y as linear function of x with slope β and y- intercept α. For each 1-unit increase in x, y increases β units β > 0 Line slopes upward ( positive relationship) β = 0 Horizontal line ( y does not depend on x ) β < 0 Line slopes downward ( negative relation)
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Example: Economic Level and CO2 Emissions OECD (Organization for Economic Development, www.oecd.org ): Advanced industrialized nations “committed to democracy and the market economy.” oecd-data file (from 2004) on p. 62 of text and at text website www.stat.ufl.edu/~aa/social/ Let y = carbon dioxide emissions (per capita, in metric tons) Ranges from 5.6 in Portugal to 22.0 in Luxembourg (U.S. = 19.8) mean = 10.4, standard deviation = 4.6 x = gross domestic product (GDP, in thousands of dollars per capita) Ranges from 19.6 in Portugal to 70.0 in Luxembourg (U.S. = 39.7) mean = 32.1, standard deviation = 9.6
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The relationship between x and y can be approximated by y = 0.42 + 0.31x. At x = 0, predicted CO2 level y = 0.42 + 0.31 x = 0.42 + 0.31(0) = 0.42 (irrelevant, because no GDP values near 0) At x = 39.7 (value for U.S.), predicted CO2 level y = 0.42 + 0.31(39.7) = 12.7 (actual = 19.8 for U.S.) For each increase of 1 thousand dollars in per capita GDP, CO2 use predicted to increase by 0.31 metric tons per capita But, this linear equation is just an approximation. The correlation between x and y for these nations was 0.64, not 1.0 (It is even less, 0.41, if we remove the outlier observation for Luxembourg.)
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Effect of variable coding? Slope and intercept depend on units of measurement. If x = GDP measured in dollars (instead of thousands of dollars), then y = 0.42 + 0.00031 x (instead of y = 0.42 + 0.31 x ) because a change of $1 has only 1/1000 the impact of a change of $1000 (so, the slope is multiplied by 0.001). If y = CO2 output in kilograms instead of metric tons (1 metric ton = 1000 kilograms), with x in dollars, then y = 1000(0.42 + 0.00031 x ) = 420 + 0.31x Suppose x changes from U.S. dollars to British pounds and 1 pound = 2 dollars. What happens?
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Probabilistic Models In practice, the relationship between y and x is not “perfect” because y is not completely determined by x .
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