1_Correlation and Regression

# 1_Correlation and Regression - Outline Advanced Topics in...

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Fal 2007, C. Staudhammer Advanced Topics in Forest Biometrics - FOR6934 Review of correlation and Regression Fal 2007, C. Staudhammer Outline ± Correlation ± Simple Linear Regression (SLR) ² Formulae and degrees of freedom ² Hypothesis tests ² Assumptions ± Multiple Linear Regression ² Formulae and df ² Hypothesis tests ² Evaluating model fit ² Influence diagnostics ² Multicollinearity ² Model selection Fal 2007, C. Staudhammer Uses ± Often used to evaluate associations ± Continuous X and Y ± Common in mensurative (empirical observation) studies ± Used to characterize relationships between variables ± Used for parameter estimation for rates such as growth, survival Fal 2007, C. Staudhammer Correlation ± Used to evaluate the degree of association between two continuous variables x, y Birdwatcher Participation Shorebird Density Fal 2007, C. Staudhammer Causation? Fal 2007, C. Staudhammer Causation? Birdwatcher Participation

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Fal 2007, C. Staudhammer Correlation ∑∑ = 2 2 y x xy r ± r represents the linear association between x and y ± s r is the standard error of the correlation ± Usually used to imply associations rather than cause/effect 2 1 2 = n r s r Fal 2007, C. Staudhammer Correlation ± r can range from: ² -1 (complete negative relation between x, y) ² 0 (no relation) ² 1 (complete positive relation between x, y) ² Hypothesis test required to test whether r is significantly different from zero Fal 2007, C. Staudhammer Hypotheses and Testing r s r t = H o : ρ =0 H a : ρ≠ 0 Reject if: v t t ), 2 ( α = Where v = n-2 Fal 2007, C. Staudhammer Simple Linear Regression ± Attempt to find the linear relationship between x and y Model: y i = β 0 + 1 x i + ε i (population) y i = b 0 + b 1 x i + e i (sample) Predictions: ŷ i = b 0 + b 1 x i Errors: i i i y y e ˆ = * To find estimates of b o and b 1 , solve using least squares Fal 2007, C. Staudhammer Least Squares Fit Properties: 1. Always passes through ) , ( y x 2. Sum of residuals is zero, i.e., Σ e i =0 3. Sum of squared residuals is minimized in fit (least squares estimate) Fal 2007, C. Staudhammer 0.0 4.0 8.0 12.0 16.0 20.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 dbh h e i g h t i i y y ˆ i i y y ˆ i i y y ˆ i i y y ˆ
Fal 2007, C. Staudhammer Partitioning Variance Residual Regression 1 2 1 2 1 2 ) ˆ ( ) ˆ ( ) ( SS SS SS y y y y y y Total n i i i n i i n i i + = + = = = = Degrees of freedom?!?!?!

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## This note was uploaded on 07/23/2011 for the course FOR 6934 taught by Professor Staff during the Spring '08 term at University of Florida.

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1_Correlation and Regression - Outline Advanced Topics in...

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