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LECTURE 6

LECTURE 6. CURVILINEAR RELATIONSHIPS AND TRANSFORMATIONS. POWER POLYNOMIALS. Types of curve functions of interest: Quadratic Cubic. C 1. 3 2 1 0 -1 -2 -3. 3 2 1 0 -1 -2 -3. C 3. 3 2 1 0 -1 -2 -3. C 2. 0 100 200 300. 0 100 200 300. 0 100 200 300.

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LECTURE 6

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  1. LECTURE 6 CURVILINEAR RELATIONSHIPS AND TRANSFORMATIONS

  2. POWER POLYNOMIALS • Types of curve functions of interest: • Quadratic • Cubic

  3. C1 3 2 1 0 -1 -2 -3 3 2 1 0 -1 -2 -3 C3 3 2 1 0 -1 -2 -3 C2 0 100 200 300 0 100 200 300 0 100 200 300 Fig. 9.2: Graphs of planned orthogonal contrasts for four interval treatments

  4. Centering and Orthogonality • Grand mean centering: subtract mean from each score x1 = x-meanx • Create quadratic variable by squaring centered variable: x2= (x1)2 This results in rx1x2 << rxx2 Reduces multicollinearity in predictors

  5. Nonlinear Transformations • Lowess lines indicate various nonlinear conditions: heteroscedasticity, nonnormal distribution of errors, nonlinear variable relationships

  6. Nonlinear Transformations • Exponential relations: y=e-x • Use logarithmic transformation: log(y) = -x y log(y) X X

  7. Heteroscedastic, Skewed • Square root transformation: y=bx1 + b0 x’ = x pr(x) pr( x) x x

  8. Arcsin transformation • Linearize proportion variables (eg. %getting particular item correct predicted by total test score ) • 2*Arcsin(p) P(item i=1) 2arcsin(p) Test score x Test score x

  9. Logit transformation • Use when p is binary outcome (pass-fail) • L= ½ ln (p/(1-p))

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