Derivation of Coefficients for Trend Analysis in Psychology: Linear and Quadratic - Prof. , Assignments of Psychology

Download Derivation of Coefficients for Trend Analysis in Psychology: Linear and Quadratic - Prof. and more Assignments Psychology in PDF only on Docsity! Psych 610 Handout #8, p. 1 Prof. Moore Derivation of coefficients for Trend Analysis (1) µi = β1(a1 + Xj) + β2(a2 + b2Xj + Xj 2) + β3(a3 + b3Xj + c3Xj 2 + Xj 3) + etc. -- Assume an experiment with 4 age groups – 8, 9, 10, and 11-year-olds. We want to do trend analysis on Age. So the ages are used as the values of X. For linear: cj = a1 + Xj (from Eq. 1). Plug in the age as Xj: c1 = a1 + 8 c1 = -9.5 + 8 = -1.5 Add c2 = a1 + 9 c2 = -9.5 + 9 = -.5 these c3 = a1 + 10 c3 = -9.5 + 10 = .5 c4 = a1 + 11 c4 = -9.5 + 11 = 1.5 ____________ ∑cj = 4a1 + 38 0 = 4a1 + 38; because ∑cj = 0 -38 = 4a1 a1 = -9.5 Now plug in –9.5 for a1 -- So, coefficients for linear are –1.5, -.5, .5, 1.5. If we multiply by 2, we get the values from Keppel Table A –4, -3, -1, 1, 3. -- Next we want to find the coefficients for quadratic trend. We want these coefficients to be orthogonal to the linear coefficients. For quadratic: cj = a2 + b2Xj + Xj 2 (from Eq. 1). Plug in the age as Xj: c1 = a2 + (b2)(8) + 8 2 c1 = 89 + (-19)(8) + 64 = 1 Add c2 = a2 + (b2)(9) + 9 2 c2 = 89 + (-19)(9) + 81 = -1 these c3 = a2 + (b2)(10) + 10 2 c3 = 89 + (-19)(10) + 100 = -1 c4 = a2 + (b2)(11) + 11 2 c4 = 89 + (-19)(11) + 121 = 1 __________________ ∑cj = 4a2 + 38b2 + 366 Now apply orthogonality constraint. ∑cjcj′ = 0.