chapter 1

Chapter 13: Nonlinear and Multiple Regression22.1ya. = α + βx, so withis1y= α + βx+ ε.1y′ = , y = α + βxy′ . The corresponding probabilistic modelb.1− 1= eyα+βx⎛ 1 ⎞⎛ 1 ⎞ln . Thus with y ′ = ln⎜−1⎟ , y = α + βx⎝ y ⎠⎝ y ⎠Y ′ = α + βx+ ε ′ , or equivalently, so ⎜ −1⎟= α + βxThe corresponding probabilistic model isY= α + β x1 + e1⋅ εwhereεε′= e .′ .α + βxc. ln ( y) = e = ln ( ln ( y) = α + βx. Thus with y′ = ( ln ( y)),y′=α + βxThe probabilistic model isεε′= e .d. This function cannot be linearized.ln .α + βxeY ′ = α + βx+ ε ′ , or equivalently, = e ⋅εY whereβxβx2 2βx223. [ ] ( )2Var ( Y ) = Var(αe⋅ε)= αe⋅Varε = α e ⋅τwhere we have set2( ε ) = τVar . If β > 0, this is an increasing function of x so we expect more spread in yfor large x than for small x, while the situation is reversed if β < 0 . It is important to realizethat a scatter plot of data generated from this model will not spread out uniformly about theexponential regression function throughout the range of x values; the spread will only beuniform on the transformed scale. Similar results hold for the multiplicative power model.24. β 0H vs H : β ≠ 100:1=a. The value of the test statistic is z = .73, with acorresponding p-value of .463. Since the p-value is greater than any sensible choice of alphawe do not reject H o . There is insufficient evidence to claim that age has a significant impacton the presence of kyphosis.407