Onset of a neurodegenerative disorder. Spinocerebellar ataxia type 1 (SCA1) is an inherited neurodegenerative disorder characterized by dysfunction of the brain. From a DNA analysis of SCA1 chromosomes, researchers discovered the presence of repeat gene sequences (Cell Biology, February 1995). In general, the more repeat sequences observed, the earlier the onset of the disease (in years of age). The following scatterplot shows this relationship for data collected on 113 individuals diagnosed with SCA1.
(a) Suppose you want to model the age y of onset of the disease as a function of number x of repeat gene sequences in SCA1 chromosomes. Propose a quadratic model for y.
(b) Will the sign of β2 in the model, part a, be positive or negative? Base your decision on the results shown in the scatterplot.
(c) The researchers reported a correlation of r = -.815 between age and number of repeats. Since r2 = (-.815)2 = .664, they concluded that about ''66% of the variability in the age of onset can be accounted for by the number of repeats.'' Does this statement apply to the quadratic model E(y) = β0 + β1x + β2x2? If not, give the equation of the model for which it does apply.