Some Hypotheses Concerning Two Phase Regression Lines

The regression of a growth measurement y on time x can often be reasonably represented by two intersecting straight lines, one being appropriate when x takes values below and the other when x takes values above a certain fixed but often unknown value corresponding to the intersection. Such regressions are here called two-phase regressions, the intersection of the phases being referred to as the changeover point, and the value of x at which it occurs being called the changeover value. Situations in which such regressions might occur include the onset of a disease resulting in a reduced growth rate; the application of a treatment having an immediate stimulating or inhibiting effect; the occurrence of an extremely hot or cold day or some other change in external conditions; physical injury of an organism. In a study of the compatibility of peach scions on plum rootstocks Garner and Hammond 1938 noted that the peach variety Hale's Early developed at constant but different rates on compatible and incompatible rootstock-scion unions up to a certain date. After that date the growth rate in the compatible case continued at a new constant rate, whilst in the incompatible case all growth then ceased. Thus for a compatible union there was a typical two-phase regression, whilst for the incompatible union a rather special case occurred in which the slope of the second phase was zero. A further example is given in Section 4 in which the date of phase change in relation to time elapsed after application of treatments is of interest. If x and y are growth measurements on two different parts of the same organism, and Huxley's allometric growth law operates (i.e. there is a linear relation between log x and log y) it has sometimes been found that sudden changes in slope occur. Reeve and Huxley 1945 have discussed this situation with reference to changes in growth equilibrium in crustacea at sexual maturity. Skellam et al. [19591 also