Some Hypotheses Concerning Two Phase Regression Lines
Some Hypotheses Concerning Two Phase Regression Lines
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作者：
P. Sprent
生长测量y对时间x的回归通常可以用两条相交的直线来合理地表示，当x取低于某个固定但通常未知的值时，一条直线是合适的，而当x取高于某个固定但通常未知的值时，另一条直线是合适的。这样的回归在这里被称为两阶段回归，阶段的交叉点被称为转换点，并且其发生的x的值被称为转换值。可能发生这种退化的情况包括疾病的发作导致生长速度降低;应用具有立即刺激或抑制作用的治疗;发生极热或极冷的天气或外部条件的一些其他变化;生物体的物理损伤。Garner和哈蒙德[1938]在一项关于桃接穗与李砧木的亲和性的研究中注意到，桃品种黑尔的早期在亲和和不亲和砧木-接穗结合上以恒定但不同的速率发育，直到一定日期。在此之后，相容情况下的生长速率继续以一个新的恒定速率增长，而在不相容情况下，所有的生长都停止了。因此，对于相容性愈合，存在典型的两阶段回归，而对于不相容性愈合，发生了相当特殊的情况，其中第二阶段的斜率为零。在第4节中给出了另一个例子，其中与应用治疗后经过的时间相关的相变日期是有意义的。如果x和y是同一生物体两个不同部分的生长测量值，并且赫胥黎异速生长定律起作用（即log x和log y之间存在线性关系），则有时会发现斜率发生突然变化。Reeve和Huxley [1945]讨论了甲壳纲动物在性成熟时生长平衡的变化。Ellam等人[1959年]还
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