Two-phase regression

Segmented regression, also known as piecewise regression or "broken-stick regression", is a method in regression analysis in which the independent variable is partitioned into intervals and a separate line segment is fit to each interval. Segmented regression analysis can also be performed on multivariate data by partitioning the various independent variables. Segmented regression is useful when the independent variables, clustered into different groups, exhibit different relationships between the variables in these regions. The boundaries between the segments are breakpoints.

Segmented linear regression is segmented regression whereby the relations in the intervals are obtained by linear regression.

Segmented linear regression with two segments separated by a breakpoint can be useful to quantify an abrupt change of the response function (Yr) of a varying influential factor (x). The breakpoint can be interpreted as a critical, safe, or threshold value beyond or below which (un)desired effects occur. The breakpoint can be important in decision making

The figures illustrate some of the results and regression types obtainable.

A segmented regression analysis is based on the presence of a set of ( y, x ) data, in which y is the dependent variable and x the independent variable.

The least squares method applied separately to each segment, by which the two regression lines are made to fit the data set as closely as possible while minimizing the sum of squares of the differences (SSD) between observed (y) and calculated (Yr) values of the dependent variable, results in the following two equations:

where:

The data may show many types or trends, see the figures.

The method also yields two correlation coefficients (R):

and

where:

and

In the determination of the most suitable trend, statistical tests must be performed to ensure that this trend is reliable (significant).

When no significant breakpoint can be detected, one must fall back on a regression without breakpoint.

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