Segmented 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, two segments

1st limb horizontal

1st limb sloping up

1st limb sloping down

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:

• Yr = A₁.x + K₁     for x < BP (breakpoint)
• Yr = A₂.x + K₂     for x > BP (breakpoint)

where:

Yr is the expected (predicted) value of y for a certain value of x;

A₁ and A₂ are regression coefficients (indicating the slope of the line segments);

K₁ and K₂ are regression constants (indicating the intercept at the y-axis).

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

The method also yields two correlation coefficients (R):

• $R_1 ^ 2 = 1 - \frac{\sum (y - Y_r) ^ 2 }{ \sum (y - Y_{a1})^2}$     for x < BP (breakpoint)

and

• $R_2 ^ 2 = 1 - \frac{\sum (y - Y_r) ^ 2 }{ \sum (y - Y_{a2})^2}$     for x > BP (breakpoint)

where:

$\sum (y - Y_r) ^2$ is the minimized SSD per segment

and

Y_a1 and Y_a2 are the average values of y in the respective segments.

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.

Example

File:MUSTARD.JPG

Segmented linear regression, type 3b

For the blue figure at the right that gives the relation between yield of mustard (Yr = Ym, t/ha) and soil salinity (x = Ss, expressed as electric conductivity of the soil solution EC in dS/m) it is found that:

BP = 4.93, A₁ = 0, K₁ = 1.74, A₂ = −0.129, K₂ = 2.38, R₁² = 0.0035 (insignificant), R₂² = 0.395 (significant) and:

• Ym = 1.74 t/ha                        for Ss < 4.93 (breakpoint)
• Ym = −0.129 Ss + 2.38 t/ha     for Ss > 4.93 (breakpoint)

indicating that soil salinities < 4.93 dS/m are safe and soil salinities > 4.93 dS/m reduce the yield @ 0.129 t/ha per unit increase of soil salinity.

The figure also shows confidence intervals and uncertainty as elaborated hereunder.

Test procedures

Example time series, type 5

The following statistical tests are used to determine the type of trend:

1. significance of the breakpoint (BP) by expressing BP as a function of regression coefficients A₁ and A₂ and the means Y₁ and Y₂ of the y-data and the means X₁ and X₂ of the x data (left and right of BP), using the laws of propagation of errors in additions and multiplications to compute the standard error (SE) of BP, and applying Student's t-test
2. significance of A₁ and A₂ applying Student's t-distribution and the standard error SE of A₁ and A₂
3. significance of the difference of A₁ and A₂ applying Student's t-distribution using the SE of their difference.
4. significance of the difference of Y₁ and Y₂ applying Student's t-distribution using the SE of their difference.

In addition, use is made of the correlation coefficient of all data (Ra), the coefficient of determination or coefficient of explanation, confidence intervals of the regression functions, and Anova analysis.

The coefficient of determination for all data (Cd), that is to be maximized under the conditions set by the significance tests, is found from:

• $C_d=1-{\sum (y-Y_r)^2\over\sum (y-Y_a)^2}$

where Yr is the expected (predicted) value of y according to the former regression equations and Ya is the average of all y values.

The Cd coefficient ranges between 0 (no explanation at all) to 1 (full explanation, perfect match).
In a pure, unsegmented, linear regression, the values of Cd and Ra² are equal. In a segmented regression, Cd needs to be significantly larger than Ra² to justify the segmentation.

The optimal value of the breakpoint may be found such that the Cd coefficient is maximum.

See also

• Chow test
• Simple regression
• Linear regression
• Ordinary least squares
• Multivariate adaptive regression splines
• Local regression
• Regression discontinuity design
• SegReg (software) for segmented regression