Segmented regression

Key takeaways

Segmented regression belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Segmented regression to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Segmented regression from memory before moving on to harder problems.

Reference excerpt

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

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 − ∑ ( y − Y r ) 2 ∑ ( y − Y a 1 ) 2 {\displaystyle R_{1}^{2}=1-{\frac {\sum (y-Y_{r})^{2}}{\sum (y-Y_{a1})^{2}}}} for x < BP (breakpoint) and

R 2 2 = 1 − ∑ ( y − Y r ) 2 ∑ ( y − Y a 2 ) 2 {\displaystyle R_{2}^{2}=1-{\frac {\sum (y-Y_{r})^{2}}{\sum (y-Y_{a2})^{2}}}} for x > BP (breakpoint) where:

∑ ( y − Y r ) 2 {\displaystyle \sum (y-Y_{r})^{2}} is the minimized SSD per segment and

Ya1 and Ya2 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

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

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

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 significance of A1 and A2 applying Student's t-distribution and the standard error SE of A1 and A2 significance of the difference of A1 and A2 applying Student's t-distribution using the SE of their difference. significance of the difference of Y1 and Y2 applying Student's t-distribution using the SE of their difference. A more formal statistical approach to test for the existence of a breakpoint, is via the pseudo score test which does not require estimation of the segmented line. 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:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Segmented regression

Start with the simplest possible case. Write down what Segmented regression claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Segmented regression before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Segmented regression ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Segmented regression

In research: Segmented regression appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Segmented regression in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Segmented regression is common in secondary-school and first-year university syllabi. It links to neighbouring topics Regression models, so understanding it makes those chapters shorter.
In everyday life: Look for Segmented regression outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Segmented regression in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Segmented regression means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Segmented regression out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Segmented regression in simple terms?

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 multiva…

Why does Segmented regression matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Segmented regression?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Segmented regression.