What is the broken stick model?

The broken stick model describes a set of individual curves by a linear mixed model using second-order linear B-splines. The model can be used to

  • smooth growth curves by a series of connected straight lines;
  • align irregularly observed curves to a common age grid;
  • create synthetic curves at a user-specified set of break ages;
  • estimate the time-to-time correlation matrix;
  • predict future observations.

The user specifies a set of break ages at which the straight lines connect. Each individual obtains an estimate at each break age, so the set of estimates of the individual form a smoothed version of the observed trajectory.

What are the main model assumptions?

The main assumptions of the broken stick model are:

  • The development between the break ages follows a straight line;
  • Broken stick estimates follow a common multivariate normal distribution;

In order to conform to the assumption of multivariate normality, the user may fit the broken stick model on suitably transformed data that yield the standard normal (\(Z\)) scale.

Why should I want to use the broken stick model?

Three unique features of the broken stick model are:

  • Modular: Issues related to nonlinearities of the growth curves in the observed scale can be treated separately, i.e., outside the broken stick model;
  • Local: A given data point will contribute only to the estimates corresponding to the closest break ages;
  • Exportable: The broken stick model can be exported and reused for prediction for new data in alternative computing environments.

What is in the package?

The brokenstick package contains functions to fit, predict and plot data. See the reference page for an overview.


Development of the brokenstick package was kindly supported by the Healthy Birth Growth and Development knowledge integration (HBGDki) program of the Bill & Melinda Gates Foundation.

Further reading

  • Main functions
    • Plot trajectories
    • Orginal scale and \(Z\)-score scale
    • 1-line model
    • 2-line broken stick model
    • 9-line broken stick model
    • Prediction
    • Subject-level analysis
  • Broken Stick Model for Irregular Longitudinal Data
    • Irregular observation times
    • Literature overview
    • Definition of the model
    • Interpretation of the model
    • Estimation by lmer and kr methods
    • Software overview
    • brokenstick() for model fitting
    • predict() for trajectory plotting
    • Conversion back and forth to the \(Z\)-score scale
    • Predict growth curve of new subjects
    • Assess the quality of the model
    • Knot placement strategies
    • Critical periods
    • Time-to-time correlations
    • Profile analysis
    • Curve interpolation
    • Multiple imputation
    • Curve matching
    • Discussion
  • Perfect model
    • Properties of the perfect model
    • Estimating time-to-time correlations
  • Help for old friends
    • Properties of the perfect model
    • Estimating time-to-time correlations