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Transformations to additivity in measurement error models

R S Eckert1, R J Carroll, N Wang

  • 1Lilly Research Laboratories, Eli Lilly and Company, Indianapolis, Indiana 46285, USA.

Biometrics
|March 1, 1997
PubMed
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This study introduces a new measurement error model, h(W) = h(X) + U, where a monotone transformation h(.) corrects for non-additive errors. This approach improves modeling when the true covariate X is unobservable, using a substitute W.

Area of Science:

  • Statistics
  • Biostatistics
  • Econometrics

Background:

  • Modeling relationships between a response Y and covariate X is common.
  • Direct observation of covariate X can be difficult, expensive, or impossible.
  • Substitute variables W are often used, typically assuming additive measurement error (W = X + U).

Purpose of the Study:

  • To propose a new model, h(W) = h(X) + U, for situations where additive measurement error is unreasonable.
  • To introduce the concept that measurement error may be additive in a transformed scale.
  • To present methods for selecting an appropriate monotone transformation function h(.) from a family H.

Main Methods:

  • Proposing a new model: h(W) = h(X) + U, where h(.) is a monotone transformation.
  • Developing two strategies for selecting the transformation family H:

Related Experiment Videos

  • - Minimizing correlation between mean and standard deviation of replicated W's.
  • - Ensuring transformed errors (U's) fit a specified distribution.
  • Utilizing parametric power transformations and cubic spline families for H.

    • Main Results:

    • Demonstrated the application of the new model through several data examples.
    • Illustrated how the proposed transformation methods can be practically implemented.
    • Showcased the flexibility of the model in handling various data characteristics.

    Conclusions:

    • The proposed model h(W) = h(X) + U offers a more flexible alternative to traditional additive measurement error models.
    • The selection methods for transformation families provide practical tools for data analysis.
    • The approach is effective in improving the modeling of relationships when direct covariate measurement is not feasible.