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Adjusted Confidence Intervals for a Bounded Parameter

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Abstract

It is well known that the regular likelihood ratio test of a bounded parameter is not valid if the boundary value is being tested. This is the case for testing the null value of a scalar variance component. Although an adjusted test of variance component has been suggested to account for the effect of its lower bound of zero, no adjustment of its interval estimate has ever been proposed. If left unadjusted, the confidence interval of the variance may still contain zero when the adjusted test rejects the null hypothesis of a zero variance, leading to conflicting conclusions. In this research, we propose two ways to adjust the confidence interval of a parameter subject to a lower bound, one based on the Wald test and the other on the likelihood ratio test. Both are compatible to the adjusted test and parametrization-invariant. A simulation study and two examples are given in the framework of ACDE models in twin studies.

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Notes

  1. All upper and lower limits of CIs in this article are non-inclusive except for two cases: (1) a zero lower limit in both adjusted CIs and (2) a lower limit in Wald test adjusted CI that takes the value of the middle point between the observed and the boundary values.

  2. This assumption is satisfied for most models as long as θ0 is not an natural boundary and no nuisance parameter is on or close to their boundaries. See "Discussion" for the issues of a natural boundary and boundary nuisance parameters.

  3. The “=” sign in the constraint can be replaced by “ ≤ ” if the constrained optimization algorithm requires that the starting value satisfy the constraint. This is also the case for constraints in Eqs. 10 and 11

  4. i.e. the probability that the true value is not covered by the CI.

  5. Non-convergent replications were handled by manually adjusting the starting values until convergence was reached.

  6. See “Discussion” section for the distinction between natural and attainable boundaries.

  7. Similarly, a regular CI for an unbounded parameter may not be valid when another parameter is close to its boundary.

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Acknowledgments

This research is supported by the National Institute of Drug Abuse training grant R25DA026119 to the second author. We thank Gregory Carey, Sophie van der Sluis and Conor Dolan for their helpful comments that have led to the improvement of the style of this article.

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Authors and Affiliations

  1. Department of Psychology, Boston College, 300 McGuinn, 140 Commonwealth Avenue, Chestnut Hil, MA, 02467, USA

    Hao Wu

  2. Department of Psychiatry and Virginia Institute for Psychiatric and Behavior Genetics, Virginia Commonwealth University, E. 800 Leigh St, Richmond, VA, 23219, USA

    Michael C. Neale

Authors
  1. Hao Wu
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  2. Michael C. Neale
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Correspondence to Hao Wu.

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Edited by Gitta Lubke.

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Wu, H., Neale, M.C. Adjusted Confidence Intervals for a Bounded Parameter. Behav Genet 42, 886–898 (2012). https://doi.org/10.1007/s10519-012-9560-z

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  • DOI: https://doi.org/10.1007/s10519-012-9560-z

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