Confidence Intervals for the Partial Correlation Coefficient under ML Method 1

In large samples, we can apply a typical normal theory-based confidence interval (CI) for pyx|c:

Given the small-sample limitations of Wald-type CIs and the inherent lack of variance stabilization, Lyles et al. (2001b) suggested profile-likelihood-based intervals for the crude correlation pyx and gave empirical evidence suggesting improvements in overall coverage and coverage balance relative to the CIs in Equation 6.8. However, a computationally faster and simpler approach would be to apply Fisher's z-transformation here, despite the fact that left censoring in Y and X precludes the argument that it fully stabilizes the variance.

This leads to the following alternative CI, based on back-transforming the limits of a standard Wald interval for the parameter pyx|c = 0.5 ln^-+py^:

where

Although SE (byx|c) could be obtained by reparameterizing the likelihood based on the Type 1-4 contributions directly in terms of pyx|c, a virtually identical approach is to base it on a familiar delta method-based approximation:

In an effort to further improve upon the CI based on Equations 6.9 and 6.-0 in the crude correlation case, Li et al. (2005) proposed an alternate pivotal quantity by further refining a Taylor series expansion for pyx - pyx. Applying this refinement in the partial correlation case considered here, one would use Equation 6.9 to back-transform the limits of the following alternative CI for pyx|c:

where

In Section 6.4, we explore the coverage and coverage balance properties of CIs based on Equations 6.8 through 6.-- empirically using simulated data.

 
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