Nonparametric Multivariate Inference on Shift Parameters

Rationale and Objectives

Materials and Methods

Our method uses estimating equations derived from an existing rank-based estimator of the area under the receiver operating characteristic curve (AUC). This existing AUC estimator is adjusted for stratification and for covariates. We use the adjusted AUC estimator to construct a family of tests by shifting manifest variables in one of the treatment groups by an arbitrary constant. The null hypothesis for these tests is that the AUC is half. We report the set of shift values for which the null hypothesis is not rejected as the resulting confidence region.

Results

Simulated data show performance consistent with the distributional approximations used by the proposed methodology. This methodology is applied to two examples. In the first example, the mean difference in PSA levels between advanced and nonadvanced prostate cancer patients is estimated, controlling for Gleason score. In the second example, to assess the degree to which age and baseline tumor size are prognostic factors for breast cancer recurrence, differences in age and tumor size between subjects with recurrent and nonrecurrent breast cancer, stratified on Tamoxifen treatment and adjusting for tumor grade, are estimated.

Conclusions

The proposed methodology provides a nonparametric method for a statistic measuring adjusted AUC to be used to estimate shift between two manifest variables.

Serum prostate-specific antigen (PSA) and biopsy Gleason score are prognostic factors frequently used for patients screened for prostate cancer . Prognostic factors are important in clinical practice because they “have a direct impact on therapeutic planning” .

Consider a study evaluating the prognostic value of prostate-specific antigen (PSA), in the presence of Gleason score, in differentiating between early and advanced prostate cancer . This data set features subjects divided into two groups (early versus advanced cancer), with one manifest variable (PSA), one covariate (Gleason score), and one stratification variable (hospital, taking three values). These data were analyzed using Bayesian regression techniques ; readers with a radiological or medical physics background might be particularly interested in these references. Our aim is to learn about the difference between median values for PSA for subjects in the two groups. We refer to such differences in location as a shift. This problem has been well studied in the case when the separate location estimates are approximately normally distributed. Our method applies in more general situations, in which normality assumptions are not needed. We use estimating equation techniques in conjunction with the statistic of Kawaguchi et al to provide an estimate of the shift in the manifest variable coinciding with group assignment. We generate a confidence interval for the shift in PSA scores between the two outcome groups, adjusting for Gleason score. Our method generalizes to the case of multiple manifest variables, and provides a confidence region for the vector of shift parameters for all of these manifest variables. In the case with a single manifest variable and without our covariate adjustment, this model has been previously used .

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Materials and methods

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Results

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Table 1

Univariate Confidence Intervals for Shift in Age and Tumor Size between Recurrent and Non-recurrent Subjects

Level Variable Lower Bound Upper Bound 0.900 Age −2.990 4.000 0.900 Size 0.400 0.899 0.950 Age −3.990 4.990 0.950 Size 0.301 0.900 0.975 Age −4.000 5.000 0.975 Size 0.301 0.900

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Discussion

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Acknowledgments

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Appendix

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(

)

−

)

⊤

−

(

)

−

)

≤

}

represents a confidence region for the offsets. Here χ2r,α χ

2 represents the 1−α 1

−

α quantile for the χ2r χ

2 distribution, and 1 is a vector with r components whose entries are all 1.

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Nonparametric Multivariate Inference on Shift Parameters

Rationale and Objectives

Materials and Methods

Results

Conclusions

Materials and methods

Results

Discussion

Acknowledgments

Appendix

References

Further Reading

A Brief History of Free-Response Receiver Operating Characteristic Paradigm Data Analysis

An Additive Selection of Markers to Improve Diagnostic Accuracy Based on a Discriminatory Measure

An Examination of Data Confidentiality and Disclosure Issues Related to Publication of Empirical ROC Curves