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(((((<<<8t<#S#U#U#U#U#U#U#$%.(y#!(y#((4#((S#S#!h" <"?##0#"(j( "(("y#y##(: Benchmark dose calculation for ordered categorical responses with multiple endpoints
ChuChih Chen
Division of Biostatistics and Bioinformatics
Institute of Population Health Sciences
National Health Research Institutes
Taiwan
The use of benchmark dose (BMD) calculations for dichotomous or continuous responses is well established in the risk assessment of cancer and noncancer endpoints. In some cases, responses to exposure are categorized in terms of ordinal severity effects such as none, mild, adverse, and severe. Such responses can be assessed using categorical regression (CATREG) analysis. However, while CATREG has been employed to compare the benchmark approach and the noadverseeffectlevel (NOAEL) approach in determining a reference dose, the utility of CATREG for risk assessment remains unclear. In our previous project results, we propose a CATREG model to extend the BMD approach to ordered categorical responses by modeling severity levels as censored interval limits of a standard normal distribution. The BMD is calculated as a weighted average of the BMDs obtained at dichotomous cutoffs for each adverse severity level above the critical effect, with the weights being proportional to the reciprocal of the expected loss at the cutoff under the normal probability model. This approach provides a link between the current BMD procedures for dichotomous and continuous data.
In this paper, we extend the previous results to study further the case of benchmark dose calculation for ordered responses with multiple endpoints. The common approach for benchmark dose calculation in the case of multiple endpoints is to select the most critical endpoint or the response that is most sensitive. However, the most critical endpoint may not be available for all toxic chemicals, and the most sensitive endpoint may not necessarily representative of the overall toxic effects. Furthermore, the responses of different endpoints may also vary corresponding to different ranges of exposure dose. Therefore, a more appropriate approach is to propose a dose calculation method that can accommodate multiple endpoints simultaneously and take into account their correlations as well. To be able to compare the health effects of multiple endpoints, they need to be standardized to be comparable, which will result in ordered categorical data with different severity levels for each of the endpoints before being able to be integrated for benchmark dose derivation. For benchmark dose calculation with multiple endpoints, BudtzJrgensen (2007) proposed a structured equation model approach by representing the multiple endpoints by their factor loadings of latent variable using principal components analysis (PCA). He then derived the benchmark dose corresponding to the latent variable of the principal factor. In this study, similar to BudtzJrgensen's approach, we propose benchmark dose calculation to the converted ordered categorical data of the multiple endpoints. Buntine and Pettu's multinomial PCA is applied for the derivation. The latent variable of the principal component of the categorical data of the multiple endpoints is distributed as a Dirichlet distribution, with the ordered categorical data expressed as the latent variable multiplied by the factor loading matrix. The method of benchmark dose calculation for ordered categorical responses is then applied to the resultant latent variable.
Keywords: Dirichlet distribution; factor loading; latent variable; tolerable daily intake; principal component analysis
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