skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: A Note Comparing Component-Slope, Scheffé, and Cox Parameterizations of the Linear Mixture Experiment Model

Abstract

A mixture experiment involves combining two or more components in various proportions and collecting data on one or more responses. A linear mixture model may adequately represent the relationship between a response and mixture component proportions and be useful in screening the mixture components. The Scheffé and Cox parameterizations of the linear mixture model are commonly used for analyzing mixture experiment data. With the Scheffé parameterization, the fitted coefficient for a component is the predicted response at that pure component (i.e., single-component mixture). With the Cox parameterization, the fitted coefficient for a mixture component is the predicted difference in response at that pure component and at a pre-specified reference composition. This paper presents a new component-slope parameterization, in which the fitted coefficient for a mixture component is the predicted slope of the linear response surface along the direction determined by that pure component and at a pre-specified reference composition. The component-slope, Scheffé, and Cox parameterizations of the linear mixture model are compared and their advantages and disadvantages are discussed.

Authors:
Publication Date:
Research Org.:
Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
882377
Report Number(s):
PNNL-SA-45662
TRN: US200614%%16
DOE Contract Number:
AC05-76RL01830
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Applied Statistics, 33(4):397-403; Journal Volume: 33; Journal Issue: 4
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; MIXTURES; MATHEMATICAL MODELS; PARAMETRIC ANALYSIS; RESPONSE FUNCTIONS; Mixture component effects; Scheffé linear mixture model; Cox linear mixture model; Component-slope linear mixture model

Citation Formats

Piepel, Gregory F. A Note Comparing Component-Slope, Scheffé, and Cox Parameterizations of the Linear Mixture Experiment Model. United States: N. p., 2006. Web. doi:10.1080/02664760500449170.
Piepel, Gregory F. A Note Comparing Component-Slope, Scheffé, and Cox Parameterizations of the Linear Mixture Experiment Model. United States. doi:10.1080/02664760500449170.
Piepel, Gregory F. Mon . "A Note Comparing Component-Slope, Scheffé, and Cox Parameterizations of the Linear Mixture Experiment Model". United States. doi:10.1080/02664760500449170.
@article{osti_882377,
title = {A Note Comparing Component-Slope, Scheffé, and Cox Parameterizations of the Linear Mixture Experiment Model},
author = {Piepel, Gregory F.},
abstractNote = {A mixture experiment involves combining two or more components in various proportions and collecting data on one or more responses. A linear mixture model may adequately represent the relationship between a response and mixture component proportions and be useful in screening the mixture components. The Scheffé and Cox parameterizations of the linear mixture model are commonly used for analyzing mixture experiment data. With the Scheffé parameterization, the fitted coefficient for a component is the predicted response at that pure component (i.e., single-component mixture). With the Cox parameterization, the fitted coefficient for a mixture component is the predicted difference in response at that pure component and at a pre-specified reference composition. This paper presents a new component-slope parameterization, in which the fitted coefficient for a mixture component is the predicted slope of the linear response surface along the direction determined by that pure component and at a pre-specified reference composition. The component-slope, Scheffé, and Cox parameterizations of the linear mixture model are compared and their advantages and disadvantages are discussed.},
doi = {10.1080/02664760500449170},
journal = {Journal of Applied Statistics, 33(4):397-403},
number = 4,
volume = 33,
place = {United States},
year = {Mon May 01 00:00:00 EDT 2006},
month = {Mon May 01 00:00:00 EDT 2006}
}
  • A paper by Foglio Bonda et al. published previously in this journal (2016, Vol. 83, pp. 175–183) discussed the use of mixture experiment design and modeling methods to study how the proportions of three components in an extemporaneous oral suspension affected the mean diameter of drug particles (Z ave). The three components were itraconazole (ITZ), Tween 20 (TW20), and Methocel® E5 (E5). This commentary addresses some errors and other issues in the previous paper, and also discusses an improved model relating proportions of ITZ, TW20, and E5 to Z ave. The improved model contains six of the 10 terms inmore » the full-cubic mixture model, which were selected using a different cross-validation procedure than used in the previous paper. In conclusion, compared to the four-term model presented in the previous paper, the improved model fit the data better, had excellent cross-validation performance, and the predicted Z ave of a validation point was within model uncertainty of the measured value.« less
  • Cloud properties have been simulated with a new double-moment microphysics scheme under the framework of the single column version of NCAR CAM3. For comparisons, the same simulation was made with the standard single-moment microphysics scheme of CAM3. Results from both simulations were compared favorably with observations during the Tropical Warm Pool- International Cloud Experiment by US Department of Energy Atmospheric Radiation Program in terms of the temporal variation and vertical distribution of cloud fraction and cloud condensate. Major differences between the two simulations are in the magnitude and distribution of ice water content within the mixed-phase cloud during the monsoonmore » period, though the total frozen water (snow plus ice) content is similar. The ice mass content in the mixed-phase cloud from the new scheme is larger than that from the standard scheme, and extends 2 km further downward, which are closer to observations. The dependence of the frozen water mass fraction in total condensate on temperature from the new scheme is also closer to available observations. Outgoing longwave radiation (OLR) at the top of the atmosphere (TOA) from the simulation with the new scheme is in general larger than that with the standard scheme, while the surface downward longwave radiation is similar. Sensitivity tests suggest that different treatments of the ice effective radius contribute significantly to the difference in the TOA OLR in addition to cloud water path. The deep convection process affects both TOA OLR and surface downward longwave radiation. The over-frequently-triggered deep convention process in the model is not the only mechanism for the excess middle and high level clouds. Further evaluation especially for ice cloud properties based on in-situ data is needed.« less
  • Here, cloud properties have been simulated with a new double-moment microphysics scheme under the framework of the single-column version of NCAR Community Atmospheric Model version 3 (CAM3). For comparison, the same simulation was made with the standard single-moment microphysics scheme of CAM3. Results from both simulations compared favorably with observations during the Tropical Warm Pool–International Cloud Experiment by the U.S. Department of Energy Atmospheric Radiation Measurement Program in terms of the temporal variation and vertical distribution of cloud fraction and cloud condensate. Major differences between the two simulations are in the magnitude and distribution of ice water content within themore » mixed-phase cloud during the monsoon period, though the total frozen water (snow plus ice) contents are similar. The ice mass content in the mixed-phase cloud from the new scheme is larger than that from the standard scheme, and ice water content extends 2 km further downward, which is in better agreement with observations. The dependence of the frozen water mass fraction on temperature from the new scheme is also in better agreement with available observations. Outgoing longwave radiation (OLR) at the top of the atmosphere (TOA) from the simulation with the new scheme is, in general, larger than that with the standard scheme, while the surface downward longwave radiation is similar. Sensitivity tests suggest that different treatments of the ice crystal effective radius contribute significantly to the difference in the calculations of TOA OLR, in addition to cloud water path. Numerical experiments show that cloud properties in the new scheme can respond reasonably to changes in the concentration of aerosols and emphasize the importance of correctly simulating aerosol effects in climate models for aerosol-cloud interactions. Further evaluation, especially for ice cloud properties based on in-situ data, is needed.« less
  • No abstract prepared.