# 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}

}