Logistic Regression (Mixture)

Introduction

The process of separating mixtures of compounds is called chromatography. Ultra high performance liquid chromatography (UHPLC) is the gold standard for commercially available chromatography techniques, but it uses very expensive equipment to provide ultra high pressure.

You are a member of a group of researchers who are working to create a polymeric column that will have a pore size and skeleton thickness to achieve the same efficiency as UHPLC techniques but with lower pressure and therefore much lower cost.

Problems:
  • A formulation does not always create a polymer that is completely solid, which we will call a “homogenous” column.
  • The pores are note always interconnected, meaning the compound to be separated will not flow through the column.
First Step:
  • Design and experiment to model how the blending properties of water, monomer and surfactant effect the probability of creating an acceptable column, meaning homogenous and capable of flow.

Design

The emulsion used to create the column is a mixutre of a water, a monomer, surfactant and an initiator.

  • The monomer and water are two immiscible fluids, and the surfactant is a chemical agent that reduces the surface tension of the two fluids.
  • The monomer is what eventually becomes the solid structure of the monolith, and the water droplets in the mixture are replaced during the polymerization process by pores. The idea is to create a mixture where the water is well dispersed throughout the monomer in tiny droplets.

The researchers determined the constraints on the mixture components based on literature and experience, but are still somewhat arbitrary:

  Component Lower Bound Upper Bound
A Water 70 86
B Monomer 10 25
C Surfactant 2 8
D Initiator* 0.02 0.25
E Salt 0.8 0.8
    Total = 100 wt %

*Initiator varies from 0.2% of monomer to 1.0% of monomer; i.e. 0.002 \(\leq\) D/B \(\leq\) 0.01

Having a binary response changes how we build the design:

  • Estimating a model for a response with two complementary outcomes requires a larger design than is required for modeling a continuous numeric response.
  • Logistic reqgression depends on having independent input factors. The collinearity in this constrained mixture requires having a larger design.
  • For a binary response it is good to have a space filling design.
  • Replicates aren’t necessary when the response is binary.

Because of the binary response we need to change the optimal design defaults provided by Design-Expert.

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Defaults

We will:

  • Add “Additional model points” to overcome collinearity.
  • Add “Lack-of-fit points” to fill space.
  • Reduce “Replicate points” to zero.

Build the Design

1. Choose a 5-component Optimal Mixture design and enter the components and levels:

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Entering components and levels

2. Click on Edit constraints… and enter the constraint for “Initiator/Monomer” ratio:

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Entering the constraint

Click OK and then Next >>

3. Enter 10 Additional model points, 40 Lack-of-fit points, and 0 Replicate points:

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Specifying the design

Click Next

  1. Enter two responses:
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Entering responses

  1. Click Finish

Loading the Data

Rather than entering response data into this new design, you can load a design with pre-existing data by clicking on the Help, Tutorial Data menu and selecting Polymer Column.

Analyze the Design

Both homogeneous and flow are binary responses: homogeneous (1) or not (0); flow (1) or none (0). Analyze using logistic regression.

Note

Try choosing a quadratic model and reducing it. Use backward selection with the AICc criterion.

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homogeneous - ANOVA

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homogeneous - Diagnostics

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homogeneous - Model Graphs

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flow - ANOVA

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flow- Diagnostics

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flow - Model Graphs

Optimize

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Maximize both homogeneous and flow