Question

Question: Glass as a waste encapsulant. Because glass is not subject to radiation damage, encapsulation of waste in glass is considered to be one of the most promising solutions to the problem of low-level nuclear waste in the environment. However, chemical reactions may weaken the glass. This concern led to a study undertaken jointly by the Department of Materials Science and Engineering at the University of Florida and the U.S. Department of Energy to assess the utility of glass as a waste encapsulant. Corrosive chemical solutions (called corrosion baths) were prepared and applied directly to glass samples containing one of three types of waste (TDS-3A, FE, and AL); the chemical reactions were observed over time. A few of the key variables measured were

y = Amount of silicon (in parts per million) found in solution at end of experiment. (This is both a measure of the degree of breakdown in the glass and a proxy for the amount of radioactive species released into the environment.)

x₁ = Temperature (°C) of the corrosion bath

x₂ = 1 if waste type TDS-3A, 0 if not

x₃ = 1 if waste type FE, 0 if not

(Waste type AL is the base level.) Suppose we want to model amount y of silicon as a function of temperature (x₁) and type of waste (x₂, x₃).

a. Write a model that proposes parallel straight-line relationships between amount of silicon and temperature, one line for each of the three waste types.

b. Add terms for the interaction between temperature and waste type to the model of part a.

c. Refer to the model of part b. For each waste type, give the slope of the line relating amount of silicon to temperature.

e. Explain how you could test for the presence of temperature–waste type interaction.

Short answer

Answer

a. A model that proposes parallel straight-line relationships between amount of silicon and temperature can be written as$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3$.

b. A model that proposes relationships between amount of silicon and temperature and waste types with interaction between temperature and waste types can be written as $y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_1{x}_2+{\beta }_5{x}_1{x}_3$.

c. For AL waste type, the slope of the line will be β₁. For TSA-3A waste type, the slope of the line will be localid="1651563388536" $({\beta }_1+{\beta }_4).$For FE waste type, the slope of the line will be$({\beta }_1+{\beta }_5).$.

d. The presence of temperature-waste type interaction can be tested by doing hypothesis testing on β parameters indicating interaction terms.

Step-by-step solution

Model 

A model that proposes parallel straight-line relationships between amount of silicon and temperature can be indicated by a model where there is no interaction amongst the variables in the model.

Mathematically, it can be written as $y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3$.

Interaction model

A model that proposes relationships between amount of silicon and temperature and waste types with interaction between temperature and waste types can be written as$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_1{x}_2+{\beta }_5{x}_1{x}_3$.

Slope of the line for each waste type

For the three waste types; TDS-3A, FE, and AL, two variables are introduced in the model x₂ for TSA-3A and x₃ for FE. Therefore, Al indicates the base levels.

For AL waste type, the slope of the line will be

$$\begin{gathered} y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_1{x}_2+{\beta }_5{x}_1{x}_3 \\ y={\beta }_0+{\beta }_1{x}_1+{\beta }_2\left(0\right)+{\beta }_3\left(0\right)+{\beta }_4{x}_1\left(0\right)+{\beta }_5\left(0\right) \\ y={\beta }_0+{\beta }_1{x}_1 \end{gathered}$$

The slope of the line is β₁.

For TSA-3A waste type, the slope of the line will be

role="math" localid="1651554382496" $$\begin{gathered} y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_1{x}_2+{\beta }_5{x}_1{x}_3 \\ y={\beta }_0+{\beta }_1{x}_1+{\beta }_2\left(1\right)+{\beta }_3\left(0\right)+{\beta }_4{x}_1\left(0\right)+{\beta }_5{x}_1\left(0\right)for{x}_2=1,and{x}_3=0 \\ y=({\beta }_0+{\beta }_2)+({\beta }_1+{\beta }_4)x_1 \end{gathered}$$

$Theslopeofthelineis({\beta }_1+{\beta }_4).$

For FE waste type, the slope of the line will be

$$\begin{gathered} y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_1{x}_2+{\beta }_5{x}_1{x}_3 \\ y={\beta }_0+{\beta }_1{x}_1+{\beta }_2\left(1\right)+{\beta }_3\left(0\right)+{\beta }_4{x}_1\left(0\right)+{\beta }_5{x}_1\left(1\right)for{x}_2=0,and{x}_3=1 \\ y=({\beta }_0+{\beta }_3)+({\beta }_1+{\beta }_5)x_1 \end{gathered}$$

$Theslopeofthelineis({\beta }_1+{\beta }_5).$

Hypotheses testing

The presence of temperature-waste type interaction can be tested by doing hypothesis testing on β parameters indicating interaction terms.