### Linear Regression Modelling

After all of the significant factors, and interactions have been identified in the factorial design, the data can be fit to a regression model

| (37) |

Where Y is the response variable, x_{k} are the independent variables, or factors, β_{k} are the unknown parameter
coefficients, and ϵ is the error term. Determining the β coefficients will allow for us to describe the relationship
between independent variables, and the response variable (dependent variable) through an approximate
function. Since there are multiple phases that compose the 2032Nb system, a set of regressions functions will
need to be determined to appropriately describe the system. After these functions have been approximated, a
model for optimizing the system can be drawn, either through statistical or linear programming methods.

Using linear regression models to fit the factorial design data, assumes that the data fits linearly with
the response variable. This assumption neglects any significant interactions that were uncovered
in the factorial design, as they would be regarded as polynomial terms. However, a linear model
may still be used by approximating these polynomial terms as new variables. For example if the
significant terms in the regression model were Si,Nb, and Si × Nb, the regression function would be
Y = β_{0} + β_{1}Si + β_{2}Nb + β_{3}NbSi + ϵ. If we let x_{3} = NbSi this modifies the equation from containing two
independent variables to incorporating three independent variables, and can now be considered a linear
function.

The linear coefficient can be solved for using the Least Squares Method described in the next section. The
regression function should then be tested for significance, and how well if fits with the original data. The easiest
way to compare the fit of the approximated function is to calculate the coefficient of multiple determination , or
R^{2} value. R^{2} is calculated as

| (38) |

The R^{2} value will be a fraction of how much of the model accounts for the variability in the original data. For
example if R^{2} = 0.95, 95% of the variability of the response data is accounted for in the regression model. There
are some short-comings of the R^{2} value as it keeps improving as more terms are added to the model. This is
compensated for in the adjusted R^{2} value, but for the purposes of this study only the regular R^{2} value will be
reported.

##### 1.2. Least Squares Method

The least squares method is an effective method in solving a system of linear equations, and is used in both regression modelling, and fitting experimental data to Gibbs energy models. If a set of linear equations are described as

| (39) |

Where x_{ij} is the ith observation of the k total independent variables x_{j}, y_{i} is the dependent variable of
observation i, β_{j} is the coefficient for the jth term, and ϵ is the error between the calculated ŷ_{i} and the
measured y_{i} values [26]. The best fits for the β_{j} coefficients are determined by the minimization of the least
squares function,

| (40) |

which can be expressed as

| (41) |

Rearranging for y_{i} a set of n equations can be solved for providing the best estimates for β_{j}.

### 2. References

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### 3.Glossary *

- constituent
- Element, or species, that occupies a specific sublattice of a specific phase. A phase can also
be considered as a constituent of the total system.. 5, 14
- end-member
- The final chemical formula of a stable or metastable phase whose sublattice(s) are
occupied by single constituents. For example M23C6 is an end member of (Cr,Ni,Fe,Nb)
_{23}C_{6}.. 8 - factor
- The independent variable of a factorial design. 23
- main effect
- How much the change in an individual factor effects the change in the response variable of
a factorial design.. 24
- replicate
- Independent repetition of a treatment in a factorial experiment. 26

- response variable
- The dependent variable of a factorial experiment, or a regression model.. 24
- ThermoCalc
- A computational thermodynamics program that can calculate equilibrium phase diagrams
for multicomponent systems, as well as Scheil simulations, and various thermodynamic properties
(C
_{p}, ΔH_{m}, ΔG_{m}etc...). 3 - treatment
- A specific level of a factor in a factorial design.. 24

### 4. Acronyms *

### 5. Nomenclature *

- β
_{j} - the effect of the ith level of factor ‘B’
- ϵ
_{ijk} - random error component
- ∀
- for all instances of ...
- ∈
- in a set ...
- μ
- Overall mean effect
- μ
_{i} - Chemical potential of component or end-member i
- ψ
_{k} - the effect of the ith level of factor ‘C’
- τ
_{i} - the effect of the ith level of factor ‘A’
- G
- total Gibbs energy; G = ∑
_{α}m^{α}·G_{m}^{α} - G
_{i}^{α} - partial Gibbs energy of component i in phase α; G
_{i}^{α}=_{T,P,Nj} - G
_{m}^{α} - integral molar Gibbs energy of a phase
- I
_{i} - constituent array of order i
- L
_{I} - interaction parameter of compound I
- m
^{α} - fraction of a phase
- N
_{i} - moles of component i
- R
- gas constant, 8.314Jmol
^{-1}K^{-1} - R
^{2} - coefficient of multiple determination
- S
_{m}^{α} - molar entropy of a phase
- T
- Temperature (K)
- x
_{i} - total mol fraction of component i; x
_{i}= ∑_{α}m^{α}· x_{i}^{α} - x
_{i}^{α} - mole fraction of component i in phase α