Model Components

Model terms

Main effect

Term that contains only 1 variable, with power of 1, i.e., is neither an interaction term nor a higher order term.

A model with only the main effect terms is called a first-order model

Interaction

A term that contains two or more variables.

For a model $E(y)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2$,

• $E(y)$ increases by ${\beta }_1+{\beta }_3{x}_2$ for every 1 unit increase in $x_2$
• $E(y)$ increases by ${\beta }_1+{\beta }_3{x}_1$ for every 1 unit increase in $x_1$

Higher-order

A term with a non-linear variable, e.g., a quadratic or qubic variable.

Model with qualitative variables

Data with categorical variables can be modelled by forming dummy variables. For a categorical variable with $n$ levels, there can be $n-1$ dummy variables formed.

Dummy variable from 2 levels

Suppose we want to insert a variable $x_1$ of “gender” into the model, with possible categories “Male” and “Female”. This process is called coding, which is illustrated in the following:

$$x_1\{\begin{array}{ll}1,& \text{Felame}\\ 0,& \text{Male}\end{array}$$

Values coded to 0 is called the “baseline”. In this case, the baseline category is “Male”

Dummy variable from 3 levels

Suppose a categorical variable with levels “Physics”, “Math”, “Biology”. There are $n-1=2$ dummy variables:

$$x_1\{\begin{array}{ll}1,& \text{Math}\\ 0,& \text{Otherwise}\end{array}{x}_2\{\begin{array}{ll}1,& \text{Biology}\\ 0,& \text{Otherwise}\end{array}$$

In this case, the baseline is “Physics”

Others

Complete second order model

Model that contains all the possible terms at the 2nd degree

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2+{\beta }_4{x}_1^2+{\beta }_5{x}_2^2$$

Multiplicative (log) model

$$\ln (y)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\cdots +{\beta }_k{x}_k+ϵ$$

$(e^{{\beta }_i}-1)\times 100\%$ reprsents the percentage of change in $y$ for every 1 unit increase of $x_i$