This tutorial includes the explanation of what a linear mixed model is, how to structure its statistical model, data example, as well as steps for linear mixed models in SPSS. Definition of Linear Mixed Models Linear mixed models (LMMs) are statistical models used to analyze data that have both fixed and random effects. They are … Read more
This tutorial explains what data type (including numerical data and categorical data) is and how to summarize different types of data. Data Type Broadly speaking, data can be categorized into two types: categorical and numerical. Categorical data refers to variables that have a finite number of categories or groups. Examples of categorical data include gender … Read more
This tutorial explains why and how linear regression can be viewed as an orthogonal projection on 2 and 3-dimensional spaces. Projection with 2 Dimensions Suppose that both X0 and Y have 2 dimensions (e.g., 2 observations from 2 participants). It is worth pointing out that, when talking about dimensions here, we refer to the number … Read more
This tutorial explains how mean can be viewed as an orthogonal projection onto a subspace defined by the span of an all 1’s vector (i.e., basis vector). Suppose that \( \vec{y} \in \mathbb{R}^n \) and \( L \subset \mathbb{R}^n\) is the span defined by the space of vector \( \vec{x} \), namely, \( \vec{x}=\left[\begin{array}{ccc}1\\1\\ …\\ … Read more
This tutorial explains what an orthogonal projection is in linear algebra. Further, it provides proof that the difference between a vector and a subspace is orthogonal to that subspace. Let’s define two vectors, \(\vec{X} \) and \(\vec{Y} \), and we want to find the shortest distance between \(\vec{Y} \) and the subspace defined by the … Read more
Two Orthogonal Vectors Definition: Two vectors are orthogonal if they are perpendicular to each other. That is, the dot product of the two vectors is zero. The following is an example of two orthonormal vectors. \( \vec{V_1} =\left[\begin{array}{ccc}1\\0\\-1\end{array}\right]\), \( \vec{V_2} =\left[\begin{array}{ccc}1\\3\\1\end{array}\right] \) That is, \( (1 \times 1) + (0 \times 3) +(-1 \times 1) … Read more
This tutorial focuses on interaction between a categorial variable and a continuous variable in linear regression. Note that, in this tutorial, we limit the the categorical variable to be 2 levels. (For a categrocial variable with 3 levels, please refer to my another tutotrial on interaction and coding in linear regression .) Coding Note In … Read more
This tutorial explains the differences between dummy coding and contrast coding in linear regression using R code examples. It is worth pointing out that, this tutorial focuses on the categorical independent variable has 3 levels. Short Note Note that, in R, the default reference group in dummy coding uses the first item in an alphabetical … Read more
You can change the reference level in dummy coding in R by using the following R code. contr.treatment(total_levels, base = Number_reference_level) Step 1: Prepare Data The following R code generates a sample data. X Y 1 1 -0.56047565 2 2 -0.23017749 3 3 1.55870831 4 1 0.07050839 5 2 0.12928774 6 3 1.71506499 7 1 … Read more
“Dummy” or “treatment” coding is to create dichotomous variables where each level of the categorical variable is contrasted to a specified reference level. Basic Syntax of Dummy and Contrast Coding 1. Dummy Coding The following is the syntax to do dummy coding in R. contr.treatment( number_of_level_of_X ) 2 3 1 0 0 2 1 0 3 … Read more