user Columbia and Cornell are two schools in the Ivy League directly offering MS in Business Analytics (MSBA). Further, UPenn offers a dual program within its MBA program with MS in Data Science. Other than these 3 universities, all other Ive League universities do not offer MSBA, even though Harvard offers a certificate (NOT degree) in … Read more
This tutorial discusses how dummy and contrast codings impact p-values in SPSS for linear regressions. Single Categorical Variable We can start with only one Y (numerical data, or continuous data) and one X (categorical data). We keep it simple to only have 4 observations. The cell means are (3+4)/2=3.5 vs. (5+6)/2=5.5, and the difference is … 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
This tutorial explains the difference between space and subspace. What is space? The most simple example of space is the two-dimensional space, \( \mathbb{R}^2 \). You can visualize it as the xy-coordinate plane. \( \mathbb{R}^2 \) is a vector space that contains all possible two-dimensional vectors \( \vec{v} = (x, y) \). Similarly, \( \mathbb{R}^3 … Read more
A linear subspace or vector subspace is a vector space that is a subset of some larger vector space. To be considered a linear subspace, a vector set needs to meet the following 3 requirements. For instance, if we say that \( V \) is a subset of \( \mathbb{R}^n \) and in order to … Read more
This tutorial provides examples to explain vectors and arrows in space (vector visualization). It includes 2 and 3-dimension vectors as well as vector addition and subtraction. Example 1: 2 dimension vectors \( \vec{V_1} =(3, 2 ) = \left[\begin{array}{ccc}3\\2 \end{array}\right]\) Note that, \( (3, 2 ) \) and \( \left[\begin{array} {ccc} 3\\ 2 \end{array} \right]\) are … 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