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
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
This tutorial shows how you can plot multiple t-distribution bell-shaped curves in R. In short, it is a combination of dt(), plot(), and lines(). Example 1 of plotting multiple t-distribution bell-shaped curves The following is the R code to plot two t-distribution bell-shaped curves, with degree freedom of 1 and 5. Example 2 of plotting … Read more
This tutorial compares t-distribution and normal distribution by explaining the similarities and connections between t-distribution and normal distribution. Similarities between t-distribution and normal distribution There are a few similarities between t-distribution and normal distribution. The following figure shows the t-distribution density function curve and the standard normal curve. As we can see, as the sample … Read more
This tutorial shows the steps of simulating a dataset for logistic regression R. Logistic regression is based on the following link function. In particular, the following are the steps for simulating a dataset for logistic regression in R. Step 1: Generate Xs Suppose that we have 2 Xs in the logistic regression, and the following … Read more
The 2 popular Python packages for hypothesis testing are scipy.stats and statsmodels. The following includes a brief introduction to each of them, along with simple examples. Introduction of Scipy.stats Based on scipy.stats official webpage, the module contains a large number of probability distributions, summary and frequency statistics, correlation functions and statistical tests, and more. Example … Read more