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
Definition of Quartile A quartile is a statistic describing how a set of data points are divided into 4 groups. Quartiles split a set of data by using 3 points: the lower quartile (Q1), the median (Q2), and the upper quartile (Q3). Together with the minimum and maximum values, 3 quartiles split the data set … Read more
Descriptive statistics aim to summarize the characteristics of a given data set. In contrast, inferential statistics aim to use a sample of data to draw inferences about the whole population (i.e., hypothesis testing). Types of Descriptive Statistics 1. Measures of Central Tendency Central tendency is used to describe where the center of a dataset is located. Mean, … Read more
A population is the entire group of individuals about whom you want to draw conclusions. In contrast, a sample is the subset of the same entire group. Example 1 of sample and population You would like to study if students like online courses at your university. Suppose your university has 10K students; thus, these 10K students … Read more
This tutorial shows the formula for population variance and the steps for calculating population variance by hand. Formula Population variance is the measure of the variability of a population. The following is the formula for population variance. where, Population vs. Sample Data The following is the population of a set of data. It has 11 … Read more
Sample variance is the measure of the variability in a given sample. A sample is a set of observations that are a subset of a population. Sample Variance Formula The following is the formula for sample variance. where, Data Example The following is a sample of 6 students with math scores. Calculating sample variance by … Read more