Introduction to Basic Regression Analysis for Beginners

Regression analysis predicts a dependent variable based on an independent variable using the method of least squares.

### Introduction to Basic Regression Analysis for Non-Statistics Majors
* 📊 Regression analysis is often referred to as a "**statistical flower**," highlighting its significance in data analysis, yet many still lack a **full understanding** of it.
* 🔍 The term "**regression**" implies a return to something, and its primary purpose is to **predict the dependent variable** based on independent variables from given data.
* 📈 Simple regression analysis involves **one independent variable**, while multiple regression analysis includes **two or more independent variables**, but there is always **one dependent variable**.
* 💵 For example, in examining the relationship between **income** (independent variable) and **expenditure** (dependent variable), we can visualize this through a **scatter plot of data points**.
* 📏 The goal of regression analysis is to establish a **trend line that can predict spending based on income**, even if there are **no data points for certain income levels**.
* 📝 The equation of the trend line follows the format **y = ax + b**, where '**y**' is the dependent variable, '**x**' is the independent variable, '**a**' is the y-intercept, and '**b**' is the slope, indicating **how much 'y' changes with 'x'**.
* 📉 To find the most accurate trend line, we use the method of **least squares**, which **minimizes the sum of the squares of the errors**, ensuring the trend line is the **best fit for the data**.
* 📚 Ultimately, understanding that **regression analysis aims to predict outcomes** and that **Ordinary Least Squares (OLS) is the foundational method** will provide a solid basis for further exploration in statistics.

Hello, I would like to talk about basic regression analysis for non-statistics majors in this first session. Today, we will start discussing what regression analysis is. Many people generally refer to regression analysis as a statistical flower, which is a fitting expression, as it signifies its importance. However, surprisingly, while regression analysis is widely used, many people still do not fully understand it. 

So, in this session, we will begin with some very basic concepts about what regression analysis is. In English, it is referred to as regression. The term implies going back to something. The meaning is essentially the same. So, why do we use the term regression? This is actually a very important point. Regression analysis involves going back to something, but what exactly are we going back to?

어디를 향해 가느냐가 중요한 거죠. 아, 이야기를 분석하는 회사나 먼저 우리가 알아야 할 것은 꽤 재미있게 본 세계의 목적이 무엇이냐는 것입니다. 여기는 생의 목적을 아주 단순하게 표현하자면 이렇습니다. 주어진 데이터에서 독립변수를 가지고 종속변수를 예측하기 위한 그것 자체가 목적입니다. 일반적으로 단순 회귀 분석이라고 하는 것은 독립변수가 한 개인 경우를 말하고, 다중 회귀 분석은 독립변수가 두 개 이상인 경우입니다. 물론 단순이든 다중이든 종속변수는 무조건 하나여야 합니다. 자, 예를 한번 들어보죠. 예를 들면 우리가 수입과 지출의 관계를 알고 싶습니다. 직수입이 독립변수이고 지출이 종속변수입니다. 왜냐하면 수입이 있기 때문에 지출이 있는 것이지, 지출이 있기 때문에 수입이 있는 게 아니죠. 그래서 항상 이야기하지만...

The independent variable is the cause, and the dependent variable is the result. We construct our logic in this way and then perform regression analysis. For example, let's assume we have about 34 data points, and the data pertains to income and expenditure. That is, we have the income and expenditure of a person. Of course, they might take out a loan, but logically, there aren't many cases where expenditure exceeds income. Now, if we plot these data points on the x-axis and y-axis, we might call it a scatter plot. If we plot the data, let's assume it looks like this. Now, we will conduct regression analysis with this data. The purpose of the analysis is this: if a person's income is 100 million won, how much would their expenditure be? The problem is, as you can see from the scatter plot on the left, if we assume that 100 million won is around this point, there were no data points for people with an income of 100 million won in the data we have.

Now, what we can do is predict the spending of someone whose income is 100 million won through regression analysis. To make this prediction, we need a trend line. What is a trend line? It is essentially a line that we find by analyzing the data we have. We look for a line that best explains these data points. This is actually the main goal of regression analysis. The reason is that if we know this trend line, we can see how, from the independent variable of 100 million won, it rises to a certain point, which represents spending. Of course, this will be based on our data analysis to predict the spending. So, how can we reasonably find this trend line? The method for finding this trend line is regression analysis, which is its basic purpose. Let's delve deeper. This trend line is a straight line.

If we follow the equation of a straight line that we learned in middle school, what we need to determine the trend is the equation of the line, which is expressed as y = ax + b. Here, y is the dependent variable, which represents our spending, and x is the independent variable. That's right. We always place x on the horizontal axis and y on the vertical axis, making x the independent variable and y the dependent variable. Both y and x are given values in our data, and the points are plotted accordingly. What we need are the values of a and b. Do you understand this? In conclusion, through the data we have, our goal is to find this equation of the line, which is represented by y = a + bx. At this point, y and x are our data, with y being the dependent variable and x the independent variable. So, all we need to know is the values of a and b.

Finding the values of a and b is what regression analysis is all about. Now, to be more precise, this is something we also learned in middle school. Statistics applies the same principles. At this point, the value of a is where the y-axis intersects with this line, which is the point of intersection. This point is referred to as the y-intercept, which is represented when x is 0 and gives us the y value. That's right, it is well expressed here. Next, b is referred to as the slope. The slope indicates the change in y relative to the change in x. Do you understand? If it seems a bit difficult, consider this: a, known as the y-intercept, is the point where it intersects the y-axis, while b, the slope, indicates how much y increases when x increases. This change can be illustrated using a triangle to represent the difference.

Here, the meaning of increase refers to the change in x and y. It can get quite complicated, so in some cases, we fix x at 1 to analyze the situation. What that means is, if x is 1 and y increases by this much, then that is the amount of increase we are referring to. This is expressed as b in our terminology. We will discuss this further next time. 

Now, we have a different expression that appears here. When y is represented with something that looks like a hat, we refer to this as y hat. The term y hat and y refers to this line. This line represents the relationship where y depends on x. But how can we express these points? We can simply express these points with u and y in LaTeX. This represents the line.

What are the young subjects? There is an error, which we expressed as error. Here, this refers to the fifth degree. Now, let's take a closer look. Very rigorously, since we have 34 corresponding points for y, we can attach a lowercase 'i' to this. This is the same as what we used earlier with variance; ultimately, this 'i' refers to individual values. In other words, since there are four responses from individuals, we would have values from y1 to y34. This doesn't hold much significance. 

Now, if we look more closely at the error mentioned earlier, we can expand these points. Using these points, we don't know how the 2-3 regression line comes out, but for now, let's consider it this way. If we were to describe these data points with the most rational trend line, the issue is that there are numerous errors in the linear and total responses.

If we think about it here, the error in the second degree of the equation can be improved by measuring. The value we derived from the line of the chest is exactly the fifth degree. So, the fifth degree equation is yi - y. In other words, it's y - y. Now, what is the most representative trend line? It's very simple: it would be the trend line with the smallest total error. However, there is one problem. What is that problem? The errors can be both positive and negative. In this case, the error is positive, while in another case, it will be negative. Therefore, if we simply add up all the errors, they will cancel each other out. To derive the most rational trend line, we need to minimize the total error. To put it more accurately, we need to minimize the sum of the squares of the errors. Why? Because...

If you simply mix plus and minus, it will all get blurred together. Therefore, if we completely remove the penalties, everything becomes processed. By using this method of combining all the plastic, we can minimize the sum of the squares of the errors. The trend line with the smallest sum of squares of the errors will likely be the most accurate and rational solution. The concept of the sum of the squares of the errors seems familiar; we have seen it in variance. This is a very similar concept. So, how do we calculate this? How do we find the trend line that minimizes the sum of the fifth power of the errors? In statistics, this is called the method of least squares. It's a method for minimizing the sum of the squares. To derive the line represented by y = a + bx, we use the method of least squares. The core of this method is to find the trend line where the sum of the squares of the errors is minimized.

Oh, how do I obtain this? The oven, this is really... the sum of the squares of the deviations. For those who want to see the proof of that, I recommend looking at a statistics book; it will be much easier. This lecture is not intended for those who are statistics majors or for a high-level course, so I won't go into complex proofs or demonstrations. I will only discuss the basic logic, concepts, and objectives. 

To reiterate, the purpose of analysis is to find a trend line for a given set of data. The most rational and accurate method to find this trend line is to use the method of least squares, which minimizes the sum of the squares of the errors. So, let's summarize: the purpose of this regression analysis is to predict the dependent variable based on the independent variable from the given data, and for that, we need to derive a linear form of the recommendation.

The equation of the trend line is represented as y = a + px, where a is the y-intercept and p is the slope. To find the trend line, we use the method of least squares. This method essentially minimizes the sum of the squares of the differences. This is the best method, and it has already been proven, so you can trust it. The line we derive using this method is precisely the regression line we want. According to this equation, the line will pass through the mean values. In fact, the regression analysis initially asks where it is going back to; it refers to regression to the mean. In English, this method is called Ordinary Least Squares, often abbreviated as OLS. Ordinary Least Squares is the most common and simplest method.

There are dozens, even hundreds of types of regressions. In fact, if you read the book The Matrix, you'll find that the entire book is essentially about liberation. However, the starting point for all these regression analyses is OLS (Ordinary Least Squares). The important thing is to understand the purpose of regression analysis and, ultimately, where regression analysis leads. If you grasp just this, you'll be fine. So, this is the first time for our analysis. Today, we will stop here.

10-1 회귀분석이란? - 회귀분석 (1) -

통알못을 위한 통계튜브 
- 통통튜브 -

통알못(통계를 알지 못하는 사람)을 위한 통계튜브입니다.
이제 책도 웹/모바일에서 보실 수 있습니다.
 
통알못을 위한 기초통계
1권: https://who4u78.github.io/book1/
2권: https://who4u78.github.io/book2/

이런분들에게 적합합니다.
   1. p값이 0.05보다 큰지 작은지만 말할 수 있는 분
   2. 그런데 p값이 뭔지 모르는 분
   3. 논문작성을 위한 통계를 아무리 들어도 어렵기만 하고 이해가 안되는 분
   4. 대학원에서 통계 강의 들을 때마다 C학점을 벗어나지 못하는 분
   5. 논문을 쓰려고 설문지도 돌리고 데이터도 다 코딩했는데 이제 어떻게 해야할지 모르겠는 분
   6. 통계가 배우고 싶은데 아무리 공부해 봐도 어려운 증명만 나와 무슨 말인지 모르겠는 분
   7. 문송한데 통계까지 죄송할까봐 아예 통계는 들여다 볼 엄두도 못 내는 분

어려운 수학 하나 없이 최대한 산수 수준에서 문송한 분들을 위한 통계 강의
개념을 이해해야 통계가 보이는데, 무조건 계산하고 풀어서 답을 맞추라고 하니 통계가 어려웠습니다

통계를 보다 쉽게 잘 이해하도록 돕기위해 만든 통계 채널입니다

통계가 왜 어려웠는지에서부터 기초적인 것들부터 시작해서 통계의 핵심을 마스터 합니다.
처음부터 차근차근 들어보세요. 
통계가 쉬워집니다.

중고등학생들도 쉽게 통계를 이해하고 분석할 수 있습니다!!
빅데이터 분석을 하고 싶은데 통계를 몰라서 못하겠다고요?
여기서 부터 시작해 보세요!! 
천리 길도 한 걸음부터!! 
하루에 한 클립씩만 차근 차근 봅시다!!

사용 통계 소프트웨어
1. JAMOVI
  - 다운로드: https://www.jamovi.org/download.html
2. Bluesky Statistics
  - 다운로드: https://www.blueskystatistics.com




Sapientia a Dei

Regression analysis is a statistical method used to predict a dependent variable based on one or more independent variables.

The term "regression" implies going back to something, and its purpose is to analyze the relationship between variables.

There are two types of regression analysis: simple regression (with one independent variable) and multiple regression (with two or more independent variables), both aiming to predict a single dependent variable.

Predicting Spending Through Regression Analysis

Regression analysis is used to predict spending based on an income of 100 million won by finding a trend line that best explains the data points.

The equation of the trend line is expressed as y = ax + b, where y is the dependent variable (spending) and x is the independent variable (income).

The values of a (y-intercept) and b (slope) are crucial for determining the trend, with a representing the point of intersection on the y-axis and b indicating the change in y relative to the change in x.

Analyzing the Relationship Between x and y

The increase in y is analyzed by fixing x at 1, leading to the expression of the increase as b.

The relationship between y and x is represented by y hat, with individual values denoted as y1 to y34.

To derive the most rational trend line, the total error needs to be minimized, specifically by minimizing the sum of the squares of the errors.

The Method of Least Squares in Regression Analysis

The method of least squares is used to minimize the sum of the squares of the errors, helping to find the most accurate trend line for a given set of data.

This method is similar to the concept of variance and is used to derive the linear equation y = a + bx.

The goal of regression analysis is to predict the dependent variable based on the independent variable from the given data.

Understanding the Trend Line and Regression Analysis

The trend line is represented by the equation y = a + px, with a as the y-intercept and p as the slope.

The method of least squares is used to derive the trend line, minimizing the sum of the squares of the differences.

Ordinary Least Squares (OLS) is the most common and simplest method for regression analysis, serving as the starting point for various types of regressions.