Technology & Engineering
Simple Linear Regression Model
Simple Linear Regression Model is a statistical method used to model the relationship between a single independent variable and a dependent variable. It assumes a linear relationship between the variables and aims to find the best-fitting line to make predictions. The model is often used in various fields, including technology and engineering, to analyze and predict the behavior of systems and processes.
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7 Key excerpts on "Simple Linear Regression Model"
- Andrew F. Hayes(Author)
- 2020(Publication Date)
- Routledge(Publisher)
Regression cannot tell the researcher whether the relationship between the outcome and the predictor is causal. That is a research design issue, not a statistical one. The two variables may not be causally related, but that doesn’t mean you can’t conduct a regression analysis, because regression analysis is simply a means of assessing correlation and quantifying association. We will define simple linear regression as a procedure for generating a mathemat-ical model estimating the outcome variable Y from a single independent or predictor variable, X . The resulting model is often called a regression model , and we use the term simple to mean that there is only a single X variable in the model. In Chapter 13, I introduce multiple linear regression , where there can be more than one X variable. The goal of simple linear regression is to generate a mathematical model of the rela-tionship between X and Y that “best fits” the data or that produces estimations for Y from X that are as close as possible to the actual Y data. I will frequently use the term “model” to refer to a regression model. Fitting a regression model to a data set is sometimes called generating a model , running a regression , or regressing Y on X . 275 12.1. The Simple Linear Regression Model 12.1.1 The Simple Regression Line You probably can recall from high school mathematics that any straight line can be represented with a mathematical equation. Figure 12.2 displays a number of lines in two-dimensional space, along with their equations. A linear equation is defined by two pieces of information: the Y-intercept and the slope . The Y -intercept is the value of Y when X = 0, and the slope quantifies how much Y changes with each one-unit increase in X . Consider the equation in Figure 12.2 defined as Y = 3 . 5 + 0 . 5 X . The first number in this equation is the Y -intercept. Notice that if you set X to 0 and do the math, then Y = 3 .
eBook - PDFHandbook of Regression and Modeling
Applications for the Clinical and Pharmaceutical Industries
- Daryl S. Paulson(Author)
- 2006(Publication Date)
- Chapman and Hall/CRC(Publisher)
2 Simple Linear Regression Simple linear regression analysis provides bivariate statistical tools essential to the applied researcher in many instances. Regression is a methodology that is grounded in the relationship between two quantitative variables ( y, x ) such that the value of y (dependent variable) can be predicted based on the value of x (independent variable). Determining the mathematical relationship between these two variables, such as exposure time and lethality or wash time and log 10 microbial reductions, is very common in applied research. From a mathematical perspective, two types of relationships must be discussed: (1) a functional relationship and (2) a statistical relationship. Recall that, math-ematically, a functional relationship has the form y ¼ f ( x ), where y is the resultant value, on the function of x ( f ( x )), and f ( x ) is any set of mathematical procedure or formula such as x þ 1, 2 x þ 10, or 4 x 3 2 x 2 þ 5 x – 10, or log 10 x 2 þ 10, and so on. Let us look at an example in which y ¼ 3 x . Hence, y x 3 1 6 2 9 3 Graphing the function y on x , we have a linear graph (Figure 2.1). Given a particular value of x , y is said to be determined by x . A statistical relationship, unlike a mathematical one, does not provide an exact or perfect data fit in the way that a functional one does. Even in the best of conditions, y is composed of the estimate of x , as well as some amount of unexplained error or disturbance called statistical error, e . That is, ^ y ¼ f ( x ) þ e : So, using the previous example, y ¼ 3 x , now ^ y ¼ 3 x þ e ( ^ y indicates that ^ y estimates y , but is not exact, as in a mathematical function). They differ by 25 some random amount termed e (Figure 2.2). Here, the estimates of y on x do not fit the data estimate precisely.
eBook - PDF- Iain Pardoe(Author)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 2 SIMPLE LINEAR REGRESSION In the preceding chapter, we considered univariate data, that is, datasets consisting of measurements of just a single variable on a sample of observations. In this chapter, we consider two variables measured on a sample of observations, that is, bivariate data. In particular, we will learn about simple linear regression, a technique for analyzing bivariate data which can help us to understand the linear association between the two variables, to see how a change in one of the variables is associated with a change in the other variable, and to estimate or predict the value of one of the variables knowing the value of the other variable. We will use the statistical thinking concepts from Chapter 1 to accomplish these goals. After reading this chapter you should be able to: • Define a Simple Linear Regression Model as a linear association between a quantitative response variable and a quantitative predictor variable. • Express the value of an observed response variable as the sum of a deterministic linear function of the corresponding predictor value plus a random error. • Use statistical software to apply the least squares criterion to estimate the sample simple linear regression equation by minimizing the residual sum of squares. • Interpret the intercept and slope of an estimated simple linear regression equation. • Calculate and interpret the regression standard error in simple linear regression. • Calculate and interpret the coefficient of determination in simple linear regression. 39 Applied Regression Modeling, Third Edition. Iain Pardoe. © 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc. Book companion site: www.wiley.com/go/pardoe/AppliedRegressionModeling3e 40 SIMPLE LINEAR REGRESSION • Understand the relationship between the coefficient of determination and the corre- lation coefficient. • Conduct and draw conclusions from hypothesis tests for the regression parameters in simple linear regression.
eBook - PDFBest Fit Lines & Curves
And Some Mathe-Magical Transformations
- Alan R. Jones(Author)
- 2018(Publication Date)
- Routledge(Publisher)
An act of reflecting and recalling memories from an earlier stage of life, or an alleged previous life Simple and Multiple Linear Regression 4 114 | Simple and Multiple Linear Regression 2. A statistical technique that determines the ‘Best Fit’ relationship between two or more variables Although we may be tempted to consider Regression as a means of entering a trance like state where we can reflect on life before we became estimators, it is, of course, the second definition that is of relevance to us here. However, this definition for me (although correct) does not fully convey the process and power of Regression Anal-ysis as it misses the all-important element of what determines ‘best fit’. Instead, we will revise our definition of Regression to be: Definition 4.1 Regression Analysis Regression Analysis is a systematic procedure for establishing the Best Fit rela-tionship of a predefined form between two or more variables, according to a set of Best Fit criteria. Note that Regression only assumes that there is a relationship between two or more variables; it does not imply causation. It also assumes that there is a continuous relation-ship between the dependent variable, i.e. the one we are trying to predict or model, and at least one of the independent variables used as a driver or predictor. One of the primary outputs from a Regression Analysis is the Regression Equa-tion, which we would typically use to interpolate or extrapolate in order to generate an estimate for defined input values (drivers). The technique has a very wide range of applications in business and can be used to identify a pattern of behaviour between one or more estimate drivers (the independent variables) and the thing or entity we want to estimate (the dependent variable). Examples might include the relationship between cost and a range of physical parameters, sales forecasts and levels of marketing budgets, Learning Curves, Time Series .
eBook - PDF- D.R. Helsel, R.M. Hirsch(Authors)
- 1993(Publication Date)
- Elsevier Science(Publisher)
This chapter will present the assumptions, computation and applications of linear regression, as well as its limitations and common misapplications by the water resources community. 222 Statistical Methods in Water Resources Ordinary Least Squares (OLS), commonly referred to as linear regression, is a very important tool for the statistical analysis of water resources data. It is used to describe the covariation between some variable of interest and one or more other variables. Regression is performed in order to 1) learn something about the relationship between the two variables, or 2) remove a portion of the variation in one variable (a portion that is not of interest) in order to gain a better understanding of some other, more interesting, portion of the variation, or variable, for which more data are available. 3) estimate or predict values of one variable based on knowledge of another This chapter deals with the relationship between one continuous variable of interest, called the response variable, and one other variable - the explanatory variable. The name simple linear regression is applied because one explanatory variable is the simplest case of regression models. The case of multiple explanatory variables is dealt with in Chapter 11 - multiple regression. 9.1 The Linear Regression Model The model for simple linear regression is: i=1,2, .... , n Yi = PO + Plxi + Ei where yi xi PO is the intercept P1 is the slope ~i n is the sample size. is the ith observation of the response (or dependent) variable is the ith observation of the explanatory (or independent) variable is the random error or residual for the ith observation, and The error around the linear model ~i is a random variable. That is, its magnitude is not controlled by the analyst, but arises from the natural variability inherent in the system. ~i has a mean of zero, and a constant variance o2 which does not depend on x.
eBook - PDFPredictive Analytics
Parametric Models for Regression and Classification Using R
- Ajit C. Tamhane(Author)
- 2020(Publication Date)
- Wiley(Publisher)
Chapter 2 Simple linear regression and correlation One of the simplest and yet a commonly occurring data analytic problem is exploring the relationship between two numerical variables. In many applications, one of the variables may be regarded as a response variable and the other as a predictor variable and the goal is to find the best fitting relationship between the two. For example, it may be of interest to predict the amount of sales from advertising dollars or the reduction in tumor size from the amount of radiation exposure. This is referred to as a regression problem . In other applications, there is no such distinction between the two variables, and it is of interest to simply assess the strength of relationship between them. For example, it may be of interest to assess the relationship between the average summer temperature and the average winter temperature in a given region of the country or the amounts of sales in two divisions of a company. This is referred to as a correlation problem . We study both these problems in this chapter. We will use the following two data sets to illustrate the methods introduced in this chapter: Example 2.1 (Bacteria Counts: Data) Chatterjee and Hadi (2012) gave the data shown in Table 2.1 on the number of surviving bacteria (in hundreds) exposed to 200 kv X-rays for 15 six-minute intervals. The main question of interest is how do the bacteria decay with time, in particular, does the exponential decay law apply and if so what is the decay rate? Predictive Analytics: Parametric Models for Regression and Classification Using R , First Edition. Ajit C. Tamhane. c 2021 John Wiley & Sons, Inc. Published 2021 by John Wiley & Sons, Inc. Companion website: www.wiley.com/go/tamhane/predictiveanalytics 7 8 Chapter 2. Simple linear regression and correlation Table 2.1.
eBook - PDF- Dawn A. Willoughby(Author)
- 2016(Publication Date)
- Wiley(Publisher)
C H A P T E R • 8 Simple Linear Regression OBJECTIVES This chapter explains how to: • distinguish between independent and dependent variables • understand a Simple Linear Regression Model • find the equation of a line of best fit • use a regression equation to - calculate predicted values - interpret the y-intercept and gradient KEY TERMS dependent variable explanatory variable extrapolation gradient independent variable interpolation least squares method line of best fit linear equation regression line residual response variable simple linear regression y-intercept Introduction In the previous chapter, we explored the way in which correlation can be used to describe the linear relationship between two quantitative variables, but we found that it is limited to describing the direction and strength of the relationship. Using simple linear regression, we can investigate the relationship between the two variables in more detail by determining how a change in the value of one variable affects the value of the other variable. Most importantly, we can use this regression model to form an equation that allows us to predict the value of one variable for a known value of the other. Before introducing the concepts that result in creating a Simple Linear Regression Model, we will review linear equations as a reminder about the meaning of the y-intercept and the gradient. Later in the chapter, you will learn how to determine the equation of the line of best fit for a data set, and gain an understanding of the interpretation and use of the equation. Independent and Dependent Variables In the context of regression, we need to distinguish between independent and dependent variables. An independent variable, or explanatory variable, is used to predict the value of a dependent variable. We sometimes use the term ‘explanatory variable’ because a change in this variable helps to explain a change in a dependent variable.
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