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Simple Linear Regression
Coefficient of determination, Method of least squares, Market model, Regression, Residual, Residual plots, Slope, Standardized residuals, y- intercept, \( \bar{y} \), y i , and \( \hat{y} \).
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Simple Linear Regression
- First Online: 20 January 2023
Cite this chapter
- Daniel P. McGibney 10
Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 337))
546 Accesses
In Albert Einstein’s quote above, he stresses the paramount importance of simplicity. In regression analysis, focusing on only two variables demonstrates the concepts simply. Thus, in Chap. 2 , we calculated the least squares line by using two variables. We also plotted scatterplots and calculated correlation coefficients to further assess the linear relationship. From this analysis, we obtained a detailed understanding of the relationship between two variables. Upon understanding a linear relationship, other more complicated processes become easier to grasp.
If you can’t explain it simply, you don’t understand it well enough. —Albert Einstein
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McGibney, D.P. (2023). Simple Linear Regression. In: Applied Linear Regression for Business Analytics with R. International Series in Operations Research & Management Science, vol 337. Springer, Cham. https://doi.org/10.1007/978-3-031-21480-6_4
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Linear Regression in Medical Research
Patrick schober.
From the * Department of Anesthesiology, Amsterdam UMC, Vrije Universiteit Amsterdam, Amsterdam, the Netherlands
Thomas R. Vetter
† Department of Surgery and Perioperative Care, Dell Medical School at the University of Texas at Austin, Austin, Texas.
Related Article, see p 110
Linear regression is used to quantify the relationship between ≥1 independent (predictor) variables and a continuous dependent (outcome) variable.
In this issue of Anesthesia & Analgesia , Müller-Wirtz et al 1 report results of a study in which they used linear regression to assess the relationship in a rat model between tissue propofol concentrations and exhaled propofol concentrations (Figure.
Table 2 given in Müller-Wirtz et al, 1 showing the estimated relationships between tissue (or plasma) propofol concentrations and exhaled propofol concentrations. The authors appropriately report the 95% confidence intervals as a measure of the precision of their estimates, as well as the coefficient of determination ( R 2 ). The presented values indicate, for example, that (1) the exhaled propofol concentrations are estimated to increase on average by 4.6 units, equal to the slope (regression) coefficient, for each 1-unit increase of plasma propofol concentration; (2) the “true” mean increase could plausibly be expected to lie anywhere between 3.6 and 5.7 units as indicated by the slope coefficient’s confidence interval; and (3) the R 2 suggests that about 71% of the variability in the exhaled concentration can be explained by its relationship with plasma propofol concentrations.
Linear regression is used to estimate the association of ≥1 independent (predictor) variables with a continuous dependent (outcome) variable. 2 In the most simple case, thus referred to as “simple linear regression,” there is only one independent variable. Simple linear regression fits a straight line to the data points that best characterizes the relationship
between the dependent ( Y ) variable and the independent ( X ) variable, with the y -axis intercept ( b 0 ), and the regression coefficient being the slope ( b 1 ) of this line:
A model that includes several independent variables is referred to as “multiple linear regression” or “multivariable linear regression.” Even though the term linear regression suggests otherwise, it can also be used to model curved relationships.
Linear regression is an extremely versatile technique that can be used to address a variety of research questions and study aims. Researchers may want to test whether there is evidence for a relationship between a categorical (grouping) variable (eg, treatment group or patient sex) and a quantitative outcome (eg, blood pressure). The 2-sample t test and analysis of variance, 3 which are commonly used for this purpose, are essentially special cases of linear regression. However, linear regression is more flexible, allowing for >1 independent variable and allowing for continuous independent variables. Moreover, when there is >1 independent variable, researchers can also test for the interaction of variables—in other words, whether the effect of 1 independent variable depends on the value or level of another independent variable.
Linear regression not only tests for relationships but also quantifies their direction and strength. The regression coefficient describes the average (expected) change in the dependent variable for each 1-unit change in the independent variable for continuous independent variables or the expected difference versus a reference category for categorical independent variables. The coefficient of determination, commonly referred to as R 2 , describes the proportion of the variability in the outcome variable that can be explained by the independent variables. With simple linear regression, the coefficient of determination is also equal to the square of the Pearson correlation between the x and y values.
When including several independent variables, the regression model estimates the effect of each independent variable while holding the values of all other independent variables constant. 4 Thus, linear regression is useful (1) to distinguish the effects of different variables on the outcome and (2) to control for other variables—like systematic confounding in observational studies or baseline imbalances due to chance in a randomized controlled trial. Ultimately, linear regression can be used to predict the value of the dependent outcome variable based on the value(s) of the independent predictor variable(s).
Valid inferences from linear regression rely on its assumptions being met, including
- the residuals are the differences between the observed values and the values predicted by the regression model, and the residuals must be approximately normally distributed and have approximately the same variance over the range of predicted values;
- the residuals are also assumed to be uncorrelated. In simple language, the observations must be independent of each other; for example, there must not be repeated measurements within the same subjects. Other techniques like linear mixed-effects models are required for correlated data 5 ; and
- the model must be correctly specified, as explained in more detail in the next paragraph.
Whereas Müller-Wirtz et al 1 used simple linear regression to address their research question, researchers often need to specify a multivariable model and make choices on which independent variables to include and on how to model the functional relationship between variables (eg, straight line versus curve; inclusion of interaction terms).
Variable selection is a much-debated topic, and the details are beyond the scope of this Statistical Minute. Basically, variable selection depends on whether the purpose of the model is to understand the relationship between variables or to make predictions. This is also predicated on whether there is informed a priori theory to guide variable selection and on whether the model needs to control for variables that are not of primary interest but are confounders that could distort the relationship between other variables.
Omitting important variables or interactions can lead to biased estimates and a model that poorly describes the true underlying relationships, whereas including too many variables leads to modeling the noise (sampling error) in the data and reduces the precision of the estimates. Various statistics and plots, including adjusted R 2 , Mallows C p , and residual plots are available to assess the goodness of fit of the chosen linear regression model.
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Simple linear regression estimates the coe fficients b and b of a linear model which predicts the. value of a single dependent variable ( y) against a single independent variable ( x) in the form ...
9.1. THE MODEL BEHIND LINEAR REGRESSION 217 0 2 4 6 8 10 0 5 10 15 x Y Figure 9.1: Mnemonic for the simple regression model. than ANOVA. If the truth is non-linearity, regression will make inappropriate predictions, but at least regression will have a chance to detect the non-linearity.
In contrast, if a regressor violates the underlying regression assumption, then it becomes difficult to generalize a model. Table 1-1. Linear Regression Assumptions Assumption Description Linearity there must be a linear relationship between an independent variable and a dependent variable. We verify this assumption using a pairwise scatter plot.
The most basic regression relationship is a simple linear regression. In this case, E ( Y | X) = μ ( X) = β0 + β1X, a line with intercept β0 and slope β1. We can interpret this as Y having a ...
Simple linear regression refers to linear regression with one predictor variable and one response variable. In the previous chapter, our primary focus entailed the calculation and fundamental programming concepts of linear regression. The goal of this chapter is to guide the reader through some of the understanding and analysis details of ...
dict the values of one variable using the values of the other. One of. the simplest prediction methods is linear regression, in which we. attempt to find a 'best line' through the data points ...
Linear regression is commonly used in applied research. We will explore how to use linear regression for causal effect estimation. To build intuition, we focus on the application of simple linear regression. Linear Regression and Causality. Regression conditional expectation function of Y given X. E(Y j X) = f (X) = >X.
Regression allows you to estimate how a dependent variable changes as the independent variable (s) change. Simple linear regression example. You are a social researcher interested in the relationship between income and happiness. You survey 500 people whose incomes range from 15k to 75k and ask them to rank their happiness on a scale from 1 to ...
The most elementary type of regression model is the simple linear regression model, which can be expressed by the following equation: y t = β 1 + β 2 X t + u t. (1.01) The subscript t is used to index the observations of a sample. The total number of observations, also called the sample size, will be denoted by n.
1 Statistical Analysis 6: Simple Linear Regression Research question type: When wanting to predict or explain one variable in terms of another What kind of variables? Continuous (scale/interval/ratio) Common Applications: Numerous applications in finance, biology, epidemiology, medicine etc. Example 1: A dietetics student wants to look at the relationship between calcium intake and knowledge about
The simple linear regression model has 2 coefficients ( β0 and β1) as shown in Eq. (4.1). In simple linear regression, it is always the case that the degrees of freedom for the SSE are n − 2. From the approximation of the variance, an approximation of the standard deviation can be found by taking the square root:
Home environment and reading achievement research has been largely dominated by a focus on early reading acquisition, while research on the relationship between home environments and reading success with preadolescents (Grades 4-6) has been largely overlooked. There are other limitations as well. Clarke and Kurtz-Costes (1997) argued that prior ...
A simple model A linear model that predicts demand: predicted peak demand = 1 (high temperature) + 2 60 65 70 75 80 85 90 95 1.5 2 2.5 3 High Temperature (F) Peak Hourly Demand (GW) Observed data Linear regression prediction Parameters of model: 1; 2 2R ( 1 = 0:046, 2 = 1:46) Stefano Ermon Machine Learning 1: Linear Regression March 31, 2016 8 / 25
Simple linear regression: Simple linear regression is a model with a single regressor x that has a. relationship with a response y that is a straight line. This simple linear regression. model can be expressed as. y = β0 + β1x + ε. where the intercept β0 and the slope β1 are unknown constants and ε is a random.
SIMPLE LINEAR REGRESSION Documents prepared for use in course B01.1305, New York University, Stern School of Business Fictitious example, n = 10. Page 3 This shows the arithmetic for fitting a simple linear regression. Summary of simple regression arithmetic page 4 This document shows the formulas for simple linear regression, including
Linear regression is used to estimate the association of ≥1 independent (predictor) variables with a continuous dependent (outcome) variable. 2 In the most simple case, thus referred to as "simple linear regression," there is only one independent variable. Simple linear regression fits a straight line to the data points that best characterizes the relationship
Regression is a statistical term used for describing models that estimate the relationships among variables. Linear regression models study the relationship between a single dependent variable Y an...
In both cases, we still use the term 'linear' because we assume that the response variable is directly related to a linear combination of the explanatory variables. The equation for multiple linear regression has the same form as that for simple linear regression but has more terms: = 0 +. 1 +. 2 + ⋯ +.
This paper is meant to provide an illustration of this amended recommendation. The data and accompanying analysis presented in this paper provide both a meaningful example of data analysis using simple linear regression and a story of remarkable success of international cooperation addressing a global environmental problem.
The aim of this paper is to assess the significant impact of using quantile analysis in multiple fields of scientific research . Here, we focus on estimating conditional quantile functions when the errors follow a GARMA (Generalized Auto-Regressive Moving Average) model. Our key theoretical contribution involves identifying the Quantile-Regression (QR) coefficients within the context of GARMA ...
The multiple linear regression formula as follows: (2) Y i = b 0 + b 1 X 1 + ⋯ + b i X i + ⋯ + b k X k + ɛ, i = 1, 2, …, k In Eq. (2), Y i is the dependent variable observed in multiple linear regression, X i 's are the independent variables, b's are the regression coefficients, ɛ is a fixed number. Using the least squares method ...
This paper proposes a regression-based P&O MPPT method to find GMPP speedily, avoiding being stuck in local MPP. Generated PVMs on a flexible Python environment have been used in linear and non-linear regression analyses. Regression analyses narrowed search areas to improve the speed of finding MPPT for preventing falling into local maxima.