## 2 9 Simple Linear Regression Examples STAT 462

Posted on August 17, 2022

They help us understand the distribution of the data points and the presence of outliers. In Simple Linear Regression (SLR), we will have a single input how to calculate cost per unit variable based on which we predict the output variable. Where in Multiple Linear Regression (MLR), we predict the output based on multiple inputs.

- Specifically, the interpretation of βj is the expected change in y for a one-unit change in xj when the other covariates are held fixed—that is, the expected value of the partial derivative of y with respect to xj.
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- It is important to interpret the slope of the line in the context of the situation represented by the data.
- Numerous extensions have been developed that allow each of these assumptions to be relaxed (i.e. reduced to a weaker form), and in some cases eliminated entirely.

However, it is never possible to include all possible confounding variables in an empirical analysis. For example, a hypothetical gene might increase mortality and also cause people to smoke more. For this reason, randomized controlled trials are often able to generate more compelling evidence of causal relationships than can be obtained using regression analyses of observational data. When controlled experiments are not feasible, variants of regression analysis such as instrumental variables regression may be used to attempt to estimate causal relationships from observational data.

## What is Regression Analysis?

This lesson introduces the concept and basic procedures of simple linear regression. If you have more than one independent variable, use multiple linear regression instead. Utilizing a linear regression model will permit you to find whether a connection between variables exists by any means. To see precisely what that relationship is and whether one variable causes another, you will require extra examination and statistical analysis. The use of regression for parametric inference assumes that the errors (ε) are (1) independent of each other and (2) normally distributed with the same variance for each level of the independent variable. The errors (residuals) are greater for higher values of x than for lower values.

The visualization step for multiple regression is more difficult than for simple regression, because we now have two predictors. One option is to plot a plane, but these are difficult to read and not often published. The standard errors for these regression coefficients are very small, and the t statistics are very large (-147 and 50.4, respectively). For both parameters, there is almost zero probability that this effect is due to chance. Because we only have one independent variable and one dependent variable, we don’t need to test for any hidden relationships among variables.

The slope is negative because the line slants down from left to right, as it must for two variables that are negatively correlated, reflecting that one variable decreases as the other increases. When the correlation is positive, β 1is positive, and the line slants up from left to right. For each of these deterministic relationships, the equation exactly describes the relationship between the two variables.

## Linear Regression Example¶

In finance, regression analysis is used to calculate the Beta (volatility of returns relative to the overall market) for a stock. The size of the correlation \(r\) indicates the strength of the linear relationship between \(x\) and \(y\). Values of \(r\) close to –1 or to +1 indicate a stronger linear relationship between \(x\) and \(y\). Linear regression is widely used in biological, behavioral and social sciences to describe possible relationships between variables. Some of the more common estimation techniques for linear regression are summarized below. Let’s see if there’s a linear relationship between income and happiness in our survey of 500 people with incomes ranging from $15k to $75k, where happiness is measured on a scale of 1 to 10.

## Step 3: Perform the linear regression analysis

Even when you see a strong pattern in your data, you can’t know for certain whether that pattern continues beyond the range of values you have actually measured. Therefore, it’s important to avoid extrapolating beyond what the data actually tell you. If we instead fit a curve to the data, it seems to fit the actual pattern much better.

Simple linear regression gets its adjective “simple,” because it concerns the study of only one predictor variable. In contrast, multiple linear regression, which we study later in this course, gets its adjective “multiple,” because it concerns the study of two or more predictor variables. Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line.

Note that the numbers in red are the coefficients that the analysis provided. Trend lines are sometimes used in business analytics to show changes in data over time. Trend lines are often used to argue that a particular action or event (such as training, or an advertising campaign) caused observed changes at a point in time.

The author clearly demonstrates effective methods of regression analysis with examples that contain the types of data irregularities commonly encountered in the real world. This newest edition also offers a brand-new, easy to read chapter on the freely available statistical software package R. In the newly revised sixth edition of Regression Analysis By Example Using R, distinguished statistician Dr Ali S. Hadi delivers an expanded and thoroughly updated discussion of exploratory data analysis using regression analysis in R. The book provides in-depth treatments of regression diagnostics, transformation, multicollinearity, logistic regression, and robust regression.

## Multiple regression: biking, smoking, and heart disease

There are several ways to find a regression line, but usually the least-squares regression line is used because it creates a uniform line. Residuals, also called “errors,” measure the distance from the actual value of \(y\) and the estimated value of \(y\). The Sum of Squared Errors, when set to its minimum, calculates the points on the line of best fit. Regression lines can be used to predict values within the given set of data, but should not be used to make predictions for values outside the set of data.

## Simple linear regression in R

This is a simple technique, and does not require a control group, experimental design, or a sophisticated analysis technique. However, it suffers from a lack of scientific validity in cases where other potential changes can affect the data. Based on these residuals, we can say that our model meets the assumption of homoscedasticity.

To describe the linear association between quantitative variables, a statistical procedure called regression often is used to construct a model. Regression is used to assess the contribution of one or more “explanatory” variables (called independent variables) to one “response” (or dependent) variable. It also can be used to predict the value of one variable based on the values of others. When there is only one independent variable and when the relationship can be expressed as a straight line, the procedure is called simple linear regression. A fitted linear regression model can be used to identify the relationship between a single predictor variable xj and the response variable y when all the other predictor variables in the model are “held fixed”. Specifically, the interpretation of βj is the expected change in y for a one-unit change in xj when the other covariates are held fixed—that is, the expected value of the partial derivative of y with respect to xj.

The equations that you used to estimate the intercept and slope determine a line of “best fit” by minimizing the sum of the squared residuals. Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms “least squares” and “linear model” are closely linked, they are not synonymous.