linear regression

(noun)

an approach to modeling the relationship between a scalar dependent variable yyy and one or more explanatory variables denoted xxx.

Examples of linear regression in the following topics:

  • The Equation of a Line

    • In statistics, linear regression can be used to fit a predictive model to an observed data set of yyy and xxx values.
    • In statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable.
    • Simple linear regression fits a straight line through the set of nnn points in such a way that makes the sum of squared residuals of the model (that is, vertical distances between the points of the data set and the fitted line) as small as possible.
    • Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.
    • If the goal is prediction, or forecasting, linear regression can be used to fit a predictive model to an observed data set of yyy and XXX values.
  • Introduction to inference for linear regression

    • In this section we discuss uncertainty in the estimates of the slope and y-intercept for a regression line.
    • However, in the case of regression, we will identify standard errors using statistical software.
    • This video introduces consideration of the uncertainty associated with the parameter estimates in linear regression.
  • Evaluating Model Utility

    • Multiple regression is beneficial in some respects, since it can show the relationships between more than just two variables; however, it should not always be taken at face value.
    • It is easy to throw a big data set at a multiple regression and get an impressive-looking output.
    • But many people are skeptical of the usefulness of multiple regression, especially for variable selection, and you should view the results with caution.
    • You should examine the linear regression of the dependent variable on each independent variable, one at a time, examine the linear regressions between each pair of independent variables, and consider what you know about the subject matter.
    • You should probably treat multiple regression as a way of suggesting patterns in your data, rather than rigorous hypothesis testing.
  • Slope and Intercept

    • A simple example is the equation for the regression line which follows:
    • Linear regression is an approach to modeling the relationship between a scalar dependent variable yyy and one or more explanatory (independent) variables denoted XXX.
    • The case of one explanatory variable is called simple linear regression.
    • For more than one explanatory variable, it is called multiple linear regression.
    • (This term should be distinguished from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable).
  • Polynomial Regression

    • The goal of polynomial regression is to model a non-linear relationship between the independent and dependent variables.
    • Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(y ∣ x)E(y\ | \ x)E(y ∣ x) is linear in the unknown parameters that are estimated from the data.
    • For this reason, polynomial regression is considered to be a special case of multiple linear regression.
    • This is similar to the goal of non-parametric regression, which aims to capture non-linear regression relationships.
    • Explain how the linear and nonlinear aspects of polynomial regression make it a special case of multiple linear regression.
  • Checking the Model and Assumptions

    • These assumptions are similar to those of standard linear regression models.
    • Linearity.
    • Fortunately, slight deviations from linearity will not greatly affect a multiple regression model.
    • In effect, residuals appear clustered and spread apart on their predicted plots for larger and smaller values for points along the linear regression line; the mean squared error for the model will be incorrect.
    • Paraphrase the assumptions made by multiple regression models of linearity, homoscedasticity, normality, multicollinearity and sample size.
  • Introduction to Linear Regression

    • In simple linear regression, we predict scores on one variable from the scores on a second variable.
    • In simple linear regression, the topic of this section, the predictions of Y when plotted as a function of X form a straight line.
    • Linear regression consists of finding the best-fitting straight line through the points.
    • The regression equation is
    • Of course, if the relationship between X and Y is not linear, a different shaped function could fit the data better.
  • Estimating and Making Inferences About the Slope

    • The purpose of a multiple regression is to find an equation that best predicts the YYY variable as a linear function of the XXX variables.
    • You use multiple regression when you have three or more measurement variables.
    • The purpose of a multiple regression is to find an equation that best predicts the YYY variable as a linear function of the XXXvariables.
    • When the purpose of multiple regression is prediction, the important result is an equation containing partial regression coefficients (slopes).
    • A graphical representation of a best fit line for simple linear regression.
  • Predictions and Probabilistic Models

    • Best-practice advice here is that a linear-in-variables and linear-in-parameters relationship should not be chosen simply for computational convenience, but that all available knowledge should be deployed in constructing a regression model.
    • A scatterplot shows a linear relationship between a quantitative explanatory variable xxx and a quantitative response variable yyy.
    • A good rule of thumb when using the linear regression method is to look at the scatter plot of the data.
    • This graph is a visual example of why it is important that the data have a linear relationship.
    • Each of these four data sets has the same linear regression line and therefore the same correlation, 0.816.
  • Multiple Regression Models

    • Multiple regression is used to find an equation that best predicts the YYY variable as a linear function of the multiple XXX variables.
    • You use multiple regression when you have three or more measurement variables.
    • The purpose of a multiple regression is to find an equation that best predicts the YYY variable as a linear function of the XXX variables.
    • Multiple regression is a statistical way to try to control for this; it can answer questions like, "If sand particle size (and every other measured variable) were the same, would the regression of beetle density on wave exposure be significant?
    • As you are doing a multiple regression, there is also a null hypothesis for each XXX variable, meaning that adding that XXX variable to the multiple regression does not improve the fit of the multiple regression equation any more than expected by chance.
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