multiple regression

(noun)

regression model used to find an equation that best predicts the $Y$ variable as a linear function of multiple $X$ variables

Related Terms

Examples of multiple regression in the following topics:

  • Multiple Regression Models

    • Multiple regression is used to find an equation that best predicts the $Y$ variable as a linear function of the multiple $X$ variables.
    • You use multiple regression when you have three or more measurement variables.
    • One use of multiple regression is prediction or estimation of an unknown $Y$ value corresponding to a set of $X$ values.
    • Multiple regression would give you an equation that would relate the tiger beetle density to a function of all the other variables.
    • As you are doing a multiple regression, there is also a null hypothesis for each $X$ variable, meaning that adding that $X$ variable to the multiple regression does not improve the fit of the multiple regression equation any more than expected by chance.

  • 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 probably treat multiple regression as a way of suggesting patterns in your data, rather than rigorous hypothesis testing.
    • For example, let's say you did a multiple regression on vertical leap in children five to twelve years old, with height, weight, age, and score on a reading test as independent variables.

  • Using the Model for Estimation and Prediction

    • Standard multiple regression involves several independent variables predicting the dependent variable.
    • Standard multiple regression is the same idea as simple linear regression, except now we have several independent variables predicting the dependent variable.
    • We would use standard multiple regression in which gender and weight would be the independent variables and height would be the dependent variable.
    • Consequently, the first independent variable is no longer uniquely predictive and would not be considered significant in multiple regression.
    • Multiple regression is the same idea as single regression, except we deal with more than one independent variables predicting the dependent variable.

  • Estimating and Making Inferences About the Slope

    • The purpose of a multiple regression is to find an equation that best predicts the $Y$ variable as a linear function of the $X$ 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 $Y$ variable as a linear function of the $X$variables.
    • When the purpose of multiple regression is prediction, the important result is an equation containing partial regression coefficients (slopes).
    • When the purpose of multiple regression is understanding functional relationships, the important result is an equation containing standard partial regression coefficients, like this:

  • Introduction to multiple regression

    • The principles of simple linear regression lay the foundation for more sophisticated regression methods used in a wide range of challenging settings.
    • In Chapter 8, we explore multiple regression, which introduces the possibility of more than one predictor, and logistic regression, a technique for predicting categorical outcomes with two possible categories.
    • Multiple regression extends simple two-variable regression to the case that still has one response but many predictors (denoted x1 , x2 , x3 , ...).
    • Multiple regression will help us answer these and other questions.
    • Multiple regression also allows for categorical variables with many levels, though we do not have any such variables in this analysis, and we save these details for a second or third course.

  • Checking the Model and Assumptions

    • There are a number of assumptions that must be made when using multiple regression models.
    • When working with multiple regression models, a number of assumptions must be made.
    • The following are the major assumptions with regard to multiple regression models:
    • Fortunately, slight deviations from linearity will not greatly affect a multiple regression model.
    • Paraphrase the assumptions made by multiple regression models of linearity, homoscedasticity, normality, multicollinearity and sample size.

  • Some Pitfalls: Estimability, Multicollinearity, and Extrapolation

    • Some problems with multiple regression include multicollinearity, variable selection, and improper extrapolation assumptions.
    • Until recently, any review of literature on multiple linear regression would tend to focus on inadequate checking of diagnostics because, for years, linear regression was used inappropriately for data that were really not suitable for it.
    • Despite the fact that automated stepwise procedures for fitting multiple regression were discredited years ago, they are still widely used and continue to produce overfitted models containing various spurious variables.
    • If the method assumes the data are smooth, then a non-smooth regression function will be poorly extrapolated.
    • Examine how the improper choice of explanatory variables, the presence of multicollinearity between variables, and extrapolation of poor quality can negatively effect the results of a multiple linear regression.

  • Introduction to Logistic regression

    • In this section we introduce logistic regression as a tool for building models when there is a categorical response variable with two levels.
    • Logistic regression is a type of generalized linear model (GLM) for response variables where regular multiple regression does not work very well.
    • Second, we model the parameter of the distribution using a collection of predictors and a special form of multiple regression.
    • This video covers some basic concepts regarding logistic regression.

  • Polynomial Regression

    • For this reason, polynomial regression is considered to be a special case of multiple linear regression.
    • Although polynomial regression is technically a special case of multiple linear regression, the interpretation of a fitted polynomial regression model requires a somewhat different perspective.
    • This is similar to the goal of non-parametric regression, which aims to capture non-linear regression relationships.
    • An advantage of traditional polynomial regression is that the inferential framework of multiple regression can be used.
    • Explain how the linear and nonlinear aspects of polynomial regression make it a special case of multiple linear regression.

  • Slope and Intercept

    • In the regression line equation the constant $m$ is the slope of the line and $b$ is the $y$-intercept.
    • A simple example is the equation for the regression line which follows:
    • 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).