Multiple regression in R
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Multiple regression analyzes the relationship between multiple independent variables and one dependent variable. The dependent variable is modeled as a function of the independent variables with corresponding coefficients. It is commonly used in econometrics and financial analysis to model the linear relationship between explanatory and response variables. The assumptions are that there is a linear relationship between dependent and independent variables and that the independent variables are not highly correlated with each other. Model accuracy is assessed using R-squared and residual standard error, with R-squared representing the proportion of variance in the dependent variable that can be predicted from the independent variables.
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Chapter 15Multiple Regression and Model BuildingCo
Chapter 15
Multiple Regression and Model Building
Copyright ©2018 McGraw-Hill Education. All rights reserved.
1
Chapter Outline
15.1 The Multiple Regression Model and the Least Squares Point Estimate
15.2 R2 and Adjusted R2
15.3 Model Assumptions and the Standard Error
15.4 The Overall F Test
15.5 Testing the Significance of an Independent Variable
15.6 Confidence and Prediction Intervals
15-2
2
Chapter Outline Continued
15.7 The Sales Representative Case: Evaluating Employee Performance
15.8 Using Dummy Variables to Model Qualitative Independent Variables (Optional)
15.9 Using Squared and Interaction Variables (Optional)
15.10 Multicollinearity, Model Building and Model Validation (Optional)
15.11 Residual Analysis and Outlier Detection in Multiple Regression (Optional)
15-3
3
15.1 The Multiple Regression Model and the Least Squares Point Estimate
Simple linear regression used one independent variable to explain the dependent variable
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15-4
4
The Multiple Regression Model
The linear regression model relating y to x1, x2,…, xk is y = β0 + β1x1 + β2x2 +…+ βkxk +
µy = β0 + β1x1 + β2x2 +…+ βkxk is the mean value of the dependent variable y when the values of the independent variables are x1, x2,…, xk
β0, β1, β2,… βk are the unknown regression parameters relating the mean value of y to x1, x2,…, xk
is an error term that describes the effects on y of all factors other than the independent variables x1, x2,…, xk
LO15-1
15-5
5
The Least Squares Estimates and Point Estimation and Prediction
Estimation/prediction equation
ŷ = b0 + b1x1 + b2x2 + … + bkxk
is the point estimate of the mean value of the dependent variable when the values of the independent variables are x1, x2,…, xk
It is also the point prediction of an individual value of the dependent variable when the values of the independent variables are x1, x2,…, xk
b0, b1, b2,…, bk are the least squares point estimates of the parameters β0, β1, β2,…, βk
x1, x2,…, xk are specified values of the independent predictor variables x1, x2,…, xk
LO15-1
15-6
6
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Figure 15.4 (a)
15-7
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8
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Linear Regression
linear regression is a linear approach for modelling a predictive relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables), which are measured without error. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. If the explanatory variables are measured with error then errors-in-variables models are required, also known as measurement error models.
In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis.
Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.[4] This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.
Linear regression has many practical uses. Most applications fall into one of the following two broad categories:
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If the goal is to explain variation in the response variable that can be attributed to variation in the explanatory variables, linear regression analysis can be applied to quantify the strength of the relationship between the response and the explanatory variables, and in particular to determine whether some explanatory variables may have no linear relationship with the response at all, or to identify which subsets of explanatory variables may contain redundant information about the response.
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Chapter 15Multiple Regression and Model BuildingCo
Chapter 15
Multiple Regression and Model Building
Copyright ©2018 McGraw-Hill Education. All rights reserved.
1
Chapter Outline
15.1 The Multiple Regression Model and the Least Squares Point Estimate
15.2 R2 and Adjusted R2
15.3 Model Assumptions and the Standard Error
15.4 The Overall F Test
15.5 Testing the Significance of an Independent Variable
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- 2. Definition Multiple regression generally explains the relationship between multiple independent or predictor variables and one dependent or criterion variable. A dependent variable is modeled as a function of several independent variables with corresponding coefficients, along with the constant term. y = b1x1 + b2x2 + … + bnxn + c.
- 3. Why to use it? . The goal of multiple linear regression (MLR) is to model the linear relationship between the explanatory (independent) variables and response (dependent) variable.
- 4. Where? Assumptions? MLR is used extensively in econometrics and financial inference. There is a linear relationship between the dependent variables and the independent variables. The independent variables are not too highly correlated with each other.
- 5. Model Accuracy As we have seen in simple linear regression, the overall quality of the model can be assessed by examining the R-squared (R2) and Residual Standard Error (RSE). R-squared: In multiple linear regression, the R2 represents the correlation coefficient between the observed values of the outcome variable (y) and the fitted (i.e., predicted) values of y. For this reason, the value of R will always be positive and will range from zero to one. R2 represents the proportion of variance, in the outcome variable y, that may be predicted by knowing the value of the x variables. An R2 value close to 1 indicates that the model explains a large portion of the variance in the outcome variable
- 6. From our Model In our example, with youtube and facebook predictor variables, the adjusted R2 = 0.89, meaning that “89% of the variance in the measure of sales can be predicted by youtube and facebook advertising budgets. The model is better than the simple linear model with only youtube (Chapter simple-linear-regression), which had an adjusted R2 of 0.61.