10 Questions
In linear regression, the fit, 𝑦̂, is a non-linear combination of prescribed functions of the explanatory variables.
False
The carriers are the explanatory variables in a linear regression model.
False
Residual plots are used to visualize the relationships between the response variable and the explanatory variables.
False
A point cloud pattern in a residual plot suggests that there is a simple residual relationship between the residuals and the carrier.
False
A sloping band pattern in a residual plot indicates that a quadratic term could be inserted into the model.
False
The purpose of residual plots is to evaluate the goodness of fit of a linear regression model.
True
Exploratory data analysis is used to construct an initial model for linear regression.
True
The carrier x in a residual plot is necessarily one of the explanatory variables included in the regression.
False
A residual plot can reveal systematic relationships between the response variable and an explanatory variable that has not been accounted for in the fit.
True
The primary goal of residual plots is to identify patterns in the residuals that can inform model improvement.
True
Study Notes
Examining Residuals
- In exploratory data analysis, residuals are used to uncover and magnify patterns in the data, without anticipating a specific model.
- Residual plots are used extensively to suggest improvements to the fit and to see how the technique acts on the data to give the fit.
Criteria for a Good Fit
- A small residual sum of squares and a high multiple correlation coefficient (R²) are less effective but more familiar criteria for a good fit.
- In exploratory data analysis, robust and resistant techniques are used with weaker demands on errors.
Residuals as Batches
- To examine residuals, patterns in the residuals that are consequences of the assumptions and fitting procedure are first identified.
- The least-squares fit is used to obtain the residuals, which are then plotted to identify patterns and unusual points.
Expected Learning Outcomes
- Illustrate the importance of examining residuals.
- Construct and interpret residual plots.
- Formulate changes in the model based on diagnostics on residuals.
Residuals and Fit
- The fit embodies some of the major patterns in the data, and the residuals provide more detail.
- The goal is to put as much of the pattern in the data into the fit as possible.
Residual Plots
- Residual plots help identify patterns in the residuals, unusual points, and the residual distribution.
- Plots of residuals against carriers can reveal systematic relationships between the response and explanatory variables that have not been accounted for in the fit.
- Four patterns may appear in residual plots: point cloud, sloping band, and others.
Understanding regression analysis, including residual sum of squares, multiple correlation coefficient, and exploratory data analysis techniques.
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