On the Interpretation of Regression Plots

By Cook, R. Dennis | Journal of the American Statistical Association, March 1994 | Go to article overview

On the Interpretation of Regression Plots


Cook, R. Dennis, Journal of the American Statistical Association


A framework is developed for the interpretation of regression plots, including plots of the response against selected covariates, residual plots, added-variable plots, and detrended added-variable plots. It is shown that many of the common interpretations associated with these plots can be misleading. The framework also allows for the generalization of standard plots and the development of new plotting paradigms. A paradigm for graphical exploration of regression problems is sketched.

KEY WORDS: Added-variable plots; Elliptically contoured distributions; Nonlinear models; Regression graphics; Residual plots.

1. PRELUDE

The ability to interpret correctly the various plots that arise in a regression analysis is always useful, particularly in situations where a parsimoniously parameterized true model is unavailable. This ability has become increasingly important in recent years with the rapid development of software that provides easy access to modern graphical methods such as spinning, brushing, and animation.

A full interpretation of any regression plot requires two distinct but related tasks. Out of necessity the first task is to characterize the plot itself. Characterizing a two-dimensional scatterplot is relatively easy. Accurately characterizing a three-dimensional scatterplot is possible on a routine basis, albeit with more effort. It is also possible to obtain useful insights into higher-dimensional plots, but for the most part their interpretation must rely on lower-dimensional constructions. Having characterized the plot at hand, the second task is to use this information to form conclusions that can guide the remaining analysis. A key point here is that the first task is data analytic in nature, whereas the second is inferential. Together they comprise the interpretation of the plot.

Detecting a fan-shaped pattern in the usual plot of residuals versus fitted values from a first-order ordinary least squares (OLS) regression, for example, leads to the data analytic observation that the residual variance changes monotonically in the direction of the fitted values. The second step comes when this conclusion is used to infer heteroscedastic errors and to justify a weighted regression. As a second example, Cook and Weisberg (1989) suggested that a saddle-shaped pattern in certain detrended added-variable plots can be used to infer the need for interaction terms. Although the regression literature is replete with this sort of advice, bridging the connection between the data analytic characterization of a plot and the subsequent inference often requires a leap of faith regarding the behavior of the true regression.

In this article I study the interpretation of regression plots, concentrating on aspects of both what to look for and how to infer from what is found. The requirements of the first task can often be met easily in practice, although this is certainly not the case at the level of generality of the main results that follow. The interpretation of any plot depends on the range of possibilities that is allowed for the true regression. Inferring heteroscedastic errors from a fan-shaped pattern in a plot of residuals versus fitted values, for example, is appropriate only under certain restrictions (Sec. 7). In Section 3 I describe an essentially nonrestrictive regression model that will be used to guide plot interpretation.

It turns out that the behavior of the covariates is critical to the interpretation of regression plots in a largely nonparametric setting. This aspect of the problem is studied in Sections 4 and 5. Plots of the response versus selected covariates, added-variable plots, detrended added-variable plots, and some extensions are studied in Section 6. A discussion of residual plots is presented in Section 7. A paradigm for graphical exploration of a regression problem is sketched in Section 8, and concluding comments are given in Section 9. …

The rest of this article is only available to active members of Questia

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this article

This article has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this article

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited article

On the Interpretation of Regression Plots
Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Help
Full screen

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Style
    Citations are available only to our active members.
    Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

    1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

    Cited passage

    Thanks for trying Questia!

    Please continue trying out our research tools, but please note, full functionality is available only to our active members.

    Your work will be lost once you leave this Web page.

    Buy instant access to save your work.

    Already a member? Log in now.

    Author Advanced search

    Oops!

    An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.