Prospective Analysis of Logistic Case-Control Studies

By Carroll, R. J.; Wang, Suojin et al. | Journal of the American Statistical Association, March 1995 | Go to article overview

Prospective Analysis of Logistic Case-Control Studies


Carroll, R. J., Wang, Suojin, Wang, C. Y., Journal of the American Statistical Association


[Mathematical Expression Omitted].

Let [[Psi].sub.12] = [[Psi].sub.22]([center dot], [Theta]) be the vector of size 2M whose (dM + m)th element equals I(D = d) {I(Z = m) - [[Theta].sub.2,md]}.

Let [Mathematical Expression Omitted]. Denote the logistic argument in (A.12) by [H.sub.*](Z, X, [Theta]) and write [Mathematical Expression Omitted].

At the end of this section, the estimating equation is shown to be unbiased, the particular method being to condition on all [Mathematical Expression Omitted] or, equivalently, on all the [Delta]'s. In addition, with 1. INTRODUCTION

In a classical prospective logistic regression study, a random sample from a source population is taken and the status of a binary outcome D is ascertained, along with the values of covariates (Z, X), these being related via the logistic regression model

[Mathematical Expression Omitted],

where H([center dot]) is the logistic distribution function. The classical case-control study (choice-based sample in econometrics) begins with the model (1), but instead uses retrospective sampling. Specifically, one first obtains a set of cases (D = 1) and controls (D = 0), and then samples from within the cases and controls to observe the covariates. The analysis of case-control studies of this type was described by Prentice and Pyke (1979), who showed that if one ignored the case-control sampling scheme and analyzed the data as if it came from a prospective sampling scheme, then the resulting estimates of ([[Theta].sub.11], [[Theta].sub.12]) are consistent and the usual standard errors are asymptotically correct.

For prospective logistic regression studies, many other types of analyses and sampling schemes are possible. Here are a few examples:

* One might replace the classical logistic regression parameter estimates by robust methods of estimation (Copas 1988; Carroll and Pederson 1993, Kunsch, Stefanski, and Carroll 1989).

* When X is measured with error, there is a large literature dealing with techniques for measurement error corrections in logistic regression (e.g., Carroll and Stefanski 1994; Rosner, Willett, and Spiegelman 1989; Satten and Kupper 1993; Stefanski and Carroll, 1987).

* In problems with partially missing data, one can use likelihood techniques (Little and Rubin 1987) or unbiased estimating equations due to Robins, Rotnitzky, and Zhao (1994).

Although the prospective analyses of these prospective techniques have been worked out, there is to date no corresponding general theory for whether they even lead to consistent estimates when applied to case-control studies and, if they do, whether these prospectively calculated standard errors are asymptotically correct in case-control studies. Our aim is to provide one version of such a theory, and in particular to answer the question: When can prospective analyses be used in case-control studies without having to adjust for the retrospective sampling structure?

We will show that, in general, using prospectively derived standard errors is at worst asymptotically conservative; that is, the standard errors are at worst too large. In addition, we derive a simple sufficient condition guaranteeing that prospective standard errors are asymptotically correct.

In the Appendix we sketch an informal argument derived from a semiparametric perspective that suggests that prospectively computed standard errors are retrospectively correct whenever the distribution of (Z, X) is left unrestricted. Much of this article is a formalization of this argument, along with consideration of cases that are not so easily categorized. The key feature of our analysis is that we start with a general class of unbiased estimating equations, instead of working with specific examples. The results allow for general patterns of missing data as well as for stratified studies. The asymptotic distribution theory is almost trivial to derive in this general framework, thus facilitating the identification of a simple sufficient condition for checking whether prospectively derived standard errors are asymptotically correct. …

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

Sign up now for a free, 1-day trial and receive full access to:

  • Questia's entire collection
  • Automatic bibliography creation
  • More helpful research tools like notes, citations, and highlights
  • Ad-free environment

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.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

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 article

Prospective Analysis of Logistic Case-Control Studies
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?

Full screen

matching results for page

Cited passage

Style
Citations are available only to our active members.
Sign up now 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.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.