How can I correct for measurement error in the dependent variable in a logit regression?

I'm running a binary logit regression where I know the dependent variable is miscoded in a small percentage of cases. So I'm trying to estimate $\beta$ in this model:

$prob(y_i) = 1/(1 + e^{-z_i})$

$z_i = \alpha + X_i\beta$

But instead of the vector $Y$, I have $\tilde{Y}$, which includes some random errors (i.e. $y_i = 1$, but $\tilde{y_i} = 0$, or vice versa, for some $i$).

Is there a (reasonably) simple correction for this problem?

I know that logit has some nice properties in case-control studies. It seems likely that something similar applies here, but I haven't been able to find a good solution.

A few other constraints: this is a text-mining application, so the dimensions of $X$ are large (in the thousands or tens of thousands). This may rule out some computationally intensive procedures.

Also, I don't care about correctly estimating $\alpha$, only $\beta$.

This situation is often referred to as misclassification error. This paper my help you correctly estimating $\beta$. EDIT: I found relevant-looking papers using http://www.google.com/search?q=misclassification+of+dependent+variable+logistic.

• According to the abstract, this paper seems to deal with an "error-prone binary covariate": that is, with misclassified independent variables only. – whuber Apr 11 '11 at 15:03
• Actually the abstract deals with both: "For outcome misclassification, we argue that a likelihood-based analysis is the cleanest and the most preferable approach. In the case of covariate misclassification, we combine [....] – rolando2 Apr 11 '11 at 16:00

You can either estimate a parametric model of the error using MLE, or you can use a semi-paramteric approach based on something like the maximal rank correlation (MRC) estimator. Computationally, MRC is prohibitive for large samples, so it looks like MLE is the right approach for me.

Thanks to GaBorgulya for some good, prompt direction, especially on the term "misclassification error."

Here are some good sources on the topic:

The basic model, exactly as described in the original problem

Ungated version of the same

A more complicated, but more general model

A nice overview