help firthlogit                                          Version 1.0 2008-07-11

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Title

        firthlogit -- Penalized maximum likelihood logistic regression

Syntax

        firthlogit depvar [indepvars] [if] [in] [weight] [, options]


    options               Description
    -------------------------------------------------------------------------
    Main
      level(#)            set confidence level; default is prevailing setting
                            (see creturn)
      or                  report odds ratios
      maximize_options    maximization options
    -------------------------------------------------------------------------
    by may be used with firthlogit; see by.
    fweights are allowed with firthlogit; see weight.


Description

    firthlogit fits logistic models by penalized maximum likelihood
    regression. The method originally was proposed to reduce bias in maximum
    likelihood estimates in generalized linear models. It also has utility in
    logistic regression in circumstances in which "separation" is
    problematic.


Options

        +------+
    ----+ Main +-------------------------------------------------------------

    level set confidence level; default is the Stata level setting.

    or is another reporting option, displaying coefficients as odds ratios.

    maximize_options many of the conventional ml options are available, the
        most important of which is constraint(), which is used in penalized
        likelihood ratio tests.  Options not available with d0 estimators are
        not available; see ml.


Remarks

    Firth (1993) suggested a modification of the score equations in order to
    reduce bias seen in generalized linear models.  Heinze and Schemper
    (2002) suggested using Firth's method to overcome the problem of
    "separation" in logistic regression, a condition in the data in which
    maximum likelihood estimates tend to infinity (become inestimable). The
    method allows convergence to finite estimates in cases of separation in
    logistic regression.

    The method penalizes the log-likelihood with one-half of the logarithm of
    the determinant of the information matrix.  firthlogit uses ml with a d0
    log-likelihood estimator program. d0 estimators use numerical
    derivatives, and so are slower and slightly less accurate than linear
    form lf, d1 or d2 estimator types.  Nevertheless, differences in standard
    errors of the estimates between firthlogit and other software packages
    are very minor. At least one of the latter uses the unpenalized Hessian
    in the Newton-Raphson algorithm in order to avoid resorting to numerical
    derivatives there.

    When the method is used in fitting logistic models in datasets giving
    rise to separation, the affected estimate is typically approaching a
    boundary condition.  As a result, the likelihood profile is often
    asymmetric under these conditions; Wald tests and confidence intervals
    are liable to be inaccurate. In these circumstances, Heinze and coworkers
    recommend using likelihood ratio tests and profile likelihood confidence
    intervals in lieu of Wald-based statistics. Calculation of likelihood
    ratio test statistics with the method is done differently by Heinze and
    coworkers from what is conventionally done: instead of omitting the
    variable of interest and refitting the reduced model, the coefficient of
    interest is constrained to zero and left in the model in order to allow
    its contributing to the penalization. The test statistic is then computed
    as twice the difference in penalized log likelihood values of the
    unconstrained and constrained models by lrtest in a manner directly
    analogous to that of conventional likelihood ratio tests.

    The penalization that allows for convergence to finite estimates in
    conditions of separation also allows convergence to finite estimates with
    very sparse data.  In these circumstances, the penalization tends to
    over-correct for bias.


Examples

    . webuse hiv1

    . firthlogit hiv cd4 cd8

    . firthlogit, or

    . estimates store Full

    . constraint define 1 cd4

    . firthlogit hiv cd4 cd8, constraint(1)

    . lrtest Full .


References

    Firth, D. 1993. Bias reduction of maximum likelihood estimates.
    Biometrika 80:27-38.

    Heinze, G. and Schemper, M. 2002. A solution to the problem of separation
    in logistic regression. Statistics in Medicine 21:2409-19.


Acknowledgements

    Jeff Pitblado provided a valuable pointer in displaying the results
    correctly. The command is named so as to acknowledge David Firth as the
    source of the method.  Note that Professor Firth is not otherwise
    associated with or responsible for this command: contact the author
    (below) to report bugs or other problems with the command.


Author

    Joseph Coveney jcoveney@bigplanet.com


Also see

    Manual:  [R] exlogistic