R: Bias reduction in Binomial-response GLMs

brglm {brglm} | R Documentation

## Bias reduction in Binomial-response GLMs

### Description

Fits binomial-response GLMs using the bias-reduction method developed in Firth (1993) for the removal of the leading (O(n^{-1})) term from the asymptotic expansion of the bias of the maximum likelihood estimator. Fitting is performed using pseudo-data representations, as described in Kosmidis (2007, Chapter 5). For estimation in binomial-response GLMs, the bias-reduction method is an improvement over traditional maximum likelihood because:

- the bias-reduced estimator is second-order unbiased and has smaller variance than the maximum likelihood estimator and

- the resultant estimates and their corresponding standard errors are always finite while the maximum likelihood estimates can be infinite (in situations where complete or quasi separation occurs).

### Usage

    brglm(formula, family = binomial, data, weights, subset, na.action,
          start = NULL, etastart, mustart, offset,
          control.glm = glm.control1(...), model = TRUE, method = "brglm.fit",
          pl = FALSE, x = FALSE, y = TRUE, contrasts = NULL,
          control.brglm = brglm.control(...), ...)

    brglm.fit(x, y, weights = rep(1, nobs), start = NULL, etastart = NULL,
              mustart = NULL, offset = rep(0, nobs), family = binomial(),
              control = glm.control(), control.brglm = brglm.control(),
              intercept = TRUE, pl = FALSE)

### Arguments

`formula` | as in `glm`.
`family` | as in `glm`. `brglm` currently supports only the `"binomial"` family with links `"logit"`, `"probit"`, `"cloglog"`, `"cauchit"`.
`data` | as in `glm`.
`weights` | as in `glm`.
`subset` | as in `glm`.
`na.action` | as in `glm`.
`start` | as in `glm`.
`etastart` | as in `glm`.
`mustart` | as in `glm`.
`offset` | as in `glm`.
`control.glm` | `control.glm` replaces the `control` argument in `glm` but essentially does the same job. It is a list of parameters to control `glm.fit`. See the documentation of `glm.control1` for details.
`control` | same as in `glm`. Only available to `brglm.fit`.
`intercept` | as in `glm`.
`model` | as in `glm`.
`method` | the method to be used for fitting the model. The default method is `"brglm.fit"`, which uses either the modified-scores approach to estimation or maximum penalized likelihood (see the `pl` argument below). The standard `glm` methods `"glm.fit"` for maximum likelihood and `"model.frame"` for returning the model frame without any fitting, are also accepted.
`pl` | a logical value indicating whether the model should be fitted using maximum penalized likelihood, where the penalization is done using Jeffreys invariant prior, or using the bias-reducing modified scores. It is only used when `method = "brglm.fit"`. The default value is `FALSE` (see also the Details section).
`x` | as in `glm`.
`y` | as in `glm`.
`contrasts` | as in `glm`.
`control.brglm` | a list of parameters for controlling the fitting process when `method = "brglm.fit"`. See documentation of `brglm.control` for details.
`...` | further arguments passed to or from other methods.

### Details

`brglm.fit` is the workhorse function for fitting the model using either the bias-reduction method or maximum penalized likelihood. If `method = "glm.fit"`, usual maximum likelihood is used via `glm.fit`.

The main iteration of `brglm.fit` consists of the following steps:

1. Calculate the diagonal components of the hat matrix (see `gethats` and `hatvalues`).

2. Obtain the pseudo-data representation at the current value of the parameters (see `modifications` for more information).

3. Fit a local GLM, using `glm.fit` on the pseudo data.

4. Adjust the quadratic weights to agree with the original binomial totals.

Iteration is repeated until either the iteration limit has been reached or the sum of the absolute values of the modified scores is less than some specified positive constant (see the `br.maxit` and `br.epsilon` arguments in `brglm.control`).

The default value (`FALSE`) of `pl`, when `method = "brglm.fit"`, results in estimates that are free of any O(n^{-1}) terms in the asymptotic expansion of their bias. When `pl = TRUE` bias-reduction is again achieved but generally not at such order of magnitude. In the case of logistic regression the value of `pl` is irrelevant since maximum penalized likelihood and the modified-scores approach coincide for natural exponential families (see Firth, 1993).

For other language related details see the details section in `glm`.

### Value

`brglm` returns an object of class `"brglm"`. A `"brglm"` object inherits first from `"glm"` and then from `"lm"` and is a list containing the following components:

`coefficients` | as in `glm`.
`residuals` | as in `glm`.
`fitted.values` | as in `glm`.
`effects` | as in `glm`.
`R` | as in `glm`.
`rank` | as in `glm`.
`qr` | as in `glm`.
`family` | as in `glm`.
`linear.predictors` | as in `glm`.
`deviance` | as in `glm`.
`aic` | as in `glm` (see Details).
`null.deviance` | as in `glm`.
`iter` | as in `glm`.
`weights` | as in `glm`.
`prior.weights` | as in `glm`.
`df.residual` | as in `glm`.
`df.null` | as in `glm`.
`y` | as in `glm`.
`converged` | as in `glm`.
`boundary` | as in `glm`.
`ModifiedScores` | the vector of the modified scores for the parameters at the final iteration. If `pl = TRUE` they are the derivatives of the penalized likelihood at the final iteration.
`FisherInfo` | the Fisher information matrix evaluated at the resultant estimates. Only available when `method = "brglm.fit"`.
`hats` | the diagonal elements of the hat matrix. Only available when `method = "brglm.fit"`
`nIter` | the number of iterations that were required until convergence. Only available when `method = "brglm.fit"`.
`cur.model` | a list with components `ar` and `at` which contains the values of the additive modifications to the responses (`y`) and to the binomial totals (`prior.weights`) at the resultant estimates (see `modifications` for more information). Only available when `method = "brglm.fit"`.
`model` | as in `glm`.
`call` | as in `glm`.
`formula` | as in `glm`.
`terms` | as in `glm`.
`data` | as in `glm`.
`offset` | as in `glm`.
`control.glm` | as `control` in the result of `glm`.
`control.brglm` | the `control.brglm` argument that was passed to `brglm`. Only available when `method = "brglm.fit"`.
`method` | the method used for fitting the model.
`contrasts` | as in `glm`.
`xlevels` | as in `glm`.
`pl` | logical having the same value with the `pl` argument passed to `brglm`. Only available when `method = "brglm.fit"`.

### Warnings

1. It is not advised to use methods associated with model comparison (`add1`, `drop1`, `anova`, etc.) on objects of class `"brglm"`. Model comparison when estimation is performed using the modified scores or the penalized likelihood is an on-going research topic and will be implemented as soon as it is concluded.

2. The use of Akaike's information criterion (AIC) for model selection when `method = "brglm.fit"` is controversial. AIC was developed under the assumptions that (i) estimation is by maximum likelihood and (ii) that estimation is carried out in a parametric family of distributions that contains the “true” model. At least the first assumption is not valid when using `method = "brglm.fit"`. However, since the MLE is asymptotically unbiased, asymptotically the modified-scores approach is equivalent to maximum likelihood. A more appropriate information criterion seems to be Konishi's generalized information criterion (see Konishi & Kitagawa, 1996, Sections 3.2 and 3.3), which will be implemented in a future version.

### Note

1. Supported methods for objects of class `"brglm"` are:

- `print`through `print.brglm`.

- `summary`through `summary.brglm`.

- `coefficients`inherited from the `"glm"` class.

- `vcov`inherited from the`"glm"` class.

- `predict`inherited from the`"glm"` class.

- `residuals`inherited from the`"glm"` class.

- and other methods that apply to objects of class `"glm"`

2. A similar implementation of the bias-reduction method could be done for every GLM, following Kosmidis (2007) (see also Kosmidis and Firth, 2009). The full set of families and links will be available in a future version. However, bias-reduction is not generally beneficial as it is in the binomial family and it could cause inflation of the variance (see Firth, 1993).

3. Basically, the differences between maximum likelihood, maximum penalized likelihood and the modified scores approach are more apparent in small sample sizes, in sparse data sets and in cases where complete or quasi-complete separation occurs. Asymptotically (as n goes to infinity), the three different approaches are equivalent to first order.

4. When an offset is not present in the model, the modified-scores based estimates are usually smaller in magnitude than the corresponding maximum likelihood estimates, shrinking towards the origin of the scale imposed by the link function. Thus, the corresponding estimated asymptotic standard errors are also smaller.

The same is true for the maximum penalized likelihood estimates when for example, the logit (where the maximum penalized likelihood and modified-scores approaches coincide) or the probit links are used. However, generally the maximum penalized likelihood estimates do not shrink towards the origin. In terms of mean-value parameterization, in the case of maximum penalized likelihood the fitted probabilities would shrink towards the point where the Jeffreys prior is maximized or equivalently where the quadratic weights are simultaneously maximized (see Kosmidis, 2007).

5. Implementations of the bias-reduction method for logistic regressions can also be found in the logistf package. In addition to the obvious advantage of `brglm` in the range of link functions that can be used (`"logit"`, `"probit"`, `"cloglog"` and `"cauchit"`), `brglm` is also more efficient computationally. Furthermore, for any user-specified link function (see the Example section of `family`), the user can specify the corresponding pseudo-data representation to be used within `brglm` (see `modifications` for details).

### Author(s)

Ioannis Kosmidis, i.kosmidis@ucl.ac.uk

### References

Bull, S. B., Lewinger, J. B. and Lee, S. S. F. (2007). Confidence intervals for multinomial logistic regression in sparse data. Statistics in Medicine 26, 903–918.

Firth, D. (1992) Bias reduction, the Jeffreys prior and GLIM. In Advances in GLIM and statistical modelling: Proceedings of the GLIM 92 conference, Munich, Eds. L.~Fahrmeir, B.~Francis, R.~Gilchrist and G.Tutz, pp. 91–100. New York: Springer.

Firth, D. (1992) Generalized linear models and Jeffreys priors: An iterative generalized least-squares approach. In Computational Statistics I, Eds. Y. Dodge and J. Whittaker. Heidelberg: Physica-Verlag.

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, 2409–2419.

Konishi, S. and Kitagawa, G. (1996). Generalised information criteria in model selection. Biometrika 83, 875–890.

Kosmidis, I. (2007). Bias reduction in exponential family nonlinear models. PhD Thesis, Department of Statistics, University of Warwick.

Kosmidis, I. and Firth, D. (2009). Bias reduction in exponential family nonlinear models. Biometrika 96, 793–804.

### See Also

`glm`, `glm.fit`

### Examples

    ## Begin Example
    data(lizards)
    # Fit the GLM using maximum likelihood
    lizards.glm <- brglm(cbind(grahami, opalinus) ~ height + diameter +
                      light + time, family = binomial(logit), data=lizards,
                      method = "glm.fit")
    # Now the bias-reduced fit:
    lizards.brglm <- brglm(cbind(grahami, opalinus) ~ height + diameter +
                      light + time, family = binomial(logit), data=lizards,
                      method = "brglm.fit")
    lizards.glm
    lizards.brglm
    # Other links
    update(lizards.brglm, family = binomial(probit))
    update(lizards.brglm, family = binomial(cloglog))
    update(lizards.brglm, family = binomial(cauchit))
    # Using penalized maximum likelihood
    update(lizards.brglm, family = binomial(probit), pl = TRUE)
    update(lizards.brglm, family = binomial(cloglog), pl = TRUE)
    update(lizards.brglm, family = binomial(cauchit), pl = TRUE)

[Package brglm version 0.5-9 Index]