R: Bias reduction for multinomial response models using the...

brmultinom {brglm2} | R Documentation

## Bias reduction for multinomial response models using the Poisson trick.

### Description

`brmultinom` is a wrapper of `brglmFit` that fits multinomial regression models using implicit and explicit bias reduction methods. See Kosmidis & Firth (2011) for details.

### Usage

    brmultinom(
      formula,
      data,
      weights,
      subset,
      na.action,
      contrasts = NULL,
      ref = 1,
      model = TRUE,
      x = TRUE,
      control = list(...),
      ...
    )

### Arguments

`formula` | a formula expression as for regression models, of the form `response ~ predictors`. The response should be a factor or a matrix with K columns, which will be interpreted as counts for each of K classes. A log-linear model is fitted, with coefficients zero for the first class. An offset can be included: it should be a numeric matrix with K columns if the response is either a matrix with K columns or a factor with K >= 2 classes, or a numeric vector for a response factor with 2 levels. See the documentation of `formula()` for other details.
`data` | an optional data frame in which to interpret the variables occurring in `formula`.
`weights` | optional case weights in fitting.
`subset` | expression saying which subset of the rows of the data should be used in the fit. All observations are included by default.
`na.action` | a function to filter missing data.
`contrasts` | a list of contrasts to be used for some or all of the factors appearing as variables in the model formula.
`ref` | the reference category to use for multinomial regression. Either an integer, in which case levels(response)[ref] is used as a baseline, or a character string. Default is 1.
`model` | logical. If true, the model frame is saved as component `model` of the returned object.
`x` | should the model matrix be included with in the result (default is `TRUE`).
`control` | a list of parameters for controlling the fitting process. See `brglmControl` for details.
`...` | arguments to be used to form the default 'control' argument if it is not supplied directly.

### Details

The models `brmultinom` handles are also known as baseline-category logit models (see, Agresti, 2002, Section 7.1), because they model the log-odds of every category against a baseline category. The user can control which baseline (or reference) category is used via the `ref`. By default `brmultinom` uses the first category as reference.

The maximum likelihood estimates for the parameters of baseline-category logit models have infinite components with positive probability, which can result in problems in their estimation and the use of inferential procedures (e.g. Wad tests). Albert and Andreson (1984) have categorized the possible data patterns for such models into the exclusive and exhaustive categories of complete separation, quasi-complete separation and overlap, and showed that infinite maximum likelihood estimates result when complete or quasi-complete separation occurs.

The adjusted score approach to bias reduction that `brmultinom` implements (`type = "AS_mean"`) is an alternative to maximum likelihood that results in estimates with smaller asymptotic bias that are also *always* finite, even in cases of complete or quasi-complete separation.

`brmultinom` is a wrapper of `brglmFit` that fits multinomial logit regression models through the 'Poisson trick' (see, for example, Palmgren, 1981; Kosmidis & Firth, 2011).

The implementation relies on the construction of an 'extended' model matrix for the log-linear model and constraints on the sums of the Poisson means. Specifically, a log-linear model is fitted on a Kronecker product (https://en.wikipedia.org/wiki/Kronecker_product) of the original model matrix `X` implied by the formula, augmented by `nrow(X)` dummy variables.

The extended model matrix is sparse, and the Matrix package is used for its effective storage.

While `brmultinom` can be used for analyses using multinomial regression models, the current implementation is more of a 'proof of concept' and is not expected to scale well with either of `nrow(X)`, `ncol(X)` or the number of levels in the categorical response.

### Author(s)

Ioannis Kosmidis ioannis.kosmidis@warwick.ac.uk

### References

Kosmidis I, Kenne Pagui E C, Sartori N (2020). Mean and median bias reduction in generalized linear models. *Statistics and Computing*, **30**, 43-59 https://doi.org/10.1007/s11222-019-09860-6

Agresti A (2002). *Categorical data analysis* (2nd edition). Wiley Series in Probability and Statistics. Wiley.

Albert A, Anderson J A (1984). On the Existence of Maximum Likelihood Estimates in Logistic Regression Models. *Biometrika*, **71** 1–10, https://doi.org/10.2307/2336390

Kosmidis I, Firth D (2011). Multinomial logit bias reduction via the Poisson log-linear model. *Biometrika*, **98**, 755-759 http://dx.doi.org/10.1093/biomet/asr026

Palmgren, J (1981). The Fisher Information Matrix for Log Linear Models Arguing Conditionally on Observed Explanatory Variables. *Biometrika*, **68**, 563-566 https://doi.org/10.1093/biomet/68.2.563

### See Also

`multinom`, `bracl` for adjacent category logit models for ordinal responses

### Examples

    data("housing", package = "MASS")

    # Maximum likelihood using nnet::multinom
    houseML1nnet <- nnet::multinom(Sat ~ Infl + Type + Cont, weights = Freq,
                                   data = housing)
    # Maximum likelihood using brmultinom with baseline category 'Low'
    houseML1 <- brmultinom(Sat ~ Infl + Type + Cont, weights = Freq,
                           data = housing, type = "ML", ref = 1)
    # The estimates are numerically the same as houseML0
    all.equal(coef(houseML1nnet), coef(houseML1), tolerance = 1e-04)

    # Maximum likelihood using brmultinom with 'High' as baseline
    houseML3 <- brmultinom(Sat ~ Infl + Type + Cont, weights = Freq,
                          data = housing, type = "ML", ref = 3)
    # The fitted values are the same as houseML1
    all.equal(fitted(houseML3), fitted(houseML1), tolerance = 1e-10)

    # Bias reduction
    houseBR3 <- update(houseML3, type = "AS_mean")
    # Bias correction
    houseBC3 <- update(houseML3, type = "correction")

[Package brglm2 version 0.7.0 Index]