# glmrob: Robust Fitting of Generalized Linear Models In robustbase: Basic Robust Statistics

## Description

`glmrob` is used to fit generalized linear models by robust methods. The models are specified by giving a symbolic description of the linear predictor and a description of the error distribution. Currently, robust methods are implemented for ```family = binomial```, ` = poisson`, ` = Gamma` and ` = gaussian`.

## Usage

 ```1 2 3 4``` ```glmrob(formula, family, data, weights, subset, na.action, start = NULL, offset, method = c("Mqle", "BY", "WBY", "MT"), weights.on.x = c("none", "hat", "robCov", "covMcd"), control = NULL, model = TRUE, x = FALSE, y = TRUE, contrasts = NULL, trace.lev = 0, ...) ```

## Arguments

 `formula` a `formula`, i.e., a symbolic description of the model to be fit (cf. `glm` or `lm`). `family` a description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family `function` or the result of a call to a family function. (See `family` for details of family functions.) `data` an optional data frame containing the variables in the model. If not found in `data`, the variables are taken from `environment(formula)`, typically the environment from which `glmrob` is called. `weights` an optional vector of weights to be used in the fitting process. `subset` an optional vector specifying a subset of observations to be used in the fitting process. `na.action` a function which indicates what should happen when the data contain `NA`s. The default is set by the `na.action` setting in `options`. The “factory-fresh” default is `na.omit`. `start` starting values for the parameters in the linear predictor. Note that specifying `start` has somewhat different meaning for the different `method`s. Notably, for `"MT"`, this skips the expensive computation of initial estimates via sub samples, but needs to be robust itself. `offset` this can be used to specify an a priori known component to be included in the linear predictor during fitting. `method` a character string specifying the robust fitting method. The details of method specification are given below. `weights.on.x` a character string (can be abbreviated), a `function` or `list` (see below), or a numeric vector of length `n`, specifying how points (potential outliers) in x-space are downweighted. If `"hat"`, weights on the design of the form √{1-h_{ii}} are used, where h_{ii} are the diagonal elements of the hat matrix. If `"robCov"`, weights based on the robust Mahalanobis distance of the design matrix (intercept excluded) are used where the covariance matrix and the centre is estimated by `cov.rob` from the package MASS. Similarly, if `"covMcd"`, robust weights are computed using `covMcd`. The default is `"none"`. If `weights.on.x` is a `function`, it is called with arguments `(X, intercept)` and must return an n-vector of non-negative weights. If it is a `list`, it must be of length one, and as element contain a function much like `covMcd()` or `cov.rob()` (package MASS), which computes multivariate location and “scatter” of a data matrix `X`. `control` a list of parameters for controlling the fitting process. See the documentation for `glmrobMqle.control` for details. `model` a logical value indicating whether model frame should be included as a component of the returned value. `x, y` logical values indicating whether the response vector and model matrix used in the fitting process should be returned as components of the returned value. `contrasts` an optional list. See the `contrasts.arg` of `model.matrix.default`. `trace.lev` logical (or integer) indicating if intermediate results should be printed; defaults to `0` (the same as `FALSE`). `...` arguments passed to `glmrobMqle.control` when `control` is `NULL` (as per default).

## Details

`method="model.frame"` returns the `model.frame()`, the same as `glm()`.
`method="Mqle"` fits a generalized linear model using Mallows or Huber type robust estimators, as described in Cantoni and Ronchetti (2001) and Cantoni and Ronchetti (2006). In contrast to the implementation described in Cantoni (2004), the pure influence algorithm is implemented.
`method="WBY"` and `method="BY"`, available for logistic regression (`family = binomial`) only, call `BYlogreg(*, initwml= . )` for the (weighted) Bianco-Yohai estimator, where `initwml` is true for `"WBY"`, and false for `"BY"`.
`method="MT"`, currently only implemented for `family = poisson`, computes an “[M]-Estimator based on [T]ransformation”, by Valdora and Yohai (2013), via (hidden internal) `glmrobMT()`; as that uses `sample()`, from R version 3.6.0 it depends on `RNGkind(*, sample.kind)`. Exact reproducibility of results from R versions 3.5.3 and earlier, requires setting `RNGversion("3.5.0")`.

`weights.on.x= "robCov"` makes sense if all explanatory variables are continuous.

In the cases,where `weights.on.x` is `"covMcd"` or `"robCov"`, or list with a “robCov” function, the mahalanobis distances `D^2` are computed with respect to the covariance (location and scatter) estimate, and the weights are `1/sqrt(1+ pmax.int(0, 8*(D2 - p)/sqrt(2*p)))`, where `D2 = D^2` and `p = ncol(X)`.

## Value

`glmrob` returns an object of class `"glmrob"` and is also inheriting from `glm`.
The `summary` method, see `summary.glmrob`, can be used to obtain or print a summary of the results.
The generic accessor functions `coefficients`, `effects`, `fitted.values` and `residuals` (see `residuals.glmrob`) can be used to extract various useful features of the value returned by `glmrob()`.

An object of class `"glmrob"` is a list with at least the following components:

 `coefficients` a named vector of coefficients `residuals` the working residuals, that is the (robustly “huberized”) residuals in the final iteration of the IWLS fit. `fitted.values` the fitted mean values, obtained by transforming the linear predictors by the inverse of the link function. `w.r` robustness weights for each observations; i.e., `residuals` * `w.r` equals the psi-function of the Preason's residuals. `w.x` weights used to down-weight observations based on the position of the observation in the design space. `dispersion` robust estimation of dispersion paramter if appropriate `cov` the estimated asymptotic covariance matrix of the estimated coefficients. `tcc` the tuning constant c in Huber's psi-function. `family` the `family` object used. `linear.predictors` the linear fit on link scale. `deviance` NULL; Exists because of compatipility reasons. `iter` the number of iterations used by the influence algorithm. `converged` logical. Was the IWLS algorithm judged to have converged? `call` the matched call. `formula` the formula supplied. `terms` the `terms` object used. `data` the `data argument`. `offset` the offset vector used. `control` the value of the `control` argument used. `method` the name of the robust fitter function used. `contrasts` (where relevant) the contrasts used. `xlevels` (where relevant) a record of the levels of the factors used in fitting.

## Author(s)

Andreas Ruckstuhl ("Mqle") and Martin Maechler

## References

Eva Cantoni and Elvezio Ronchetti (2001) Robust Inference for Generalized Linear Models. JASA 96 (455), 1022–1030.

Eva Cantoni (2004) Analysis of Robust Quasi-deviances for Generalized Linear Models. Journal of Statistical Software, 10, http://www.jstatsoft.org/v10/i04

Eva Cantoni and Elvezio Ronchetti (2006) A robust approach for skewed and heavy-tailed outcomes in the analysis of health care expenditures. Journal of Health Economics 25, 198–213.

S. Heritier, E. Cantoni, S. Copt, M.-P. Victoria-Feser (2009) Robust Methods in Biostatistics. Wiley Series in Probability and Statistics.

Marina Valdora and Víctor J. Yohai (2013) Robust estimators for Generalized Linear Models. In progress.

`predict.glmrob` for prediction; `glmrobMqle.control`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84``` ```## Binomial response -------------- data(carrots) Cfit1 <- glm(cbind(success, total-success) ~ logdose + block, data = carrots, family = binomial) summary(Cfit1) Rfit1 <- glmrob(cbind(success, total-success) ~ logdose + block, family = binomial, data = carrots, method= "Mqle", control= glmrobMqle.control(tcc=1.2)) summary(Rfit1) Rfit2 <- glmrob(success/total ~ logdose + block, weights = total, family = binomial, data = carrots, method= "Mqle", control= glmrobMqle.control(tcc=1.2)) coef(Rfit2) ## The same as Rfit1 ## Binary response -------------- data(vaso) Vfit1 <- glm(Y ~ log(Volume) + log(Rate), family=binomial, data=vaso) coef(Vfit1) Vfit2 <- glmrob(Y ~ log(Volume) + log(Rate), family=binomial, data=vaso, method="Mqle", control = glmrobMqle.control(tcc=3.5)) coef(Vfit2) # c = 3.5 ==> not much different from classical ## Note the problems with tcc <= 3 %% FIXME algorithm ??? Vfit3 <- glmrob(Y ~ log(Volume) + log(Rate), family=binomial, data=vaso, method= "BY") coef(Vfit3)## note that results differ much. ## That's not unreasonable however, see Kuensch et al.(1989), p.465 ## Poisson response -------------- data(epilepsy) Efit1 <- glm(Ysum ~ Age10 + Base4*Trt, family=poisson, data=epilepsy) summary(Efit1) Efit2 <- glmrob(Ysum ~ Age10 + Base4*Trt, family = poisson, data = epilepsy, method= "Mqle", control = glmrobMqle.control(tcc= 1.2)) summary(Efit2) ## 'x' weighting: (Efit3 <- glmrob(Ysum ~ Age10 + Base4*Trt, family = poisson, data = epilepsy, method= "Mqle", weights.on.x = "hat", control = glmrobMqle.control(tcc= 1.2))) try( # gives singular cov matrix: 'Trt' is binary factor --> # affine equivariance and subsampling are problematic Efit4 <- glmrob(Ysum ~ Age10 + Base4*Trt, family = poisson, data = epilepsy, method= "Mqle", weights.on.x = "covMcd", control = glmrobMqle.control(tcc=1.2, maxit=100)) ) ##--> See example(possumDiv) for another Poisson-regression ### -------- Gamma family -- data from example(glm) --- clotting <- data.frame( u = c(5,10,15,20,30,40,60,80,100), lot1 = c(118,58,42,35,27,25,21,19,18), lot2 = c(69,35,26,21,18,16,13,12,12)) summary(cl <- glm (lot1 ~ log(u), data=clotting, family=Gamma)) summary(ro <- glmrob(lot1 ~ log(u), data=clotting, family=Gamma)) clotM5.high <- within(clotting, { lot1 <- 60 }) op <- par(mfrow=2:1, mgp = c(1.6, 0.8, 0), mar = c(3,3:1)) plot( lot1 ~ log(u), data=clotM5.high) plot(1/lot1 ~ log(u), data=clotM5.high) par(op) ## Obviously, there the first observation is an outlier with respect to both ## representations! cl5.high <- glm (lot1 ~ log(u), data=clotM5.high, family=Gamma) ro5.high <- glmrob(lot1 ~ log(u), data=clotM5.high, family=Gamma) with(ro5.high, cbind(w.x, w.r))## the 5th obs. is downweighted heavily! plot(1/lot1 ~ log(u), data=clotM5.high) abline(cl5.high, lty=2, col="red") abline(ro5.high, lwd=2, col="blue") ## result is ok (but not "perfect") ```