vglm {VGAM} | R Documentation

## Fitting Vector Generalized Linear Models

### Description

`vglm` is used to fit vector generalized linear models (VGLMs). This is a large class of models that includes generalized linear models (GLMs) as a special case.

### Usage

    vglm(formula, family, data = list(), weights = NULL, subset = NULL,
         na.action = na.fail, etastart = NULL, mustart = NULL,
         coefstart = NULL, control = vglm.control(...), offset = NULL,
         method = "vglm.fit", model = FALSE, x.arg = TRUE, y.arg = TRUE,
         contrasts = NULL, constraints = NULL, extra = list(),
         form2 = NULL, qr.arg = FALSE, smart = TRUE, ...)

### Arguments

In the following, M is the number of linear predictors.

`formula` | a symbolic description of the model to be fit. The RHS of the formula is applied to each linear predictor. Different variables in each linear predictor can be chosen by specifying constraint matrices.
`family` | a function of class `"vglmff"` (see `vglmff-class`) describing what statistical model is to be fitted. This is called a ``VGAM family function''. See `CommonVGAMffArguments` for general information about many types of arguments found in this type of function.
`data` | an optional data frame containing the variables in the model. By default the variables are taken from `environment(formula)`, typically the environment from which `vglm` is called.
`weights` | an optional vector or matrix of (prior) weights to be used in the fitting process. If `weights` is a matrix, then it must be in matrix-band form, whereby the first M columns of the matrix are the diagonals, followed by the upper-diagonal band, followed by the band above that, etc. In this case, there can be up to M(M+1) columns, with the last column corresponding to the (1,M) elements of the weight matrices.
`subset` | an optional logical 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 of `options`, and is `na.fail` if that is unset. The ``factory-fresh'' default is `na.omit`.
`etastart` | starting values for the linear predictors. It is a M-column matrix. If M=1 then it may be a vector.
`mustart` | starting values for the fitted values. It can be a vector or a matrix. Some family functions do not make use of this argument.
`coefstart` | starting values for the coefficient vector.
`control` | a list of parameters for controlling the fitting process. See `vglm.control` for details.
`offset` | a vector or M-column matrix of offset values. These are a priori known and are added to the linear predictors during fitting.
`method` | the method to be used in fitting the model. The default (and presently only) method `vglm.fit` uses iteratively reweighted least squares (IRLS).
`model` | a logical value indicating whether the model frame should be assigned in the `model` slot.
`x.arg, y.arg` | logical values indicating whether the model matrix and response vector/matrix used in the fitting process should be assigned in the `x` and `y` slots. Note the model matrix is the LM model matrix; to get the VGLM model matrix type `model.matrix(vglmfit)` where `vglmfit` is a `vglm` object.
`contrasts` | an optional list. See the `contrasts.arg` of `model.matrix.default`.
`constraints` | an optional list of constraint matrices. The components of the list must be named with the term it corresponds to (and it must match in character format exactly). Each constraint matrix must have M rows, and be of full-column rank. By default, constraint matrices are the M by M identity matrix unless arguments in the family function itself override these values. If `constraints` is used it must contain all the terms; an incomplete list is not accepted.
`extra` | an optional list with any extra information that might be needed by the VGAM family function.
`form2` | The second (optional) formula. If argument `xij` is used (see `vglm.control`) then `form2` needs to have all terms in the model. Also, some VGAM family functions such as `micmen` use this argument to input the regressor variable. If given, the slots `@Xm2` and `@Ym2` may be assigned. Note that smart prediction applies to terms in `form2` too.
`qr.arg` | logical value indicating whether the slot `qr`, which returns the QR decomposition of the VLM model matrix, is returned on the object.
`smart` | logical value indicating whether smart prediction (`smartpred`) will be used.
`...` | further arguments passed into `vglm.control`.

### Details

A vector generalized linear model (VGLM) is loosely defined as a statistical model that is a function of M linear predictors. The central formula is given by

eta_j = beta_j^T x

where x is a vector of explanatory variables (sometimes just a 1 for an intercept), and beta_j is a vector of regression coefficients to be estimated. Here, j=1,...,M, where M is finite. Then one can write eta=(eta_1,...,eta_M)^T as a vector of linear predictors.

Most users will find `vglm` similar in flavour to `glm`. The function `vglm.fit` actually does the work.

### Value

An object of class `"vglm"`, which has the following slots. Some of these may not be assigned to save space, and will be recreated if necessary later.

`extra` | the list `extra` at the end of fitting.
`family` | the family function (of class `"vglmff"`).
`iter` | the number of IRLS iterations used.
`predictors` | a M-column matrix of linear predictors.
`assign` | a named list which matches the columns and the (LM) model matrix terms.
`call` | the matched call.
`coefficients` | a named vector of coefficients.
`constraints` | a named list of constraint matrices used in the fitting.
`contrasts` | the contrasts used (if any).
`control` | list of control parameter used in the fitting.
`criterion` | list of convergence criterion evaluated at the final IRLS iteration.
`df.residual` | the residual degrees of freedom.
`df.total` | the total degrees of freedom.
`dispersion` | the scaling parameter.
`effects` | the effects.
`fitted.values` | the fitted values, as a matrix. This is often the mean but may be quantiles, or the location parameter, e.g., in the Cauchy model.
`misc` | a list to hold miscellaneous parameters.
`model` | the model frame.
`na.action` | a list holding information about missing values.
`offset` | if non-zero, a M-column matrix of offsets.
`post` | a list where post-analysis results may be put.
`preplot` | used by `plotvgam`, the plotting parameters may be put here.
`prior.weights` | initially supplied weights (the `weights` argument). Also see `weightsvglm`.
`qr` | the QR decomposition used in the fitting.
`R` | the R matrix in the QR decomposition used in the fitting.
`rank` | numerical rank of the fitted model.
`residuals` | the working residuals at the final IRLS iteration.
`rss` | residual sum of squares at the final IRLS iteration with the adjusted dependent vectors and weight matrices.
`smart.prediction` | a list of data-dependent parameters (if any) that are used by smart prediction.
`terms` | the `terms` object used.
`weights` | the working weight matrices at the final IRLS iteration. This is in matrix-band form.
`x` | the model matrix (linear model LM, not VGLM).
`xlevels` | the levels of the factors, if any, used in fitting.
`y` | the response, in matrix form.

This slot information is repeated at `vglm-class`.

### Note

This function can fit a wide variety of statistical models. Some of these are harder to fit than others because of inherent numerical difficulties associated with some of them. Successful model fitting benefits from cumulative experience. Varying the values of arguments in the VGAM family function itself is a good first step if difficulties arise, especially if initial values can be inputted. A second, more general step, is to vary the values of arguments in `vglm.control`. A third step is to make use of arguments such as `etastart`, `coefstart` and `mustart`.

Some VGAM family functions end in `"ff"` to avoid interference with other functions, e.g., `binomialff`, `poissonff`, `gaussianff`, `gammaff`. This is because VGAM family functions are incompatible with `glm` (and also `gam` in the gam library and `gam` in the mgcv library).

The smart prediction (`smartpred`) library is incorporated within the VGAM library.

The theory behind the scaling parameter is currently being made more rigorous, but it it should give the same value as the scale parameter for GLMs.

In Example 5 below, the `xij` argument to illustrate covariates that are specific to a linear predictor. Here, `lop`/`rop` are the ocular pressures of the left/right eye (artificial data). Variables `leye` and `reye` might be the presence/absence of a particular disease on the LHS/RHS eye respectively. See `vglm.control` and `fill` for more details and examples.

### Author(s)

Thomas W. Yee

### References

Yee, T. W. and Hastie, T. J. (2003) Reduced-rank vector generalized linear models. Statistical Modelling, 3, 15–41.

Yee, T. W. and Wild, C. J. (1996) Vector generalized additive models. Journal of the Royal Statistical Society, Series B, Methodological, 58, 481–493.

Yee, T. W. (2008) The `VGAM` Package. R News, 8, 28–39.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

### See Also

`vglm.control`, `vglm-class`, `vglmff-class`, `smartpred`, `vglm.fit`, `fill`, `rrvglm`, `vgam`. Methods functions include `coef.vlm`, `predict.vglm`, `summary.vglm`, `AIC.vglm`, etc.

### Examples

    # Example 1. Dobson (1990) Page 93: Randomized Controlled Trial :
    counts = c(18,17,15,20,10,20,25,13,12)
    outcome = gl(3,1,9)
    treatment = gl(3,3)
    print(d.AD <- data.frame(treatment, outcome, counts))
    vglm.D93 = vglm(counts ~ outcome + treatment, family=poissonff)
    summary(vglm.D93)

    # Example 2. Multinomial logit model
    pneumo = transform(pneumo, let=log(exposure.time))
    vglm(cbind(normal, mild, severe) ~ let, multinomial, pneumo)

    # Example 3. Proportional odds model
    fit3 = vglm(cbind(normal,mild,severe) ~ let, trace=TRUE,
                cumulative(parallel=TRUE, reverse=TRUE), pneumo)
    coef(fit3, matrix=TRUE)
    constraints(fit3)
    model.matrix(fit3, type="lm") # LM model matrix
    model.matrix(fit3)            # Larger VGLM (or VLM) model matrix

    # Example 4. Bivariate logistic model
    fit4 = vglm(cbind(nBnW, nBW, BnW, BW) ~ age, binom2.or, coalminers)
    coef(fit4, matrix=TRUE)
    fit4@y  # Response are proportions
    weights(fit4, type="prior")

    # Example 5. The use of the xij argument (simple case).
    # The constraint matrix for 'op' has one column.
    nn = 1000
    eyesdat = round(data.frame(lop = runif(nn),
                               rop = runif(nn),
                                op = runif(nn)), dig=2)
    eyesdat = transform(eyesdat, eta1 = -1+2*lop,
                                 eta2 = -1+2*lop)
    eyesdat = transform(eyesdat,
                        leye = rbinom(nn, size=1, prob=logit(eta1, inv=TRUE)),
                        reye = rbinom(nn, size=1, prob=logit(eta2, inv=TRUE)))
    fit5 = vglm(cbind(leye,reye) ~ op,
                binom2.or(exchangeable=TRUE, zero=3),
                data=eyesdat, trace=TRUE,
                xij = list(op ~ lop + rop + fill(lop)),
                form2 = ~  op + lop + rop + fill(lop))
    coef(fit5)
    coef(fit5, matrix=TRUE)

[Package VGAM version 0.7-9 Index]