# Fitting Vector Generalized Linear Models `vglm` fits vector generalized linear models (VGLMs). This very large class of models includes generalized linear models (GLMs) as a special case. ```r vglm(formula, family = stop("argument 'family' needs to be assigned"), data = list(), weights = NULL, subset = NULL, na.action, 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 = TRUE, smart = TRUE, ...) ``` ## Arguments - `formula`: a symbolic description of the model to be fit. The RHS of the formula is applied to each linear predictor. The effect of different variables in each linear predictor can be controlled by specifying constraint matrices---see `constraints` below. - `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. The argument name `"family"` is used loosely and for the ease of existing `glm` users; there is no concept of a formal ``error distribution'' for VGLMs. Possibly the argument name should be better `"model"` but unfortunately that name has already been taken. - `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 fixed and known) weights to be used in the fitting process. If the `VGAM` family function handles multiple responses ($Q > 1$ of them, say) then `weights` can be a matrix with $Q$ columns. Each column matches the respective response. If it is a vector (the usually case) then it is recycled into a matrix with $Q$ columns. The values of `weights` must be positive; try setting a very small value such as `1.0e-8` to effectively delete an observation. Currently the `weights` argument supports sampling weights from complex sampling designs via `svyVGAM`. Some details can be found at [https://CRAN.R-project.org/package=svyVGAM](https://CRAN.R-project.org/package=svyVGAM). - `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` which is known as **complete case analysis** and applied to both sides of the formula. - `etastart`: optional starting values for the linear predictors. It is a $M$-column matrix with the same number of rows as the response. If $M = 1$ then it may be a vector. Note that `etastart` and the output of `predict(fit)` should be comparable. Here, `fit` is the fitted object. Almost all `VGAM` family functions are self-starting. - `mustart`: optional starting values for the fitted values. It can be a vector or a matrix; if a matrix, then it has the same number of rows as the response. Usually `mustart` and the output of `fitted(fit)` should be comparable. Most family functions do not make use of this argument because it is not possible to compute all $M$ columns of `eta` from `mu`. - `coefstart`: optional starting values for the coefficient vector. The length and order must match that of `coef(fit)`. - `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/additive 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 LM matrix and response vector/matrix used in the fitting process should be assigned in the `x` and `y` slots. Note that the model matrix is the LM matrix; to get the VGLM 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 (labelled) with the term it corresponds to (and it must match in character format **exactly**---see below). There are two types of input: `"lm"`-type and `"vlm"`-type. The former is a subset of the latter. The former has a matrix for each term of the LM matrix. The latter has a matrix for each column of the big VLM matrix. After fitting, the `constraints` extractor function may be applied; it returns the `"vlm"`-type list of constraint matrices by default. If `"lm"`-type are returned by `constraints` then these can be fed into this argument and it should give the same model as before. If the `constraints` argument is used then the family function's `zero` argument (if it exists) needs to be set to `NULL`. This avoids what could be a probable contradiction. Sometimes setting other arguments related to constraint matrices to `FALSE` is also a good idea, e.g., `parallel = FALSE`, `exchangeable = FALSE`. Properties: 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, e.g., `parallel` (see `CommonVGAMffArguments`). If `constraints` is used then it must contain **all** the terms; an incomplete list is not accepted. As mentioned above, the labelling of each constraint matrix must match exactly, e.g., `list("s(x2,df=3)"=diag(2))` will fail as `as.character(~ s(x2,df=3))` produces white spaces: `"s(x2, df = 3)"`. Thus `list("s(x2, df = 3)" = diag(2))` is needed. See Example 6 below. More details are given in Yee (2015; Section 3.3.1.3) which is on p.101. Note that the label for the intercept is `"(Intercept)"`. - `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 and can be estimated by Fisher scoring. The central formula is given by $$ \eta_j = \beta_j^T x%eta_j = beta_j^T x $$ 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, $j=1,\ldots,M$, where $M$ is finite. Then one can write $eta=(eta_1,\ldots,\eta_M)^T$ as a vector of linear predictors. Most users will find `vglm` similar in flavour to `glm`. The function `vglm.fit` actually does the work. ## Returns 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. - **ResSS**: 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`. ## References Yee, T. W. (2015). Vector Generalized Linear and Additive Models: With an Implementation in R. New York, USA: **Springer**. 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. (2014). Reduced-rank vector generalized linear models with two linear predictors. **Computational Statistics and Data Analysis**, 71 , 889--902. Yee, T. W. (2008). The `VGAM` Package. **R News**, 8 , 28--39. ## Author(s) Thomas W. Yee ## 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`. This is because `VGAM` family functions are incompatible with `glm` (and also `gam()` in `gam` 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 `fill1` for more details and examples. ## WARNING See warnings in `vglm.control`. Also, see warnings under `weights` above regarding sampling weights from complex sampling designs. ## See Also `vglm.control`, `vglm-class`, `vglmff-class`, `smartpred`, `vglm.fit`, `fill1`, `rrvglm`, `vgam`. Methods functions include `add1.vglm`, `anova.vglm`, `AICvlm`, `coefvlm`, `confintvglm`, `constraints.vlm`, `drop1.vglm`, `fittedvlm`, `hatvaluesvlm`, `hdeff.vglm`, `Influence.vglm`, `linkfunvlm`, `lrt.stat.vlm`, `score.stat.vlm`, `wald.stat.vlm`, `nobs.vlm`, `npred.vlm`, `plotvglm`, `predictvglm`, `residualsvglm`, `step4vglm`, `summaryvglm`, `lrtest_vglm`, `update`, `TypicalVGAMfamilyFunction`, etc. ## Examples ```r # Example 1. See help(glm) (d.AD <- data.frame(treatment = gl(3, 3), outcome = gl(3, 1, 9), counts = c(18,17,15,20,10,20,25,13,12))) vglm.D93 <- vglm(counts ~ outcome + treatment, poissonff, data = d.AD, trace = TRUE) 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, propodds, pneumo) coef(fit3, matrix = TRUE) constraints(fit3) model.matrix(fit3, type = "lm") # LM model matrix model.matrix(fit3) # Larger VGLM (or VLM) matrix # Example 4. Bivariate logistic model fit4 <- vglm(cbind(nBnW, nBW, BnW, BW) ~ age, binom2.or, coalminers) coef(fit4, matrix = TRUE) depvar(fit4) # 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)), digits = 2) eyesdat <- transform(eyesdat, eta1 = -1 + 2 * lop, eta2 = -1 + 2 * lop) eyesdat <- transform(eyesdat, leye = rbinom(nn, 1, prob = logitlink(eta1, inv = TRUE)), reye = rbinom(nn, 1, prob = logitlink(eta2, inv = TRUE))) head(eyesdat) fit5 <- vglm(cbind(leye, reye) ~ op, binom2.or(exchangeable = TRUE, zero = 3), data = eyesdat, trace = TRUE, xij = list(op ~ lop + rop + fill1(lop)), form2 = ~ op + lop + rop + fill1(lop)) coef(fit5) coef(fit5, matrix = TRUE) constraints(fit5) fit5@control$xij head(model.matrix(fit5)) # Example 6. The use of the 'constraints' argument. as.character(~ bs(year,df=3)) # Get the white spaces right clist <- list("(Intercept)" = diag(3), "bs(year, df = 3)" = rbind(1, 0, 0)) fit1 <- vglm(r1 ~ bs(year,df=3), gev(zero = NULL), data = venice, constraints = clist, trace = TRUE) coef(fit1, matrix = TRUE) # Check ```

vglm function