# Common VGAM Family Function Arguments Here is a description of some common and typical arguments found in many `VGAM` family functions, e.g., `zero`, `lsigma`, `isigma`, `gsigma`, `eq.mean`, `nsimEI` and `parallel`. ```r TypicalVGAMfamilyFunction(lsigma = "loglink", isigma = NULL, zero = NULL, gsigma = exp(-5:5), eq.mean = FALSE, parallel = TRUE, imethod = 1, vfl = FALSE, Form2 = NULL, type.fitted = c("mean", "quantiles", "Qlink", "pobs0", "pstr0", "onempstr0"), percentiles = c(25, 50, 75), probs.x = c(0.15, 0.85), probs.y = c(0.25, 0.50, 0.75), multiple.responses = FALSE, earg.link = FALSE, ishrinkage = 0.95, nointercept = NULL, whitespace = FALSE, bred = FALSE, lss = TRUE, oim = FALSE, nsimEIM = 100, byrow.arg = FALSE, link.list = list("(Default)" = "identitylink", x2 = "loglink", x3 = "logofflink", x4 = "multilogitlink", x5 = "multilogitlink"), earg.list = list("(Default)" = list(), x2 = list(), x3 = list(offset = -1), x4 = list(), x5 = list()), Thresh = NULL, nrfs = 1) ``` ## Arguments - `lsigma`: Character. Link function applied to a parameter and not necessarily a mean. See `Links` for a selection of choices. If there is only one parameter then this argument is often called `link`. - `isigma`: Optional initial values can often be inputted using an argument beginning with `"i"`. For example, `"isigma"` and `"ilocation"`, or just `"init"` if there is one parameter. A value of `NULL` means a value is computed internally, i.e., a **self-starting** `VGAM` family function. If a failure to converge occurs make use of these types of arguments. - `zero`: An important argument, either an integer vector, or a vector of character strings. If an integer, then it specifies which linear/additive predictor is modelled as **intercept-only**. That is, the regression coefficients are set to zero for all covariates except for the intercept. If `zero` is specified then it may be a vector with values from the set $\{1,2,\ldots,M\}$. The value `zero = NULL` means model **all** linear/additive predictors as functions of the explanatory variables. Here, $M$ is the number of linear/additive predictors. Technically, if `zero` contains the value $j$ then the $j$th row of every constraint matrix (except for the intercept) consists of all 0 values. Some `VGAM` family functions allow the `zero` argument to accept negative values; if so then its absolute value is recycled over each (usual) response. For example, `zero = -2` for the two-parameter negative binomial distribution would mean, for each response, the second linear/additive predictor is modelled as intercepts-only. That is, for all the $k$ parameters in `negbinomial` (this `VGAM` family function can handle a matrix of responses). Suppose `zero = zerovec` where `zerovec` is a vector of negative values. If $G$ is the usual $M$ value for a univariate response then the actual values for argument `zero` are all values in `c(abs(zerovec), G + abs(zerovec), 2*G + abs(zerovec), ... )` lying in the integer range $1$ to $M$. For example, setting `zero = -c(2, 3)` for a matrix response of 4 columns with `zinegbinomial` (which usually has $G = M = 3$ for a univariate response) would be equivalent to `zero = c(2, 3, 5, 6, 8, 9, 11, 12)`. This example has $M = 12$. Note that if `zerovec` contains negative values then their absolute values should be elements from the set `1:G`. Note: `zero` may have positive and negative values, for example, setting `zero = c(-2, 3)` in the above example would be equivalent to `zero = c(2, 3, 5, 8, 11)`. The argument `zero` also accepts a character vector (for `VGAM` 1.0-1 onwards). Each value is fed into `grep` with `fixed = TRUE`, meaning that wildcards `"*"` are not useful. See the example below---all the variants work; those with `LOCAT` issue a warning that that value is unmatched. Importantly, the parameter names are `c("location1", "scale1", "location2", "scale2")` because there are 2 responses. Yee (2015) described `zero` for only numerical input. Allowing character input is particularly important when the number of parameters cannot be determined without having the actual data first. For example, with time series data, an ARMA($p$,$q$) process might have parameters $\theta_1,\ldots,\theta_p$ which should be intercept-only by default. Then specifying a numerical default value for `zero` would be too difficult (there are the drift and scale parameters too). However, it is possible with the character representation: `zero = "theta"` would achieve this. In the future, most `VGAM` family functions might be converted to the character representation---the advantage being that it is more readable. When programming a `VGAM` family function that allows character input, the variable `predictors.names` must be assigned correctly. 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`. - `gsigma`: Grid-search initial values can be inputted using an argument beginning with `"g"`, e.g., `"gsigma"`, `"gshape"` and `"gscale"`. If argument `isigma` is inputted then that has precedence over `gsigma`, etc. If the grid search is 2-dimensional then it is advisable not to make the vectors too long as a nested `for` loop may be used. Ditto for 3-dimensions etc. Sometimes a `".mux"` is added as a suffix, e.g., `gshape.mux`; this means that the grid is created relatively and not absolutely, e.g., its values are multipled by some single initial estimate of the parameter in order to create the grid on an absolute scale. Some family functions have an argument called `gprobs.y`. This is fed into the `probs` argument of `quantile` in order to obtain some values of central tendency of the response, i.e., some spread of values in the middle. when `imethod = 1` to obtain an initial value for the mean Some family functions have an argument called `iprobs.y`, and if so, then these values can overwrite `gprobs.y`. - `eq.mean`: Logical. Constrain all the means to be equal? This type of argument is simpler than `parallel` because only a single `TRUE` or `FALSE` can be assigned and not a formula. Thus if `TRUE` then it will be enforced over all variables. - `parallel`: A logical, or a simple formula specifying which terms have equal/unequal coefficients. The formula must be simple, i.e., additive with simple main effects terms. Interactions and nesting etc. are not handled. To handle complex formulas use the `constraints` argument (of `vglm` etc.); however, there is a lot more setting up involved and things will not be as convenient. Here are some examples. 1. `parallel = TRUE ~ x2 + x5` means the parallelism assumption is only applied to $X_2$, $X_5$ and the intercept. 2. `parallel = TRUE ~ -1` and `parallel = TRUE ~ 0` mean the parallelism assumption is applied to **no** variables at all. Similarly, `parallel = FALSE ~ -1` and `parallel = FALSE ~ 0` mean the parallelism assumption is applied to **all** the variables including the intercept. 3. `parallel = FALSE ~ x2 - 1` and `parallel = FALSE ~ x2 + 0` applies the parallelism constraint to all terms (including the intercept) except for $X_2$. 4. `parallel = FALSE ~ x2 * x3` probably will not work. Instead, expand it out manually to get `parallel = FALSE ~ x2 + x3 + x2:x3`, and that should work. That's because the main formula processes or expands the `"*"` operator but `parallel` does not. 5. To check whether `parallel` has done what was expected, type `coef(fit, matrix = TRUE)` or `constraints(fit)` for confirmation. This argument is common in `VGAM` family functions for categorical responses, e.g., `cumulative`, `acat`, `cratio`, `sratio`. For the proportional odds model (`cumulative`) having parallel constraints applied to each explanatory variable (except for the intercepts) means the fitted probabilities do not become negative or greater than 1. However this parallelism or proportional-odds assumption ought to be checked. - `nsimEIM`: Some `VGAM` family functions use simulation to obtain an approximate expected information matrix (EIM). For those that do, the `nsimEIM` argument specifies the number of random variates used per observation; the mean of `nsimEIM` random variates is taken. Thus `nsimEIM` controls the accuracy and a larger value may be necessary if the EIMs are not positive-definite. For intercept-only models (`y ~ 1)` the value of `nsimEIM` can be smaller (since the common value used is also then taken as the mean over the observations), especially if the number of observations is large. Some `VGAM` family functions provide two algorithms for estimating the EIM. If applicable, set `nsimEIM = NULL` to choose the other algorithm. - `imethod`: An integer with value `1` or `2` or `3` or ... which specifies the initialization method for some parameters or a specific parameter. If failure to converge occurs try the next higher value, and continue until success. For example, `imethod = 1` might be the method of moments, and `imethod = 2` might be another method. If no value of `imethod` works then it will be necessary to use arguments such as `isigma`. For many `VGAM` family functions it is advisable to try this argument with all possible values to safeguard against problems such as converging to a local solution. `VGAM` family functions with this argument usually correspond to a model or distribution that is relatively hard to fit successfully, therefore care is needed to ensure the global solution is obtained. So using all possible values that this argument supplies is a good idea. `VGAM` family functions such `genpoisson2` recycle `imethod` to be of length 2 corresponding to the 2 parameters. In the future, this feature will be extended to other family functions to confer more flexibility. - `Form2`: Formula. Using applied to models with $M=2$. Specifies the terms for $\eta_2$ and the other terms belong to $\eta_1$. It is a way to partition the X matrix into two sets of covariates, where they are assigned to each $\eta_j$ separately. This argument sets up constraint matrices `rbind(0, 1)` for terms in `Form2` and `rbind(1, 0)` for `setdiff(formula, Form2)` so to speak. Note that sometimes this argument is only accessed if `vfl = TRUE`. Arguments such as `Form1` and `Form3` are also possible in `VGAM` family functions because the $\eta_j$ which is likely to be modelled more simply is chosen for convenience. - `vfl`: A single logical. This stands for **variance--variance factored loglinear** (VFL) model. If `TRUE` then usually some other argument such as `Form2` or `parallel` is used to partition the main `vglm` `formula` into two sets of covariates. For some families such as `negbinomial` this enables overdispersion to be modelled conveniently via a loglinear model, given the mean. It is necessary to read the online help regarding each `VGAM` family function because each one may different from others. To fit some VFL models it is necessary to make a copy of existing covariates by creating new variable names and then adding it to the main formula. A good question is: why is `vfl` necessary? Wouldn't `Form2` be sufficient? Setting `vfl = TRUE` enables some internal checking such as avoiding conflicts. For example, it is often necessary to set `zero = NULL` and `parallel = FALSE`, otherwise there would be contradictions. - `type.fitted`: Character. Type of fitted value returned by the `fitted()` methods function. The first choice is always the default. The available choices depends on what kind of family function it is. Using the first few letters of the chosen choice is okay. See `fittedvlm` for more details. The choice `"Qlink"` refers to quantile-links, which was introduced in December 2018 in `VGAMextra` 0.0-2 for several 1-parameter distributions. Here, either the `loglink` or `logitlink` or `identitylink` of the quantile is the link function (and the choice is dependent on the support of the distribution), and link functions end in `"Qlink"`. A limited amount of support is provided for such links, e.g., `fitted(fit)` are the fitted quantiles, which is the same as `predict(fit, type = "response")`. However, `fitted(fit, percentiles = 77)` will not work. - `percentiles`: Numeric vector, with values between 0 and 100 (although it is not recommended that exactly 0 or 100 be inputted). Used only if `type.fitted = "quantiles"` or `type.fitted = "percentiles"`, then this argument specifies the values of these quantiles. The argument name tries to reinforce that the values lie between 0 and 100. See `fittedvlm` for more details. - `probs.x, probs.y`: Numeric, with values in (0, 1). The probabilites that define quantiles with respect to some vector, usually an `x` or `y` of some sort. This is used to create two subsets of data corresponding to `low' and `high' values of x or y. Each value is separately fed into the `probs` argument of `quantile`. If the data set size is small then it may be necessary to increase/decrease slightly the first/second values respectively. - `lss`: Logical. This stands for the ordering: location, scale and shape. Should the ordering of the parameters be in this order? Almost all `VGAM` family functions have this order by default, but in order to match the arguments of existing R functions, one might need to set `lss = FALSE`. For example, the arguments of `weibullR` are scale and shape, whereas `rweibull` are shape and scale. As a temporary measure (from `VGAM` 0.9-7 onwards but prior to version 1.0-0), some family functions such as `sinmad` have an `lss` argument without a default. For these, setting `lss = FALSE` will work. Later, `lss = TRUE` will be the default. Be careful for the `dpqr`-type functions, e.g., `rsinmad`. - `Thresh`: **Thresholds** is another name for the intercepts, e.g., for categorical models. They may be constrained by functions such as `CM.equid` and `CM.qnorm`. The string `"CM."` and the `Thresh` argument is pasted and then that function is called to obtain the constraint matrix for the intercept term. So `Thresh = "free"`, `Thresh = "equid"`, `Thresh = "qnorm"`, `Thresh = "symm0"`, `Thresh = "symm1"` etc. are possibilities. Families that use this include `multinomial`, `cratio`, `sratio`, `cumulative`, `acat`. - `whitespace`: Logical. Should white spaces (`" "`) be used in the labelling of the linear/additive predictors? Setting `TRUE` usually results in more readability but it occupies more columns of the output. - `oim`: Logical. Should the observed information matrices (OIMs) be used for the working weights? In general, setting `oim = TRUE` means the Newton-Raphson algorithm, and `oim = FALSE` means Fisher-scoring. The latter uses the EIM, and is usually recommended. If `oim = TRUE` then `nsimEIM` is ignored. - `nrfs`: Numeric, a value in $[0, 1]$. Experimental argument for allowing a mixture of Newton-Raphson and Fisher scoring. The working weights are taken as a linear combination of the two. If `nrfs = 0` then Newton-Raphson is used, i.e., the OIM is wholly used. If `nrfs = 1` then Fisher scoring is used, i.e., the EIM is wholly used. If convergence is successful then one might expect Newton-Raphson to be faster than Fisher scoring because the former has an order-2 convergence rate while the latter has a linear rate. , - `multiple.responses`: Logical. Some `VGAM` family functions allow a multivariate or vector response. If so, then usually the response is a matrix with columns corresponding to the individual response variables. They are all fitted simultaneously. Arguments such as `parallel` may then be useful to allow for relationships between the regressions of each response variable. If `multiple.responses = TRUE` then sometimes the response is interpreted differently, e.g., `posbinomial` chooses the first column of a matrix response as success and combines the other columns as failure, but when `multiple.responses = TRUE` then each column of the response matrix is the number of successes and the `weights` argument is of the same dimension as the response and contains the number of trials. - `earg.link`: This argument should be generally ignored. - `byrow.arg`: Logical. Some `VGAM` family functions that handle multiple responses have arguments that allow input to be fed in which affect all the responses, e.g., `imu` for initalizing a `mu` parameter. In such cases it is sometime more convenient to input one value per response by setting `byrow.arg = TRUE`; then values are recycled in order to form a matrix of the appropriate dimension. This argument matches `byrow` in `matrix`; in fact it is fed into such using `matrix(..., byrow = byrow.arg)`. This argument has no effect when there is one response. - `ishrinkage`: Shrinkage factor $s$ used for obtaining initial values. Numeric, between 0 and 1. In general, the formula used is something like $s*mu + (1-s)*y$ where $mu$ is a measure of central tendency such as a weighted mean or median, and $y$ is the response vector. For example, the initial values are slight perturbations of the mean towards the actual data. For many types of models this method seems to work well and is often reasonably robust to outliers in the response. Often this argument is only used if the argument `imethod` is assigned a certain value. - `nointercept`: An integer-valued vector specifying which linear/additive predictors have no intercepts. Any values must be from the set {1,2, ,$M$}. A value of `NULL` means no such constraints. - `bred`: Logical. Some `VGAM` family functions will allow bias-reduction based on the work by Kosmidis and Firth. Sometimes half-stepping is a good idea; set `stepsize = 0.5` and monitor convergence by setting `trace = TRUE`. - `link.list, earg.list`: Some `VGAM` family functions (such as `normal.vcm`) implement models with potentially lots of parameter link functions. These two arguments allow many such links and extra arguments to be inputted more easily. One has something like `link.list = list` `("(Default)" = "identitylink", x2 = "loglink", x3 = "logofflink")` and `earg.list = list` `("(Default)" = list(), x2 = list(), x3 = "list(offset = -1)")`. Then any unnamed terms will have the default link with its corresponding extra argument. Note: the `multilogitlink` link is also possible, and if so, at least two instances of it are necessary. Then the last term is the baseline/reference group. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm` and `vgam`. ## Warning The `zero` argument is supplied for convenience but conflicts can arise with other arguments, e.g., the `constraints` argument of `vglm` and `vgam`. See Example 5 below for an example. If not sure, use, e.g., `constraints(fit)` and `coef(fit, matrix = TRUE)` to check the result of a fit `fit`. The arguments `zero` and `nointercept` can be inputted with values that fail. For example, `multinomial(zero = 2, nointercept = 1:3)` means the second linear/additive predictor is identically zero, which will cause a failure. Be careful about the use of other potentially contradictory constraints, e.g., `multinomial(zero = 2, parallel = TRUE ~ x3)`. If in doubt, apply `constraints()` to the fitted object to check. `VGAM` family functions with the `nsimEIM` may have inaccurate working weight matrices. If so, then the standard errors of the regression coefficients may be inaccurate. Thus output from `summary(fit)`, `vcov(fit)`, etc. may be misleading. Changes relating to the `lss` argument have very important consequences and users must beware. Good programming style is to rely on the argument names and not on the order. ## Details Full details will be given in documentation yet to be written, at a later date! A general recommendation is to set `trace = TRUE` whenever any model fitting is done, since monitoring convergence is usually very informative. ## References Yee, T. W. (2015). Vector Generalized Linear and Additive Models: With an Implementation in R. New York, USA: **Springer**. Kosmidis, I. and Firth, D. (2009). Bias reduction in exponential family nonlinear models. **Biometrika**, 96 , 793--804. Miranda-Soberanis, V. F. and Yee, T. W. (2019). New link functions for distribution--specific quantile regression based on vector generalized linear and additive models. **Journal of Probability and Statistics**, 5 , 1--11. ## See Also `Links`, `vglm`, `vgam`, `vglmff-class`, `UtilitiesVGAM`, `multilogitlink`, `multinomial`, `VGAMextra`. ## Author(s) T. W. Yee ## Examples ```r # Example 1 cumulative() cumulative(link = "probitlink", reverse = TRUE, parallel = TRUE) # Example 2 wdata <- data.frame(x2 = runif(nn <- 1000)) wdata <- transform(wdata, y = rweibull(nn, shape = 2 + exp(1 + x2), scale = exp(-0.5))) fit <- vglm(y ~ x2, weibullR(lshape = logofflink(offset = -2), zero = 2), data = wdata) coef(fit, mat = TRUE) # Example 3; multivariate (multiple) response ## Not run: ndata <- data.frame(x = runif(nn <- 500)) ndata <- transform(ndata, y1 = rnbinom(nn, exp(1), mu = exp(3+x)), # k is size y2 = rnbinom(nn, exp(0), mu = exp(2-x))) fit <- vglm(cbind(y1, y2) ~ x, negbinomial(zero = -2), ndata) coef(fit, matrix = TRUE) ## End(Not run) # Example 4 ## Not run: # fit1 and fit2 are equivalent fit1 <- vglm(ymatrix ~ x2 + x3 + x4 + x5, cumulative(parallel = FALSE ~ 1 + x3 + x5), cdata) fit2 <- vglm(ymatrix ~ x2 + x3 + x4 + x5, cumulative(parallel = TRUE ~ x2 + x4), cdata) ## End(Not run) # Example 5 udata <- data.frame(x2 = rnorm(nn <- 200)) udata <- transform(udata, x1copy = 1, # Copy of the intercept x3 = runif(nn), y1 = rnorm(nn, 1 - 3*x2, sd = exp(1 + 0.2*x2)), y2 = rnorm(nn, 1 - 3*x2, sd = exp(1))) args(uninormal) fit1 <- vglm(y1 ~ x2, uninormal, udata) # This is okay fit2 <- vglm(y2 ~ x2, uninormal(zero = 2), udata) # This is okay fit4 <- vglm(y2 ~ x2 + x1copy + x3, uninormal(zero = NULL, vfl = TRUE, Form2 = ~ x1copy + x3 - 1), udata) coef(fit4, matrix = TRUE) # VFL model # This creates potential conflict clist <- list("(Intercept)" = diag(2), "x2" = diag(2)) fit3 <- vglm(y2 ~ x2, uninormal(zero = 2), data = udata, constraints = clist) # Conflict! coef(fit3, matrix = TRUE) # Shows that clist[["x2"]] was overwritten, constraints(fit3) # i.e., 'zero' seems to override the 'constraints' arg # Example 6 ('whitespace' argument) pneumo <- transform(pneumo, let = log(exposure.time)) fit1 <- vglm(cbind(normal, mild, severe) ~ let, sratio(whitespace = FALSE, parallel = TRUE), pneumo) fit2 <- vglm(cbind(normal, mild, severe) ~ let, sratio(whitespace = TRUE, parallel = TRUE), pneumo) head(predict(fit1), 2) # No white spaces head(predict(fit2), 2) # Uses white spaces # Example 7 ('zero' argument with character input) set.seed(123); n <- 1000 ldata <- data.frame(x2 = runif(n)) ldata <- transform(ldata, y1 = rlogis(n, loc = 5*x2, scale = exp(2))) ldata <- transform(ldata, y2 = rlogis(n, loc = 5*x2, scale = exp(1*x2))) ldata <- transform(ldata, w1 = runif(n)) ldata <- transform(ldata, w2 = runif(n)) fit7 <- vglm(cbind(y1, y2) ~ x2, # logistic(zero = "location1"), # location1 is intercept-only # logistic(zero = "location2"), # logistic(zero = "location*"), # Not okay... all is unmatched # logistic(zero = "scale1"), # logistic(zero = "scale2"), # logistic(zero = "scale"), # Both scale parameters are matched logistic(zero = c("location", "scale2")), # All but scale1 # logistic(zero = c("LOCAT", "scale2")), # Only scale2 is matched # logistic(zero = c("LOCAT")), # Nothing is matched # trace = TRUE, # weights = cbind(w1, w2), weights = w1, data = ldata) coef(fit7, matrix = TRUE) ```

CommonVGAMffArguments function