# Fitting Reduced-Rank Vector Generalized Linear Models (RR-VGLMs) and Doubly Constrained RR-VGLMs (DRR-VGLMs) A **reduced-rank vector generalized linear model** (RR-VGLM) is fitted. RR-VGLMs are VGLMs but some of the constraint matrices are estimated. **Doubly constrained** RR-VGLMs (DRR-VGLMs) can also be fitted, and these provide structure for the two other outer product matrices. ```r rrvglm(formula, family = stop("'family' is unassigned"), data = list(), weights = NULL, subset = NULL, na.action = na.fail, etastart = NULL, mustart = NULL, coefstart = NULL, control = rrvglm.control(...), offset = NULL, method = "rrvglm.fit", model = FALSE, x.arg = TRUE, y.arg = TRUE, contrasts = NULL, constraints = NULL, extra = NULL, qr.arg = FALSE, smart = TRUE, ...) ``` ## Arguments - `formula, family`: See `vglm`. - `weights, data`: See `vglm`. - `subset, na.action`: See `vglm`. - `etastart, mustart, coefstart`: See `vglm`. - `control`: a list of parameters for controlling the fitting process. See `rrvglm.control` for details. - `offset, model, contrasts`: See `vglm`. - `method`: the method to be used in fitting the model. The default (and presently only) method `rrvglm.fit` uses iteratively reweighted least squares (IRLS). - `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. - `constraints`: See `vglm`. - `extra, smart, qr.arg`: See `vglm`. - ``...``: further arguments passed into `rrvglm.control`. ## Details In this documentation, $M$ is the number of linear predictors. For RR-VGLMs, the central formula is given by $$ \eta = B_1^T x_1 + A \nu%eta = B_1^T x_1 + A nu $$ where $x1$ is a vector (usually just a 1 for an intercept), $x2$ is another vector of explanatory variables, and $nu = C^T x_2$ is an $R$-vector of latent variables. Here, $eta$ is a vector of linear predictors, e.g., the $m$th element is $eta_m = log(E[Y_m])$ for the $m$th Poisson response. The dimension of $\eta$ is $M$ by definition. The matrices $B_1$, $A$ and $C$ are estimated from the data, i.e., contain the regression coefficients. For ecologists, the central formula represents a **constrained linear ordination** (CLO) since it is linear in the latent variables. It means that the response is a monotonically increasing or decreasing function of the latent variables. For identifiability it is common to enforce **corner constraints** on A : by default, for RR-VGLMs, the top $R$ by $R$ submatrix is fixed to be the order-$R$ identity matrix and the remainder of A is estimated. And by default, for DRR-VGLMs, there is also an order-$R$ identity matrix embedded in A because the RRR must be **separable** (this is so that any existing structure in A is preserved). The underlying algorithm of RR-VGLMs is iteratively reweighted least squares (IRLS) with an optimizing algorithm applied within each IRLS iteration (e.g., alternating algorithm). In theory, any `VGAM` family function that works for `vglm` and `vgam` should work for `rrvglm` too. The function that actually does the work is `rrvglm.fit`; it is essentially `vglm.fit` with some extra code. ## Returns For RR-VGLMs, an object of class `"rrvglm"`, which has the the same slots as a `"vglm"` object. The only difference is that the some of the constraint matrices are estimates rather than known. But `VGAM` stores the models the same internally. The slots of `"vglm"` objects are described in `vglm-class`. For DRR-VGLMs, an object of class `"drrvglm"`. ## References Yee, T. W. and Hastie, T. J. (2003). Reduced-rank vector generalized linear models. **Statistical Modelling**, 3 , 15--41. Yee, T. W. (2004). A new technique for maximum-likelihood canonical Gaussian ordination. **Ecological Monographs**, 74 , 685--701. Anderson, J. A. (1984). Regression and ordered categorical variables. **Journal of the Royal Statistical Society, Series B, Methodological**, 46 , 1--30. 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., Frigau, L. and Ma, C. (2024). Heaping and seeping, GAITD regression and doubly constrained reduced rank vector generalized linear models, in smoking studies. **In preparation**. ## Author(s) Thomas W. Yee ## Note The arguments of `rrvglm` are in general the same as those of `vglm` but with some extras in `rrvglm.control`. The smart prediction (`smartpred`) library is packed with the `VGAM` library. In an example below, a rank-1 **stereotype** (reduced-rank multinomial logit) model of Anderson (1984) is fitted to some car data. The reduced-rank regression is performed, adjusting for two covariates. Setting a trivial constraint matrix (`diag(M)`) for the latent variable variables in $x2$ avoids a warning message when it is overwritten by a (common) estimated constraint matrix. It shows that German cars tend to be more expensive than American cars, given a car of fixed weight and width. If `fit <- rrvglm(..., data = mydata)` then `summary(fit)` requires corner constraints and no missing values in `mydata`. Sometimes the estimated variance-covariance matrix of the parameters is not positive-definite; if this occurs, try refitting the model with a different value for `Index.corner`. For **constrained quadratic ordination** (CQO) see `cqo` for more details about QRR-VGLMs. With multiple binary responses, one must use `binomialff(multiple.responses = TRUE)` to indicate that the response is a matrix with one response per column. Otherwise, it is interpreted as a single binary response variable. To fit DRR-VGLMs see the arguments `H.A.thy` and `H.C` in `rrvglm.control`. DRR-VGLMs provide structure to the A and C matrices via constraint matrices. So instead of them being general unstructured matrices, one can make specified elements to be identically equal to 0, for example. This gives greater control over what is modelled as a latent variable, e.g., in a health study, if one subset of the covariates are physical variables and the remainder are psychological variables then a rank-2 model might have each latent variable a linear combination of each of the types of variables separately. Incidentally before I forget, since `Corner = TRUE`, then the differences between the `@H.A.thy` and `@H.A.alt` slots are due to `Index.corner`, which specifies which rows of A are not estimated. However, in the alternating algorithm, it is more efficient to estimate the entire A , bar (effectively) rows `str0`, and then normalize it. In contrast, optimizing over the subset of A to be estimated is slow. In the `@misc` slot are logical components `is.drrvglm` and `is.rrvglm`. Only one is `TRUE`. If `is.rrvglm` then (full) corner constraints are used. If `is.drrvglm` then **restricted corner constraints** (RCCs) are used and the reduced rank regression (RRR) must be **separable**. The case `is.rrvglm` means that `H.A.thy` is a `vector("list", Rank)` with `H.A.thy[[r]] <- diag(M)` assigned to all $r=1,\ldots,R$. Because DRR-VGLMs are implemented only for separable problems, this means that all columns of `H.A.thy[[s]]` are orthogonal to all columns from `H.A.try[[t]]`, for all $s$ and $t$. DRR-VGLMs are proposed in Yee et al. (2024) in the context of GAITD regression for heaped and seeped survey data. ## See Also `rrvglm.control`, `summary.drrvglm`, `lvplot.rrvglm` (same as `biplot.rrvglm`), `rrvglm-class`, `grc`, `cqo`, `vglmff-class`, `vglm`, `vglm-class`, `smartpred`, `rrvglm.fit`. Special family functions include `negbinomial` `zipoisson` and `zinegbinomial`. (see Yee (2014) and what was formerly in `COZIGAM`). Methods functions include `Coef.rrvglm`, `calibrate.rrvglm`, etc. Data include `crashi`. ## Examples ```r ## Not run: # Example 1: RR NB with Var(Y) = mu + delta1 * mu^delta2 nn <- 1000 # Number of observations delta1 <- 3.0 # Specify this delta2 <- 1.5 # Specify this; should be greater than 1 a21 <- 2 - delta2 mydata <- data.frame(x2 = runif(nn), x3 = runif(nn)) mydata <- transform(mydata, mu = exp(2 + 3 * x2 + 0 * x3)) mydata <- transform(mydata, y2 = rnbinom(nn, mu = mu, size = (1/delta1)*mu^a21)) plot(y2 ~ x2, mydata, pch = "+", col = 'blue', las = 1, main = paste0("Var(Y) = mu + ", delta1, " * mu^", delta2)) rrnb2 <- rrvglm(y2 ~ x2 + x3, negbinomial(zero = NULL), data = mydata, trace = TRUE) a21.hat <- (Coef(rrnb2)@A)["loglink(size)", 1] beta11.hat <- Coef(rrnb2)@B1["(Intercept)", "loglink(mu)"] beta21.hat <- Coef(rrnb2)@B1["(Intercept)", "loglink(size)"] (delta1.hat <- exp(a21.hat * beta11.hat - beta21.hat)) (delta2.hat <- 2 - a21.hat) # delta1.hat: # exp(a21.hat * predict(rrnb2)[1,1] - predict(rrnb2)[1,2]) summary(rrnb2) # Obtain a 95 percent CI for delta2: se.a21.hat <- sqrt(vcov(rrnb2)["I(latvar.mat)", "I(latvar.mat)"]) ci.a21 <- a21.hat + c(-1, 1) * 1.96 * se.a21.hat (ci.delta2 <- 2 - rev(ci.a21)) # The 95 percent CI Confint.rrnb(rrnb2) # Quick way to get it # Plot the abundances and fitted values vs the latent variable plot(y2 ~ latvar(rrnb2), data = mydata, col = "blue", xlab = "Latent variable", las = 1) ooo <- order(latvar(rrnb2)) lines(fitted(rrnb2)[ooo] ~ latvar(rrnb2)[ooo], col = "red") # Example 2: stereotype model (RR multinomial logit model) data(car.all) scar <- subset(car.all, is.element(Country, c("Germany", "USA", "Japan", "Korea"))) fcols <- c(13,14,18:20,22:26,29:31,33,34,36) # These are factors scar[, -fcols] <- scale(scar[, -fcols]) # Stdze all numerical vars ones <- CM.ones(3) # matrix(1, 3, 1) clist <- list("(Intercept)" = diag(3), Width = ones, Weight = ones, Disp. = diag(3), Tank = diag(3), Price = diag(3), Frt.Leg.Room = diag(3)) set.seed(111) fit <- rrvglm(Country ~ Width + Weight + Disp. + Tank + Price + Frt.Leg.Room, multinomial, data = scar, Rank = 2, trace = TRUE, constraints = clist, noRRR = ~ 1 + Width + Weight, # Uncor = TRUE, Corner = FALSE, # orig. Index.corner = c(1, 3), # Less correlation Bestof = 3) fit@misc$deviance # A history of the fits Coef(fit) biplot(fit, chull = TRUE, scores = TRUE, clty = 2, Ccex = 2, ccol = "blue", scol = "orange", Ccol = "darkgreen", Clwd = 2, main = "1=Germany, 2=Japan, 3=Korea, 4=USA") ## End(Not run) ```

rrvglm function