# Summarizing Reduced Rank Vector Generalized Linear Model (RR-VGLM) and Doubly constrained RR-VGLM Fits These functions are all `methods` for class `"drrvglm"` or `"summary.drrvglm"` objects, or for class `"rrvglm"` or `"summary.rrvglm"` objects. ```r ## S3 method for class 'drrvglm' summary(object, correlation = FALSE, dispersion = NULL, digits = NULL, numerical = TRUE, h.step = 0.005, omit123 = FALSE, omit13 = FALSE, fixA = FALSE, presid = FALSE, signif.stars = getOption("show.signif.stars"), nopredictors = FALSE, eval0 = TRUE, ...) ## S3 method for class 'summary.drrvglm' show(x, digits = NULL, quote = TRUE, prefix = "", signif.stars = NULL) ## S3 method for class 'rrvglm' summary(object, correlation = FALSE, dispersion = NULL, digits = NULL, numerical = TRUE, h.step = 0.005, omit123 = FALSE, omit13 = FALSE, fixA = TRUE, presid = FALSE, signif.stars = getOption("show.signif.stars"), nopredictors = FALSE, upgrade = FALSE, ...) ## S3 method for class 'summary.rrvglm' show(x, digits = NULL, quote = TRUE, prefix = "", signif.stars = NULL) ``` ## Arguments - `object`: an object of class `"drrvglm"` or `"rrvglm"`, a result of a call to `rrvglm`. - `x`: an object of class `"summary.drrvglm"` or `"summary.rrvglm"`, a result of a call to `summary.drrvglm` or `summary.rrvglm`. - `dispersion`: used mainly for GLMs. Not really implemented in `VGAM` so should not be used. - `correlation`: See `summaryvglm`. - `digits`: See `summaryvglm`. - `signif.stars`: See `summaryvglm`. - `presid, quote`: See `summaryvglm`. - `nopredictors`: See `summaryvglm`. - `upgrade`: Logical. Upgrade `object` to `drrvglm-class`? Treating the object as a DRR-VGLM has advantages since the framework is larger. The code for ordinary RR-VGLMs was written a long time ago so it is a good idea to check that both give the same answer. - `numerical`: Logical, use a finite difference approximation for partial derivatives? If `FALSE` then theoretical formulas are used (however this option may no longer be implemented). - `h.step`: Numeric, positive and close to 0. If `numerical` then this is the forward step for each finite difference approximation. That is, it plays the role of $h$ in $(f(x+h)-f(x))/h$ for some function $f$. If the overall variance-covariance matrix is not positive-definite, varying this argument can make a difference, e.g., increasing it to `0.01` is recommended. - `fixA`: Logical, if `TRUE` then the largest block matrix is for B1 and C , else it is for A and B1 . This should not make any difference because both estimates of B1 should be extremely similar, including the SEs. - `omit13`: Logical, if `TRUE` then the (1,3) block matrix is set to O . That is, A and C are assumed to asymptotically uncorrelated. Setting this `TRUE` is an option when V (see below) is not positive-definite. If this fails, another option that is often better is to set `omit123 = TRUE`. - `omit123`: Logical. If `TRUE` then **two** block matrices are set to O (blocks (1,2) and (1,3), else blocks (1,3) and (2,3), depending on `fixA`), This will almost surely result in an overall variance-covariance matrix that is positive-definite, however, the SEs will be biased. This argument is more extreme than `omit13`. - `prefix`: See `summaryvglm`. - `eval0`: Logical. Check if V is positive-definite? That is, all its eigenvalues are positive. - ``...``: Logical argument `check.2` might work here. If `TRUE` then some quantities are printed out, for checking and debugging. ## Details Most of this document concerns DRR-VGLMs but also apply equally well to RR-VGLMs as a special case. The overall variance-covariance matrix The overall variance-covariance matrix (called V below) is computed. Since the parameters comprise the elements of the matrices A , B1 and C (called here block matrices 1, 2, 3 respectively), and an alternating algorithm is used for estimation, then there are two overlapping submodels that are fitted within an IRLS algorithm. These have blocks 1 and 2, and 2 and 3, so that B1 is common to both. They are combined into one large overall variance-covariance matrix. Argument `fixA` specifies which submodel the B1 block is taken from. Block (1,3) is the most difficult to compute and numerical approximations based on first derivatives are used by default for this. Sometimes the computed V is not positive-definite. If so, then the standard errors will be `NA`. To avoid this problem, try varying `h.step` or refitting the model with a different `Index.corner`. Argument `omit13` and `omit123` can also be used to give approximate answers. If V is not positive-definite then this may indicate that the model does not fit the data very well, e.g., `Rank` is not a good value. Potentially, there are many ways why the model may be ill-conditioned. Try several options and set `trace = TRUE` to monitor convergence---this is informative about how well the model and data agree. How can one fit an ordinary RR-VGLM as a DRR-VGLM? If one uses corner constraints (default) then one should input `H.A` as a list containing `Rank` `diag(M)` matrices---one for each column of A . Then since `Corner = TRUE` by default, then `object@H.A.alt` has certain columns deleted due to corner constraints. In contrast, `object@H.A.thy` is the `H.A` that was inputted. FYI, the `alt` suffix indicates the alternating algorithm, while the suffix `thy` stands for **theory**. ## Returns `summarydrrvglm` returns an object of class `"summary.drrvglm"`. ## References Chapter 5 of: Yee, T. W. (2015). Vector Generalized Linear and Additive Models: With an Implementation in R. New York, USA: **Springer**. Sections 5.2.2 and 5.3 are particularly relevant. ## Author(s) T. W. Yee. ## Note Note that `vcov` methods exist for `rrvglm-class` and `drrvglm-class` objects. Sometimes this function can take a long time and this is because numerical derivatives are computed. ## Warning DRR-VGLMs are a recent development so it will take some time to get things totally ironed out. RR-VGLMs were developed a long time ago and are more well-established, however they have only recently been documented here. ## See Also `rrvglm`, `rrvglm.control`, `vcovdrrvglm`, `CM.free`, `summaryvglm`, `summary.rrvglm-class`, `summary.drrvglm-class`. ## Examples ```r ## Not run: # Fit a rank-1 RR-VGLM as a DRR-VGLM. set.seed(1); n <- 1000; S <- 6 # S must be even myrank <- 1 rdata <- data.frame(x1 = runif(n), x2 = runif(n), x3 = runif(n), x4 = runif(n)) dval <- ncol(rdata) # Number of covariates # Involves x1, x2, ... a rank-1 model: ymatrix <- with(rdata, matrix(rpois(n*S, exp(3 + x1 - 0.5*x2)), n, S)) H.C <- vector("list", dval) # Ordinary "rrvglm" for (i in 1:dval) H.C[[i]] <- CM.free(myrank) names(H.C) <- paste0("x", 1:dval) H.A <- list(CM.free(S)) # rank-1 rfit1 <- rrvglm(ymatrix ~ x1 + x2 + x3 + x4, poissonff, rdata, trace = TRUE) class(rfit1) dfit1 <- rrvglm(ymatrix ~ x1 + x2 + x3 + x4, poissonff, rdata, trace = TRUE, H.A = H.A, # drrvglm H.C = H.C) # drrvglm class(dfit1) Coef(rfit1) # The RR-VGLM is the same as Coef(dfit1) # the DRR-VGLM. max(abs(predict(rfit1) - predict(dfit1))) # 0 abs(logLik(rfit1) - logLik(dfit1)) # 0 summary(rfit1) summary(dfit1) ## End(Not run) ```

summarydrrvglm function