# Generally Altered, Inflated, Truncated and Deflated Negative Binomial Regression Fits a generally altered, inflated truncated and deflated negative binomial regression by MLE. The GAITD combo model having 7 types of special values is implemented. This allows mixtures of negative binomial distributions on nested and/or partitioned support as well as a multinomial logit model for (nonparametric) altered, inflated and deflated values. ```r gaitdnbinomial(a.mix = NULL, i.mix = NULL, d.mix = NULL, a.mlm = NULL, i.mlm = NULL, d.mlm = NULL, truncate = NULL, zero = c("size", "pobs", "pstr", "pdip"), eq.ap = TRUE, eq.ip = TRUE, eq.dp = TRUE, parallel.a = FALSE, parallel.i = FALSE, parallel.d = FALSE, lmunb.p = "loglink", lmunb.a = lmunb.p, lmunb.i = lmunb.p, lmunb.d = lmunb.p, lsize.p = "loglink", lsize.a = lsize.p, lsize.i = lsize.p, lsize.d = lsize.p, type.fitted = c("mean", "munbs", "sizes", "pobs.mlm", "pstr.mlm", "pdip.mlm", "pobs.mix", "pstr.mix", "pdip.mix", "Pobs.mix", "Pstr.mix", "Pdip.mix", "nonspecial", "Numer", "Denom.p", "sum.mlm.i", "sum.mix.i", "sum.mlm.d", "sum.mix.d", "ptrunc.p", "cdf.max.s"), gpstr.mix = ppoints(7) / 3, gpstr.mlm = ppoints(7) / (3 + length(i.mlm)), imethod = 1, mux.init = c(0.75, 0.5, 0.75, 0.5), imunb.p = NULL, imunb.a = imunb.p, imunb.i = imunb.p, imunb.d = imunb.p, isize.p = NULL, isize.a = isize.p, isize.i = isize.p, isize.d = isize.p, ipobs.mix = NULL, ipstr.mix = NULL, ipdip.mix = NULL, ipobs.mlm = NULL, ipstr.mlm = NULL, ipdip.mlm = NULL, byrow.aid = FALSE, ishrinkage = 0.95, probs.y = 0.35, nsimEIM = 500, cutoff.prob = 0.999, eps.trig = 1e-7, nbd.max.support = 4000, max.chunk.MB = 30) ``` ## Arguments - `truncate`: See `gaitdpoisson`. - `a.mix, i.mix, d.mix`: See `gaitdpoisson`. - `a.mlm, i.mlm, d.mlm`: See `gaitdpoisson`. - `lmunb.p, lmunb.a, lmunb.i, lmunb.d`: Link functions pertaining to the mean parameters. See `gaitdpoisson` where `llambda.p` etc. are the equivalent. - `lsize.p, lsize.a, lsize.i, lsize.d`: Link functions pertaining to the `size` parameters. See `NegBinomial`. - `eq.ap, eq.ip, eq.dp`: See `gaitdpoisson`. These apply to both `munb` and `size` parameters simultaneously. See `NegBinomial` also. - `parallel.a, parallel.i, parallel.d`: See `gaitdpoisson`. - `type.fitted`: See `gaitdpoisson`. - `gpstr.mix, gpstr.mlm`: See `gaitdpoisson`. - `imethod, ipobs.mix, ipstr.mix, ipdip.mix`: See `gaitdpoisson` and `CommonVGAMffArguments`. - `ipobs.mlm, ipstr.mlm, ipdip.mlm`: See `gaitdpoisson`. - `mux.init`: Numeric, of length 4. General downward multiplier for initial values for the sample proportions (MLEs actually). See `gaitdpoisson`. The fourth value corresponds to `size`. - `imunb.p, imunb.a, imunb.i, imunb.d`: See `gaitdpoisson`; `imunb.p` is similar to `ilambda.p`, etc. - `isize.p, isize.a, isize.i, isize.d`: See `gaitdpoisson`; `isize.p` is similar to `ilambda.p`, etc. - `probs.y, ishrinkage`: See `CommonVGAMffArguments` for information. - `byrow.aid`: Details are at `Gaitdpois`. - `zero`: See `gaitdpoisson` and `CommonVGAMffArguments`. - `nsimEIM, cutoff.prob, eps.trig`: See `negbinomial`. - `nbd.max.support, max.chunk.MB`: See `negbinomial`. ## Details The GAITD--NB combo model is the pinnacle of GAITD regression for counts because it potentially handles underdispersion, equidispersion and overdispersion relative to the Poisson, as well as alteration, inflation, deflation and truncation at arbitrary support points. In contrast, `gaitdpoisson` cannot handle overdispersion so well. The GAITD--NB is so flexible that it can accommodate up to seven modes. The full GAITD--NB--NB--MLM--NB-MLM--NB-MLM combo model may be fitted with this family function. There are seven types of special values and all arguments for these may be used in a single model. Here, the MLM represents the nonparametric while the NB refers to the negative binomial mixtures. The defaults for this function correspond to an ordinary negative binomial regression so that `negbinomial` is called instead. While much of the documentation here draws upon `gaitdpoisson`, there are additional details here because the NBD is a **two** parameter distribution that handles **overdispersion** relative to the Possion. Consequently, this family function is exceeding flexible and there are many more pitfalls to avoid. The order of the linear/additive predictors is best explained by an example. Suppose a combo model has `length(a.mix) > 3` and `length(i.mix) > 3`, `length(d.mix) > 3`, `a.mlm = 3:5`, `i.mlm = 6:9` and `d.mlm = 10:12`, say. Then `loglink(munb.p)` and `loglink(size.p)` are the first two. The third is `multilogitlink(pobs.mix)` followed by `loglink(munb.a)` and `loglink(size.a)` because `a.mix` is long enough. The sixth is `multilogitlink(pstr.mix)` followed by `loglink(munb.i)` and `loglink(size.i)` because `i.mix` is long enough. The ninth is `multilogitlink(pdip.mix)` followed by `loglink(munb.d)` and `loglink(size.d)` because `d.mix` is long enough. Next are the probabilities for the `a.mlm` values. Then are the probabilities for the `i.mlm` values. Lastly are the probabilities for the `d.mlm` values. All the probabilities are estimated by one big MLM and effectively the `"(Others)"` column of left over probabilities is associated with the nonspecial values. These might be called the **nonspecial baseline probabilities** (NBP) or reserve probabilities. The dimension of the vector of linear/additive predictors here is $M=21$. Apart from the order of the linear/additive predictors, the following are (or should be) equivalent: `gaitdnbinomial()` and `negbinomial()`, `gaitdnbinomial(a.mix = 0)` and `zanegbinomial(zero = "pobs0")`, `gaitdnbinomial(i.mix = 0)` and `zinegbinomial(zero = "pstr0")`, `gaitdnbinomial(truncate = 0)` and `posnegbinomial()`. Likewise, if `a.mix` and `i.mix` are assigned a scalar then it effectively moves that scalar to `a.mlm` and `i.mlm` because there is no parameters such as `munb.i` being estimated. Thus `gaitdnbinomial(a.mix = 0)` and `gaitdnbinomial(a.mlm = 0)` are the effectively same, and ditto for `gaitdnbinomial(i.mix = 0)` and `gaitdnbinomial(i.mlm = 0)`. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, `rrvglm` and `vgam`. The `fitted.values` slot of the fitted object, which should be extracted by the generic function `fitted`, returns the mean $mu$ by default. See the information above on `type.fitted`. ## References Yee, T. W. and Ma, C. (2024). Generally altered, inflated, truncated and deflated regression. **Statistical Science**, 39 (in press). ## Warning See `gaitdpoisson`. Also, having `eq.ap = TRUE`, `eq.ip = TRUE` and `eq.dp = TRUE` is often needed to obtain initial values that are good enough because they borrow strength across the different operators. It is usually easy to relax these assumptions later. This family function is under constant development and future changes will occur. ## Author(s) T. W. Yee ## Note If `length(a.mix)` is 1 then effectively this becomes a value of `a.mlm`. If `length(a.mix)` is 2 then an error message will be issued (overfitting really). If `length(a.mix)` is 3 then this is almost overfitting too. Hence `length(a.mix)` should be 4 or more. Ditto for `length(i.mix)` and `length(d.mix)`. See `gaitdpoisson` for notes about numerical problems that can easily arise. With the NBD there is even more potential trouble that can occur. In particular, good initial values are more necessary so it pays to experiment with arguments such as `imunb.p` and `isize.p`, as well as fitting an intercept-only model first before adding covariates and using `etastart`. Currently `max.support` is missing because only `Inf` is handled. This might change later. ## See Also `Gaitdnbinom`, `dgaitdplot`, `multinomial`, `rootogram4`, `specials`, `plotdgaitd`, `spikeplot`, `meangaitd`, `KLD`, `gaitdpoisson`, `gaitdlog`, `gaitdzeta`, `multilogitlink`, `multinomial`, `goffset`, `Trunc`, `negbinomial`, `CommonVGAMffArguments`, `simulate.vlm`. ## Examples ```r ## Not run: i.mix <- c(5, 10, 12, 16) # Inflate these values parametrically i.mlm <- c(14, 15) # Inflate these values a.mix <- c(1, 6, 13, 20) # Alter these values tvec <- c(3, 11) # Truncate these values pstr.mlm <- 0.1 # So parallel.i = TRUE pobs.mix <- pstr.mix <- 0.1; set.seed(1) gdata <- data.frame(x2 = runif(nn <- 1000)) gdata <- transform(gdata, munb.p = exp(2 + 0.0 * x2), size.p = exp(1)) gdata <- transform(gdata, y1 = rgaitdnbinom(nn, size.p, munb.p, a.mix = a.mix, i.mix = i.mix, pobs.mix = pobs.mix, pstr.mix = pstr.mix, i.mlm = i.mlm, pstr.mlm = pstr.mlm, truncate = tvec)) gaitdnbinomial(a.mix = a.mix, i.mix = i.mix, i.mlm = i.mlm) with(gdata, table(y1)) fit1 <- vglm(y1 ~ 1, crit = "coef", trace = TRUE, data = gdata, gaitdnbinomial(a.mix = a.mix, i.mix = i.mix, i.mlm = i.mlm, parallel.i = TRUE, eq.ap = TRUE, eq.ip = TRUE, truncate = tvec)) head(fitted(fit1, type.fitted = "Pstr.mix")) head(predict(fit1)) t(coef(fit1, matrix = TRUE)) # Easier to see with t() summary(fit1) spikeplot(with(gdata, y1), lwd = 2) plotdgaitd(fit1, new.plot = FALSE, offset.x = 0.2, all.lwd = 2) ## End(Not run) ```

gaitdnbinomial function