# Specify an Ordered Categorical Family in a CGAM Formula This is a subroutine to specify an ordered catergorical family in a cgam formula. It set things up to a routine called cgam.polr. This is learned from the polr routine in the MASS package, which fits a logistic or probit regression model to an ordered categorical response. Currently only the logistic regression model is allowed. ```r Ord(link = "identity") ``` ## Arguments - `link`: The link function. Users don't need specify this term. ## Details See the polr section in the official manual of the MASS package ([https://cran.r-project.org/package=MASS](https://cran.r-project.org/package=MASS)) for details. ## Returns - **muhat**: The estimated expected value of a latent variable. - **zeta**: Estimated cut-points defining the intervals of a latent variable such that the latent variable is between two adjacent cut-points is equivalent to that the ordered categorical response is in a category. ## Author(s) Xiyue Liao ## References Agresti, A. (2002) **Categorical Data**. Second edition. Wiley. ## See Also `mental` ## Examples ```r ## Not run: # Example 1. # generate the predictor and the latenet variable n <- 500 set.seed(123) x <- runif(n, 0, 1) yst <- 5*x^2 + rlogis(n) # generate observed ordered response, which has levels 1, 2, 3. cts <- quantile(yst, probs = seq(0, 1, length = 4)) yord <- cut(yst, breaks = cts, include.lowest = TRUE, labels = c(1:3), Ord = TRUE) y <- as.numeric(levels(yord))[yord] # regress y on x under the shape-restriction: the latent variable is "increasing-convex" # w.r.t x ans <- cgam(y ~ s.incr.conv(x), family = Ord) # check the estimated cut-points ans$zeta # check the estimated expected value of the latent variable head(ans$muhat) # check the estimated probabilities P(y = k), k = 1, 2, 3 head(fitted(ans)) # check the estimated latent variable plot(x, yst, cex = 1, type = "n", ylab = "latent variable") cols <- topo.colors(3) for (i in 1:3) { points(x[y == i], yst[y == i], col = cols[i], pch = i, cex = 0.7) } for (i in 1:2) { abline(h = (ans$zeta)[i], lty = 4, lwd = 1) } lines(sort(x), (5*x^2)[order(x)], lwd = 2) lines(sort(x), (ans$muhat)[order(x)], col = 2, lty = 2, lwd = 2) legend("topleft", bty = "n", col = c(1, 2), lty = c(1, 2), c("true latent variable", "increasing-convex fit"), lwd = c(1, 1)) ## End(Not run) ## Not run: # Example 2. mental impairment data set # mental impairment is an ordinal response with 4 categories recorded as 1, 2, 3, and 4 # two covariates are life event index and socio-economic status (high = 1, low = 0) data(mental) table(mental$mental) # model the relationship between the latent variable and life event index as increasing # socio-economic status is included as a binary covariate fit.incr <- cgam(mental ~ incr(life) + ses, data = mental, family = Ord) # check the estimated probabilities P(mental = k), k = 1, 2, 3, 4 probs.incr <- fitted(fit.incr) head(probs.incr) ## End(Not run) ```

Ord function