# Parametric Monotone/Quadratic vs Smooth Constrained Test Performs a hypothesis test comparing a parametric model (e.g., linear, quadratic, warped plane) to a constrained smooth alternative under shape restrictions. ```r testpar( formula0, formula, data, family = gaussian(link = "identity"), ps = NULL, edfu = NULL, nsim = 1000, multicore = getOption("cgam.parallel"), method = "testpar.fit", arp = FALSE, p = NULL, space = "E", ... ) ``` ## Arguments - `formula0`: A model formula representing the null hypothesis (e.g., linear or quadratic). - `formula`: A model formula under the alternative hypothesis, possibly with shape constraints. - `data`: A data frame or environment containing the variables in the model. - `family`: A description of the error distribution and link function to be used in the model (e.g., `gaussian()`). Gaussian and binomial are now included. - `ps`: User-defined penalty term. If `NULL`, optimal values are estimated. - `edfu`: User-defined unconstrained effective degrees of freedom. - `nsim`: Number of simulations to perform to get the mixture distribution of the test statistic. (default is 1000). - `multicore`: Logical. Whether to enable parallel computation (default uses global option `cgam.parallel`). - `method`: A character string (default is `"testpar.fit"`), currently unused. - `arp`: Logical. If `TRUE`, uses autoregressive structure estimation. - `p`: Integer vector or value specifying AR(p) order. Ignored if `arp = FALSE`. - `space`: Character. Either `"Q"` for quantile-based knots or `"E"` for evenly spaced knots. Default is `"E"`. - `...`: Additional arguments passed to internal functions. ## Returns A list containing test statistics, fitted values, coefficients, and other relevant objects. ## Examples ```r # Example 1: Linear vs. monotone alternative set.seed(123) n <- 100 x <- sort(runif(n)) y <- 2 * x + rnorm(n) dat <- data.frame(y = y, x = x) ans <- testpar(formula0 = y ~ x, formula = y ~ s.incr(x), multicore = FALSE, nsim = 10) ans$pval summary(ans) ## Not run: # Example 2: Binary response: linear vs. monotone alternative set.seed(123) n <- 100 x <- runif(n) eta <- 8 * x - 4 mu <- exp(eta) / (1 + exp(eta)) y <- rbinom(n, size = 1, prob = mu) ans <- testpar(formula0 = y ~ x, formula = y ~ s.incr(x), family = binomial(link = "logit"), nsim = 10) summary(ans) xs = sort(x) ord = order(x) par(mfrow = c(1,1)) plot(x, y, cex = .2) lines(xs, binomial(link="logit")$linkinv(ans$etahat0)[ord], col=4, lty=4) lines(xs, binomial(link="logit")$linkinv(ans$etahat)[ord], col=2, lty=2) lines(xs, mu[ord]) legend("topleft", legend = c("H0 fit (etahat0)", "H1 fit (etahat)", "True mean"), col = c(4, 2, 1), lty = c(4, 2, 1), bty = "n", cex = 0.8) ## End(Not run) # Example 3: Bivariate warped-plane surface test set.seed(1) n <- 100 x1 <- runif(n) x2 <- runif(n) mu <- 4 * (x1 + x2 - x1 * x2) eps <- rnorm(n) y <- mu + eps # options(cgam.parallel = TRUE) #allow parallel computation ans <- testpar(formula0 = y ~ x1 * x2, formula = y ~ s.incr.incr(x1, x2), nsim = 10) summary(ans) par(mfrow = c(1, 2)) persp(ans$k1, ans$k2, ans$etahat0_surf, th=-50, main = 'H0 fit') persp(ans$k1, ans$k2, ans$etahat_surf, th=-50, main = 'H1 fit') ## Not run: # Example 4: AR(1) error, quadratic vs smooth convex set.seed(123) n = 100 x = runif(n) mu = 1+x+2*x^2 eps = arima.sim(n = n, list(ar = c(.4)), sd = 1) y = mu + eps ans = testpar(y~x^2, y~s.conv(x), arp=TRUE, p=1, nsim=10) ans$pval ans$phi #autoregressive coefficient estimate ## End(Not run) ## Not run: # Example 5: Three additive components with shape constraints n <- 100 x1 <- runif(n) x2 <- runif(n) x3 <- runif(n) z <- rnorm(n) mu1 <- 3 * x1 + 1 mu2 <- x2 + 3 * x2^2 mu3 <- -x3^2 mu <- mu1 + mu2 + mu3 y <- mu + rnorm(n) ans <- testpar(formula0 = y ~ x1 + x2^2 + x3^2 + z, formula = y ~ s.incr(x1) + s.incr.conv(x2) + s.decr.conc(x3) + z, family = "gaussian", nsim = 10) ans$pval summary(ans) ## End(Not run) ```

testpar function