# Fit a semiparametric ANCOVA model with a local polynomial smoother Fit a semiparametric ANCOVA model with a local polynomial smoother. The specific model considered here is y_ij= g_i + m(x_ij) + e_ij, where the parametric part of the model, g_i, is a factor variable; the nonparametric part of the model, m(.), is a nonparametric smooth function; e_ij are independent identically distributed errors. The errors e_ij do not have to be independent N(0, sigma^2) errors. The errors can be heteroscedastic, i.e., e_ij = sigma_i(x_ij) * u_ij, where u_ij are independent identically distributed errors with mean 0 and variance 1. The model is fitted by the direct estimation method (Speckman, 1988), or by the backfitting method (Buja, Hastie and Tibshirani, 1989; Hastie and Tibshirani, 1990). ```r loess.ancova(x, y, group, degree = 2, criterion = c("aicc", "gcv"), family = c("gaussian", "symmetric"), method=c("Speckman", "Backfitting"), iter = 10, tol = 0.01, user.span = NULL, plot = FALSE, data.points = FALSE, legend.position = "topright", ...) ``` ## Arguments - `x`: a vector or two-column matrix of covariate values. - `y`: a vector of response values. - `group`: a vector of group indicators that has the same length as y. - `degree`: the degree of the local polynomials to be used. It can ben 0, 1 or 2. - `criterion`: the criterion for automatic smoothing parameter selection: ``aicc'' denotes bias-corrected AIC criterion, ``gcv'' denotes generalized cross-validation. - `family`: if ``gaussian'' fitting is by least-squares, and if ``symmetric'' a re-descending M estimator is used with Tukey's biweight function. - `method`: if ``Speckman'' the direct estimation method by Speckman (1988) will be used, and if ``Backfitting'' The model is fitted by the backfitting method (Buja, Hastie and Tibshirani, 1989; Hastie and Tibshirani, 1990). - `iter`: the number of iterations. - `tol`: the number of tolerance in the iterations. - `user.span`: the user-defined parameter which controls the degree of smoothing. If it is not specified, the smoothing parameter will be selected by ``aicc'' or ``gcv'' criterion. - `plot`: if TRUE (when x is one-dimensional), the fitted curves for all groups will be generated; if TRUE (when x is two-dimensional), only the smooth component in the model will be plotted. - `data.points`: if TRUE, the data points will be displayed in the plot. - `legend.position`: the position of legend in the plot: ``topright'', ``topleft'', ``bottomright'', ``bottomleft'', etc. - ``...``: control parameters. ## Details Fit a local polynomial regression with automatic smoothing parameter selection. The predictor x can either one-dimensional or two-dimensional. ## Returns a list of a vector of the parametric estimates and an object of class ``loess''. ## References Speckman, P. (1988). Kernel Smoothing in Partial Linear Models. **Journal of the Royal Statistical Society. Series B (Methodological)**, 50, 413--436. Buja, A., Hastie, T. J. and Tibshirani, R. J. (1989). Linear smoothers and additive models (with discussion). **Annals of Statistics**, 17, 453--555. Hastie, T. J. and Tibshirani, R. J. (1990). **Generalized Additive Models**. Vol. 43 of Monographs on Statistics and Applied Probability, Chapman and Hall, London. ## Author(s) X.F. Wang wangx6@ccf.org ## See Also `loess`. ## Examples ```r ## Fit semiparametric ANCOVA model set.seed(555) n1 <- 80 x1 <- runif(n1,min=0, max=3) sd1 <- 0.3 e1 <- rnorm(n1,sd=sd1) y1 <- 3*cos(pi*x1/2) + 6 + e1 n2 <- 75 x2 <- runif(n2, min=0, max=3) sd2 <- 0.2 e2 <- rnorm(n2, sd=sd2) y2 <- 3*cos(pi*x2/2) + 3 + e2 n3 <- 90 x3 <- runif(n3, min=0, max=3) sd3 <- 0.3 e3 <- rnorm(n3, sd=sd3) y3 <- 3*cos(pi*x3/2) + e3 data.bind <- rbind(cbind(x1,y1,1), cbind(x2,y2,2),cbind(x3,y3,3)) data.bind <- data.frame(data.bind) colnames(data.bind)=c('x','y','group') x <- data.bind[,1] y <- data.bind[,2] group <- data.bind[,3] loess.ancova(x,y,group, plot=TRUE, data.points=TRUE) ## Fit semiparametric ANCOVA model when the predictor is two-dimensional n1 <- 100 x11 <- runif(n1,min=0, max=3) x12 <- runif(n1,min=0, max=3) sd1 <- 0.2 e1 <- rnorm(n1,sd=sd1) y1 <- sin(2*x11) + sin(2*x12) + e1 n2 <- 100 x21 <- runif(n2, min=0, max=3) x22 <- runif(n2, min=0, max=3) sd2 <- 0.25 e2 <- rnorm(n2, sd=sd2) y2 <- sin(2*x21) + sin(2*x22) + 1 + e2 n3 <- 120 x31 <- runif(n3, min=0, max=3) x32 <- runif(n3, min=0, max=3) sd3 <- 0.25 e3 <- rnorm(n3, sd=sd3) y3 <- sin(2*x31) + sin(2*x32) + 3 + e3 data.bind <- rbind(cbind(x11, x12 ,y1,1), cbind(x21, x22, y2,2),cbind(x31, x32,y3,3)) data.bind <- data.frame(data.bind) colnames(data.bind)=c('x1','x2', 'y','group') loess.ancova(data.bind[,c(1,2)], data.bind$y, data.bind$group, plot=TRUE) ```

loess.ancova function