# Test the equality of nonparametric curves or surfaces based on L2 distance Test the equality of nonparametric curves or surfaces based on L2 distance. The specific model considered here is y_ij= m_i(x_ij) + e_ij, where m_i(.), are nonparametric smooth functions; 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. We are interested in the problem of testing the equality of the regression curves (when x is one-dimensional) or surfaces (when x is two-dimensional), H_0: m_1(.) = m_2(.) = ... v.s. H_1: otherwise The problem can also be viewed as the test of the equality in the one-sample problem for functional data. ```r T.L2(x, ...) ## Default S3 method: T.L2(x, y, group, B = 200, degree = 1, criterion = c("aicc", "gcv"), family = c("gaussian", "symmetric"), m = 225, user.span = NULL, ...) ``` ## 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. - `B`: the number of bootstrap replicates. Usually this will be a single positive integer. - `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. - `m`: the number of the sampling points for the Monte-Carlo integration. - `user.span`: the user-defined parameter which controls the degree of smoothing. - ``...``: some control parameters can also be supplied directly. ## Details A wild bootstrap algorithm is applied to test the equality of nonparametric curves or surfaces based on L2 distance. ## Returns An object of class ``fANCOVA''. ## References Dette, H., Neumeyer, N. (2001). Nonparametric analysis of covariance. **Annals of Statistics**. 29, 1361--1400. Wang. X.F. and Ye, D. (2010). On nonparametric comparison of images and regression surfaces. **Journal of Statistical Planning and Inference**. 140, 2875--2884. ## Author(s) X.F. Wang wangx6@ccf.org ## See Also `T.aov`, `T.var`, `loess.as`, `loess.ancova`. ## Examples ```r ## Nonparametric test the equality of multiple regression curves ## Simulate data sets n1 <- 100 x1 <- runif(n1,min=0, max=3) sd1 <- 0.2 e1 <- rnorm(n1,sd=sd1) y1 <- sin(2*x1) + e1 n2 <- 100 x2 <- runif(n2, min=0, max=3) sd2 <- 0.25 e2 <- rnorm(n2, sd=sd2) y2 <- sin(2*x2) + 1 + e2 n3 <- 120 x3 <- runif(n3, min=0, max=3) sd3 <- 0.25 e3 <- rnorm(n3, sd=sd3) y3 <- sin(2*x3) + 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') t1 <- T.L2(data.bind$x, data.bind$y, data.bind$group, degree=2) t1 plot(t1) plot(t1, test.statistic=FALSE) ######## ## Nonparametric test the equality for regression surfaces ## Simulate data sets n1 <- 100 x11 <- runif(n1,min=0, max=3) x12 <- runif(n1,min=0, max=3) sd1 <- 0.2 e1 <- rnorm(n1,sd=sd1) y1 <- sin(2*x11) + sin(2*x12) + e1 n2 <- 100 x21 <- runif(n2, min=0, max=3) x22 <- runif(n2, min=0, max=3) sd2 <- 0.25 e2 <- rnorm(n2, sd=sd2) y2 <- sin(2*x21) + sin(2*x22) + 1 + e2 n3 <- 120 x31 <- runif(n3, min=0, max=3) x32 <- runif(n3, min=0, max=3) sd3 <- 0.25 e3 <- rnorm(n3, sd=sd3) y3 <- sin(2*x31) + sin(2*x32) + e3 data.bind <- rbind(cbind(x11, x12 ,y1,1), cbind(x21, x22, y2,2),cbind(x31, x32,y3,3)) data.bind <- data.frame(data.bind) colnames(data.bind)=c('x1','x2', 'y','group') T.L2(data.bind[,c(1,2)], data.bind$y, data.bind$group, degree=2) ```

T.L2 function