# T.L2: Test the equality of nonparametric curves or surfaces based... In fANCOVA: Nonparametric Analysis of Covariance

## Description

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.

## Usage

 ```1 2 3 4``` ```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.

## Value

An object of class “fANCOVA”.

## Author(s)

X.F. Wang [email protected]

## 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.

`T.aov`, `T.var`, `loess.as`, `loess.ancova`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61``` ```## 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) ```