T.var: 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 variance estimators. 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

T.var(x, ...)
## Default S3 method:
T.var(x, y, group, B = 200, degree = 1, criterion = c("aicc", "gcv"),
		family = c("gaussian", "symmetric"), 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.
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 variance estimators.

Value

An object of class “fANCOVA”.

Author(s)

X.F. Wang [e-mail]

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.

See Also

T.L2, T.aov, loess.as, loess.ancova.

Examples

## 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.var(data.bind$x, data.bind$y, data.bind$group, degree=2, criterion="gcv")
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.var(data.bind[,c(1,2)], data.bind$y, data.bind$group)

Example output

fANCOVA 0.6-1 loaded

	Test the equality of curves based on variance estimators

Comparing 3 nonparametric regression curves 
Local polynomial regression with automatic smoothing parameter selection via GCV is used for curve fitting. 
Wide-bootstrap algorithm is applied to obtain the null distribution. 

Null hypothesis: there is no difference between the 3 curves.
T =  0.1963     p-value =  0.004975 


	Test the equality of surfaces based on variance estimators

Comparing 3 nonparametric regression surfaces 
Local polynomial regression with automatic smoothing parameter selection via AICC is used for surface fitting. 
Wide-bootstrap algorithm is applied to obtain the null distribution. 

Null hypothesis: there is no difference between the 3 surfaces.
T =  0.2139     p-value =  0.004975 

fANCOVA documentation built on Nov. 13, 2020, 5:07 p.m.