Test the equality of nonparametric curves or surfaces based on L2 distance

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

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

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 wangx6@ccf.org

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.aov, T.var, 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.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)

Test the equality of nonparametric curves or surfaces based on an ANOVA-type statistic

Description

Test the equality of nonparametric curves or surfaces based on an ANOVA-type statistic. 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.aov(x, ...)
## Default S3 method:
T.aov(x, y, group, B = 200, degree = 1, criterion = c("aicc", "gcv"),
		family = c("gaussian", "symmetric"), tstat = c("DN", "YB"),
		user.span = NULL, ...)

Arguments

Details

A wild bootstrap algorithm is applied to test the equality of nonparametric curves or surfaces based on an ANOVA-type statistic.

Value

An object of class “fANCOVA”

Author(s)

X.F. Wang wangx6@ccf.org

References

Dette, H., Neumeyer, N. (2001). Nonparametric analysis of covariance. Annals of Statistics. 29, 1361–1400.

Young, S.G. and Bowman, A.W. (1995). Nonparametric analysis of covariance. Biometrics. 51, 920–931.

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.var, 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.aov(data.bind$x, data.bind$y, data.bind$group)
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.aov(data.bind[,c(1,2)], data.bind$y, data.bind$group)

Test the equality of nonparametric curves or surfaces based on variance estimators

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

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 wangx6@ccf.org

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)

US national population

Description

US national population by four groups from 1900 to 1979. The four groups are: Age 0; Age 20; Age 40; Age 60.

Usage

data(USpopu)

Format

A data frame with 320 observations on 3 variables.

References

https://www.census.gov/data/tables/time-series/demo/popest/pre-1980-national.html, U.S. Census Bureau, National Intercensal Tables: 1900-1990. Last Revised: November 30, 2016

See Also

T.L2, T.aov, T.var.

Examples

data(USpopu)
t1 <- T.L2(USpopu$year, USpopu$population, USpopu$age, degree=2)
t1
plot(t1)
plot(t1, test.statistic=FALSE, legend.position="topleft")

Fit a semiparametric ANCOVA model with a local polynomial smoother

Description

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).

Usage

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

Details

Fit a local polynomial regression with automatic smoothing parameter selection. The predictor x can either one-dimensional or two-dimensional.

Value

a list of a vector of the parametric estimates and an object of class “loess”.

Author(s)

X.F. Wang wangx6@ccf.org

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.

See Also

loess.

Examples

## 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)

Fit a local polynomial regression with automatic smoothing parameter selection

Description

Fit a local polynomial regression with automatic smoothing parameter selection. Two methods are available for the selection of the smoothing parameter: bias-corrected Akaike information criterion (aicc); and generalized cross-validation (gcv).

Usage

loess.as(x, y, degree = 1, criterion = c("aicc", "gcv"), 
		family = c("gaussian", "symmetric"), user.span = NULL, 
		plot = FALSE, ...)

Arguments

Details

Fit a local polynomial regression with automatic smoothing parameter selection. The predictor x can either one-dimensional or two-dimensional.

Value

An object of class “loess”.

Author(s)

X.F. Wang wangx6@ccf.org

References

Cleveland, W. S. (1979) Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association. 74, 829–836.

Hurvich, C.M., Simonoff, J.S., and Tsai, C.L. (1998), Smoothing Parameter Selection in Nonparametric Regression Using an Improved Akaike Information Criterion. Journal of the Royal Statistical Society B. 60, 271–293.

Golub, G., Heath, M. and Wahba, G. (1979). Generalized cross validation as a method for choosing a good ridge parameter. Technometrics. 21, 215–224.

See Also

loess, loess.ancova, T.L2, T.aov, T.var.

Examples

## Fit Local Polynomial Regression with Automatic Smoothing Parameter Selection
n1 <- 100
x1 <- runif(n1,min=0, max=3)
sd1 <- 0.2
e1 <- rnorm(n1,sd=sd1)
y1 <- sin(2*x1) + e1

(y1.fit <- loess.as(x1, y1, plot=TRUE))

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

(y2.fit <- loess.as(cbind(x21, x22), y2, plot=TRUE))

Plot a fANCOVA Object

Description

To plot a fANCOVA object

Usage

## S3 method for class 'fANCOVA'
plot(x, test.statistic=TRUE, main="", n=256, legend.position="topright", ...)

Arguments

Details

This function is to plot a fANCOVA object. The plot will be generated only if the predictor x is one-dimensional. if “test.statistic=TRUE”, a density plot of the test statistic under null hypothesis will be generated; if “test.statistic=FALSE”, the estimated curves for all groups are drawn.

Author(s)

X.F. Wang wangx6@ccf.org

See Also

T.L2, T.aov, T.var.

Generate one or multiple bootstrap samples of regression residuals using the wild bootstrap method

Description

Generate bootstrap samples using the wild bootstrap method introduced by Wu (1986). One of the advantages for the wild bootstrap method is that it allows for a heterogeneous variance in the residuals in regression analysis.

Usage

wild.boot(x, nboot = 1)

Arguments

Details

This function is to generate bootstrap residuals using the wild bootstrap method.

Value

a vector or a matrix.

Author(s)

X.F. Wang wangx6@ccf.org

References

Wu, C. (1986) Jackknife, bootstrap and other resampling methods in regression analysis (with discussion). Annals of Statistics. 14, 1261–1350.

Mammen, E. (1991). Bootstrap, wild bootstrap, and asymptotic normality. Probability Theory and Related Fields. 93, 439–455.

See Also

T.L2, T.aov, T.var.

Examples

n <- 1000
x <- runif(n, min=0, max=1)
## generate heteroscedastic error variances
sig.x <- sqrt(exp(x)/2.5-0.4)
err <- sapply(sig.x, function(x) rnorm(1, sd=x))
x2 <- x^2
y <- 10+3*x+2*x2 +err
plot(x,y)
fit <- lm(y ~ x + x2)
## obtain 12 samples of the wild bootstrap residuals 
res.boot <- wild.boot(fit$res, nboot=12)
## obtain 12 samples of the wild bootstrap responses
y.boot <- matrix(rep(fit$fit,time=12), ncol=12) + res.boot
## plot the 12 wild bootstrap samples
## The wild bootstrap method keeps the patterns of variance heterogeneity 
## in the orginal sample.
par(mfrow=c(4,3))
for (i in 1:12) plot(x, y.boot[,i])