Test against a Parametric Function

Description

Given a response and predictors, the null hypothesis of a parametric regression function is tested versus a large-dimensional alternative in the form of a union of polyhedral convex cones.

Details

The doubconetest function is the generic version. The user provides an irreducible constraint matrix that defines two convex cones; the intersection of the cones is the null space of the matrix. The function provides a p-value for the test that the expected value of a vector is in the null space using the double-cone alternative.

Given a vector y and a design matrix X, the agconst function performs a test of the null hypothesis that the expected value of y is constant versus the alternative that it is monotone (increasing or decreasing) in each of the predictors.

The function partlintest performs a test of a linear model versus a partial linear model, using a double-cone alternative.

Author(s)

Mary C Meyer and Bodhisattva Sen Maintainer: Mary C Meyer <meyer@stat.colostate.edu>

References

TBA

Sub-clinical ADHD behaviors and classroom functioning in school-age children

Description

Observations on children aged 9-11 in classroom settings, for a study on the effects of sub-clinical hyperactive and inattentive behaviors on social and academic functioning.

Usage

data(adhd)

Format

A data frame with 686 observations on the following 4 variables.

sex

1=boy; 2=girl

ethn

1=Colombian, 2=African American, 3=Hispanic American, 5=European American

hypb

Classroom hyperactive behavior level

fcn

A measure of social and academic functioning

Source

Brewis, A.A. Schmidt, K.L., and Meyer, M.C. (2000) ADHD-type behavior and harmful dysfunction in childhood: a cross-cultural model, American Anthropologist, 102(4), pp823-828.

Examples

data(adhd)
plot(adhd$hypb,adhd$fcn) 

Test null hypothesis of constant regression function against a general, high-dimensional alternative

Description

Given a response and 1-3 predictors, the function will test the null hypothesis that the response and predictors are not related (i.e., regression function is constant), against the alternative that the regression function is monotone in each of the predictors. For one predictor, the alternative set is a double cone; for two predictors the alternative set is a quadruple cone, and an octuple cone alternative is used when there are three predictors.

Usage

agconst(y, xmat, nsim = 1000)

Arguments

Details

For one predictor, the set of non-decreasing regression functions can be described by an n-dimensional convex polyhedral cone, and the set of non-increasing regression functions is the "opposite" cone. The one-dimensional null space is the intersection of these cones. For two predictors, the alternative set consists of four cones, defined by combinations of increasing/decreasing assumptions, and for three predictors we have eight cones.

Value

Author(s)

Mary C Meyer and Bodhisattva Sen

References

TBA

See Also

doubconetest,partlintest

Examples

	n=100
	x1=runif(n);x2=runif(n);xmat=cbind(x1,x2)
	mu=1:n;for(i in 1:n){mu[i]=20*max(x1[i]-2/3,x2[i]-2/3,0)^2}
	x1g=1:21/22;x2g=x1g
	par(mar=c(1,1,1,1))
	y=mu+rnorm(n)
	ans=agconst(y,xmat,nsim=0)
	grfit=matrix(nrow=21,ncol=21)
	for(i in 1:21){for(j in 1:21){
			if(sum(x1>=x1g[i]&x2>=x2g[j])>0){
				if(sum(x1<=x1g[i]&x2<=x2g[j])>0){
					f1=min(ans$thetahat[x1>=x1g[i]&x2>=x2g[j]])
					f2=max(ans$thetahat[x1<=x1g[i]&x2<=x2g[j]])
					grfit[i,j]=(f1+f2)/2
				}else{
					grfit[i,j]=min(ans$thetahat)
				}
			}else{grfit[i,j]=max(ans$thetahat)}
	}}
	persp(x1g,x2g,grfit,th=-50,tick="detailed",xlab="x1",ylab="x2",zlab="mu")
##to get p-value for test against constant function:
#	ans=agconst(y,xmat,nsim=1000)
#	ans$pval

Kentucky Derby Winner Speed

Description

The Speeds of the Winning Horses in the Kentucky Derby, 1896-2012

Usage

data(derby)

Format

A data frame with 117 observations on the following 4 variables.

speed

winning speed

year

year of race

cond

track condition with levels fast good heav mudd slop slow

name

Name of the winning horse

Source

www.kentuckyderby.com

Examples

data(derby)
n=length(derby$year)
track=1:n*0+1
track[derby$cond=="good"]=2
track[derby$cond=="fast"]=3
plot(derby$year,derby$speed,col=track)

Test for a vector being in the null space of a double cone

Description

Given an n-vector y and the model y=m+e, and an m by n "irreducible" matrix amat, test the null hypothesis that the vector m is in the null space of amat.

Usage

doubconetest(y, amat, nsim = 1000)

Arguments

Details

The matrix amat defines a polyhedral convex cone of vectors x such that amat%*%x>=0, and also the opposite cone amat%*%x<=0. The linear space C is those x such that amat%*%x=0. The function provides a p-value for the null hypothesis that m=E(y) is in C, versus the alternative that it is in one of the two cones defined by amat.

Value

Author(s)

Mary C Meyer and Bodhisattva Sen

References

TBA, Meyer, M.C. (1999) An Extension of the Mixed Primal-Dual Bases Algorithm to the Case of More Constraints than Dimensions, Journal of Statistical Planning and Inference, 81, pp13-31.

See Also

agconst,partlintest

Examples

## test against a constant function
n=100
x=1:n/n
mu=4-5*(x-1/2)^2
y=mu+rnorm(n)
amat=matrix(0,nrow=n-1,ncol=n)
for(i in 1:(n-1)){amat[i,i]=-1;amat[i,i+1]=1}
ans=doubconetest(y,amat)
ans$pval
plot(x,y,col="slategray");lines(x,mu,lty=3,col=3)
lines(x,ans$p1,col=2)
lines(x,ans$p2,col=4)

Tests linear versus partial linear model

Description

Given a response y, a predictor x, and covariates z, the model y=m(x) +b'z +e is considered, where e is a mean-zero random error. There are three options for the null hypothesis: h0=0 tests m(x) is constant; h0=1 tests m(x) is linear, and h0=2 tests m(x) is quadratic. The (respective) alternatives are: m(x) is increasing or decreasing, m(x) is convex or concave, and m(x) is hyper-convex or hyper-concave (referring to the third derivative of m).

Usage

partlintest(x, y, zmat, h0 = 0, nsim = 1000)

Arguments

Details

For the constant null hypothesis, the alternative fit is either the monotone increasing or monotone decreasing fit – whichever minimizes the sum of squared residuals. For the linear null hypothsis, the alternative fit is either convex or concave, and for the quadratic null hypothesis, the alternative fit is constrained so that the third derivative is either positive or negative over the range of x-values.

Value

Author(s)

Mary C Meyer and Bodhisattva Sen

References

TBA

See Also

agconst,doubconetest

Examples

data(derby)
n=length(derby$speed)
zmat=matrix(0,nrow=n,ncol=2);zvec=1:n*0+1
zmat[derby$cond=="good",1]=1;zvec[derby$cond=="good"]=2
zmat[derby$cond=="fast",2]=1;zvec[derby$cond=="fast"]=3
ans=partlintest(derby$year,derby$speed,zmat,h0=2)
ans$pval
par(mar=c(4,4,1,1));par(mfrow=c(1,2))
plot(derby$year,derby$speed,col=zvec,pch=zvec)
points(derby$year,ans$p0,pch=20,col=zvec)
title("Null fit")
legend(1980,51.6,pch=3:1,col=3:1,legend=c("fast","good","slow"))
plot(derby$year,derby$speed,col=zvec,pch=zvec)
points(derby$year,ans$p1,pch=20,col=zvec)
title("Alternative fit")

data(adhd)
n=length(adhd$sex)
zmat=matrix(0,nrow=n,ncol=2)
zmat[adhd$sex==1,1]=1
zmat[adhd$ethn<5,2]=1
ans=partlintest(adhd$hypb,adhd$fcn,zmat,h0=1)
ans$pval
cols=c("pink3","lightskyblue3")
plot(adhd$hypb,adhd$fcn,col=cols[zmat[,1]+1],pch=zmat[,2]+1,
xlab="Hyperactive behavior level",ylab="Social and Academic Function Score")
cols2=c(2,4)
points(adhd$hypb,ans$p1,col=cols2[zmat[,1]+1],pch=20)