| Title: | Test Against Parametric Regression Function |
|---|---|
| Description: | Performs hypothesis tests concerning a regression function in a least-squares model, where the null is a parametric function, and the alternative is the union of large-dimensional convex polyhedral cones. See Bodhisattva Sen and Mary C Meyer (2016) <doi:10.1111/rssb.12178> for more details. |
| Authors: | Mary C Meyer, Bodhisattva Sen |
| Maintainer: | Mary C Meyer <[email protected]> |
| License: | GPL-2 | GPL-3 |
| Version: | 1.1 |
| Built: | 2024-11-28 06:36:37 UTC |
| Source: | CRAN |
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.
| Package: | DoubleCone |
| Type: | Package |
| Version: | 1.0 |
| Date: | 2013-10-24 |
| License: | GPL-2 | GPL-3 |
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.
Mary C Meyer and Bodhisattva Sen Maintainer: Mary C Meyer <[email protected]>
TBA
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.
data(adhd)data(adhd)
A data frame with 686 observations on the following 4 variables.
sex1=boy; 2=girl
ethn1=Colombian, 2=African American, 3=Hispanic American, 5=European American
hypbClassroom hyperactive behavior level
fcnA measure of social and academic functioning
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.
data(adhd) plot(adhd$hypb,adhd$fcn)data(adhd) plot(adhd$hypb,adhd$fcn)
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.
agconst(y, xmat, nsim = 1000)agconst(y, xmat, nsim = 1000)
y |
A numeric response vector, length n |
xmat |
an n by k design matrix, full column rank, where k=1,2, or 3. |
nsim |
The number of data sets simulated under the null hypothesis, to estimate the null distribution of the test statistic. The default is 1000, make this larger if a more precise p-value is desired. |
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.
pval |
The p-value for the test: H0: constant regression function |
p1 through p8 |
monotone fits – only p1 and p2 are returned for one predictor, etc. |
thetahat |
The least-squares alternative fit – i.e., the projection onto the multiple-cone alternative |
Mary C Meyer and Bodhisattva Sen
TBA
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$pvaln=100 x1=runif(n);x2=runif(n);xmat=cbind(x1,x2) mu=1:n;for(i in 1:n){mu[i]=20*max(x1[i]-2/3,x2[i]-2/3,0)^2} x1g=1:21/22;x2g=x1g par(mar=c(1,1,1,1)) y=mu+rnorm(n) ans=agconst(y,xmat,nsim=0) grfit=matrix(nrow=21,ncol=21) for(i in 1:21){for(j in 1:21){ if(sum(x1>=x1g[i]&x2>=x2g[j])>0){ if(sum(x1<=x1g[i]&x2<=x2g[j])>0){ f1=min(ans$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
The Speeds of the Winning Horses in the Kentucky Derby, 1896-2012
data(derby)data(derby)
A data frame with 117 observations on the following 4 variables.
speedwinning speed
yearyear of race
condtrack condition with levels fast good heav mudd slop slow
nameName of the winning horse
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$speed,col=track)data(derby) n=length(derby$year) track=1:n*0+1 track[derby$cond=="good"]=2 track[derby$cond=="fast"]=3 plot(derby$year,derby$speed,col=track)
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.
doubconetest(y, amat, nsim = 1000)doubconetest(y, amat, nsim = 1000)
y |
a vector of length n |
amat |
an m by n "irreducible" matrix |
nsim |
number of simulations to approximate null distribution – default is 1000, but choose more if a more precise p-value is desired |
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.
pval |
The p-value for the test |
p0 |
The least-squares fit under the null hypothesis |
p1 |
The least-squares fit to the "positive" cone |
p2 |
The least-squares fit to the "negative" cone |
Mary C Meyer and Bodhisattva Sen
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.
## 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$p2,col=4)## test against a constant function n=100 x=1:n/n mu=4-5*(x-1/2)^2 y=mu+rnorm(n) amat=matrix(0,nrow=n-1,ncol=n) for(i in 1:(n-1)){amat[i,i]=-1;amat[i,i+1]=1} ans=doubconetest(y,amat) ans$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)
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).
partlintest(x, y, zmat, h0 = 0, nsim = 1000)partlintest(x, y, zmat, h0 = 0, nsim = 1000)
x |
a vector of length n; this is the main predictor of interest |
y |
a vector of length n; this is the response |
zmat |
an n by k matrix of covariates, should be full column rank . |
h0 |
An indicator of what null hypothesis is to be tested: h0=0 for the null hypothesis: m(x) is constant; h0=1 tests m(x) is linear, and h0=2 tests m(x) is quadratic. |
nsim |
The number of simulations used in creating the null distribution of the test statistic. The default is nsim=1000, if a more precise p-value is desired, make nsim larger. |
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.
pval |
The p-value for the test |
p0 |
The null hypothesis fit |
p1 |
The "positive" fit |
p2 |
The "negative" fit |
Mary C Meyer and Bodhisattva Sen
TBA
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)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)