Permutation F-test for functional linear regression.

Description

Fperm.fd creates a null distribution for a test of no effect in functional linear regression. It makes generic use of fRegress and permutes the yfdPar input.

Usage

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Fperm.fd(yfdPar, xfdlist, betalist, wt=NULL, nperm=200,
         argvals=NULL, q=0.05, plotres=TRUE, ...)

Arguments

yfdPar

the dependent variable object. It may be an object of three possible classes:

  • vector if the dependent variable is scalar.

  • fd a functional data object if the dependent variable is functional.

  • fdPar a functional parameter object if the dependent variable is functional, and if it is necessary to smooth the prediction of the dependent variable.

xfdlist

a list of length equal to the number of independent variables. Members of this list are the independent variables. They be objects of either of these two classes:

  • a vector if the independent dependent variable is scalar.

  • a functional data object if the dependent variable is functional.

In either case, the object must have the same number of replications as the dependent variable object. That is, if it is a scalar, it must be of the same length as the dependent variable, and if it is functional, it must have the same number of replications as the dependent variable.

betalist

a list of length equal to the number of independent variables. Members of this list define the regression functions to be estimated. They are functional parameter objects. Note that even if corresponding independent variable is scalar, its regression coefficient will be functional if the dependent variable is functional. Each of these functional parameter objects defines a single functional data object, that is, with only one replication.

wt

weights for weighted least squares, defaults to all 1.

nperm

number of permutations to use in creating the null distribution.

argvals

If yfdPar is a fd object, the points at which to evaluate the point-wise F-statistic.

q

Critical upper-tail quantile of the null distribution to compare to the observed F-statistic.

plotres

Argument to plot a visual display of the null distribution displaying the qth quantile and observed F-statistic.

...

Additional plotting arguments that can be used with plot.

Details

An F-statistic is calculated as the ratio of residual variance to predicted variance. The observed F-statistic is returned along with the permutation distribution.

If yfdPar is a fd object, the maximal value of the pointwise F-statistic is calculated. The pointwise F-statistics are also returned.

The default of setting q = 0.95 is, by now, fairly standard. The default nperm = 200 may be small, depending on the amount of computing time available.

If argvals is not specified and yfdPar is a fd object, it defaults to 101 equally-spaced points on the range of yfdPar.

Value

A list with the following components:

pval

the observed p-value of the permutation test.

qval

the qth quantile of the null distribution.

Fobs

the observed maximal F-statistic.

Fnull

a vector of length nperm giving the observed values of the permutation distribution.

Fvals

the pointwise values of the observed F-statistic.

Fnullvals

the pointwise values of of the permutation observations.

pvals.pts

pointwise p-values of the F-statistic.

qvals.pts

pointwise qth quantiles of the null distribution

fRegressList

the result of fRegress on the observed data

argvals

argument values for evaluating the F-statistic if yfdPar is a functional data object.

Side Effects

a plot of the functional observations

Source

Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data Analysis, 2nd ed., Springer, New York.

See Also

fRegress Fstat.fd

Examples

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##
## 1.  yfdPar = vector
##
annualprec <- log10(apply(
    CanadianWeather$dailyAv[,,"Precipitation.mm"], 2,sum))

#  set up a smaller basis using only 40 Fourier basis functions
#  to save some computation time

smallnbasis <- 40
smallbasis  <- create.fourier.basis(c(0, 365), smallnbasis)
tempfd      <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"Temperature.C"],
                       smallbasis)$fd
constantfd <- fd(matrix(1,1,35), create.constant.basis(c(0, 365)))

xfdlist <- vector("list",2)
xfdlist[[1]] <- constantfd
xfdlist[[2]] <- tempfd[1:35]

betalist   <- vector("list",2)
#  set up the first regression function as a constant
betabasis1 <- create.constant.basis(c(0, 365))
betafd1    <- fd(0, betabasis1)
betafdPar1 <- fdPar(betafd1)
betalist[[1]] <- betafdPar1

nbetabasis  <- 35
betabasis2  <- create.fourier.basis(c(0, 365), nbetabasis)
betafd2     <- fd(matrix(0,nbetabasis,1), betabasis2)

lambda        <- 10^12.5
harmaccelLfd365 <- vec2Lfd(c(0,(2*pi/365)^2,0), c(0, 365))
betafdPar2    <- fdPar(betafd2, harmaccelLfd365, lambda)
betalist[[2]] <- betafdPar2

# Should use the default nperm = 200
# but use 10 to save test time for illustration
F.res2 = Fperm.fd(annualprec, xfdlist, betalist, nperm=10)

##
## 2.  yfdpar = Functional data object (class fd)
##
# The very simplest example is the equivalent of the permutation
# t-test on the growth data.

# First set up a basis system to hold the smooths

# cut this example to reduce test time on CRAN
if(!CRAN()){

Knots  <- growth$age
norder <- 6
nbasis <- length(Knots) + norder - 2
hgtbasis <- create.bspline.basis(range(Knots), nbasis, norder, Knots)

# Now smooth with a fourth-derivative penalty and a very small smoothing
# parameter

Lfdobj <- 4
lambda <- 1e-2
growfdPar <- fdPar(hgtbasis, Lfdobj, lambda)

hgtfd <- smooth.basis(growth$age,
                      cbind(growth$hgtm,growth$hgtf),growfdPar)$fd

# Now set up factors for fRegress:

cbasis = create.constant.basis(range(Knots))

maleind = c(rep(1,ncol(growth$hgtm)),rep(0,ncol(growth$hgtf)))

constfd = fd( matrix(1,1,length(maleind)),cbasis)
maleindfd = fd( matrix(maleind,1,length(maleind)),cbasis)

xfdlist = list(constfd,maleindfd)

# The fdPar object for the coefficients and call Fperm.fd

betalist = list(fdPar(hgtbasis,2,1e-6),fdPar(hgtbasis,2,1e-6))

# Should use nperm = 200 or so,
# but use 10 to save test time
Fres = Fperm.fd(hgtfd,xfdlist,betalist,nperm=10)
}

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