localnulltest: Local variable selection

Description

Learn whether covariate effects are zero at given coordinates using Bayesian model selection or information criteria.

Use coef to extract estimates and posterior probabilities for local effects.

Usage

localnulltest(y, x, z, x.adjust, localgridsize=100, localgrid,
nbaseknots=20, nlocalknots=c(5,10,15), basedegree=3, cutdegree=0,
usecutbasis=TRUE, priorCoef=normalidprior(taustd=1),
priorGroup=normalidprior(taustd=1), priorDelta=modelbbprior(),
mc.cores=min(4,length(nlocalknots)), return.mcmc=FALSE, verbose=FALSE,
...)
localnulltest_fda(y, x, z, x.adjust, function_id,
Sigma='AR/MA', localgridsize=100, localgrid, nbaseknots=20,
nlocalknots=c(5,10,15), basedegree=3, cutdegree=0, usecutbasis=TRUE,
priorCoef=momprior(), priorGroup=groupmomprior(),
priorDelta=modelbbprior(), mc.cores=min(4,length(nlocalknots)),
return.mcmc=FALSE, verbose=FALSE, ...)
localnulltest_givenknots(y, x, z, x.adjust, localgridsize=100,
localgrid, nbaseknots=20, nlocalknots=10, basedegree=3, cutdegree=0,
usecutbasis=TRUE, priorCoef=normalidprior(taustd=1),
priorGroup=normalidprior(taustd=1), priorDelta=modelbbprior(),
verbose=FALSE, ...)
localnulltest_fda_givenknots(y, x, z, x.adjust, function_id,
Sigma='AR/MA', localgridsize=100, localgrid, nbaseknots=20,
nlocalknots=10, basedegree=3, cutdegree=0, usecutbasis=TRUE,
priorCoef=normalidprior(taustd=1), priorGroup=normalidprior(taustd=1),
priorDelta=modelbbprior(), verbose=FALSE, ...)

Value

Object of class localtest, which extends a list with elements

covareffects: Estimated local covariate effects at different z values, 0.95 posterior intervals and posterior probability for the existence of an effect
pplocalgrid: Posterior probabilities for the existence of an effect for regions of z values. Do not use these unless you know what you're doing
covareffects.mcmc: MCMC output used to build covareffects. Only returned if return.mcmc=TRUE
ms: Objects of class msfit returned by modelSelection
pp_localknots: Posterior probability for each resolution level (value of nlocalknots)
Sigma: Input parameter
nlocalknots: Input parameter
basedegree: Input parameter
cutdegree: Input parameter
knots: Input parameters
regionbounds: List with region bounds defined by the local testing knots at each resolution level

Arguments

y: Vector with the outcome variable
x: Numerical matrix with covariate values
z: Matrix with d-dimensional coordinates (d>=1$ for each entry in y, and d columns)
x.adjust: Optionally, further adjustment covariates to be included in the model with no testing being performed
function_id: Function identifier. It is assumed that one observes multiple functions over z, this is the identifier of each individual function
Sigma: Error covariance. By default 'identity', other options are 'MA', 'AR' or 'AR/MA' (meaning that BIC is used to choose between MA and AR). Alternatively the user can supply a function such that Sigma(z[i,],z[j,]) returns the within-function cov(y[i,], y[j,])
localgridsize: Local test probabilities will be returned for a grid of z values of size localgridsize for each dimension
localgrid: Regions at which tests will be performed. Defaults to dividing each [min(z[,i]), max(z[,i])] into 10 equal intervals. If provided, localgrid must be a list with one entry for each z[,i], containing a vector with the desired grid for that z[,i]
nbaseknots: Number of knots for the spline approximation to the baseline effect of x on y
nlocalknots: Number of knots for the basis capturing the local effects
basedegree: Degree of the spline approximation to the baseline
cutdegree: Degree of the cut spline basis used for testing
usecutbasis: If FALSE, then the basis is not cut and a standard spline basis is returned (not recommended unless you know what you're doing)
priorCoef: Prior on the coefficients, passed on to modelSelection
priorGroup: Prior on grouped coefficients, passed on to modelSelection
priorDelta: Prior on the models, passed on to modelSelection
mc.cores: If package parallel is available on your system and nlocalknots has several entries defining several resolution levels, they will be run in parallel on mc.cores
return.mcmc: Set to TRUE to return the MCMC output from modelSelection
verbose: If TRUE some progress information is printed
...: Other arguments to be passed on to modelSelection, e.g. family='binomial' for logistic regression

Author

David Rossell

Details

Local variable selection considers the model

$$y_i= \beta_0(z_i) + sum_{j=1}^p \beta_j(z_i, x_i) + e_i$$

$\beta_0(z_i)$ is the baseline mean

$\beta_j(z_i,x_i)$ is local effect of covariate j at coordinate z_i

$e_i$ a Gaussian error term assumed either independent or with a covariance structure given by Sigma. If assuming independence it is possible to consider alternatives to Gaussianity, e.g. set family='binomial' for logistic regression or family='poisson' for Poisson regression

Note: a sum-to-zero type constraint is set on $\beta_1(z_i,x_i)$ so that it defines a deviation from the baseline mean $\beta_0(z_i)$

We model $\beta_0$ using B-splines of degree basedegree with nbaseknots knots. We model $\beta_j$ using B-splines of degree cutdegree with nlocalknots. Using cutdegree=0 runs fastest is usually gives similar inference than higher degrees, and is hence recommended by default.

Examples

Run this code


#Simulate outcome and 2 covariates
#Covariate 1 has local effect for z>0
#Covariate 2 has no effect for any z

truemean= function(x,z) {
    ans= double(nrow(x))
    group1= (x[,1]==1)
    ans[group1]= ifelse(z[group1] <=0, cos(z[group1]), 1)
    ans[!group1]= ifelse(z[!group1]<=0, cos(z[!group1]), 1/(z[!group1]+1)^2)
    return(ans)
}

n= 1000
x1= rep(0:1,c(n/2,n/2))
x2= x1 + rnorm(n)
x= cbind(x1,x2)
z= runif(n,-3,3)
m= truemean(x,z)
y= truemean(x,z) + rnorm(n, 0, .5)

#Run localnulltest with 10 knots
fit0= localnulltest(y, x=x, z=z, nlocalknots=10, niter=1000)

#Estimated covariate effects and posterior probabilities
b= coef(fit0)
b

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