Extract_Eigencomp_fDA {fPASS}

Extract/estimate eigenfunction from a sparse functional or longitudinal design by simulating from a large number of subjects.

Description

The function Extract_Eigencomp_fDA() computes the eigenfunctions and the covariance of the shrinkage scores required to conduct the projection-based test of mean function between two groups of longitudinal data or sparsely observed functional data under a random irregular design, as developed by Wang (2021).

Usage

Extract_Eigencomp_fDA(
  nobs_per_subj,
  obs.design,
  mean_diff_fnm,
  cov.type = c("ST", "NS"),
  cov.par,
  sigma2.e,
  missing_type = c("nomiss", "constant"),
  missing_percent = 0,
  eval_SS = 5000,
  alloc.ratio = c(1, 1),
  fpca_method = c("fpca.sc", "face"),
  work.grid = NULL,
  nWgrid = ifelse(is.null(work.grid), 101, length(work.grid)),
  data.driven.scores = FALSE,
  mean_diff_add_args = list(),
  fpca_optns = list()
)

Arguments

nobs_per_subj

The number of observations per subject. Each element of it must be greater than 3. It could also be a vector to indicate that the number of observation for each is randomly varying between the elements of the vector, or a scalar to ensure that the number of observations are same for each subject. See examples.

obs.design

The sampling design of the observations. Must be provided as a list with the following elements. If the design is longitudinal (e.g. a clinical trial where there is pre-specified schedule of visit for the participants) it must be a named list with elements design, visit.schedule and visit.window, where obs.design$design must be specified as 'longitudinal', visit.schedule specifying schedule of visits (in months or days or any unit of time), other than the baseline visit and visit.window denoting the maximum time window for every visit. For functional design (where the observation points are either densely observed within a compact interval or under a sparse random design), the argument must be provided as a named list with elements design and fun.domain, where obs.design$design must be specified as 'functional' and obs.design$fun.domain must be specified as a two length vector indicating the domain of the function. See Details on the specification of arguments section below more details.

mean_diff_fnm

The name of the function that output of the difference of the mean between the two groups at any given time. It must be supplied as character, so that match.fun(mean_diff_fnm) returns a valid function, that takes a vector input, and returns a vector of the same length of the input.

cov.type

The type of the covariance structure of the data, must be either of 'ST' (stationary) or 'NS' (non-stationary). This argument along with the cov.par argument must be specified compatibly to ensure that the function does not return an error. See the details of cov.par argument.

cov.par

The covariance structure of the latent response trajectory. If cov.type == 'ST' then, cov.par must be specified a named list of two elements, var and cor, where var is the common variance of the observations, which must be a positive number; and cor specifies the correlation structure between the observations. cov.par$cor must be specified in the form of the nlme::corClasses specified in R package nlme. Check the package documentation for more details for each of the correlation classes. The cov.par$cor must be a corStruct class so it can be passed onto the nlme::corMatrix() to extract the subject-specific covariance matrix. If cov.type='NS' then, cov.par must be a named list of two elements, cov.obj and eigen.comp, where only one of the cov.par$cov.obj or cov.par$eigen.comp must be non-null. This is to specify that the covariance structure of the latent trajectory can be either provided in the form of covariance function or in the form of eigenfunction and eigenvalues (Spectral decomposition). If the cov.par$cov.obj is specified, then it must be a bivariate function, with two arguments. Alternatively, if the true eigenfunctions are known, then the user can specify that by specifying cov.par$eigen.comp. In this case, the cov.par$eigen.comp must be a named list with two elements, eig.obj and eig.val, where cov.par$eigen.comp$eig.val must be positive vector and cov.par$eigen.comp$eig.obj must be a vectorized function so that its evaluation at a vector of time points returns a matrix of dimension r by length(cov.par$eigen.comp$eig.val), with r being the length of time points.

sigma2.e

Measurement error variance, should be set as zero or a very small number if the measurement error is not significant.

missing_type

The type of missing in the number of observations of the subjects. Can be one of 'nomiss' for no missing observations or 'constant' for constant missing percentage at every time point. The current version of package only supports missing_type = 'constant'.

missing_percent

The percentage of missing at each observation points for each subject. Must be supplied as number between [0, 0.8], as missing percentage more than 80% is not practical. If nobs_per_subj is supplied as vector, then missing_type is forced to set as 'nomiss' and missing_percent = 0, because the missing_type = 'constant' has no meaning if the number of observations are varying between the subject at the first, typically considered in the case of sparse random functional design.

eval_SS

The sample size based on which the eigencomponents will be estimated from data. To compute the theoretical power of the test we must make sure that we use a large enough sample size to generate the data such that the estimated eigenfunctions are very close to the true eigenfunctions and that the sampling design will not have much effect on the loss of precision. Default value 5000.

alloc.ratio

The allocation ratio of samples in the each group. Note that the eigenfunctions will still be estimated based on the total sample_size, however, the variance of the shrinkage scores (which is required to compute the power function) will be estimated based on the allocation of the samples in each group. Must be given as vector of length 2. Default value is set at c(1, 1), indicating equal sample size.

fpca_method

The method by which the FPCA is computed. Must be one of 'fpca.sc' and 'face'. If fpca_method == 'fpca.sc' then the eigencomponents are estimated using the function refund::fpca.sc(). However, since the refund::fpca.sc() function fails to estimate the correct shrinkage scores, and throws NA values when the measurement errors is estimated to be zero, we wrote out a similar function where we corrected those error in current version of refund::fpca.sc(). Check out the fpca_sc() function for details. If fpca_method == 'face', then the eigencomponents are estimated using face::face.sparse() function.

work.grid

The working grid in the domain of the functions, where the eigenfunctions and other covariance components will be estimated. Default is NULL, then, a equidistant grid points of length nWgrid will be internally created to as the default work.grid.

nWgrid

The length of the work.grid in the domain of the function based on which the eigenfunctions will be estimated. Default value is 101. If work.grid is specified, then nWgrid must be null, and vice-versa.

data.driven.scores

Indicates whether the scores are estimated from the full data, WITHOUT assuming the mean function is unknown, rather the mean function is estimated using mgcv::gam() function.

mean_diff_add_args

Additional arguments to be passed to group difference function specified in the argument mean_diff_fnm.

fpca_optns

Additional options to be passed onto either of fpca_sc() or face::face.sparse() function in order to estimate the eigencomponents. It must be a named list with elements to be passed onto the respective function, depending on the fpca_method. The names of the list must not match either of c('data', 'newdata', 'argvals.new') for fpca_method == 'face' and must not match either of c('ydata', 'Y.pred') for fpca_method == 'fpca.sc'.

Details

The function can handle data from wide variety of covariance structure, can be parametric, or non-parametric. Additional with traditional stationary structures assumed for longitudinal data (see nlme::corClasses), the user can specify any other non-stationary covariance function in the form of either a covariance function or in terms of eigenfunctions and eigenvalues. The user have a lot of flexibility into tweaking the arguments nobs_per_subject, obs.design, and cov.par to compute the eigencomponents under different sampling design and covariance process of the response trajectory, and for any arbitrary mean difference function. Internally, using the sampling design and the covariance structure specified, we generate a large data with large number of subjects, and estimate the eigenfunctions and the covariance of the estimated shrinkage scores by means of functional principal component analysis (fPCA). We put the option of using two most commonly used softwares for fPCA in the functional data literature, refund::fpca.sc() and face::face.sparse(). However, since the refund::fpca.sc() do not compute the shrinkage scores correctly, especially when the measurement error variance is estimated to be zero, we made a duplicate version of that function in our package, where we write out the scoring part on our own. The new function is named as fpca_sc(), please check it out.

Value

A list with the elements listed below.

1. mean_diff_vec - The evaluation of the mean function at the working grid.

2. est_eigenfun - The evaluation of the estimated eigenfunctions at the working grid.

3. est_eigenval - Estimated eigen values.

4. working.grid - The grid points at which mean_diff_vec and est_eigenfun are evaluated.

5. fpcCall - The exact call of either of the fpca_sc() or face::face.sparse() used to compute the eigencomponents.

6. scores_var1 - Estimated covariance of the shrinkage scores for the treatment group.

7. scores_var2 - Estimated covariance of the shrinkage scores for the placebo group.

8. pooled_var - Pooled covariance of the scores combining both the groups. This is required if the user wants to compute the power of Hotelling T statistic under equal variance assumption.

If data.driven.scores == TRUE additional components are returned

1. scores_1 - Estimated shrinkage scores for all the subjects in treatment group.

2. scores_2 - Estimated shrinkage scores for all the subjects in placebo group.

The output of this function is designed such a way the user can directly input the output obtained from this function into the arguments of Power_Proj_Test_ufDA() function to obtain the power and the sample size right away. The function PASS_Proj_Test_ufDA does the same, it is essentially a wrapper ofExtract_Eigencomp_fDA() and Power_Proj_Test_ufDA() together.

Specification of key arguments

If obs.design$design == 'functional' then a dense grid of length, specified by ngrid (typically 101/201) is internally created, and the observation points will be randomly chosen from them. The time points could also randomly chosen between any number between the interval, but then for large number of subject, fpca_sc() function will take huge time to estimate the eigenfunction. For dense design, the user must set a large value of the argument nobs_per_subj and for sparse (random) design, nobs_per_subj should be set small (and varying). On the other hand, typical to longitudinal data, if the measurements are taken at fixed time points (from baseline) for each subject, then the user must set obs.design$design == 'longitudinal' and the time points must be accordingly specified in the argument obs.design$visit.schedule. The length of obs.design$visit.schedule must match length(nobs_per_subj)-1. Internally, when obs.design$design == 'longitudinal', the function scale the visit times so that it lies between [0, 1], so the user should not specify any element named fun.domain in the list for obs.design$design == 'longitudinal'. Make sure that the mean function and the covariance function specified in the cov.par and mean_diff_fnm parameter also scaled to take argument between [0, 1]. Also, it is imperative to say that nobs_per_subj must be of a scalar positive integer for design == 'longitudinal'.

Author(s)

Salil Koner
Maintainer: Salil Koner [e-mail]

References

Wang, Qiyao (2021) Two-sample inference for sparse functional data, Electronic Journal of Statistics, Vol. 15, 1395-1423
doi:10.1214/21-EJS1802.

See Also

See Power_Proj_Test_ufDA(), refund::fpca.sc() and face::face.sparse().

Examples

# Example 1: Extract eigencomponents from stationary covariance.

set.seed(12345)
mean.diff <- function(t) {t};
obs.design <- list("design" = "longitudinal",
"visit.schedule" = seq(0.1, 0.9, length.out=7),
"visit.window" = 0.05)
cor.str <- nlme::corExp(1, form = ~ time | Subject);
sigma2 <- 1; sigma2.e <- 0.25; nobs_per_subj <- 8;
missing_type <- "constant"; missing_percent <- 0.01;
eigencomp  <- Extract_Eigencomp_fDA(obs.design = obs.design,
mean_diff_fnm = "mean.diff", cov.type = "ST",
cov.par = list("var" = sigma2, "cor" = cor.str),
sigma2.e = sigma2.e, nobs_per_subj = nobs_per_subj,
missing_type = missing_type,
missing_percent = missing_percent, eval_SS = 1000,
alloc.ratio = c(1,1), nWgrid = 201,
fpca_method = "fpca.sc", data.driven.scores = FALSE,
mean_diff_add_args = list(), fpca_optns = list(pve = 0.95))

# Example 2: Extract eigencomponents from non-stationary covariance.

alloc.ratio  <- c(1,1)
mean.diff    <- function(t) {1 * (t^3)};
eig.fun <- function(t, k) { if (k==1) {
ef <- sqrt(2)*sin(2*pi*t)
} else if (k==2) {ef <- sqrt(2)*cos(2*pi*t)}
return(ef)}
eig.fun.vec  <- function(t){cbind(eig.fun(t, 1),eig.fun(t, 2))}
eigen.comp   <- list("eig.val" = c(1, 0.5), "eig.obj" = eig.fun.vec)
obs.design   <- list(design = "functional", fun.domain = c(0,1))
cov.par      <- list("cov.obj" = NULL, "eigen.comp" = eigen.comp)
sigma2.e     <- 0.001; nobs_per_subj <- 4:7;
missing_type <- "nomiss"; missing_percent <- 0;
fpca_method  <- "fpca.sc"
eigencomp  <- Extract_Eigencomp_fDA(obs.design = obs.design,
 mean_diff_fnm = "mean.diff",
 cov.type = "NS", cov.par = cov.par,
 sigma2.e = sigma2.e, nobs_per_subj = nobs_per_subj,
 missing_type = missing_type,
 missing_percent = missing_percent, eval_SS = 1000,
 alloc.ratio = alloc.ratio, nWgrid = 201,
 fpca_method = "fpca.sc", data.driven.scores = FALSE,
 mean_diff_add_args = list(), fpca_optns = list(pve = 0.95))