Inference for High-Dimensional Repeated Measures

Provides one-group and multiple-group tests for expectation vectors in high-dimensional repeated-measures designs. Hypotheses can be specified through projection matrices or by selecting predefined effects. The multiple-group procedures cover heterogeneous and equal covariance matrices. Centered and standardized chi-square approximations are combined with exact or subsampling-based trace estimators. The functions return p-values, test statistics, estimated degrees of freedom, and convergence parameters. For further details, see Sattler and Hichert (2025) .

hdrm

The hdrm package provides inference procedures for high-dimensional repeated-measures data in one-sample (Pauly et al., 2015) and multiple-group designs (Sattler, 2021; Sattler & Pauly, 2018).

Installation

The current version can be installed with:

if (!requireNamespace("remotes", quietly = TRUE)) {
  install.packages("remotes")
}

remotes::install_github(
  "Schnieboli/hdrm",
  dependencies = TRUE
)

Because hdrm contains compiled C++ code, Windows users installing from source need the Rtools version corresponding to their version of R.

Citation

When using hdrm in a scientific publication, please cite this article Sattler & Hichert (2025). The citation can also be obtained directly in R:

citation("hdrm")

Data formats

Both main functions accept either:

• a numeric matrix with subjects in rows and repeated-measurement dimensions in columns, or
• a data.frame with columns value, subject and dimension, giving the measurements and the subject and dimension IDs.

In the grouped case, a vector specifying which subject belongs to which group must be passed to group.

Missing values in data (and group) are not allowed and will result in an error. At least two dimensions are required. The one-group method requires at least three subjects, while in the multi-group case, each group must contain at least six complete subjects.

One-Group Test

A one-group test can be performed using hdrm_single(). The functionaccepts either a numeric matrix with dimensions in rows and subjects incolumns, or a data.frame containing columns value, subject and dimension.

The package includes two example data sets: birthrates Statistisches Bundesamt (Destatis) (2024), a wide-format data set containing birth rates for the 16 German federal states from 1990 to 2023, and EEG Höller et al. (2017), a long-format data set containing EEG measurements of 160 individuals in 40 dimensions.

Wide matrix input

data("birthrates")

## transform 'birthrates' to matrix and transpose
birthrates_matrix <- t(as.matrix(birthrates))

## test whether the time profile is flat
hdrm_single(
  data = birthrates_matrix,
  hypothesis = "flat"
)

## define hypothesis = "flat" via equivalent projection matrix
d <- ncol(birthrates_matrix)
flat_projection <- diag(d) -
  matrix(1 / d, nrow = d, ncol = d)

## test whether the time profile is flat via custom hypothesis matrix
hdrm_single(
  data = birthrates_matrix,
  hypothesis = flat_projection
)

Long data.frame input

The included EEG data contain several diagnostic groups. The following example selects one group for a one-group analysis.

data("EEG")
names(EEG) # already contains columns value, dimension and subject

## select only one diagnostic group for one-group analysis
EEG_single <- EEG[EEG$group == "SCC+", ]

## test whether the time profile is flat
hdrm_single(
  data = EEG_single,
  hypothesis = "flat"
)

## define hypothesis = "flat" via equivalent projection matrix
d <- nlevels(EEG_single$dimension)
flat_projection <- diag(d) -
  matrix(1 / d, nrow = d, ncol = d)

## test whether the time profile is flat via custom hypothesis matrix
hdrm_single(
  data = EEG_single,
  hypothesis = flat_projection
)

Multiple-Group inference

hdrm_grouped() covers two different covariance settings.

In the cov.equal = TRUE case, the a * B first/second/third-trace draws are allocated across groups approximately proportionally to the numbers of available two/four/six-subject subsets.

A third-trace quantity is estimated by subsampling in every grouped analysis. Consequently, f, tau, and the p-value depend on B and the random seed even when subsampling = FALSE. With subsampling = TRUE, the test statistic itself is seed dependent as well.

Supplying seed makes a call reproducible without permanently changing the previous R random-number state.

Wide matrix input

data("birthrates")

birthrates_matrix <- t(as.matrix(birthrates))

# group states into east and west (Berlin as east)
group <- factor(c(1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2), 
                labels = c("west", "east")) 

## test for interaction effect of group and time
hdrm_grouped(
  data = birthrates_matrix,
  hypothesis = "interaction",
  group = group,
  cov.equal = FALSE,
  subsampling = FALSE,
  B = "100*N",
  seed = 3141
)

The equal-covariance method is selected explicitly:

## test for interaction effect of group and time with equal covariance assumption
hdrm_grouped(
  data = birthrates_matrix,
  hypothesis = "interaction",
  group = group,
  cov.equal = TRUE,
  B = "100*N",
  seed = 3141
)

Long data.frame input

If data is a data.frame, it must contain columns value, subject and dimension.

data("EEG")
names(EEG) # already contains columns 'value', 'subject' and 'dimension'

## test for differences between the diagnostic groups
hdrm_grouped(
  data = EEG,
  hypothesis = "whole",
  group = EEG$group,
  cov.equal = FALSE,
  subsampling = FALSE,
  B = "100*N",
  seed = 3141
)

Hypotheses

For hdrm_grouped(), the predefined hypotheses are:

• "whole": no whole-plot or group main effect,
• "sub": no subplot or dimension main effect,
• "interaction": no group-by-dimension interaction,
• "identical": identical expectation vectors across groups,
• "flat": a flat expectation profile within every group.

A custom grouped hypothesis is supplied as a named list containing projection matrices TW and TS:

## custom hypothesis for testing for effect of time
custom_hypothesis <- list(
  TW = matrix(1 / 2, nrow = 2, ncol = 2),
  TS = diag(34) - matrix(1 / 34, nrow = 34, ncol = 34)
)

## testing for time effect (equivalent to hypothesis = "sub")
hdrm_grouped(
  data = birthrates_matrix,
  hypothesis = custom_hypothesis,
  group = group,
  B = "100*N",
  seed = 3141
)

Reported p-values

Upper-tail probabilities are calculated directly. To avoid reporting an exact numerical zero, p-values smaller than machine precision are returned as .Machine$double.eps. Such a value should be interpreted as being no larger than the numerical reporting threshold.

References