nparLD: Nonparametric Analysis of Longitudinal Data

In nparLD: Nonparametric Analysis of Longitudinal Data in Factorial Experiments

Nonparametric Analysis of Longitudinal Data

Description

Performs nonparametric inference for longitudinal or repeated-measures data from crossed factorial experiments. The function can be used to test hypotheses in marginal distribution functions or hypotheses in unweighted relative marginal effects. It allows crossed whole-plot and sub-plot factors, missing values, and dependent replicate measurements.

Usage

nparLD(
  formula,
  data,
  subject,
  replicate = NULL,
  cell.weights = c("subjects", "observations"),
  effect = c("unweighted", "weighted"),
  hypothesis = c("H0F", "H0p"),
  contrast = NULL,
  sci.method = c("fisher", "multi.t"),
  Factor.Information = FALSE,
  CI.method = c("logit", "normal"),
  alpha = 0.05,
  covariance = FALSE,
  perm.test = FALSE,
  B = 1000
)

Arguments

formula	A model formula of the form response ~ factors. Factors on the right-hand side may contain crossed whole-plot and subplot factors and their interactions.
data	A data frame containing the response, subject variable, design factors, and optionally a replicate variable.
subject	Character string specifying the subject identifier.
replicate	Optional character string specifying the replicate identifier. Replicates can be dependent within subject-condition cells.
cell.weights	Character string specifying how dependent replicates are weighted. Use "subjects" for subject-level cell weighting and "observations" for observation-level cell weighting. For dependent replicates under hypothesis = "H0F", cell.weights = "subjects" averages rank scores within subject-condition cells before covariance estimation, whereas cell.weights = "observations" uses the sums of the observed rank scores within subject-condition cells and therefore gives more weight to subjects with more observed replicate measurements.The default is "subjects".
effect	Character string specifying whether weighted or unweighted relative effects are used. Use "weighted" for effects based on ordinary ranks and a sample-size weighted reference distribution. Use "unweighted" for effects based on pseudo-ranks and an equally weighted factorial-cell reference distribution. Weighted effects depend on the sample-size allocation and are mainly descriptive. Unweighted effects are the inferential target for hypothesis = "H0p". In balanced designs, both effect definitions coincide. The default is effect = "unweighted".
hypothesis	Character string specifying the hypothesis type. Use "H0F" for hypotheses in marginal distribution functions and "H0p" for hypotheses in unweighted relative marginal effects. The latter permits unequal variances or higher-order distributional differences under the null hypothesis and is related to the nonparametric Behrens-Fisher problem.
contrast	Optional list specifying a factor or interaction term for simultaneous inference. A one-element specification such as list("time") uses the standard hypothesis matrix for the selected model term, that is, the same contrast structure underlying the WTS and ATS for that term. A two-element specification such as list("time", "Tukey") or list("time", "Dunnett") applies the named multiple contrast procedure to the marginal relative effects of the selected term. A numeric vector or matrix may be supplied as the second element for user-defined contrasts.
sci.method	Character string specifying the method for simultaneous confidence intervals. Available choices are "fisher" and "multi.t". The default is "fisher".
Factor.Information	Logical. If TRUE, factor-specific relative effects, standard errors, and confidence intervals are returned. These summaries can also be displayed with plot(fit, term = ...).
CI.method	Character string specifying the method for confidence intervals for relative effects. Available choices are "logit" and "normal". The default is "logit".
alpha	Significance level for tests and confidence intervals. The default is alpha = 0.05.
covariance	Logical. If TRUE, the estimated covariance matrix used for the selected hypothesis is included in the output. The default is FALSE.
perm.test	Logical. Should the paired two-time-point permutation test be used when applicable? This option is only available for paired designs with two time points.
B	Number of permutation samples used for the paired permutation test.

Details

The function provides rank- and pseudo-rank-based procedures for factorial longitudinal data. The argument hypothesis = "H0F" specifies hypotheses in marginal distribution functions. These hypotheses compare the complete marginal distributions and are tested by rank-based procedures.

The argument hypothesis = "H0p" specifies hypotheses in unweighted relative marginal effects. These effects describe the relative position of each marginal distribution with respect to a common unweighted reference distribution and are particularly useful for effect interpretation, multiple contrast procedures, simultaneous confidence intervals, and graphical summaries.

The contrast argument computes a multiple contrast test along with simultaneous confidence intervals for the selected model terms. A one-element specification such as list("time") uses the same hypothesis matrix as the global test procedures (Wald- and ANOVA-type statistics) for the selected term. This provides simultaneous inference for the components of the corresponding global null hypothesis. A two-element specification such as list("time", "Tukey") or list("time", "Dunnett") first forms the marginal relative effects for the selected term and then applies the requested multiple contrast procedure. Thus, list("time", "Tukey") gives pairwise comparisons of the marginal time effects, whereas list("time", "Dunnett") compares the marginal time effects with the first time point. User-defined contrast vectors or matrices can be supplied as the second list element.

Missing observations are allowed. Incomplete subject-condition cells contribute where observations are available, and covariance estimation is based on the independent subject-level units. Dependent replicate measurements can be specified using the replicate argument. For hypotheses in relative marginal effects, cell.weights = "subjects" targets a typical subject-condition cell, whereas cell.weights = "observations" targets a typical replicate observation. For hypotheses in marginal distribution functions, dependent replicates are handled by averaging rank or pseudo-rank scores within subject-condition cells. In case of bivariate data (e.g., before and after measurements), the function implements a studentized permutation test for testing either hypothesis = "H0F" or hypothesis = "H0p".

Value

An object of class "nparld_fit". The object is a list containing the results of the nonparametric longitudinal analysis. The main components are:

• Design: character string describing the detected longitudinal factorial design.

• wholeplots: names of the whole-plot factors.

• subplots: names of the subplot or repeated-measures factors.

• text.ranks: character string describing the type of ranks used.

• text.hypotheses: character string describing the type of hypotheses tested.

• N.info: information on the number of subjects and observations.

• effects: a data frame with estimated relative effects. This includes the factor-level combinations, the number of contributing subjects and observations, the number of missing observations, the mean rank or pseudo-rank score, the estimated relative effect, and its standard error. For hypothesis = "H0p", confidence limits are also returned.

• factor.info: optional factor-specific relative effects, standard errors, and confidence limits for main effects and interactions, returned when Factor.Information = TRUE. These summaries can be displayed with plot(fit, term = ...).

• WTS: Wald-type test statistics, degrees of freedom, and p-values for the tested main effects and interactions.

• ATS: ANOVA-type test statistics, denominator degrees of freedom, and p-values for the tested main effects and interactions.

• MCTP: multiple contrast test results, returned when contrast is specified. These include the selected factor or interaction, the simultaneous confidence interval method, the global multiple contrast test, local contrast estimates with standard errors, simultaneous confidence limits, test statistics, p-values, degrees of freedom, and the corresponding contrast matrix.

• covariance.info: estimated covariance matrix used for the selected hypothesis, returned when covariance = TRUE.

• perm: permutation test results, returned when a permutation test is requested and applicable.

• hypothesis: the type of hypothesis tested, either "H0F" for hypotheses in marginal distribution functions or "H0p" for hypotheses in unweighted relative marginal effects.

• CI.method: method used for confidence intervals.

• .internal: internal objects used for computation and advanced post-processing.

Some components may be NULL depending on the selected hypothesis, contrast specification, and output options.

References

Akritas, M. G., & Brunner, E. (1997). A unified approach to rank tests for mixed models. Journal of Statistical Planning and Inference, 61(2), 249-277.

Brunner, E., Bathke, A.C., Konietschke, F. Rank and Pseudo-Rank Procedures for Independent Observations in Factorial Designs. Springer International Publishing, 2018.

Brunner, E., Domhof, S., & Langer, F. (2002). Nonparametric analysis of longitudinal data in factorial experiments. Wiley, New York

Brunner, E., Konietschke, F., Pauly, M., Puri, M. L. (2017). Rank-based procedures in factorial designs: Hypotheses about non-parametric treatment effects. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(5), 1463-1485.

Domhof, S., Brunner, E., Osgood, D. W. (2002). Rank procedures for repeated measures with missing values. Sociological methods & research, 30(3), 367-393.

Konietschke, F., Bathke, A. C., Hothorn, L. A., & Brunner, E. (2010). Testing and estimation of purely nonparametric effects in repeated measures designs. Computational Statistics & Data Analysis, 54(8), 1895-1905.

Konietschke, F., Hothorn, L. A., Brunner, E. (2012). Rank-based multiple test procedures and simultaneous confidence intervals. Electronic Journal of Statistics, 6, 738-759.

Rubarth, K., Pauly, M., & Konietschke, F. (2022). Ranking procedures for repeated measures designs with missing data: estimation, testing and asymptotic theory. Statistical Methods in Medical Research, 31(1), 105-118.

Rubarth, K., Sattler, P., Zimmermann, H. G., & Konietschke, F. (2021). Estimation and testing of Wilcoxon–Mann–Whitney effects in factorial clustered data designs. Symmetry, 14(2), 244.

Examples

## One repeated-measures factor: dental data
data(dental)

fit_dental <- nparLD(
  resp ~ time,
  data = dental,
  subject = "subject",
  hypothesis = "H0p"
)

fit_dental

## Crossed factorial longitudinal design: shoulder data
data(shoulder)

## Not run: 
fit_shoulder <- nparLD(
  resp ~ group1 * group2 * time,
  data = shoulder,
  subject = "subject",
  hypothesis = "H0p"
)

fit_shoulder

## End(Not run)

## Multiple contrast procedure for an interaction

## Not run: 
fit_contrast <- nparLD(
  resp ~ group1 * group2 * time,
  data = shoulder,
  subject = "subject",
  hypothesis = "H0p",
  contrast = list("group1:time")
)

fit_contrast$MCTP
print(fit_contrast$MCTP, show.matrix = TRUE)

## End(Not run)

## Missing response values
## Not run: 
dat_miss <- dental
dat_miss$resp[c(2, 7, 12)] <- NA

fit_miss <- nparLD(
  resp ~ time,
  data = dat_miss,
  subject = "subject",
  hypothesis = "H0p"
)

fit_miss

## End(Not run)

## Dependent replicate measurements: BrdU data
data(brdu)

## Not run: 
fit_brdu <- nparLD(
  resp ~ dose,
  data = brdu,
  subject = "culture",
  replicate = "replicate",
  hypothesis = "H0p",
  cell.weights = "subjects"
)

fit_brdu

## End(Not run)

## Not run: 
## Graphical displays
plot(fit_dental)
plot(fit_shoulder)
plot(fit_contrast$MCTP)

## End(Not run)

nparLD documentation built on Aug. 28, 2026, 5:06 p.m.