Title: Longitudinal Meta-Analysis with Robust Variance Estimation and Sensitivity Analysis
Version: 0.1.0
Description: Tools for longitudinal meta-analysis where studies contribute effect sizes at multiple follow-up time points. Implements robust variance estimation (RVE) with Tipton small-sample corrections following Hedges, Tipton, and Johnson (2010) <doi:10.1002/jrsm.5> and Tipton (2015) <doi:10.1037/met0000011>, time-varying sensitivity analysis via the Impact Threshold for a Confounding Variable (ITCV) following Frank (2000) <doi:10.1177/0049124100029002003>, benchmark calibration of the ITCV threshold against observed study-level covariates, spline-based nonlinear time-trend modeling with a nonlinearity test, and leave-k-out fragility analysis across the follow-up trajectory. Designed for researchers synthesising evidence from studies with repeated outcome measurement in education, psychology, health, and the social sciences.
License: MIT + file LICENSE
Encoding: UTF-8
RoxygenNote: 7.3.3
Depends: R (≥ 4.1.0)
Imports: metafor (≥ 3.8-1), splines, stats, utils
Suggests: clubSandwich (≥ 0.5.10), testthat (≥ 3.0.0), knitr, rmarkdown, ggplot2, dplyr
VignetteBuilder: knitr
Config/testthat/edition: 3
URL: https://github.com/causalfragility-lab/metaLong
BugReports: https://github.com/causalfragility-lab/metaLong/issues
NeedsCompilation: no
Packaged: 2026-03-25 21:49:15 UTC; Subir
Author: Subir Hait ORCID iD [aut, cre]
Maintainer: Subir Hait <haitsubi@msu.edu>
Repository: CRAN
Date/Publication: 2026-03-30 17:30:02 UTC

metaLong: Longitudinal Meta-Analysis with Robust Inference and Sensitivity Analysis

Description

metaLong provides a coherent workflow for synthesising evidence from studies that report outcomes at multiple follow-up time points. The package covers:

Typical workflow

meta  <- ml_meta(data, yi="g", vi="vg", study="id", time="wave")
sens  <- ml_sens(data, meta, yi="g", vi="vg", study="id", time="wave")
bench <- ml_benchmark(data, meta, sens, yi="g", vi="vg",
                       study="id", time="wave",
                       covariates=c("pub_year","n","quality"))
spl   <- ml_spline(meta, df=3)
ml_plot(meta, sens, bench, spl)

Author(s)

Maintainer: Subir Hait haitsubi@msu.edu (ORCID)

References

Frank, K. A. (2000). Impact of a confounding variable on a regression coefficient. Sociological Methods & Research, 29(2), 147-194.

Hedges, L. V., Tipton, E., & Johnson, M. C. (2010). Robust variance estimation in meta-regression with dependent effect size estimates. Research Synthesis Methods, 1(1), 39-65.

Tipton, E. (2015). Small sample adjustments for robust variance estimation with meta-regression. Psychological Methods, 20(3), 375-393.

See Also

Useful links:

Build working covariance matrix (uses clubSandwich if available, else shim)

Description

Build working covariance matrix (uses clubSandwich if available, else shim)

Usage

.build_V(vi, cluster, rho)

Arguments

vi

numeric vector of sampling variances

cluster

factor/character of study IDs

rho

within-study working correlation

Validate required columns exist in data

Description

Validate required columns exist in data

Usage

.check_cols(data, ...)

rma.mv engine: builds working V matrix, fits rma.mv, returns tidy inference

Description

Used when studies contribute multiple effects at the same time point.

Usage

.fit_intercept_mv(yi, vi, cluster, rho, small_sample, alpha, method = "REML")

rma.uni engine: fits intercept-only model, returns tidy inference

Description

Used when each study has at most one effect per time point. Stores tau2 directly from the REML estimate.

Usage

.fit_intercept_uni(yi, vi, cluster, small_sample, alpha, method = "REML")

Compute ITCV from r-scale value

Description

Compute ITCV from r-scale value

Usage

.itcv_from_r(r)

Standardise long-format data

Description

Standardise long-format data

Usage

.prep_data(data, yi, vi, study, time)

Partial r from t-stat

Description

Partial r from t-stat

Usage

.r_partial(t, df)

Convert effect to correlation scale (for ITCV)

Description

Convert effect to correlation scale (for ITCV)

Usage

.to_r_scale(theta, sy2)

Weighted mean and dispersion of y

Description

Weighted mean and dispersion of y

Usage

.weighted_stats(yi, vi)

Extract stored fitted model objects

Description

Extract stored fitted model objects

Usage

fits(x)

Arguments

x

An ml_meta object.

Value

Named list of fitted model objects, one per estimable time point.

Benchmark Calibration of Longitudinal ITCV Against Observed Covariates

Description

For each follow-up time point, regresses each observed study-level covariate on the effect sizes using RVE meta-regression, extracts the covariate's partial correlation with the outcome, and compares it to the significance-adjusted ITCV threshold from ml_sens(). A covariate that beats the threshold demonstrates that real-world confounding of at least that magnitude exists, which is direct evidence of effect fragility.

Usage

ml_benchmark(
  data,
  meta_obj,
  sens_obj,
  yi,
  vi,
  study,
  time,
  covariates,
  alpha = NULL,
  rho = 0.8,
  small_sample = TRUE,
  min_k = 3L
)

Arguments

data

Long-format data.frame.

meta_obj

Output from ml_meta().

sens_obj

Output from ml_sens().

yi, vi, study, time

Column names.

covariates

Character vector of observed moderator column names to benchmark.

alpha

Significance level (inherits from meta_obj if NULL).

rho

Working within-study correlation for V matrix.

small_sample

Logical; use CR2 + Satterthwaite?

min_k

Minimum studies required at a time point. Default 3L (one extra relative to ml_meta() because regression needs more d.f.).

Value

Object of class ml_benchmark (a data.frame) with columns:

time

Follow-up time.

covariate

Covariate name.

k

Number of studies.

r_partial

Partial correlation of covariate with effect size.

t_stat, df, p_val

RVE inference for the covariate slope.

itcv_alpha

ITCV_alpha threshold at this time point.

beats_threshold

Logical: does ⁠|r_partial| >= itcv_alpha⁠?

skip_reason

Character; reason a cell was skipped, else NA.

The "fragile_summary" attribute contains one row per time with counts.

Interpretation

If an observed covariate (e.g., publication year, sample quality, attrition rate) has ⁠|r_partial| >= ITCV_alpha(t)⁠, then an unobserved confounder with the same relationship to exposure and outcome would be sufficient to nullify the pooled effect at time t. This does not prove confounding–it calibrates the plausibility threshold.

See Also

ml_sens(), ml_meta()

Examples


dat   <- sim_longitudinal_meta(k = 15, times = c(0, 6, 12), seed = 2)
meta  <- ml_meta(dat, yi = "yi", vi = "vi", study = "study", time = "time")
sens  <- ml_sens(dat, meta, yi = "yi", vi = "vi", study = "study", time = "time")
bench <- ml_benchmark(dat, meta, sens,
                       yi = "yi", vi = "vi", study = "study", time = "time",
                       covariates = c("pub_year", "quality"))
print(bench)
plot(bench)

Leave-One-Out and Leave-k-Out Fragility Analysis

Description

Computes fragility indices for each time point by systematically removing studies and re-estimating the pooled effect. The fragility index at time t is the minimum number of studies whose removal changes the statistical conclusion (significant -> non-significant or vice versa).

Usage

ml_fragility(
  data,
  meta_obj,
  yi,
  vi,
  study,
  time,
  max_k = 5L,
  max_combinations = 500L,
  alpha = NULL,
  rho = 0.8,
  small_sample = TRUE,
  seed = NULL
)

Arguments

data

Long-format data.frame.

meta_obj

Output from ml_meta().

yi, vi, study, time

Column names.

max_k

Maximum number of studies to remove. Default 5.

max_combinations

Maximum number of combinations to test per k. Default 500. Larger values are more exhaustive but slower.

alpha

Significance level.

rho

Working correlation.

small_sample

Use CR2 + Satterthwaite?

seed

Random seed for sampling combinations. Default NULL.

Details

At each time point, studies are removed one at a time (or in combinations for the leave-k-out version) and the model is re-fit. The fragility index is the smallest k such that removing any set of k studies flips the significance of the pooled estimate. A fragility index of 1 means a single study's removal changes the conclusion.

For the leave-k-out version, a random sample of combinations is used when the number of combinations is large (controlled by max_combinations).

Value

Object of class ml_fragility (a data.frame) with columns:

time

Follow-up time.

k_studies

Number of studies at this time point.

p_original

Original p-value.

sig_original

Was the original result significant?

fragility_index

Min number of removals to flip significance. NA if not found within max_k.

fragility_quotient

fragility_index / k_studies (proportion).

study_removed

Study ID whose removal achieved the flip (leave-one-out only).

Examples


dat  <- sim_longitudinal_meta(k = 10, times = c(0, 6, 12), seed = 5)
meta <- ml_meta(dat, yi = "yi", vi = "vi", study = "study", time = "time")
frag <- ml_fragility(dat, meta, yi = "yi", vi = "vi",
                      study = "study", time = "time",
                      max_k = 1L, seed = 1)
print(frag)

Longitudinal Meta-Analysis with Robust Variance Estimation

Description

Fits a random-effects meta-analytic model at each unique time point in a long-format dataset of multi-wave effect sizes. Inference uses robust variance estimation (RVE) with optional Tipton (2015) small-sample corrections via the clubSandwich package.

Usage

ml_meta(
  data,
  yi,
  vi,
  study,
  time,
  alpha = 0.05,
  rho = 0.8,
  small_sample = TRUE,
  min_k = 2L,
  method = "REML",
  engine = c("rma.uni", "rma.mv")
)

Arguments

data

A data.frame in long format: one row per study x time point.

yi

Character. Name of the effect-size column.

vi

Character. Name of the sampling-variance column.

study

Character. Name of the study-ID column (cluster variable).

time

Character. Name of the follow-up time column (numeric).

alpha

Significance level for confidence intervals and p-values. Default 0.05.

rho

Assumed within-study correlation between effect sizes (used only when engine = "rma.mv"). Default 0.8.

small_sample

Logical. If TRUE (default), applies CR2 sandwich variance estimation with Satterthwaite degrees of freedom (Tipton, 2015). If FALSE, uses uncorrected z-based inference.

min_k

Integer. Minimum number of studies required to fit a model at a given time point. Default 2.

method

Character. Variance estimator passed to metafor. Default "REML".

engine

Character. Fitting engine: "rma.uni" (default) or "rma.mv". See section Engine choice.

Value

An object of class ml_meta (a data.frame) with one row per time point and columns: time, k, theta, se, df, t_stat, p_val, ci_lb, ci_ub, tau2, note.

Attributes:

"fits"

Named list of fitted model objects (one per time point).

"weights_by_time"

Named list of weight vectors for downstream use by ml_sens() and ml_benchmark().

"engine", "alpha", "rho", "small_sample"

Call metadata.

Engine choice

Two fitting engines are supported:

"rma.uni" (default)

metafor::rma.uni() – appropriate when each study contributes exactly one effect size per time point. Simpler, faster, and stores tau2 directly from the REML estimate.

"rma.mv"

metafor::rma.mv() with a prebuilt working covariance matrix – appropriate when studies contribute multiple effect sizes at the same time point (dependent effects within cluster). Requires the rho argument.

References

Hedges, L. V., Tipton, E., & Johnson, M. C. (2010). Robust variance estimation in meta-regression with dependent effect size estimates. Research Synthesis Methods, 1(1), 39-65.

Tipton, E. (2015). Small sample adjustments for robust variance estimation with meta-regression. Psychological Methods, 20(3), 375-393.

See Also

ml_sens(), ml_benchmark(), ml_spline()

Examples

dat <- sim_longitudinal_meta(k = 10, times = c(0, 6, 12), seed = 1)
result <- ml_meta(dat, yi = "yi", vi = "vi", study = "study", time = "time")
print(result)
plot(result)

# rma.mv engine for dependent effects

result_mv <- ml_meta(dat, yi = "yi", vi = "vi", study = "study", time = "time",
                     engine = "rma.mv", rho = 0.8)

Combined Publication-Ready Trajectory Figure

Description

Produces a multi-panel figure combining the pooled trajectory, confidence band, spline fit (if supplied), and ITCV sensitivity profile. Designed for direct inclusion in manuscripts.

Usage

ml_plot(
  meta_obj,
  sens_obj = NULL,
  bench_obj = NULL,
  spline_obj = NULL,
  frag_obj = NULL,
  ncol = NULL,
  main = NULL,
  col_effect = "#2166ac",
  col_sens = "#d73027",
  col_spline = "#1a9641",
  delta = NULL
)

Arguments

meta_obj

Output from ml_meta() (required).

sens_obj

Output from ml_sens() (optional; adds ITCV panel).

bench_obj

Output from ml_benchmark() (optional; adds benchmark marks).

spline_obj

Output from ml_spline() (optional; overlays spline).

frag_obj

Output from ml_fragility() (optional; adds fragility panel).

ncol

Number of columns in the panel layout. Default auto.

main

Overall figure title.

col_effect

Colour for the pooled effect trajectory.

col_sens

Colour for the ITCV line.

col_spline

Colour for the spline curve.

delta

Fragility benchmark line on the ITCV panel. Inherits from sens_obj if available.

Value

Invisibly returns a list of the objects passed in.

Examples

dat  <- sim_longitudinal_meta(k = 10, times = c(0, 6, 12), seed = 1)
meta <- ml_meta(dat, yi = "yi", vi = "vi", study = "study", time = "time")
sens <- ml_sens(dat, meta, yi = "yi", vi = "vi", study = "study", time = "time")
ml_plot(meta, sens_obj = sens)


spl  <- ml_spline(meta, df = 2)
ml_plot(meta, sens_obj = sens, spline_obj = spl,
        main = "Longitudinal Meta-Analysis Profile")

Time-Varying Sensitivity Analysis via Longitudinal ITCV

Description

Computes the Impact Threshold for a Confounding Variable (ITCV) at each follow-up time point using the pooled estimates and robust inference from ml_meta(). Two versions are returned: the raw ITCV (threshold to nullify the pooled effect) and the significance-adjusted ITCV_alpha (threshold to render the result non-significant under small-sample-corrected inference).

Usage

ml_sens(data, meta_obj, yi, vi, study, time, alpha = NULL, delta = 0.15)

Arguments

data

A data.frame in long format (same as passed to ml_meta()).

meta_obj

Output from ml_meta().

yi, vi, study, time

Column names (same meaning as in ml_meta()).

alpha

Significance level. Defaults to the value stored in meta_obj (or 0.05 if absent).

delta

Numeric. User-defined practical fragility benchmark: time points with ITCV_alpha(t) < delta are flagged as "practically fragile". Default 0.15.

Value

An object of class ml_sens (a data.frame) with columns:

time

Follow-up time.

theta, se, df

Copied from meta_obj.

sy

Weighted SD of observed effect sizes.

r_effect

Pooled effect on correlation scale.

itcv

Raw ITCV: confounding needed to nullify the estimate.

itcv_alpha

Significance-adjusted ITCV: confounding needed to make the result non-significant.

fragile

Logical; TRUE when itcv_alpha < delta.

Attributes include trajectory summaries (itcv_min, itcv_mean, fragile_prop) and a "fragile_times" character vector.

Mathematical background

At each time t, let \hat\theta_t be the pooled effect, s_{y,t}^2 the weighted variance of observed effect sizes, and c_t = t_{1-\alpha/2,\nu_t} \cdot \widehat{SE}(\hat\theta_t) the minimum effect still deemed significant. The correlation-scale pooled effect is

r_t = \hat\theta_t / \sqrt{\hat\theta_t^2 + s_{y,t}^2}

and the raw ITCV is \sqrt{|r_t|}. The significance-adjusted version replaces \hat\theta_t with |\hat\theta_t| - c_t.

References

Frank, K. A. (2000). Impact of a confounding variable on a regression coefficient. Sociological Methods & Research, 29(2), 147-194.

See Also

ml_meta(), ml_benchmark(), ml_plot()

Examples

dat  <- sim_longitudinal_meta(k = 10, times = c(0, 6, 12), seed = 1)
meta <- ml_meta(dat, yi = "yi", vi = "vi", study = "study", time = "time")
sens <- ml_sens(dat, meta, yi = "yi", vi = "vi", study = "study", time = "time")
print(sens)
plot(sens)

Spline-Based Nonlinear Time Trend in Longitudinal Meta-Analysis

Description

Fits a natural cubic spline meta-regression over follow-up time using the pooled time-point estimates from ml_meta(). Produces a smooth pooled trajectory with simultaneous pointwise confidence bands and tests for nonlinearity.

Usage

ml_spline(meta_obj, df = 3L, n_pred = 200L, alpha = NULL, test_linear = TRUE)

Arguments

meta_obj

Output from ml_meta().

df

Degrees of freedom for the natural cubic spline. Default 3. A value of 1 recovers a linear fit.

n_pred

Number of prediction points for the smooth curve. Default 200.

alpha

Confidence level (inherits from meta_obj if NULL).

test_linear

Logical. If TRUE, performs an F-test of nonlinearity (spline df > 1 vs linear fit). Default TRUE.

Details

The spline is fit by weighted least squares on the ml_meta() estimates, using 1 / se^2 as weights (i.e., inverse squared SE weighting to reflect the precision of each time-point estimate). This is a second-stage model.

For a fully joint spline model at the individual-effect level, users should call metafor::rma.mv() directly with mods = ~ ns(time, df). This function is primarily intended for visualisation and trajectory testing.

Value

Object of class ml_spline with elements:

pred

data.frame with time, fit, ci_lb, ci_ub for the smooth prediction grid.

coef

Spline coefficient estimates.

vcov

Coefficient covariance matrix.

r_squared

Weighted R-squared of the spline fit.

p_nonlinear

p-value for nonlinearity test (if requested).

df

Spline degrees of freedom used.

meta_obj

The original ml_meta object (for plotting).

See Also

ml_meta(), ml_plot()

Examples

dat  <- sim_longitudinal_meta(k = 10, times = c(0, 6, 12, 24), seed = 3)
meta <- ml_meta(dat, yi = "yi", vi = "vi", study = "study", time = "time")
spl  <- ml_spline(meta, df = 2)
print(spl)
plot(spl)

Simulate a Longitudinal Meta-Analytic Dataset

Description

Generates a synthetic long-format dataset suitable for testing and illustrating all metaLong functions. Studies contribute effect sizes at multiple follow-up time points with within-study correlation.

Usage

sim_longitudinal_meta(
  k = 20L,
  times = c(0, 6, 12, 24),
  mu = 0.4,
  tau = 0.2,
  v_range = c(0.02, 0.12),
  missing_prop = 0,
  add_covariates = TRUE,
  seed = NULL
)

Arguments

k

Number of studies. Default 20.

times

Numeric vector of follow-up time points. Default c(0, 6, 12, 24).

mu

Named numeric vector of true effects at each time point, or a single value (recycled). Default 0.4.

tau

Between-study SD. Default 0.2.

v_range

Two-element vector for the uniform sampling variance range. Default c(0.02, 0.12).

missing_prop

Proportion of study x time combinations to set missing (simulates unbalanced follow-up). Default 0.0.

add_covariates

Logical. If TRUE, adds study-level covariates pub_year, quality, and n for use with ml_benchmark(). Default TRUE.

seed

Random seed. Default NULL.

Details

The true effect at time t for study i is

\theta_{it} = \mu_t + u_i + \epsilon_{it}

where \mu_t is a time-varying mean effect (optionally nonlinear), u_i \sim N(0, \tau^2) is a study-level random effect, and \epsilon_{it} \sim N(0, v_{it}) is sampling error. Within-study correlation between time points is introduced through u_i.

Value

A data.frame in long format with columns:

study

Study identifier (character).

time

Follow-up time.

yi

Observed effect size.

vi

Sampling variance.

pub_year, quality, n

Study-level covariates (if add_covariates = TRUE).

Examples

dat <- sim_longitudinal_meta(k = 10, times = c(0, 6, 12), seed = 42)
head(dat)

# Nonlinear true trajectory

mu_t <- c("0" = 0.2, "6" = 0.5, "12" = 0.4, "24" = 0.1)
dat2 <- sim_longitudinal_meta(k = 10, times = c(0, 6, 12, 24), mu = mu_t,
                               missing_prop = 0.1, seed = 99)

Tidy an metaLong object into a clean data frame

Description

Tidy an metaLong object into a clean data frame

Usage

tidy(x, ...)

Arguments

x

A ml_sens or ml_benchmark object.

...

Additional arguments (unused).

Value

A tidy data.frame.

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