*! version 1.0.0 metaLong for Stata 14.1

metaLong for Stata 14.1

metaLong provides a workflow for synthesising evidence from studies that
report outcomes at multiple follow-up time points (longitudinal meta-analysis).

Core commands

ml_meta -- Pool effects at each time point using
DerSimonian-Laird tau2 and cluster-robust variance estimation.

ml_sens -- Compute time-varying Impact Threshold for a
Confounding Variable (ITCV), raw and significance-adjusted forms.

ml_benchmark -- Compare each covariate partial correlation
against the ITCV_adj threshold.

ml_spline -- Fit a restricted cubic spline meta-regression
over follow-up time with pointwise confidence bands.

ml_fragility -- Leave-one-out and leave-k-out fragility
indices across the trajectory.

metalong_plot -- Combined publication-ready trajectory,
sensitivity, and fragility figures.

sim_longmeta -- Simulate a longitudinal meta-analytic
dataset.

Typical workflow

. sim_longmeta, k(20) times(0 6 12 24) seed(42) clear

. ml_meta yi vi, study(study) time(time) saving(meta_res) replace

. ml_sens yi vi, study(study) time(time) ///

metafile(meta_res) saving(sens_res) replace

. ml_benchmark yi vi, study(study) time(time) ///

metafile(meta_res) sensfile(sens_res) ///

covariates(pub_year quality n) saving(bench_res) replace

. ml_fragility yi vi, study(study) time(time) ///

metafile(meta_res) saving(frag_res) replace

. ml_spline, metafile(meta_res) df(3) saving(spline_res) replace

. metalong_plot, metafile(meta_res) sensfile(sens_res) ///

splinefile(spline_res) fragfile(frag_res) ///

saving(figure.gph) replace

Statistical methods

Pooling. ml_meta fits an intercept-only weighted regression at
each time point using DerSimonian-Laird (DL) between-study variance tau2 and
inverse-variance random-effects weights 1/(vi + tau2). Standard errors are
cluster-robust (clustered by study), analogous to Hedges, Tipton & Johnson
(2010) RVE. With the default small-sample correction, inference uses a
t(k-1) distribution.

ITCV Sensitivity. At each time t, ml_sens computes:

r_t = theta_t / sqrt(theta_t^2 + sy2_t)

ITCV_t = sqrt(|r_t|)

ITCV_adj(t) = sqrt(|r*_t|), where r*_t uses theta* = |theta_t| - crit*se_t

A time point is "fragile" if ITCV_adj(t) < delta (default 0.15).

Benchmark. For each covariate Z and time t, ml_benchmark fits a
WLS meta-regression and computes the partial correlation
r_partial = t / sqrt(t^2 + df), compared against ITCV_adj(t).

Note on CR2 / Satterthwaite correction

Stata 14.1 does not natively implement CR2 sandwich estimation with
Satterthwaite degrees of freedom (Tipton 2015). metaLong uses CR1
cluster-robust SEs with t(k-1) degrees of freedom as the best available
small-sample approximation.

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.

Author

Subir Hait, Michigan State University

Email: haitsubi@msu.edu

ORCID: 0009-0004-9871-9677

Stata translation of the R package metaLong v0.1.0.