Trial Sequential Analysis for Meta-Analyses of Hazard Ratios

Performs Trial Sequential Analysis (TSA) for meta-analyses of time-to-event outcomes reported as hazard ratios. Implements the Schoenfeld required-events sample-size formula generalised for unequal allocation, the Diversity (D-squared) heterogeneity adjustment of Wetterslev et al. (2009), and O'Brien-Fleming-type alpha- and beta-spending trial sequential monitoring boundaries computed at the inverse-variance information based on observed study-level log-HR standard errors (not merely event counts), via a compiled (C++) recursive numerical integration engine ported from the R package 'RTSA' (Soerensen, Olsen, Lange and Gluud; an R implementation of the Trial Sequential Analysis software of the Copenhagen Trial Unit, < https://ctu.dk/tools>), with no fixed software-imposed limit on the number of looks (subject to available computational resources). Produces a cumulative Z-curve plot with efficacy, futility, and conventional significance boundaries. Methodology follows Miladinovic et al. (2013) and Wetterslev et al. (2009) .

tsahr

Trial Sequential Analysis (TSA) for meta-analyses of hazard ratios, in R.

Adapts the classical Wetterslev/Thorlund/Copenhagen Trial Unit TSA framework to time-to-event outcomes, in the spirit of Miladinovic et al. (2013). It's worth distinguishing what's established methodology versus what this package specifically contributes:

Established components (from the cited literature):

• the Schoenfeld required-events sample-size formula for time-to-event outcomes, here generalised for unequal allocation,
• the Diversity (D²) heterogeneity adjustment of Wetterslev et al. (2009),
• the general trial-sequential-monitoring (alpha/beta-spending) framework.

This package's specific implementation choices:

• an HR-specific adaptation combining the above into one workflow,
• inverse-variance information based on each study's own reported log-HR standard error (sum(1/SE^2)), used as the accrued-information measure instead of a simpler event-count approximation,
• a compiled (C++) recursive numerical integration engine, ported from RTSA, for the O'Brien-Fleming-type alpha- and beta-spending boundaries (no external group-sequential-design package, and no fixed software-imposed limit on the number of looks, subject to available computational resources) -- see ?tsa_hr, src/rtsa_core.h and inst/REVERSE_ENGINEERING_RTSA.md. The earlier R-only engine (R/obf_boundaries.R; validated against the published exact O'Brien-Fleming constant plus Monte Carlo type-I error control) is kept as a fallback that is ON by default (legacy_fallback = TRUE) and only runs if the compiled engine fails, clearly flagged in the result whenever it is used; set legacy_fallback = FALSE for confirmatory or RTSA-parity work,
• applying this monitoring framework to a cumulative random-effects meta-analysis Z-curve -- see the Caveats section below, this is an approximation shared with the official Copenhagen Trial Unit TSA software, not an exact result.

Installation

# install.packages("remotes")
remotes::install_github("tarak-dhaouadi/tsahr")

You'll also need its dependencies if you don't already have them:

install.packages(c("metafor", "readxl", "ggplot2"))

Usage

library(tsahr)

# Try it on a bundled example dataset: 20 studies by default;
# tsahr_example_data("HR_meta_2") is the 40-study version
path <- tsahr_example_data()

res <- tsa_hr(
  data              = path,
  target_HR         = 0.80,   # pre-specified anticipated HR (recommended)
  alpha_two_sided   = 0.05,
  power             = 0.80,
  allocation_source = "data"  # or "manual" with allocation_p = ...
)

summary(res)   # full results table
plot(res)      # the TSA chart

The subtitle of the chart shows the model/design summary and, on a second line, the pooled random-effects HR with its 95% CI, the p-value, tau² and I². With boundary_route = "analysis", the position and size of the "Analysis-route endpoint ... reached" label can be set with endpoint_label_x, endpoint_label_y and endpoint_label_size.

Random-effects inference

By default (re_inference = "standard") the random-effects model uses the usual normal-theory inference. re_inference = "hksj" (alias "knha") switches to the Hartung-Knapp-Sidik-Jonkman adjustment, and re_inference = "Hksj_adhoc" to its ad hoc variant in which the variance multiplier is never below 1:

res <- tsa_hr(path, target_HR = 0.80, re_inference = "hksj")
plot(res)   # the caption names the inference option

The tau² estimator (method), DARIS and the boundaries are unaffected; the pooled CI/p-value and the cumulative Z-curve (shown as the normal-equivalent of the HKSJ t statistic) change. The HKSJ statistic is undefined at the first look (cumulative$Z is NA there, and no point is drawn), and early looks generally are unstable (very few degrees of freedom) -- tsa_hr() prints a note about this under verbose = TRUE; see ?tsa_hr.

Using your own data

data can be a data.frame or a path to an .xlsx file with one row per study and (at least) these columns:

Rows should be in chronological (publication) order — TSA results are order-dependent. Either pre-sort your data yourself, or pass order_by = "<column name>" (e.g. a publication-year column) to have tsa_hr() sort it for you explicitly:

res <- tsa_hr(path, target_HR = 0.80, order_by = "Year")

Saving the plot / results

tsa_hr() does not write files itself; use standard R tools:

p <- plot(res)
ggplot2::ggsave("tsa_plot.png", p, width = 11, height = 7.5, dpi = 300)

write.csv(res$cumulative,    "tsa_cumulative_results.csv", row.names = FALSE)
write.csv(res$summary_table, "tsa_summary.csv",             row.names = FALSE)

Important caveats

• Circularity of target_HR = NA: if you don't specify target_HR, the required information size is calculated from the observed pooled effect, which is circular and can make the TSA boundary collapse to the conventional boundary almost immediately. Set target_HR to a pre-specified, clinically-anticipated effect for a standard, publication-quality TSA. See ?tsa_hr for details.
• Random-effects approximation: the plotted Z-curve is a cumulative random-effects meta-analysis, whose between-study variance is re-estimated at every step. The Lan-DeMets/O'Brien-Fleming monitoring boundaries strictly assume a fixed-information canonical process, so applying them to a random-effects Z-curve is a standard approximation (shared with the official Copenhagen Trial Unit TSA software), not an exact result.
• method changes more than just the pooled effect estimate: the heterogeneity-variance (tau^2) estimator selected via method does not change the mathematical alpha-spending function or the boundary-calculation algorithm -- those are a fixed part of the group-sequential design chosen up front (alpha_two_sided, power). However, because method changes tau^2, it also changes the pooled SE, the random-effects cumulative Z-curve, D-squared, DARIS, and the cumulative information schedule -- and therefore which study corresponds to which information fraction. So while the spending function itself is unaffected, switching, e.g., method = "DL" to method = "REML" can still change the boundary values attached to the observed looks indirectly, by changing the information schedule those looks land on, and therefore the practical timing of a boundary crossing.

References

Miladinovic B, Mhaskar R, Hozo I, Kumar A, Mahony H, Djulbegovic B. "Optimal information size in trial sequential analysis of time-to-event outcomes reveals potentially inconclusive results because of the risk of random error." J Clin Epidemiol. 2013;66(6):654-9.

Wetterslev J, Thorlund K, Brok J, Gluud C. "Estimating required information size by quantifying diversity in random-effects model meta-analyses." BMC Med Res Methodol. 2009;9:86.

Attribution and license

tsahr contains code ported from the R package RTSA (Anne Lyngholm Soerensen, Markus Harboe Olsen, Theis Lange and Christian Gluud), licensed GPL (>= 2): the C++ boundary engine (src/rtsa_core.h, src/rtsa_engine.cpp), its R orchestration (R/rtsa_engine.R) and the earlier R-only reconstruction of it (R/obf_boundaries.R). RTSA is the R version of Trial Sequential Analysis (TSA), originally developed as a stand-alone Java program by the Copenhagen Trial Unit; the RTSA manual is heavily inspired by the user manual for TSA by Kristian Thorlund, Janus Engstrøm, Jørn Wetterslev, Jesper Brok, Georgina Imberger and Christian Gluud. The original TSA software is available at https://ctu.dk/tools:

Copenhagen Trial Unit, Centre for Clinical Intervention Research, Department 3344, Rigshospitalet, DK-2100 Copenhagen Ø, Denmark. Tel. +45 3545 7171, Fax +45 3545 7101, E-mail: [email protected]

Because of that, tsahr is licensed GPL (>= 2) (as of 0.2.7.22; earlier releases were labelled MIT). See inst/COPYRIGHTS for the file-by-file provenance. If you use tsahr for boundary computations please also cite RTSA and the TSA software.