Model-based time trend adjustments in platform trials with
non-concurrent controls

This repository contains the main code and R functions to reproduce the
results of the simulation study in “On model-based time trend
adjustments in platform trials with non-concurrent controls”
[Paper]
[Preprint].

We investigated frequentist, model-based approaches to adjust for time
trends in platform trials utilizing non-concurrent controls. We
investigated conditions under which the model-based approaches can
successfully adjust for time trends in the simple case of a two-period
trial with two experimental treatments and a shared control. Assuming
that the second treatment arm is added at a later time period, we assess
the robustness of recently proposed model-based approaches to adjust for
time trends when utilizing non-concurrent controls. The following figure
illustrates the design considered:

Next, we explain the R functions we created to simulate the data and to
analyse it. After this, we describe the main simulation script and
results and briefly summarize the supplementary material and the figures
created.

R Functions

The folder functions contains all R functions necessary to reproduce
the results and figures in the main paper and supplementary material:

Functions for data generation

• trend_functions.R: Contains functions linear_trend() and
  sw_trend() that are used to generate linear or stepwise trend in
  the main paper and the function linear_trend2() that is used to
  simulate linear trend that starts in the second period in the
  supplementary material.

• data_sim_block.R: The function data_sim_block() simulates
  continuous or binary data for two-stage platform trial using block
  randomization and deterministic patient entry times, based on given
  sample sizes, effect sizes and time trend specifications. This
  function is used for data generation in the main paper.

• data_sim_brdt.R: The function data_sim_brdt() simulates
  continuous or binary data for two-stage platform trial using block
  randomization and random patient entry times.

• data_sim.R: The function data_sim() simulates continuous or
  binary data for two-stage platform trial using simple randomization
  per period and deterministic patient entry times.

• data_sim_rdt.R: The function data_sim_rdt() simulates
  continuous or binary data for two-stage platform trial using simple
  randomization per period and random patient entry times.

Functions for tests and models

• z_test.R: Contains functions z_test_pol() and z_test_sep()
  that perform pooled or separate one-sided z-test with the given
  data.

• t_test.R: Contains functions t_test_pol() and t_test_sep()
  that perform pooled or separate one-sided t-test with the given
  data.

• linear_model.R: Contains functions linear_model_a1(),
  linear_model_a1_int(), linear_model_a2(),
  linear_model_a2_int(), linear_model_b1() and linear_model_b2()
  that are used to fit different linear regression models
  (ALLTC-Linear, ALLTCI-Linear, ALLTC-Step, ALLTCI-Step, TC-Linear and
  TC-Step) to given data.

• z_prop_test.R: Contains function z_prop_pol() and
  z_prop_sep() that perform pooled or separate one-sided
  two-proportions z-test for with the given data.

• log_model.R: Contains functions logistic_model_a1(),
  logistic_model_a1_int(), logistic_model_a2(),
  logistic_model_a2_int(), logistic_model_b1(),
  logistic_model_b2(), logistic_model_sep() and
  logistic_model_pol() that are used to fit different logistic
  regression models (ALLTC-Linear, ALLTCI-Linear, ALLTC-Step,
  ALLTCI-Step, TC-Linear, TC-Step and separate and pooled logistic
  model) to given data.

Functions for simulations

• allinone_model.R: The function allinone_model() is programmed
  to call all the aforementioned functions and return the calculated
  metrics (T/F for reject H02 and bias) from all considered tests and
  models.

• allinone_sim.R: The function allinone_simsce() takes a data
  frame with different simulation scenarios as input and performs
  nsim replications for each of them to simulate the type I error
  rate/power, bias and rMSE.

• allinone_sim_par.R: The function allinone_simsce_par() is a
  parallelized version of allinone_simsce().

Main paper

Main simulation script and results

• mainpaper_script.R: Script with all scenarios considered in the
  main study. Simulates given scenarios and saves the results as .csv
  files in the folder results.

Running this script results in 100 separate .csv files, named
results_bin_trendpattern_parametrization_hypothesis_trendtype.csv or
results_cont_trendpattern_hypothesis_trendtype.csv depending on
whether the simulated data were binary or continuous.

• For binary endpoints, the given scenario is indicated in the file
  name as
  results_bin_trendpattern_parametrization_hypothesis_trendtype.csv,
  where

  • for trendpattern, we considered the options:

    • inv_1 - inverted-U trend with peak in the middle of
      period 1
    • inv_2 - inverted-U trend with peak between period 1 and
      2
    • inv_3 - inverted-U trend with peak in the middle of
      period 2
    • lin - linear trend
    • step - stepwise trend

  • for parametrization, we considered:

    • add - additive parametrization
    • mult - multiplicative parametrization

  • for hypothesis, we considered:

    • alpha - simulations of the type I error under the null
      hypothesis
    • pow - simulations of the power under the alternative
      hypothesis

  • for trendtype, we considered:

    • eq - equal time trend in all arms
    • diff_pos - positive time trend in control and arm 2,
      varying trend in arm 1
    • diff_neg - negative time trend in control and arm 2,
      varying trend in arm 1

  • additionally, OR1 refers to scenarios with OR1>1, otherwise
    OR1<1 is used

• For continuous endpoints, the given scenario is indicated in the
  file name as follows:
  results_cont_trendpattern_hypothesis_trendtype.csv, where

  • for trendpattern, we considered:

    • inv_1 - inverted-U trend with peak in the middle of
      period 1
    • inv_2 - inverted-U trend with peak between period 1 and
      2
    • inv_3 - inverted-U trend with peak in the middle of
      period 2
    • lin - linear trend
    • step - stepwise trend

  • for hypothesis, we considered:

    • alpha - simulations of the type I error under the null
      hypothesis
    • pow - simulations of the power under the alternative
      hypothesis

  • for trendtype, we considered:

    • eq - equal time trend in all arms
    • diff - time trend of lambda=0.1 in control and arm 2,
      varying trend in arm 1

Figures

• Plots_paper.Rmd: generates all figures used in the main paper
  and section E of the supplementary material and saves them in the
  folder figures.

Figures presented in the main paper have the suffix _main
(e.g. cont_response_main.png), while figures included in the
supplementary material are indicated by the suffix _supp
(e.g. cont_all_eq_bias_supp.png).

Supplementary material

The folder Supp_mat contains additional simulation results for
three-stage platform trials and additional results regarding different
randomization procedures discussed in the supplementary material.

Results that are reported in section F.1 of the supplementary material
are simulated in the Supp_mat_randomization.Rmd script and presented
in the corresponding HTML file.

For section F.2 of the supplementary material (platform trials with
three periods), there are analogous functions as for the main paper in
the subfolder functions, which were adapted for the three-stage
design. All considered three-stage scenarios can be simulated by running
the additional_sim_script.R, while the results are again saved in
the results subfolder. The files Plots_additional_sim.Rmd and the
corresponding HTML file are used for visualization of the additional
results. Figures presented in Section F.2 of the Supplementary material
are saved in the figures subfolder.

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Funding

EU-PEARL (EU Patient-cEntric clinicAl tRial
pLatforms) project has received funding from the Innovative Medicines
Initiative (IMI) 2 Joint Undertaking (JU) under grant agreement No
853966. This Joint Undertaking receives support from the European
Union’s Horizon 2020 research and innovation programme and EFPIA
andChildren’s Tumor Foundation, Global Alliance for TB Drug Development
non-profit organisation, Spring works Therapeutics Inc. This publication
reflects the authors’ views. Neither IMI nor the European Union, EFPIA,
or any Associated Partners are responsible for any use that may be made
of the information contained herein.