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ANCOVA with Combined Treatment Groups

C&SP Methodology Team

Source: vignettes/ancova_combined.Rmd
ancova_combined.Rmd

Overview

This vignette provides an high-level illustration of ANCOVA with combined treatment groups in junco.
The main purpose is to highlight the ability to construct tables with combined groups using junco function a_summarize_ancova_j().
The various options such as weighting strategies, interaction models, and alternative estimands will be described.

Clinical Context

ANCOVA is commonly used for covariate-adjusted analyses of change from baseline in clinical trials. Regulatory tables often require pooling of treatment arms while preserving the original estimand.

Example Dataset

The dataset used as an example is a small dataset with simulated values for BASE and CHG.

library(dplyr)
library(rtables)
library(junco)

set.seed(101)

adchg <- tibble(
  USUBJID = sprintf("SUBJ-%03d", 1:300),
  TRT01A = factor(rep(c("Placebo", "Low Dose", "High Dose"), length.out = 300)),
  REGION = factor(sample(c("EU", "US"), 300, replace = TRUE, prob = c(0.6, 0.4))),
  BASE = rnorm(300, 50, 10),
  CHG  = rnorm(300, 0, 8)
)

Combined Column Strategy

A combined treatment column can be added in an rtables layout, by utilizing a combodf dataframe.
As seen further in this document, adding split_fun = add_combo_levels(combodf) in split_cols_by call, will result in a combined treatment column in the final table.

combodf <- tibble::tribble(
  ~valname,   ~label,    ~levelcombo,                     ~exargs,
  "ACTIVE",   "Active", c("Low Dose", "High Dose"), list()
)

ANCOVA Model

The ANCOVA model used here is:

CHG ~ TRT01A + BASE + REGION

The model is fitted once using the original randomized treatment variable, for all columns, including the combined column.

Least-squares means approach

The recommended approach for dealing with a combined treatment group column in an ANCOVA table, is to derive the estimates for that column as contrasts (least-squares means) from the linear ANCOVA model.
This can be achieved by specifying the argument method_combo = "contrasts".

lyt <- basic_table() |>
  split_cols_by(
    "TRT01A",
    ref_group = "Placebo",
    split_fun = add_combo_levels(combodf)
  ) %>%
  add_colcounts() |>
  analyze(
    vars = "CHG",
    afun = a_summarize_ancova_j,
    var_labels = "Change from Baseline",
    extra_args = list(
      variables = list(arm = "TRT01A", covariates = c("BASE", "REGION")),
      ref_path = c("TRT01A", "Placebo"),
      method_combo = "contrasts",
      weights_combo = "equal",
      weights_emmeans = "equal",
      conf_level = 0.95      
    )
  )

build_table(lyt, adchg)
#>                                              High Dose             Low Dose               Placebo               Active       
#>                                               (N=100)               (N=100)               (N=100)               (N=200)      
#> —————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————
#> n                                               100                   100                   100                   200        
#> Mean (SD)                                  -0.76 (7.541)         -0.12 (7.490)         -1.16 (6.510)         -0.44 (7.504)   
#> Median                                         -0.27                 -0.13                 -1.60                 -0.18       
#> Min, max                                    -22.1, 16.8           -15.2, 15.6           -16.2, 17.8           -22.1, 16.8    
#> 25% and 75%-ile                             -6.51, 4.08           -5.82, 4.25           -5.68, 2.48           -6.44, 4.15    
#> Adjusted Mean (SE)                         -0.73 (0.72)          -0.03 (0.72)          -0.97 (0.73)          -0.38 (0.51)    
#> Adjusted Mean (95% CI)                  -0.73 (-2.15, 0.68)   -0.03 (-1.45, 1.40)   -0.97 (-2.41, 0.47)   -0.38 (-1.38, 0.63)
#> Difference in Adjusted Means (95% CI)   0.24 (-1.78, 2.26)    0.95 (-1.06, 2.95)                          0.59 (-1.15, 2.34) 
#>   p-value                                      0.815                 0.354                                       0.504

Weighting Strategies

Note the argument weights_combo in the above layout definition. When defining contrasts in a linear model, weights can specified to use in averaging predictions. See ?emmeans::emmeans() for more details.

The following 3 weight scenarios are available for weights_combo:

  • equal: equal contribution of arms
  • proportional: weighted by observed sample size
  • proportional_marginal: marginal weights across interaction strata

If no interaction term is included in the model, the choices proportional and proportional_marginal lead to the same weights (by observed sample size), reducing to 2 options only.
While, if an interaction, for example a term like TRT01A * REGION is included, proportional and proportional_marginal are slightly different, with proportional according to the proportions observed in the interaction strata.

Collapsing Treatment Arms

The option method_combo = "collapse" is available as alternative approach, mainly as a quick verification method, and for consistency reason with another internal system.
In this scenario, the ANCOVA model is re-fitted on pooled arms and as such changes the estimand.
While it is available as an option, our recommendation is to use the contrasts approach, and only utilize method_combo = "collapse" when explicitly specified in the SAP.

Summary

  • The method_combo = "contrasts" approach is aligned with emmeans theory
  • Combined LS means and differences are implemented as linear contrasts
  • Full covariance propagation is preserved
  • Interaction models require explicit decisions on within- and across-stratum weighting

See also [junco::a_summarize_ancova_j()].

Statistical Appendix: emmeans-based Estimation

Notation

Let:

  • μ̂k\hat{\mu}_k denote the LS mean for treatment arm kk
  • 𝛍̂=(μ̂1,…,μ̂K)T\mathbf{\hat{\mu}} = (\hat{\mu}_1, \ldots, \hat{\mu}_K)^T
  • 𝐕=Var(𝛍̂)\mathbf{V} = \mathrm{Var}(\mathbf{\hat{\mu}})

All quantities are obtained from the fitted ANCOVA model.

A. Combined LS Means

A combined treatment LS mean for a set of arms CC is defined as:

μ̂C=∑k∈Cwkμ̂k,∑k∈Cwk=1 \hat{\mu}_{C} = \sum_{k \in C} w_k \, \hat{\mu}_k , \qquad \sum_{k \in C} w_k = 1

where wkw_k are user-specified weights (e.g. equal or proportional).

Variance

The variance of the combined LS mean is:

Var(μ̂C)=𝐰T𝐕𝐰 \mathrm{Var}(\hat{\mu}_{C}) = \mathbf{w}^T \, \mathbf{V} \, \mathbf{w}

This corresponds exactly to a linear contrast implemented via emmeans::contrast().

B. Contrasts Between Treatment Groups

Simple Treatment Differences

The estimated difference between two individual treatment arms kk and rr (e.g. Active vs Placebo) is defined as the contrast:

δ̂k,r=μ̂k−μ̂r \hat{\delta}_{k,r} = \hat{\mu}_k - \hat{\mu}_r

with variance:

Var(δ̂k,r)=Var(μ̂k)+Var(μ̂r)−2Cov(μ̂k,μ̂r) \mathrm{Var}(\hat{\delta}_{k,r}) = \mathrm{Var}(\hat{\mu}_k) + \mathrm{Var}(\hat{\mu}_r) - 2 \, \mathrm{Cov}(\hat{\mu}_k, \hat{\mu}_r)

This covariance term is automatically accounted for when contrasts are derived from the joint covariance matrix 𝐕\mathbf{V}.

Differences Involving Combined Arms

For a combined treatment CC compared to reference arm rr:

δ̂C,r=μ̂C−μ̂r \hat{\delta}_{C,r} = \hat{\mu}_{C} - \hat{\mu}_r

This can be written as a single linear contrast with weight vector:

𝐰C,r=(w1,…,wK,−1) \mathbf{w}_{C,r} = (w_1, \ldots, w_K, -1)

and variance:

Var(δ̂C,r)=𝐰C,rT𝐕𝐰C,r \mathrm{Var}(\hat{\delta}_{C,r}) = \mathbf{w}_{C,r}^T \, \mathbf{V} \, \mathbf{w}_{C,r}

This ensures internally consistent standard errors, confidence intervals, and p-values for combined treatment comparisons.

C. Contrasts in Interaction Models

When the ANCOVA model includes interaction terms (e.g. TRT01A * REGION), LS means are estimated within each interaction stratum.

In this setting:

  • Treatment-specific LS means μ̂k,s\hat{\mu}_{k,s} are defined per stratum ss
  • Combined means are first formed within stratum
  • Pooling across strata is then applied via user-defined weights

Formally, the combined LS mean marginal over strata is:

μ̂C=∑sπs(∑k∈Cwk|sμ̂k,s) \hat{\mu}_{C} = \sum_{s} \pi_s \, \Big( \sum_{k \in C} w_{k|s} \, \hat{\mu}_{k,s} \Big)

where:

  • wk|sw_{k|s} are within-stratum treatment weights
  • πs\pi_s are stratum-level weights

The distinction between weights_combo = "proportional" and "proportional_marginal" determines whether wk|sw_{k|s} or πs\pi_s reflect observed sample sizes within or across strata.

This two-stage weighting clarifies how estimands change in the presence of interactions and avoids implicit, undocumented averaging.