Example for using the CER Method with adagraph • adagraph

library(adagraph)

We want to to explain how to design and test a multi-stage, multi-arm clinical trial with adaptions after the interim analysis using the CER method as described in Mehta 2025 (arXiv:2501.03197). We consider a double blind clinical trial of patients with acute exacerbation of schizophrenia where to doses of an experimental drug were compared to a matching control drug. For more details on the trial see the aformentioned paper. We are mostly interested in the variant using the CER-Method, as described in section 3.2.4.

To work with the trial in adagraph, we first need to make a cer_design object with which we save all our prespecified trial specifications. These are:

test_m, the matrix specifing the graph used for the closed testing procedure
weights, the corresponding weights for the closed testing procedure
correlation, a matrix specifing the different correlations between the hypotheses. Note that to make it clear we dont know anything about the correlation between to hypotheses we need to use the value NA, since giving 0 would still use parametric tests, only with know correlation 0.
alpha, the overall FWER we want to control for
alpha_spending_function, for specifing how much of the FWER should be spent at the interim test
t, the time information fraction at which the interim test is planned. Since for our trial, it is planned that the interim look is performed after 50% of the participants have enrolled, t is 1/2.

By default, we also use the fact that if $B_J(\chi_1)\geq 1$ for some intersection hypotheses, we can also reject that subhypotheses at the interim, as described in the beginning of section 3.2.3. If this is not the desired behaviour, set seq_bonf to FALSE.

test_m <- rbind(
  H1 = c(0, 1 / 2, 1 / 2, 0),
  H2 = c(1 / 2, 0, 0, 1 / 2),
  H3 = c(0, 1, 0, 0),
  H4 = c(1, 0, 0, 0)
)

correlation <- rbind(
  c(1, 1 / 2, NA, NA),
  c(1 / 2, 1, NA, NA),
  c(NA, NA, 1, 1 / 2),
  c(NA, NA, 1 / 2, 1)
)

as <- function(x, t) 2 - 2 * stats::pnorm(stats::qnorm(1 - x / 2) / sqrt(t))

design <- cer_design(
  correlation = correlation,
  weights = c(1 / 2, 1 / 2, 0, 0),
  alpha = 0.025,
  test_m = test_m,
  alpha_spending_f = as,
  t = 1 / 2
)

Following the description of the paper, we get after the unblinding for the interim analysis the values $p_{1,1} = 0.00445$ , $p_{2,1} = 0.0952$ , $p_{3,1} = 0.0225$ , and $p_{4,1} = 0.1104$ .

design_interim <- cer_interim_test(
  design,
  p_values = c(0.00045, 0.0952, 0.0225, 0.1104)
)

We can now perform the interim test on our design object. This returns the same object with new information about which hypotheses are rejected only calculated internally. To acces this information, we can manually look at the rej_interim value.

design_interim$rej_interim
#> [1]  TRUE FALSE FALSE FALSE

We can see that for now, only the first hypotheses is rejected. Adagraph does by itself not assume that we will change anything about our trial design, even if a hypotheses was rejected. For our example, it is decided that not only the rejected first hypotheses, but also the third hypotheses will not be further tested. This can be easily accomplished using cer_drop_hypotheses, which autmatically adjust the weights and test matrix according to the transition graph. However, this does not change the time fraction, which we calculate like specified on page 24. The weights are also further changed. Both of those adjustments can be done with cer_adapt().

reallocated_t <- (1 / (2 / 35)) / (1 / (2 / 35) + 1 / (1 / 52 + 1 / 53))
ad_t <- c(1, reallocated_t, 1, reallocated_t)

design_adj <- design_interim |>
  cer_drop_hypotheses(
    c(TRUE, FALSE, TRUE, FALSE)
  ) |>
  cer_adapt(
    weights = c(0, 0.5, 0, 0.5),
    time = ad_t
  )

Note the new testing matrix resulting from those changes:

design_adj$ad_test_m
#>    [,1] [,2] [,3] [,4]
#> H1    0    0    0    0
#> H2    0    0    0    1
#> H3    0    0    0    0
#> H4    0    1    0    0

Now that the adjustments are done, we need to calculate the adjusted bounds for the rejection of the intersection hypthoses. This ist done using cer_adapt_bounds().

design_adj_bounds <- cer_adapt_bounds(design_adj)

Now we can test for the final results, $\tilde{p}_{2,2} = 0.0111$ , and $\tilde{p}_{4,2}$ . Note that the hypthoses that we no further use are still present in the design object, they just don’t get any weight assigned. Therefore, we still need to use a vector of lenght 4 for our p-values, but can use NA as a corresponding value.

design_tested <- cer_final_test(design_adj_bounds, c(NA, 0.0111, NA, 0.0234))

design_tested$rej
#> [1]  TRUE  TRUE FALSE  TRUE

As we can see, both the hypotheses that we are still testing for are now also being rejected.