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Frequentist binary outcome designs

Source: vignettes/frequentist-binary.Rmd
frequentist-binary.Rmd

goldilocks provides two frequentist analyses for a two-arm binary outcome measured at a fixed follow-up time:

  • method = "riskdiff-fm" uses a Farrington-Manning score test.
  • method = "riskdiff-wald" retains the plug-in Wald test.

Both methods estimate the treatment-minus-control event-risk difference,

\widehat\Delta = \widehat p_{\text{treatment}} - \widehat p_{\text{control}},

and compare it with the null value or margin supplied through h0. All three alternatives are supported. For example, when events are undesirable, alternative = "less" asks whether the treatment event risk is lower than the control event risk by enough to cross the chosen threshold.

Choosing the analysis

riskdiff-fm is the more suitable choice when small samples or rare outcomes can produce zero events, all events, or little outcome variation in one arm. Its variance uses maximum likelihood event risks constrained by the null hypothesis. Under h0 = 0, an equal-arm all-zero or all-one table produces a neutral test result instead of a zero-variance error. A table with events in only one arm ordinarily retains a positive constrained variance and is analyzed in the usual way.

riskdiff-wald uses the observed arm risks directly in an unpooled plug-in variance. It is retained for designs that prespecify that analysis and for reproducing earlier results. The former method = "riskdiff" name remains a deprecated alias for "riskdiff-wald" and produces a warning. New analyses should prespecify one of the two procedures explicitly.

The Farrington-Manning calculation has no continuity correction. If its null-constrained variance is zero, goldilocks defines the score statistic as zero when the observed risk difference equals h0, positive infinity when it is above h0, and negative infinity when it is below h0.

Example design

Suppose the control event probability by 12 months is expected to be 10%, and the treatment event probability is expected to be 5%. The following small example uses the score test with a one-sided alternative:

end_of_study <- 12

result <- survival_adapt(
  hazard_treatment = prop_to_haz(0.05, endtime = end_of_study),
  hazard_control = prop_to_haz(0.10, endtime = end_of_study),
  N_total = 80,
  lambda = 10,
  interim_look = 40,
  end_of_study = end_of_study,
  alternative = "less",
  h0 = 0,
  Fn = 0.05,
  Sn = 0.90,
  prob_ha = 0.975,
  N_impute = 20,
  method = "riskdiff-fm"
)

result
#>   prob_threshold margin alternative N_treatment N_control N_enrolled N_max
#> 1          0.975      0        less          20        20         40    80
#>   post_prob_ha est_final ppp_success stop_futility stop_immediate_success
#> 1    0.9266035      -0.1           0             1                      0
#>   stop_expected_success trial_success stopping_reason decision_time
#> 1                     0         FALSE        futility      4.643332
#>   accrual_stop_time analysis_ready_time planned_completion_time
#> 1          4.643332            16.64333                16.64333
#>   followup_person_time peak_active_followup
#> 1             465.9974                   40

For a non-inferiority design, h0 can be a nonzero risk-difference margin. If an increase in undesirable events of no more than 5 percentage points is acceptable, use h0 = 0.05 with alternative = "less". The score-test variance is then calculated from event risks constrained to differ by 0.05.

Pending and missing endpoint outcomes

At an interim look, each pending endpoint is completed from the piecewise-exponential predictive model before the selected risk-difference test is applied. binary_imputation = "event-time" samples a conditional event time; binary_imputation = "bernoulli" samples the equivalent fixed-time status directly.

At the final analysis, subjects lost before complete endpoint ascertainment are excluded when imputed_final = FALSE. With imputed_final = TRUE, the package uses the selected complete-data test directly if no outcomes are missing. When outcomes require imputation, riskdiff-wald combines estimates and within-imputation variances using Rubin’s rules. This requires at least two imputations and positive total variance; zero total variance produces a non-estimability error.

Genuine final imputation is unsupported for riskdiff-fm because no validated FM pooling rule is implemented. Simulations with FM and imputed_final = TRUE require prop_loss = 0 in both arms and reject positive dropout at setup. The package does not substitute a pooled Wald test for FM.

With nonzero prop_loss, the simulator draws dropout independently of event time from an exponential distribution whose CDF at end_of_study equals prop_loss. Actual censoring by dropout can be less frequent because events can occur first. Independent censoring supports survival inference, but complete-case binary analysis can still overestimate event probabilities: early events remain observed while later endpoint outcomes can be missing. Evaluate final imputation and sensitivity to its event-time model for binary designs with dropout.

The saved results include the selected method and evaluated design arguments, providing an auditable record of the analysis specification.

Reference

Farrington CP, Manning G. Test statistics and sample size formulae for comparative binomial trials with null hypothesis of non-zero risk difference or non-unity relative risk. Statistics in Medicine. 1990; 9:1447-1454. doi:10.1002/sim.4780091208.