Sequential Sample Size for a Fixed-Width Contrast Interval

Implements the purely sequential fixed-width confidence interval procedure for a linear contrast $\psi = \sum_j c_j \mu_j$, following Chattopadhyay, Bandyopadhyay, Kelley, and Padalunkal (2025). The goal is a 100(1 - 2$\alpha$)% confidence interval $\hat\psi \pm h$ whose half-width $h$ is fixed in advance, which is what makes a noninferiority or equivalence verdict reachable by design: with bounds $\pm\delta$, equivalence can never be declared unless $h < \delta$, and targeting $h = \delta/2$ gives a truly equivalent contrast about a 90% chance of being declared equivalent at $\alpha = .05$. Because no fixed sample size can guarantee a bounded-width interval when the error variance is unknown (Dantzig, 1940), the procedure is sequential: begin with a pilot, then keep sampling, re-estimating the variance, until the stopping criterion is met. Used with pilot = TRUE the function plans the pilot and the cost-optimal allocation; with pilot = FALSE it evaluates the stopping criterion at the data in hand, in the style of mr_smd.

Usage

ss_seq_c(
  c_weights,
  half_width,
  s = NULL,
  n = NULL,
  cost = NULL,
  alpha_level = 0.05,
  quantile = c("t", "normal"),
  pilot = FALSE,
  m0 = 10
)

Arguments

c_weights: The contrast weights. The weights must sum to zero with the positive weights summing to 1 and the negative weights to -1, so that half_width is on the raw scale of the response.
half_width: The target half-width $h$ of the 100(1 - 2$\alpha$)% confidence interval, in raw units of the response. For a noninferiority or equivalence decision at bounds $\pm\delta$, the recommended target is $\delta/2$.
s: The current estimate(s) of the error standard deviation: either a single pooled value or one value per group (aligned with c_weights). Required when pilot = FALSE; optional planning values when pilot = TRUE (used only to shape the allocation).
n: The current per-group sample sizes, aligned with c_weights. Required when pilot = FALSE.
cost: Optional per-observation sampling costs, one per group, aligned with c_weights. When supplied, the reported allocation is the cost-optimal $n_j \propto |c_j|\, \sigma_j / \sqrt{\mathrm{cost}_j}$ (Chattopadhyay et al., 2025); with equal costs and standard deviations this reduces to allocation proportional to $|c_j|$.
alpha_level: One-sided rate per bound; the interval is at confidence level 1 - 2$\alpha$. Default 0.05 (a 90% interval).
quantile: "t" (default) uses the t quantile on the current error degrees of freedom in the stopping criterion, a finite-sample refinement; "normal" uses the normal quantile of the classical statement of the rule (Chow & Robbins, 1965). The normal rule stops slightly early at wide targets, the finite-sample undershoot anticipated by Woodroofe (1977); the t rule corrects it at a small cost in sample size.
pilot: TRUE to plan the pilot stage; FALSE (default) to evaluate the stopping criterion at the current data.
m0: The minimum pilot sample size per group. Default 10. A pilot that is too small makes the variance estimate that drives the early steps unstable.

Value

With pilot = TRUE, a data.frame with row pilot_n_per_group (the pilot size for each group with a nonzero weight) followed by one allocation_j row per group giving the recommended sampling proportions for the accrual stage. With pilot = FALSE, rows stop (1 = the criterion is met, stop sampling; 0 = continue), half_width_current (the half-width the interval would have now), half_width_target, N_current, N_projected (the approximate total at which the criterion would be met under the current allocation, from the normal approximation), and the allocation_j rows for the next round of sampling.

Details

The stopping criterion. Sampling stops at the first $N$ for which $$q^2 \sum_j c_j^2 s_j^2 / n_j \;\le\; h^2,$$ where $q$ is the t or normal quantile at $1-\alpha$. With a pooled $s$ and equal allocation this is the Chow and Robbins (1965) rule specialized to a contrast; the procedure is asymptotically consistent (coverage approaches 1 - 2$\alpha$) and first-order efficient (the mean stopping size approaches the oracle $n^*$ an investigator with known variance would use). See ss_seq_c_sensitivity for a Monte Carlo evaluation of both properties.

Degrees of freedom. With a pooled s the criterion uses $\nu = N - J$. With per-group s it uses the Satterthwaite approximation, which is the appropriate error law when the variances are not assumed homogeneous.

Batches. Observations may be added in batches rather than one at a time; the asymptotic properties survive batching. Re-call the function after each batch.

References

Chattopadhyay, B., Bandyopadhyay, T., Kelley, K., & Padalunkal, J. J. (2025). A sequential approach for noninferiority or equivalence of a linear contrast under cost constraints. Psychological Methods, 30(2), 425–439. doi:10.1037/met0000570

Chow, Y. S., & Robbins, H. (1965). On the asymptotic theory of fixed-width sequential confidence intervals for the mean. The Annals of Mathematical Statistics, 36(2), 457–462.

Dantzig, G. B. (1940). On the non-existence of tests of "Student's" hypothesis having power functions independent of $\sigma$. The Annals of Mathematical Statistics, 11(2), 186–192.

Mukhopadhyay, N., & de Silva, B. M. (2009). Sequential methods and their applications. CRC Press.

Woodroofe, M. (1977). Second order approximations for sequential point and interval estimation. The Annals of Statistics, 5(5), 984–995.

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Plan the pilot for a two-group contrast, target half-width 2.5
#    (bounds of 5 with the h = delta/2 rule):
ss_seq_c(c_weights = c(1, -1), half_width = 2.5, pilot = TRUE)
#>  term              value
#>  pilot_n_per_group 10   
#>  allocation_1      0.5  
#>  allocation_2      0.5  
#> 
#> Confidence level: 90%

# 2. Evaluate the stopping criterion mid-study: pooled s = 15.4 at
#    n = 60 per group. Too imprecise to stop; the projection says
#    roughly how much further to go.
ss_seq_c(c_weights = c(1, -1), half_width = 2.5,
         s = 15.4, n = c(60, 60))
#>  term               value
#>  stop               0    
#>  half_width_current 4.66 
#>  half_width_target  2.5  
#>  N_current          120  
#>  N_projected        411  
#>  allocation_1       0.5  
#>  allocation_2       0.5  
#> 
#> Confidence level: 90%

# 3. Costs differ: sampling the second group costs four times as
#    much per observation, so its share of new observations drops.
ss_seq_c(c_weights = c(1, -1), half_width = 2.5,
         s = c(15.4, 15.4), n = c(60, 60), cost = c(1, 4))
#>  term               value
#>  stop               0    
#>  half_width_current 4.66 
#>  half_width_target  2.5  
#>  N_current          120  
#>  N_projected        411  
#>  allocation_1       0.667
#>  allocation_2       0.333
#> 
#> Confidence level: 90%