Simulates the purely sequential fixed-width confidence interval
procedure of ss_seq_c under a known data generating
mechanism, reporting the distribution of the stopping sample size
and the empirical coverage of the fixed-width interval. The two
quantities to read are the ratio of the mean stopping size to the
oracle \(n^*\) (first-order efficiency: the ratio approaches 1 as
the target half-width shrinks) and the coverage (asymptotic
consistency: coverage approaches 1 - 2\(\alpha\)). The normal
quantile rule stops slightly early at wide targets, the
finite-sample undershoot anticipated by Woodroofe (1977); the
t quantile rule corrects it at a small cost in sample size.
ss_seq_c_sensitivity(
c_weights,
half_width,
true_sigma,
true_means = NULL,
alpha_level = 0.05,
quantile = c("t", "normal"),
m0 = 10,
G = 1000,
seed = NULL
)A data.frame with rows n_star (the oracle
total sample size an investigator with known \(\sigma\) would
use), mean_N, median_N, sd_N (the stopping
total across replications), ratio_mean_N_n_star,
coverage (the proportion of replications whose
\(\hat\psi_N \pm h\) interval covered the true contrast),
se_coverage (its simulation standard error), and the input
echoes half_width, true_psi (the population contrast
implied by c_weights and true_means),
true_sigma, alpha_level, and m0.
The contrast weights. The weights must sum to zero with the positive weights summing to 1 and the negative weights to -1.
The target half-width \(h\) of the 100(1 - 2\(\alpha\))% interval, in raw units of the response.
The data generating error standard deviation: a single value applied to every group.
Optional vector of data generating group means,
aligned with c_weights. Default all zero, so the true
contrast is 0.
One-sided rate per bound; the interval is at
confidence level 1 - 2\(\alpha\). Default 0.05.
"t" (default) or "normal"; see
ss_seq_c.
Pilot sample size per group. Default 10.
Number of Monte Carlo replications. Default 1000.
Optional integer seed. Default NULL (the
current RNG state is used). When supplied, the caller's RNG
state is restored on exit.
Ken Kelley [email protected]
Simulation design. Each replication samples the groups
with nonzero weights in balanced fashion (one observation per
group per step) from normal populations with common
true_sigma, starting at m0 per group, and stops at
the first step satisfying the ss_seq_c criterion with
the pooled variance estimate. This matches the equal-cost,
equal-variance case of Chattopadhyay, Bandyopadhyay, Kelley, and
Padalunkal (2025); unequal costs change the optimal allocation but
not the logic.
The oracle. With known \(\sigma\) and balanced allocation over the \(J_0\) groups with nonzero weights, the fixed-width requirement is \(n^* = z_{1-\alpha}^2\, \sigma^2 J_0 \sum_j c_j^2 / h^2\) in total. The sequential procedure spends about \(n^*\) without knowing \(\sigma\), which is its point.
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. tools:::Rd_expr_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.
Ghosh, M., Mukhopadhyay, N., & Sen, P. K. (1997). Sequential estimation. Wiley.
Woodroofe, M. (1977). Second order approximations for sequential point and interval estimation. The Annals of Statistics, 5(5), 984--995.
ss_seq_c, ss_aipe_c,
ss_power_equivalence_c
Other sequential estimation:
ss_seq_c()
# A two-group contrast, target half-width 2.5, error SD 15.67:
# the t-quantile rule stops near the oracle with near-nominal
# coverage. (G kept small here for speed; use G = 2000 or more in
# earnest.)
ss_seq_c_sensitivity(c_weights = c(1, -1), half_width = 2.5,
true_sigma = 15.67, G = 200, seed = 113)
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