Sensitivity Analysis for Sample Size Planning From the AIPE Perspective for a Polynomial Change Parameter

Quantifies how much misspecification of the population between-subject slope variance and within-subject error variance distorts an AIPE-based sample size plan for the group-by-time polynomial change parameter. On each replication the function simulates two independent groups of n subjects each, measured at \(M = f \times D + 1\) timepoints, where every subject has a true linear slope drawn from \(N(0, \mathrm{true\_variance\_trend})\) and within-subject observations have residual variance true_error_variance. Subject-level OLS slopes are computed in each group, the between-group difference in mean slopes (the change parameter \(\beta_{m1}\) that ss_aipe_pcm plans for) is estimated, and a two-group t-confidence interval on that difference (pooled standard error, \(2n - 2\) degrees of freedom) is recorded. The function only handles trend = "linear" in the simulator; for quadratic / cubic trends, the planner's closed-form solution is still available via ss_aipe_pcm.

Usage

ss_aipe_pcm_sensitivity(
  true_variance_trend = NULL,
  true_error_variance = NULL,
  estimated_variance_trend = NULL,
  estimated_error_variance = NULL,
  duration,
  frequency,
  width,
  n_per_group = NULL,
  conf_level = 0.95,
  assurance = NULL,
  G = 1000,
  print_iter = FALSE,
  save = FALSE,
  filename = "ss_aipe_pcm_sensitivity_result.csv"
)

Arguments

true_variance_trend: Population between-subject variance of the polynomial change coefficient (the data generating \(\sigma^2_{\upsilon_m}\) of Kelley & Rausch, 2011).
true_error_variance: Population within-subject error variance (\(\sigma^2_\epsilon\)).
estimated_variance_trend: Planning value of variance_trend passed to ss_aipe_pcm; supply this and estimated_error_variance, or supply n_per_group.
estimated_error_variance: Planning value of error_variance passed to ss_aipe_pcm.
duration: Study duration (in time units).
frequency: Number of measurements per unit time. Total timepoints = \(f \times D + 1\).
width: Desired full width of the CI on the between-group difference in change parameters (\(\beta_{m1}\)).
n_per_group: Number of subjects to evaluate (incompatible with the estimated-variance arguments).
conf_level: Confidence level (default 0.95).
assurance: Optional assurance probability passed to the planner.
G: Number of Monte Carlo replications.
print_iter: Logical.
save: Logical. Save per-replication CSV.
filename: Path used when save = TRUE.

Value

A data.frame with rows for mean / median / SD of the realized estimated slope difference and CI width, the proportion of intervals at or below width, tail-specific and overall non-coverage of the population slope difference (0 by construction in this simulator), and the input echoes, including assurance (present only when an assurance was supplied).

References

Kelley, K., & Rausch, J. R. (2011). Sample size planning for longitudinal models: Accuracy in parameter estimation for polynomial change parameters. Psychological Methods, 16(4), 391–405. doi:10.1037/a0023352

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapters 11 and 15.)

Author

Ken Kelley kkelley@nd.edu

Examples

# Every replication simulates two full groups of subjects, fits a
# slope for each subject, and forms a confidence interval on the
# difference in mean slopes, so the sweep is not run at example time.
# The G below is far smaller than a reported sensitivity study would
# use; the default of 1000 is the realistic setting. The call is:
# set.seed(113)
# ss_aipe_pcm_sensitivity(
#   true_variance_trend       = 0.003,
#   true_error_variance       = 0.0262,
#   estimated_variance_trend  = 0.003,
#   estimated_error_variance  = 0.0262,
#   duration  = 4, frequency = 1,
#   width     = 0.05,
#   G = 20, print_iter = FALSE
# )