Power for unbalanced designs

Define the cells

Define every cell with cell_design(). Factor values identify the cell, n is the number of subjects in its between-subject group, and m is its population mean. Name repeated-measures factors in within; all remaining factors are between-subject factors. The common population SD is specified separately with unbalanced_covariance().

For repeated-measures designs, the same n must appear on every within-subject row belonging to a given between-subject group.

unbalanced_design <- cell_design(
  group = "control",   time = "pre",  n = 22, m = 10.0,
  group = "control",   time = "post", n = 22, m = 11.0,
  group = "treatment", time = "pre",  n = 31, m = 10.1,
  group = "treatment", time = "post", n = 31, m = 12.4,
  within = "time"
)

The means define the complete effect pattern, including main effects and interactions. A single marginal SD is shared across groups and within-subject cells, preventing unequal cell sizes from being combined with unequal variances in the classical ANOVA test.

Every factor must have at least two observed levels, and every factor-level combination must be defined exactly once. If cells are missing, cell_design() reports which ones. Supply default_n and default_m together to fill missing cells automatically:

cell_design(
  group = "control",   time = "pre",  n = 22, m = 10.0,
  group = "control",   time = "post", n = 22, m = 11.0,
  group = "treatment", time = "pre",  n = 31, m = 10.1,
  within = "time",
  default_n = 31, default_m = 12.4
)

Define within-subject correlations

Use unbalanced_covariance() to define the common SD and, for repeated measures, the correlation structure. power_unbalanced() constructs one covariance matrix and uses it for every between-subject group.

If covariance is omitted, power_unbalanced() warns that it is using the common sd = 1 and, for repeated measures, correlation 0.5 for every pair. Calling unbalanced_covariance() without sd likewise warns that sd = 1 is being used. When only some correlations are named, a separate warning reports the number of undefined pairs: default_correlation applies only to those undefined pairs and does not alter explicitly supplied correlations. For a purely between-subject design, correlations do not apply and the warning mentions only the common SD.

With one within-subject factor, pair names use its levels, such as "pre:post".

power_unbalanced(
  design = unbalanced_design,
  term = "group:time",
  covariance = unbalanced_covariance(
    sd = 2,
    default_correlation = 0.5,
    correlations = c("pre:post" = 0.7)
  ),
  n_sims = 5000,
  parallel = TRUE,
  seed = 123
)

Interpret the result

The result reports simulated power, the common SD, partial eta squared from a deterministic reference dataset matching the design assumptions, and the mean, median, and 95% interval of partial eta squared across successful simulations. These effect-size summaries describe the exact allocation, means, shared variance, correlations, tested term, and sums-of-squares type supplied by the user.

Multiple within-subject factors

Name every repeated-measures factor in cell_design() and join their level combinations with underscores in correlation-pair names:

multi_within_design <- cell_design(
  group = "A", time = "pre",  cond = "control", n = 10, m = 0.0,
  group = "A", time = "pre",  cond = "treat",   n = 10, m = 0.5,
  group = "A", time = "post", cond = "control", n = 10, m = 0.2,
  group = "A", time = "post", cond = "treat",   n = 10, m = 1.0,
  group = "B", time = "pre",  cond = "control", n = 15, m = 0.0,
  group = "B", time = "pre",  cond = "treat",   n = 15, m = 0.6,
  group = "B", time = "post", cond = "control", n = 15, m = 0.3,
  group = "B", time = "post", cond = "treat",   n = 15, m = 1.4,
  within = c("time", "cond")
)

power_unbalanced(
  design = multi_within_design,
  term = "group:time:cond",
  covariance = unbalanced_covariance(
    correlations = c("pre_control:post_control" = 0.6)
  ),
  n_sims = 5000,
  seed = 123
)