Worked Examples Across Designs

Which planner fits a given prior study, and what does it need to know about that study? This vignette walks through one example of each of the eighteen ss_buc_* planners. The pattern is always the same: summarize the prior study with a single test statistic and its design, and the function returns the necessary planned-study sample size and the bias and uncertainty adjusted noncentrality parameter (Anderson, Kelley, & Maxwell, 2017; Anderson & Kelley, 2024). Every example uses 80% power and 80% assurance unless noted. For the meaning of the assurance and alpha_prior settings, see vignette("understanding-assurance", package = "BUCSS").

Independent t Test

A prior two-group study reported t = 3.00 with 20 participants per group. The returned sample size is per group.

ss_buc_independent_t(t_observed = 3, n = 20, alpha_prior = .05, alpha_planned = .05,
               assurance = .80, desired_power = .80)

term	value
necessary_n_per_group	130
total_N	260
actual_power	0.801
ncp_adjusted	1.11
smd_planning	0.349
t_observed	3
n	20
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Independent t test; Sample size unit: per group; Largest supportable assurance: .90

ss_buc_smd is the same planner under an effect size name. It still takes t, so convert a reported standardized mean difference d first: t = d / sqrt(1/n_1 + 1/n_2), with n_1 and n_2 the two group sizes.

Dependent (Paired) t Test

A prior within-pairs study of 40 pairs reported t = 3.00. The returned sample size is the number of pairs.

ss_buc_paired_t(t_observed = 3, N = 40, alpha_prior = .05, alpha_planned = .05,
                assurance = .80, desired_power = .80)

term	value
necessary_N	255
actual_power	0.801
ncp_adjusted	1.12
dz_planning	0.176
t_observed	3
N	40
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Dependent (paired) t test; Sample size unit: number of pairs; Largest supportable assurance: .90

The alias ss_buc_smd_paired serves the standardized mean difference framing of the same comparison.

One-Sample t Test

A prior study of 30 participants compared their mean with a fixed reference value, such as a scale midpoint or a published norm, and reported t = 3.20. The returned sample size is the total number of participants.

ss_buc_one_sample_t(t_observed = 3.2, N = 30, alpha_prior = .05,
                    alpha_planned = .05, assurance = .80, desired_power = .80)

term	value
necessary_N	107
actual_power	0.8
ncp_adjusted	1.5
std_mean_planning	0.273
t_observed	3.2
N	30
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: One-sample t test; Sample size unit: total; Largest supportable assurance: .93

A paired t test is a one-sample t test on the difference scores, so this planner is the paired planner under another name: for the same prior t and N the two return the same plan.

Between-Subjects ANOVA

A one-way fully between-subjects design takes the number of levels of the single factor. Here a four-group design with total N = 120 reported F = 5.00 for the omnibus effect. The returned sample size is per group.

ss_buc_one_way_anova(F_observed = 5, N = 120, levels_A = 4, alpha_prior = .05,
                     alpha_planned = .05, assurance = .80, desired_power = .80)

term	value
necessary_n_per_group	89
total_N	356
actual_power	0.802
ncp_adjusted	3.74
f_planning	0.176
F_observed	5
N	120
levels_A	4
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: One-way between-subjects ANOVA; Sample size unit: per group; Largest supportable assurance: .94

A two-way fully between-subjects design takes the levels of both factors and the effect of interest (factor_A, factor_B, or interaction). Here a 2 x 3 design with total N = 120 reported F = 5.00 for the main effect of Factor B. The returned sample size is per cell.

ss_buc_factorial_anova(F_observed = 5, N = 120, levels_A = 2, levels_B = 3,
                       effect = "factor_B", alpha_prior = .05,
                       alpha_planned = .05, assurance = .80, desired_power = .80)

term	value
necessary_n_per_cell	659
total_N	3954
actual_power	0.801
ncp_adjusted	0.293
f_planning	0.0494
F_observed	5
N	120
levels_A	2
levels_B	3
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Two-way between-subjects ANOVA (factor_B); Sample size unit: per cell; Largest supportable assurance: .83

When the effect of interest is not a standard main effect or interaction, or the design has more than two factors, use the general form and supply the degrees of freedom directly. The denominator degrees of freedom equal N minus the number of cells when the cell means are the only estimated parameters, so supplying df_denominator = N - cells (here 120 - 6 = 114) reproduces the two-way result above. A smaller df_denominator accounts for covariates or other nuisance parameters and calls for a larger sample.

ss_buc_factorial_anova_general(F_observed = 5, N = 120, cells = 6,
                               df_numerator = 2, df_denominator = 114,
                               alpha_prior = .05, alpha_planned = .05,
                               assurance = .80, desired_power = .80)

term	value
necessary_n_per_cell	659
total_N	3954
actual_power	0.801
ncp_adjusted	0.293
f_planning	0.0494
F_observed	5
N	120
cells	6
df_numerator	2
df_denominator	114
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Between-subjects ANOVA (general); Sample size unit: per cell; Largest supportable assurance: .83

Within-Subjects ANOVA

For a fully within-subjects design, supply the factor levels and the effect. Here a 2 x 3 within-subjects design with N = 60 reported F = 5.00 for the main effect of Factor B. The returned sample size is the total number of subjects.

ss_buc_rm_anova(F_observed = 5, N = 60, levels_A = 2, levels_B = 3,
                effect = "factor_B", alpha_prior = .05, alpha_planned = .05,
                assurance = .80, desired_power = .80)

term	value
necessary_N	1902
actual_power	0.8
ncp_adjusted	0.304
f_planning	0.0504
F_observed	5
N	60
levels_A	2
levels_B	3
epsilon	1
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: One- or two-way within-subjects ANOVA (factor_B); Sample size unit: total; Largest supportable assurance: .83

The general form takes the numerator degrees of freedom directly and works for any number of within-subjects factors. This example uses assurance = .50, which corrects for publication bias only.

ss_buc_rm_anova_general(F_observed = 6.5, N = 80, df_numerator = 1,
                        alpha_prior = .05, alpha_planned = .05, assurance = .50,
                        desired_power = .80)

term	value
necessary_N	256
actual_power	0.8
ncp_adjusted	2.47
f_planning	0.176
F_observed	6.5
N	80
df_numerator	1
epsilon	1
alpha_prior	0.05
alpha_planned	0.05
assurance	0.5
desired_power	0.8

Design: Within-subjects ANOVA (any number of factors); Sample size unit: total; Largest supportable assurance: .74

Split-Plot (Mixed) ANOVA

A split-plot design crosses a between-subjects factor with a within-subjects factor. Supply the levels of each and the effect (between, within, or interaction). Here a design with 2 between-subjects levels and 3 within-subjects levels and N = 60 reported F = 5.00 for the within-subjects effect. The returned sample size is per between-subjects cell.

ss_buc_mixed_anova(F_observed = 5, N = 60, levels_between = 2, levels_within = 3,
                   effect = "within", alpha_prior = .05, alpha_planned = .05,
                   assurance = .80, desired_power = .80)

term	value
necessary_n_per_group	969
total_N	1938
actual_power	0.8
ncp_adjusted	0.299
f_planning	0.0499
F_observed	5
N	60
levels_between	2
levels_within	3
epsilon	1
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Two-factor split-plot (mixed) ANOVA (within); Sample size unit: per between-subjects cell; Largest supportable assurance: .83

The general form takes the numerator degrees of freedom and the number of between-subjects groups directly. The effect is one of between_only, within_only, or between_within; supply df_num_within, the numerator degrees of freedom of the within-subjects component, only when effect = "between_within" (it is required there, and ignored with a warning otherwise). The returned sample size is per group.

ss_buc_mixed_anova_general(F_observed = 5, N = 90, df_numerator = 2,
                           num_groups = 3, effect = "between_only",
                           alpha_prior = .05, alpha_planned = .05,
                           assurance = .80, desired_power = .80)

term	value
necessary_n_per_group	1489
total_N	4467
actual_power	0.8
ncp_adjusted	0.194
f_planning	0.0465
F_observed	5
N	90
df_numerator	2
num_groups	3
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Split-plot (mixed) ANOVA (any number of factors) (between_only); Sample size unit: per group; Largest supportable assurance: .82

Departures From Sphericity

The within-subjects and split-plot planners assume sphericity by default, as the uncorrected within-subjects test does, and echo that assumption as an epsilon row of 1. When a paper reports a Greenhouse-Geisser (1959) or Huynh-Feldt (1976) correction instead, it shows fractional degrees of freedom: a three-level within-subjects factor measured on 30 subjects might be reported as F(1.73, 50.2) = 6.50. The uncorrected degrees of freedom are 2 and 58, so the correction is epsilon = 1.73/2, about .865. The denominator agrees (50.2/58 is also about .865), and 50.2/1.73 = 29.0 is one less than the 30 subjects, which confirms how many subjects the analysis used.

ss_buc_rm_anova(F_observed = 6.5, N = 30, levels_A = 3, epsilon = 1.73 / 2)

term	value
necessary_N	162
actual_power	0.802
ncp_adjusted	2
f_planning	0.183
F_observed	6.5
N	30
levels_A	3
epsilon	0.865
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: One- or two-way within-subjects ANOVA (factor_A); Sample size unit: total; Largest supportable assurance: .90

Ignoring the correction plans 92 subjects rather than 162:

ss_buc_rm_anova(F_observed = 6.5, N = 30, levels_A = 3)

term	value
necessary_N	92
actual_power	0.802
ncp_adjusted	3.21
f_planning	0.231
F_observed	6.5
N	30
levels_A	3
epsilon	1
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: One- or two-way within-subjects ANOVA (factor_A); Sample size unit: total; Largest supportable assurance: .94

The difference has two sources. The same F on fewer degrees of freedom is weaker evidence: its p value rises from .0028 to .0045, the largest supportable assurance falls from .94 to .90, and the correction for uncertainty at an assurance of .80 takes correspondingly more off. And the planned study, analyzed with the same correction, has less power at any size. Both follow the approximation of Muller and Barton (1989), in which both degrees of freedom and the noncentrality parameter are multiplied by epsilon; ncp_adjusted stays on the sphericity scale, so the two plans’ parameters are comparable. The plan is sensitive to epsilon, so it is worth seeing across the values you find plausible:

eps <- c(1, .95, .9, .85, .8, .75)
sapply(eps, function(e) {
  res <- ss_buc_rm_anova(F_observed = 6.5, N = 30, levels_A = 3, epsilon = e)
  res$value[res$term == "necessary_N"]
})
#> [1]  92 109 135 177 252 415

One epsilon describes both studies: it is a property of the covariance of the repeated measures, which the planned study shares with the prior one. When a paper reports both corrections, use the Greenhouse-Geisser value. The correction is an approximation, and the help page for ss_buc_rm_anova() reports how well it holds up in simulation. In a split-plot design it applies to the within-subjects effect and the interaction, not to the between-subjects effect, whose test is on subject means.

Multiple Regression

Anderson (2021) is a step-by-step tutorial for planning regression studies with BUCSS, and is the best starting point for the three regression planners below.

For a single regression coefficient, supply the t statistic for that coefficient, the prior sample size, and the number of predictors p. A prior study with N = 150 and 3 predictors reported t = 3.00 for one coefficient. The returned sample size is the total.

ss_buc_reg_coef(t_observed = 3, N = 150, p = 3, alpha_prior = .05,
                alpha_planned = .05, assurance = .80, desired_power = .80)

term	value
necessary_N	624
actual_power	0.8
ncp_adjusted	1.89
f2_planning	0.0126
t_observed	3
N	150
p	3
random_predictors	0
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Multiple regression: single coefficient; Predictors: fixed by design; Sample size unit: total; Largest supportable assurance: .93

For the omnibus test of the model R-squared, supply the model F statistic. Here a prior study with N = 150 and 4 predictors reported F = 5.00.

ss_buc_R2(F_observed = 5, N = 150, p = 4, alpha_prior = .05,
          alpha_planned = .05, assurance = .80, desired_power = .80)

term	value
necessary_N	234
actual_power	0.8
ncp_adjusted	7.82
R2_planning	0.0495
F_observed	5
N	150
p	4
random_predictors	0
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Multiple regression: omnibus test of model R2; Predictors: fixed by design; Sample size unit: total; Largest supportable assurance: .98

For a joint test of a subset of predictors, also supply p_joint, the number of predictors tested together. Here 2 of the 4 predictors are tested jointly.

ss_buc_reg_joint(F_observed = 5, N = 150, p = 4, p_joint = 2,
                 alpha_prior = .05, alpha_planned = .05, assurance = .80,
                 desired_power = .80)

term	value
necessary_N	3960
actual_power	0.8
ncp_adjusted	0.365
f2_planning	0.00244
F_observed	5
N	150
p	4
p_joint	2
random_predictors	0
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Multiple regression: joint test of predictors; Predictors: fixed by design; Sample size unit: total; Largest supportable assurance: .84

All three regression planners take random_predictors. By default (random_predictors = FALSE) they treat the predictors as fixed by design, as in an experiment in which the researcher sets their values, the frame of BUCSS 1.x. When the predictors are sampled along with the outcome, as in most observational research (Kelley & Maxwell, 2003), pass random_predictors = TRUE: the test statistic’s noncentrality parameter then rises and falls with the sampled spread of the tested predictors, and the planners use the exact distribution that follows (Sampson, 1974; Gatsonis & Sampson, 1989).

ss_buc_R2(F_observed = 5, N = 150, p = 4, random_predictors = TRUE)

term	value
necessary_N	239
actual_power	0.801
ncp_adjusted	7.76
R2_planning	0.0492
F_observed	5
N	150
p	4
random_predictors	1
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Multiple regression: omnibus test of model R2; Predictors: random (sampled along with the outcome); Sample size unit: total; Largest supportable assurance: .98

Planned as though they were fixed, sampled predictors make the reported actual_power overstate the planned study’s power by an amount that shrinks as 1/n for a plan of n participants and grows with the number of predictors left untested (roughly 1/n with one predictor and 4/n for a coefficient among eight: about .04 at n = 100, and less than .01 for each of the three plans above). From prior studies as large as these the two frames differ little: with random_predictors = TRUE the three plans above are 623, 239, and 3,898 rather than 624, 234, and 3,960, the joint test’s differing most because it is requested close to its assurance ceiling of .84. From a small prior study, with predictors left untested and a plan several times the prior N, the random-predictor plan can be smaller by up to about the number of untested predictors plus one, divided by the prior N: as much as 40 percent for a coefficient among eight predictors from a prior study of 20 (the help pages of ss_buc_reg_coef() and ss_buc_reg_joint() describe the pattern). With one predictor, ss_buc_reg_coef(p = 1, random_predictors = TRUE) returns exactly what ss_buc_correlation(), described in the next section, returns.

Correlation

A prior study reporting a Pearson correlation is planned from the correlation itself. A study with N = 100 reported r = .35. The returned sample size is the total.

ss_buc_correlation(r_observed = .35, N = 100, alpha_prior = .05,
                   alpha_planned = .05, assurance = .80, desired_power = .80)

term	value
necessary_N	122
actual_power	0.802
ncp_adjusted	6.71
rho_planning	0.251
r_observed	0.35
N	100
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Pearson correlation; Sample size unit: total; Largest supportable assurance: .99

The test of a correlation is arithmetically the test of a simple regression slope, so it is tempting to plan it as one. That is exact only when the predictor is fixed by design. A correlation study samples both variables, and then the statistic’s noncentrality parameter is itself random, proportional to the sampled spread of the predictor. The resulting distribution is more dispersed, and planning as though it were not overstates the power. The planner uses the exact distribution for sampled variables, so it asks for a few more participants than the regression planner does when that planner treats the predictor as fixed:

t_equivalent <- .35 * sqrt(98 / (1 - .35^2))
by_correlation <- ss_buc_correlation(t_observed = t_equivalent, N = 100)
by_regression  <- ss_buc_reg_coef(t_observed = t_equivalent, N = 100, p = 1,
                                  random_predictors = FALSE)
c(correlation = by_correlation$value[by_correlation$term == "necessary_N"],
  regression  = by_regression$value[by_regression$term == "necessary_N"])
#> correlation  regression 
#>         122         119

If your predictor really is fixed by design, the analysis is a regression slope, and ss_buc_reg_coef() with p = 1 and random_predictors = FALSE is the planner for it. vignette("correlation", package = "BUCSS") develops the comparison.

Analysis of Covariance

Covariates cost error degrees of freedom in both the prior study and the planned one, so the planner takes their number directly. A prior study with N = 120, three groups, and two covariates reported F = 6.00 for the group effect. The returned sample size is per cell.

ss_buc_ancova(F_observed = 6, N = 120, cells = 3, n_covariates = 2,
              alpha_prior = .05, alpha_planned = .05, assurance = .80,
              desired_power = .80)

term	value
necessary_n_per_cell	154
total_N	462
actual_power	0.802
ncp_adjusted	2.53
f_planning	0.145
F_observed	6
N	120
cells	3
n_covariates	2
df_numerator	2
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Analysis of covariance; Sample size unit: per cell; Largest supportable assurance: .93

With n_covariates = 0 this reproduces ss_buc_one_way_anova() exactly. For a single-degree-of-freedom contrast among the adjusted means, supply that contrast’s F and set df_numerator = 1.

Multivariate Comparison of Two Groups

When two groups are compared on several outcome variables at once, the prior study’s Hotelling T squared (Hotelling, 1931), or the F it maps to, plans the replication. A study with N = 80 comparing two groups on 3 outcomes reported F = 4.50. The returned sample size is per group.

ss_buc_manova(F_observed = 4.5, N = 80, p_variables = 3, alpha_prior = .05,
              alpha_planned = .05, assurance = .80, desired_power = .80)

term	value
necessary_n_per_group	310
total_N	620
actual_power	0.801
ncp_adjusted	1.42
D2_planning	0.0709
F_observed	4.5
N	80
p_variables	3
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Two-group multivariate comparison (Hotelling’s T squared); Sample size unit: per group; Largest supportable assurance: .88

This covers the multivariate hypothesis whose rank is one, which is what a two-group comparison gives. A hypothesis of higher rank has a set of noncentrality parameters rather than a single one, and the correction does not apply to it.

Nested Model Comparison

An effect tested by the chi square difference between two nested models, such as a constrained path in a structural equation model, is planned from that difference. Supply both models’ degrees of freedom rather than only their difference. A prior study with N = 250 fit a model with 42 degrees of freedom and a constrained version with 43, a difference of 9.50 on 1 degree of freedom.

ss_buc_chisq_diff(chisq_observed = 9.5, N = 250, df_full = 42, df_restricted = 43,
                  alpha_prior = .05, alpha_planned = .05, assurance = .80,
                  desired_power = .80)

term	value
necessary_N	726
actual_power	0.8
ncp_adjusted	2.71
rmsea_d_planning	0.104
chisq_observed	9.5
N	250
df_full	42
df_restricted	43
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Nested model chi square difference test; Sample size unit: total; Largest supportable assurance: .95

This is for the difference test, not for an omnibus model fit chi square: a publishable fit statistic is a small one, which inverts the selection the correction assumes. Requiring both models’ degrees of freedom is what lets the function enforce that. An omnibus fit chi square compares a model against the saturated model, which has zero degrees of freedom, so df_full = 0 is its signature and the function refuses it:

ss_buc_chisq_diff(chisq_observed = 9.5, N = 250, df_full = 0, df_restricted = 43)
#> Error:
#> ! 'df_full' is 0, so the full model is saturated and this comparison is the restricted model's omnibus test of model fit, not a difference test between two substantive models. This function does not apply to an omnibus fit chi square: the correction assumes a literature publishes significant results, whereas a publishable fit chi square is a small (nonsignificant) one, which inverts the selection region the method is built on. A just-identified full model also has 0 degrees of freedom; if that is your case and the comparison is a genuine difference test, the plan depends only on the number of constraints, so call with 'df_full = 1' and 'df_restricted' equal to 1 plus the number of constraints.

A just-identified full model also has zero degrees of freedom, and there the difference test is a genuine one; the message above, and the function’s help page, say how to plan that case.

Unequal Variances

When the two groups’ variances differ, the Welch test is the appropriate comparison, and planning it needs the groups’ relative spread in addition to their sizes. A prior study with 40 and 55 participants, whose second group was half again as variable as the first, reported t = 3.00.

ss_buc_welch_t(t_observed = 3, n_1 = 40, n_2 = 55, sd_ratio = 1.5,
               alpha_prior = .05, alpha_planned = .05, assurance = .80,
               desired_power = .80)

term	value
necessary_n_per_group	222
total_N	444
actual_power	0.801
ncp_adjusted	1.32
smd_planning	0.267
t_observed	3
n_1	40
n_2	55
sd_ratio	1.5
alpha_prior	0.05
alpha_planned	0.05
assurance	0.8
desired_power	0.8

Design: Welch (unequal variance) t test; Sample size unit: per group; Largest supportable assurance: .93

This planner treats the Welch statistic as noncentral t on the Welch-Satterthwaite degrees of freedom (Satterthwaite, 1946; Welch, 1947), the usual approximation in power analysis for this test. A minimum planned sample size of 10 per group is imposed because of estimation accuracy issues at smaller sample sizes: with fewer per group the approximation overstates the real test’s power, while from 10 per group it is within .006 of it at an alpha_planned of .05. At a stricter alpha_planned the power of the smallest plans is overstated by more (at 10 per group by up to about .012 at .01 and .025 at .001), and the .006 holds again from 14 and 20 per group, respectively. When the target power is reached with fewer than 10 per group, the planner returns 10.

Only the ratio enters the computation, but a paper usually reports the group standard deviations or variances rather than their ratio, so those may be given directly instead. All three calls describe the same study:

by_ratio <- ss_buc_welch_t(t_observed = 3, n_1 = 40, n_2 = 55, sd_ratio = 1.5)
by_sds <- ss_buc_welch_t(t_observed = 3, n_1 = 40, n_2 = 55, sd_1 = 8.4,
                         sd_2 = 12.6)
by_vars <- ss_buc_welch_t(t_observed = 3, n_1 = 40, n_2 = 55, var_1 = 70.56,
                          var_2 = 158.76)
size_term <- "necessary_n_per_group"
c(ratio = by_ratio$value[by_ratio$term == size_term],
  sds   = by_sds$value[by_sds$term == size_term],
  vars  = by_vars$value[by_vars$term == size_term])
#> ratio   sds  vars 
#>   222   222   222

The assumed ratio matters more than any other input here, so it is worth seeing how the plan moves across the values you find plausible:

ratios <- c(1, 1.5, 2, 2.5, 3)
sapply(ratios, function(r) {
  res <- ss_buc_welch_t(t_observed = 3, n_1 = 40, n_2 = 55, sd_ratio = r)
  res$value[res$term == "necessary_n_per_group"]
})
#> [1] 213 222 236 248 257

One sd_ratio describes both studies: the prior study, whose statistic is read under it, and the planned study. Getting it wrong can be costly. The plan built assuming equal variances (213 per group) delivers about .76 power rather than .80 if the ratio is in fact 2 in both studies. It delivers about .43 if the prior study’s groups were equally variable but the planned study’s ratio is 2, with the mean difference held fixed in units of the first group’s standard deviation. Re-running across the ratios you find plausible, as above, and planning for the least favorable one you find credible guards against the first kind of error. The second, a planned study more unequal than the prior one, is not something the prior result can speak to, and needs its own allowance.

References

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Ken Kelley, Samantha F. Anderson, and Scott E. Maxwell