Worked Examples Across Designs

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)

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, for when you are working from a standardized mean difference rather than a raw t; prefer ss_buc_independent_t when the prior t was computed from unstandardized data.

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)

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.

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)

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)

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)

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)

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)

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)

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 when the effect involves a within-subjects component. 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)

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

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)

Design: Multiple regression: single coefficient; 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)

Design: Multiple regression: omnibus test of model R2; 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)

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

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)

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:

t_equivalent <- .35 * sqrt(98 / (1 - .35^2))
c(correlation = ss_buc_correlation(t_observed = t_equivalent, N = 100)$value[1],
  regression = ss_buc_reg_coef(t_observed = t_equivalent, N = 100, p = 1)$value[1])
#> 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 is the planner for it.

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)

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 (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)

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)

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 identifies the misuse exactly 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. If you meant to compare two substantive nested models, give the full model's own degrees of freedom.

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)

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

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:

sizes <- c(
  ratio = ss_buc_welch_t(t_observed = 3, n_1 = 40, n_2 = 55, sd_ratio = 1.5)$value[1],
  sds = ss_buc_welch_t(t_observed = 3, n_1 = 40, n_2 = 55, sd_1 = 8.4, sd_2 = 12.6)$value[1],
  vars = ss_buc_welch_t(t_observed = 3, n_1 = 40, n_2 = 55, var_1 = 70.56,
                        var_2 = 158.76)$value[1])
sizes
#> 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

Planning at equal variances and being wrong is expensive: if the true ratio were 2, that plan would deliver about .43 power rather than .80. Plan for the least favorable ratio you find credible.

References

Anderson, S. F. (2021). Using prior information to plan appropriately powered regression studies: A tutorial using BUCSS. Psychological Methods, 26, 513–526. https://doi.org/10.1037/met0000366

Anderson, S. F., & Kelley, K. (2024). Sample size planning for replication studies: The devil is in the design. Psychological Methods, 29, 844–867. https://doi.org/10.1037/met0000520

Anderson, S. F., Kelley, K., & Maxwell, S. E. (2017). Sample-size planning for more accurate statistical power: A method adjusting sample effect sizes for publication bias and uncertainty. Psychological Science, 28, 1547–1562. https://doi.org/10.1177/0956797617723724