ss_buc_ancova: Necessary Sample Size to Reach Desired Power for an Analysis...

In BUCSS: Bias and Uncertainty Corrected Sample Size

Necessary Sample Size to Reach Desired Power for an Analysis of Covariance Using a Publication Bias and Uncertainty Correction Procedure

Description

ss_buc_ancova returns the necessary per-cell sample size to achieve a desired level of statistical power for a planned study testing a group effect (omnibus or contrast) while adjusting for one or more covariates, based on information obtained from a previous study. The effect from the previous study can be corrected for publication bias and/or uncertainty to provide a sample size that will achieve more accurate statistical power for a planned study, when compared to approaches that use a sample effect size at face value or rely on sample size only. The bias and uncertainty adjusted previous study noncentrality parameter is also returned, together with the planning effect size, the effect size the plan is built on, as Cohen's f.

Usage

ss_buc_ancova(
  F_observed,
  N,
  cells,
  n_covariates,
  df_numerator = cells - 1,
  alpha_prior = 0.05,
  alpha_planned = 0.05,
  assurance = 0.8,
  desired_power = 0.8
)

Arguments

F_observed	Observed F value for the effect of interest from a previous study used to plan sample size for a planned study. For a single-degree-of-freedom contrast, this is the square of the contrast's t.
N	Total sample size of the previous study.
cells	Number of cells (groups) in the between-subjects design.
n_covariates	Number of covariates adjusted for in the analysis.
df_numerator	Numerator degrees of freedom for the effect of interest. Defaults to cells - 1, the omnibus group effect; use 1 for a single-degree-of-freedom contrast.
alpha_prior	The significance level \alpha a prior result had to reach to be published (more generally, to reach you): the correction assumes you see only results with p below alpha_prior. Enter a number above 0 and at most 1. A value of 1, which may also be written "No Publication Filter", means no publication filter: every p value is at most 1, so any result would have been published whatever its p value, there is no publication bias to correct, and only the correction for uncertainty (set by assurance) remains, as for a pilot study. Anderson, Kelley, and Maxwell (2017, p. 1552) note that this level, their \alpha_P, can be set equal to 1 when publication bias is thought not to apply. The string is read without regard to case, and the result echoes it as an alpha_prior of 1.
alpha_planned	Alpha level \alpha assumed for the planned study.
assurance	Desired level of assurance: the long run proportion of planned studies whose statistical power at the true effect reaches or surpasses desired_power, when each is planned from a prior study published under the rule alpha_prior describes. The adjusted noncentrality parameter is the lower limit of a one-sided confidence interval at this level, computed from the publication-truncated distribution of the prior statistic, so assurance = .5 plans from a median unbiased estimate (correcting for publication bias only), a larger value adds the correction for uncertainty, and a value below .5 is not recommended. The term follows Kelley and Maxwell (2003), who used assurance in sample size planning for the probability that a confidence interval will be no wider than desired. With alpha_prior = 1 and assurance = .80 the plan corresponds to safeguard power, which plans for the lower limit of a two-sided 60 percent confidence interval around the prior effect size rather than for the effect size itself (see BUCSS-package for how, and for where the two differ). Enter it as a proportion in (0, 1) or as a percentage greater than 1 (for example, 80 is read as .80); a value of exactly 1 is read as 1 percent. A percentage is echoed in the result's planning-input rows as the coerced proportion (80 is echoed as .80); the same applies to desired_power.
desired_power	Desired level of statistical power for the planned study: the probability that its test rejects the null hypothesis when the effect is present (Maxwell, Delaney, & Kelley, 2027). It must exceed alpha_planned: a test's power is never below its Type I error rate, so a value at or below alpha_planned is not allowable, and the function stops rather than return a plan. Enter it as a proportion in (0, 1) or as a percentage greater than 1; a value of exactly 1 is read as 1 percent (.01), which is then refused unless alpha_planned is below .01.

Details

Researchers often use the sample effect size from a prior study as an estimate of the likely size of an expected future effect in sample size planning. However, sample effect size estimates should not usually be used at face value to plan sample size, due to both publication bias and uncertainty.

An analysis of covariance is a between-subjects analysis with additional estimated parameters. ss_buc_ancova takes the number of covariates directly and carries them through to the planned study: the prior study's error degrees of freedom are N minus the number of cells minus the number of covariates (Maxwell, Delaney, & Kelley, 2027), and the planned study's error degrees of freedom are reduced by the same number of covariates. Supplying n_covariates = 0 reproduces the corresponding ss_buc_one_way_anova or ss_buc_factorial_anova_general result exactly.

By default the omnibus group effect is planned (df_numerator = cells - 1). For a single-degree-of-freedom contrast among the adjusted means, supply the contrast's F (the square of its t) and set df_numerator = 1.

The approach uses a likelihood function of a truncated noncentral F distribution, where the truncation occurs due to small effect sizes being unobserved due to publication bias. The numerator of the likelihood function is the density of a noncentral F distribution. The denominator is the power of the test, which serves to truncate the distribution. Thus, the ratio of the numerator and the denominator is a truncated noncentral F distribution. (See Taylor & Muller, 1996, Equation 2.1; Anderson, Kelley, & Maxwell, 2017, Equation 3; and Anderson & Maxwell, 2017, for more details.)

Assurance is the proportion of times that power will be at or above the desired level, if the experiment were to be reproduced many times. For example, assurance = .5 means that power will be above the desired level half of the time, but below the desired level the other half of the time. Selecting assurance = .5 (selecting the noncentrality parameter at the 50th percentile of the likelihood distribution) results in a median unbiased estimate of the population noncentrality parameter and does not correct for uncertainty. In order to correct for uncertainty, assurance > .5 can be selected, which corresponds to selecting the noncentrality parameter associated with the (1 - assurance) quantile of the likelihood distribution.

alpha_prior is the significance level a prior result had to reach to be published: the correction assumes that only results with p values below alpha_prior reach the literature, and so reach you. Different \alpha levels represent differing amounts of publication bias. For example, setting alpha_prior = .20 assumes that studies with p values less than .20 are published, whereas those with larger p values are not, which reflects less severe publication bias than the default of .05. If the previous study of interest has not been subjected to publication bias (e.g., a pilot study), set alpha_prior = 1, which may also be written alpha_prior = "No Publication Filter"; Anderson, Kelley, and Maxwell (2017, p. 1552) note that it can be set equal to 1 when publication bias is thought not to apply. Because no p value exceeds 1, this means no publication filter at all: any result would have been published whatever its p value, so there is no publication bias to correct, and only the correction for uncertainty, set by assurance, remains.

In some cases, the corrected noncentrality parameter for a given level of assurance will be estimated to be zero. This is an indication that, at the desired level of assurance, the previous study's effect cannot be accurately estimated due to high levels of uncertainty and bias. When this happens, subsequent sample size planning is not possible with the chosen specifications, and the function stops with an informative error rather than returning a result. The error reports the largest assurance the prior result can support, which is 1 - p/\alpha, where p is the prior study's p value and \alpha is alpha_prior; at any higher assurance the corrected noncentrality parameter is driven to zero. Two remedies are available. First, users can select a lower value of assurance (e.g., .8 instead of .95), at or below the reported ceiling. Second, users can reduce the influence of publication bias by setting alpha_prior at a value greater than .05, which raises the ceiling. Note that alpha_prior is not a property of the prior study, which is fixed, but the analyst's assumption about the publication threshold its literature faced. When the ceiling is at or below the recommended floor of .5, lowering assurance cannot help and alpha_prior is the only remaining lever. It is possible to correct for uncertainty only by setting alpha_prior = 1 and choosing the desired level of assurance. We encourage users to make the adjustments as minimal as possible.

The returned actual_power is the statistical power of the plan this function reports: the power of a study of the returned sample size when the effect is the returned bias and uncertainty adjusted noncentrality parameter. Because a sample size must be a whole number, actual_power is never below the desired_power that was requested and is usually a little above it.

The covariates are treated as fixed, which is the usual convention in sample size planning for an analysis of covariance. When the covariates are themselves sampled, chance imbalance between the groups on the covariates lowers the planned study's power below the reported value, by roughly .4 to .5 times q/N at a desired power of .80 and an alpha_planned of .05 (somewhat more at a stricter alpha_planned, or when the plan has few participants per covariate), for q covariates and N planned participants in total: about .002 for the first example below (462 participants, two covariates), but about .05 for a plan of 27 participants with three covariates.

ss_buc_ancova assumes that the planned study will have equal n. If the prior N is not equally divisible by the number of cells, the per-cell size is rounded both up and down, the planned sample size is computed under each, and the larger of the two is returned, with the smaller of the two adjusted noncentrality parameters, to be conservative. The planning effect size belongs to the same balanced design as the reported noncentrality parameter, so an ordinary power analysis at it can return a sample size a little smaller than the one reported here (never a larger one), because the reported size is the larger of the two rounded designs' requirements. For a prior study with unequal group sizes, or a total sample size that does not divide evenly, the largest supportable assurance is computed at the balanced design the planner plans from, so it can be below 1 - p/\alpha at the study's own p value.

Value

An object of class bucss_plan: a data.frame with a character term column and a numeric value column, in the same shape as the sibling package DMAR's planners. The design results come first: the necessary per-cell sample size for the planned study, named for its unit (necessary_n_per_group, necessary_n_per_cell, or necessary_N); for the per-group and per-cell designs, total_N, the implied total sample size; actual_power, the power of the plan returned, that is, of the returned sample size at the returned noncentrality parameter (see Details); ncp_adjusted, the publication bias and uncertainty adjusted prior study noncentrality parameter, on the scale of the F statistic, at the prior study's sample size (for a prior N that does not divide evenly, at the balanced sample size the planner substitutes for it; see Details); and the planning effect size, f_planning, Cohen's f for the effect tested, relative to the covariate-adjusted error variance, \sqrt{\lambda/N}, where \lambda is ncp_adjusted and N the total of the balanced prior design it belongs to (see Details), so \lambda = N f^2, the convention of power analysis for between-subjects designs, and f^2/(1 + f^2) is the population partial eta squared for the effect.

A planning effect size (the row whose name ends in _planning) is an adjusted effect size: the value, in the design's usual effect size metric, used to plan the sample size given the situation at hand (the prior study's result, the publication bias assumed through alpha_prior, and the uncertainty allowed for through assurance). It is ncp_adjusted restated in that metric and is what the plan is built on, not an estimate of the population effect to report: above an assurance of .5 it is deliberately conservative (a lower confidence limit), and at .5 it is the median unbiased estimate. Because it restates ncp_adjusted, actual_power is the ordinary power of a study of the returned sample size when the population effect is the planning effect size.

Rows echoing the planning inputs follow the design results, so the assumptions travel with the result. The design, the unit the sample size is counted in, the largest assurance this prior can support (the assurance ceiling), and the planned test's degrees of freedom (df_effect, and df_error where the test has a denominator) travel on attributes. print.bucss_plan shows the design, for the regression planners the predictor frame, the unit, and the ceiling beneath the table; tidy() and glance() give the compact and wide views, the latter with every attribute as a column.

Author(s)

Ken Kelley ([e-mail]) and Samantha F. Anderson ([e-mail])

References

Anderson, S. F., & Kelley, K. (2024). Sample size planning for replication studies: The devil is in the design. Psychological Methods, 29, 844–867. \Sexpr[results=rd]{tools:::Rd_expr_doi("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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/0956797617723724")}

Anderson, S. F., & Maxwell, S. E. (2017). Addressing the "replication crisis": Using original studies to design replication studies with appropriate statistical power. Multivariate Behavioral Research, 52, 305–324. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00273171.2017.1289361")}

Kelley, K., & Maxwell, S. E. (2003). Sample size for multiple regression: Obtaining regression coefficients that are accurate, not simply significant. Psychological Methods, 8, 305–321. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1037/1082-989X.8.3.305")}

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

Taylor, D. J., & Muller, K. E. (1996). Bias in linear model power and sample size calculation due to estimating noncentrality. Communications in Statistics: Theory and Methods, 25, 1595–1610. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/03610929608831787")}

See Also

ss_buc_factorial_anova_general, which this function calls, and which takes the error degrees of freedom directly.

Examples

# Three groups, two covariates, prior N = 120: the omnibus group effect.
result <- ss_buc_ancova(F_observed = 6, N = 120, cells = 3,
  n_covariates = 2, alpha_prior = .05, alpha_planned = .05,
  assurance = .80, desired_power = .80)
result

# A single-degree-of-freedom contrast among the adjusted means.
ss_buc_ancova(F_observed = 9, N = 120, cells = 3, n_covariates = 2,
  df_numerator = 1)

# Requesting more assurance than the prior result can support stops with an
# informative error naming the largest workable assurance (here near .93):
try(ss_buc_ancova(F_observed = 6, N = 120, cells = 3, n_covariates = 2,
  assurance = .95))

BUCSS documentation built on Oct. 14, 2026, 1:07 a.m.