Necessary Sample Size to Reach Desired Power for an Independent t Test (or Standardized Mean Difference) Using a Publication Bias and Uncertainty Correction Procedure

ss_buc_independent_t returns the necessary per-group sample size to achieve a desired level of statistical power for a planned study using an independent t test, 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 a standardized mean difference.

ss_buc_smd is an alias for ss_buc_independent_t: the two are the same function. The two-group t test and the standardized mean difference (Cohen's \(d\)) describe the same comparison, so the same planner serves both framings. Prefer the test-specific name (ss_buc_independent_t) when the prior t was computed from raw (unstandardized) data, since the planner works directly from the observed t.

Usage

ss_buc_independent_t(
  t_observed,
  n,
  N,
  alpha_prior = 0.05,
  alpha_planned = 0.05,
  assurance = 0.8,
  desired_power = 0.8
)

ss_buc_smd(
  t_observed,
  n,
  N,
  alpha_prior = 0.05,
  alpha_planned = 0.05,
  assurance = 0.8,
  desired_power = 0.8
)

Arguments

t_observed: Observed t value from a previous study used to plan sample size for a planned study. Either sign is accepted: the publication rule is two-sided, so only the magnitude enters the correction (see Details).
n: Per group sample size (if equal) or the two group sample sizes of the previous study (enter either a single number or a vector of length 2).
N: Total sample size of the previous study, assumed equal across groups if specified.
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.

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-group 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 t statistic (the noncentrality parameter of the prior study's noncentral t); and the planning effect size, smd_planning, a standardized mean difference (Cohen's \(\delta\), a magnitude), \(\lambda\sqrt{2/n}\), where \(\lambda\) is ncp_adjusted and n the per-group size of the balanced prior design it belongs to (see Details).

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.

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.

The approach implemented in ss_buc_independent_t uses the observed t value and sample size from a previous study to correct the noncentrality parameter associated with the effect of interest for publication bias and/or uncertainty. This new estimated noncentrality parameter is then used to calculate the necessary per-group sample size to achieve the desired level of power in the planned study.

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. In the two-group case, this formula reduces to the density of a truncated noncentral t 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 observed t may be entered with either sign. The publication rule the correction assumes is two-sided and the truncated likelihood is symmetric in the sign of t, so only the magnitude enters the computation: t_observed = -3 plans exactly the sample size that t_observed = 3 plans. The sign of a two-group t records which group was subtracted from which, an arbitrary coding choice rather than evidence. When planning a replication, confirm that the magnitude belongs to an effect in the direction the planned study is designed to detect (Anderson & Kelley, 2024): the correction adjusts the size of the prior effect, not its direction.

If you are working from a standardized mean difference (Cohen's \(d\)) rather than a t statistic, convert it before calling: for two independent groups \(t = d / \sqrt{1/n_1 + 1/n_2}\), which equals \(d\sqrt{n/2}\) when the per-group sample sizes are equal.

ss_buc_independent_t assumes that the planned study will have equal n. Unequal n in the previous study is handled in the following way for the independent t. If the user enters an odd value for N, no information is available on the exact group sizes. The function calculates n by dividing N by 2 and both rounding up and rounding down the result, thus assuming equal n. The suggested sample size for the planned study is calculated using both of these values of n, and the function returns the larger of these two suggestions, to be conservative. If the user enters a vector for n with two different values, specific information is available on the exact group sizes. n is calculated as the harmonic mean of these two values (a measure of effective sample size). Again, this is rounded both up and down. The suggested sample size for the planned study is calculated using both of these values of n, and the function returns the larger of these two suggestions, to be conservative. The adjusted noncentrality parameter that is output is the lower of the two possibilities, again, 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. When the individual group sizes of an unequal-n previous study are known, we highly encourage entering these explicitly, especially if the sample sizes are quite discrepant, as this allows for the greatest precision in estimating an appropriate planned study n.

References

Anderson, S. F., & Kelley, K. (2024). Sample size planning for replication studies: The devil is in the design. Psychological Methods, 29, 844–867. 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. 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. 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. 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. doi:10.1080/03610929608831787

Author

Ken Kelley (kkelley@nd.edu) and Samantha F. Anderson (samantha.f.anderson@asu.edu)

Examples

result <- ss_buc_independent_t(t_observed = 3, n = 20, alpha_prior = .05,
  alpha_planned = .05, assurance = .80, desired_power = .80)
result
#>  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
result$value[result$term == "necessary_n_per_group"]
#> [1] 130

# ss_buc_smd is the same function under an effect size name
ss_buc_smd(t_observed = 3, n = 20)
#>  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

# A prior study that faced no publication filter (a pilot study, say):
# alpha_prior = "No Publication Filter" means the same as alpha_prior = 1
ss_buc_independent_t(t_observed = 3, n = 20,
  alpha_prior = "No Publication Filter")
#>  term                  value
#>  necessary_n_per_group 37   
#>  total_N               74   
#>  actual_power          0.801
#>  ncp_adjusted          2.09 
#>  smd_planning          0.661
#>  t_observed            3    
#>  n                     20   
#>  alpha_prior           1    
#>  alpha_planned         0.05 
#>  assurance             0.8  
#>  desired_power         0.8  
#> 
#> Design: Independent t test
#> Sample size unit: per group
#> Largest supportable assurance: .99

# Asking for more assurance than the prior result can support stops with an
# informative error that reports the largest workable assurance (here near
# .90, the value 1 - p/alpha_prior for this prior result):
try(ss_buc_independent_t(t_observed = 3, n = 20, assurance = .95))
#> Error : Sample size planning is not possible with these settings. After correcting the prior study's effect for publication bias and uncertainty at the requested level of assurance, the corrected noncentrality parameter is zero, which means there is not enough information in the prior result to plan the sample size as desired. With this prior result and 'alpha_prior', the largest 'assurance' that still yields a usable plan is about .90; re-run with 'assurance' at or below that value, or raise 'alpha_prior' to lift the ceiling. Re-running at a lower 'assurance' is not free: the plan it returns carries that lower assurance, not the one you first asked for, so the long run proportion of replications reaching your target power falls accordingly. 'alpha_prior' does not change the prior study, which is what it is; it is your assumption about the publication threshold the study's literature faced, so raising it toward 1 assumes less publication bias to undo (for example, 'alpha_prior = .10' assumes the literature's journals would have published a result significant at the .10 level, a more lenient policy than .05, and 'alpha_prior = 1' assumes no publication bias at all). See Anderson, Kelley, and Maxwell (2017) and Anderson and Kelley (2024).