Necessary Sample Size to Reach Desired Power for a Pearson Correlation Using a Publication Bias and Uncertainty Correction Procedure

ss_buc_correlation returns the necessary total sample size to achieve a desired level of statistical power for a test of a Pearson correlation in a planned study, 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 correlation.

Usage

ss_buc_correlation(
  r_observed,
  N,
  t_observed,
  alpha_prior = 0.05,
  alpha_planned = 0.05,
  assurance = 0.8,
  desired_power = 0.8
)

Arguments

r_observed: Observed Pearson correlation from a previous study used to plan sample size for a planned study. Either sign is accepted; only the magnitude enters the correction. Supply either r_observed or t_observed, not both.
N: Total sample size of the previous study.
t_observed: Observed t statistic for the correlation from the previous study, an alternative to r_observed.
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 total 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, \(N\rho^2/(1 - \rho^2)\) at the prior study's sample size; and the planning effect size, rho_planning, the population correlation (a magnitude), \(\sqrt{\lambda/(\lambda + N)}\), where \(\lambda\) is ncp_adjusted and N the prior sample size; the power statement that follows is for sampled variables, the exact distribution this planner uses.

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_correlation uses the observed correlation 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 total sample size to achieve the desired level of power in the planned study.

The observed correlation r with sample size N gives \(t = r\sqrt{(N - 2)/(1 - r^2)}\) on N - 2 degrees of freedom, and the planner works from that t. Supplying t_observed directly instead of r_observed is equivalent.

The approach uses a likelihood function of a truncated 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 the test statistic; the denominator is the power of the test, which serves to truncate the distribution. (See Taylor & Muller, 1996, Equation 2.1; Anderson, Kelley, & Maxwell, 2017, Equation 3; and Anderson & Maxwell, 2017, for more details.) Which density belongs in that numerator is described next; it is the density the regression planners use with random_predictors = TRUE.

Both variables are sampled, and that changes the distribution. The test of a correlation is arithmetically the test of the slope in a simple regression, so it is tempting to plan it as one. That is exact only when the predictor is fixed by design, as it is in an experiment where the researcher sets the values of X: the estimates, and the distribution of the statistic when there is no effect, are the same whether the predictor is fixed or sampled, but the distribution when there is an effect, and so the power, is not (Sampson, 1974; Gatsonis & Sampson, 1989; Kelley & Maxwell, 2003). A correlation study samples both variables. Conditional on the sampled X, the statistic is noncentral F, but its noncentrality parameter is itself random: it is proportional to the sampled spread of X, which follows a chi square distribution on N - 1 degrees of freedom. The marginal distribution is therefore a chi square mixture of noncentral F distributions, which is more dispersed than any single noncentral F, and treating the statistic as noncentral F overstates the power of the planned study in every case, by roughly 1/n for a plan of n participants: about .01 at n = 100, .02 at 50, .05 at 20, .10 at 10, and up to about .2 for the smallest plans.

ss_buc_correlation uses the mixture, which has an exact closed form (the Poisson weights of the noncentral F become negative binomial weights), for both the correction and the planned study's power. Stated for the correlation itself, it is the general sampling distribution of the multiple correlation coefficient (Fisher, 1928) with one predictor. The returned sample size is therefore exact for bivariate normal data rather than a few participants light, and it is typically 2 to 5 participants larger than the fixed-predictor calculation (1 to 3 for the smallest plans, up to about 8 for plans in the hundreds). Near the assurance ceiling the difference reverses and the mixture calls for a sample a few percent smaller, which can be hundreds of participants because plans there are very large (7,346 rather than the 7,722 of treating the predictor as fixed, for r = .50 from a prior study of 20 at an assurance of .5). If your predictor really is fixed by design, the analysis is a regression slope rather than a correlation, and ss_buc_reg_coef with p = 1 and random_predictors = FALSE is the planner for it. With one sampled predictor the regression planners reduce to this one: ss_buc_reg_coef(p = 1, random_predictors = TRUE) gives this planner's own result for the same prior t and N, exactly, as do ss_buc_R2 with p = 1 and ss_buc_reg_joint with p = 1 and p_joint = 1 given F_observed = \(t^2\); each states the planning effect size in its own metric.

The largest assurance the prior result can support is the same in either frame, since at a zero effect the mixture is the central F distribution.

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 correlation may be entered with either sign, since the publication rule the correction assumes is two-sided and only the magnitude enters the computation.

The returned ncp_adjusted is the noncentrality parameter the planning effect size implies at the prior study's sample size, \(N\rho^2/(1 - \rho^2)\), on the scale of the F statistic, as in the regression planners (the t-based planners report theirs on the scale of t instead). The planning effect size itself, the correlation sqrt(ncp_adjusted / (ncp_adjusted + N)), is returned as the rho_planning row.

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

Fisher, R. A. (1928). The general sampling distribution of the multiple correlation coefficient. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 121, 654–673. doi:10.1098/rspa.1928.0224

Gatsonis, C., & Sampson, A. R. (1989). Multiple correlation: Exact power and sample size calculations. Psychological Bulletin, 106, 516–524. doi:10.1037/0033-2909.106.3.516

Sampson, A. R. (1974). A tale of two regressions. Journal of the American Statistical Association, 69, 682–689. doi:10.1080/01621459.1974.10480189

Author

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

Examples

result <- ss_buc_correlation(r_observed = .35, N = 100, alpha_prior = .05,
  alpha_planned = .05, assurance = .80, desired_power = .80)
result
#>  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 equivalent call from the observed t statistic:
ss_buc_correlation(t_observed = .35 * sqrt(98 / (1 - .35^2)), N = 100)
#>  term          value
#>  necessary_N   122  
#>  actual_power  0.802
#>  ncp_adjusted  6.71 
#>  rho_planning  0.251
#>  t_observed    3.7  
#>  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

# Requesting more assurance than the prior result can support stops with an
# informative error naming the largest workable assurance. The correlation
# above supports .99, so a weaker prior is needed to reach the ceiling
# (here near .86):
try(ss_buc_correlation(r_observed = .27, N = 100, 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 .86; 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).