Necessary Sample Size to Reach Desired Power for a Nested Model Chi Square Difference Test Using a Publication Bias and Uncertainty Correction Procedure

ss_buc_chisq_diff returns the necessary total sample size to achieve a desired level of statistical power for a planned study whose effect of interest is tested by the chi square difference between two nested models (the likelihood ratio test used, for example, to test a constrained path in a structural equation model), 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 the root mean square error of approximation of the difference test, \(\mathrm{RMSEA}_D\) (D for difference).

Usage

ss_buc_chisq_diff(
  chisq_observed,
  N,
  df_full,
  df_restricted,
  alpha_prior = 0.05,
  alpha_planned = 0.05,
  assurance = 0.8,
  desired_power = 0.8
)

Arguments

chisq_observed: Observed chi square difference between the two nested models in the previous study.
N: Total sample size of the previous study.
df_full: Degrees of freedom of the full (less constrained) model. A saturated full model has zero, the signature of an omnibus test of model fit rather than a difference test between two substantive models; the function refuses that case (see Details, which also covers a just-identified full model).
df_restricted: Degrees of freedom of the restricted (more constrained) model. It exceeds df_full by the number of constraints imposed, which is the difference test's degrees of freedom.
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 chi square statistic, at the prior study's sample size; and the planning effect size, rmsea_d_planning, \(\mathrm{RMSEA}_D\), where the D stands for difference: the root mean square error of approximation of the chi square difference test between the two nested models (Savalei, Brace, & Fouladi, 2024), as distinct from the RMSEA of either model's fit; here \(\sqrt{\lambda/(\mathrm{df}_D N)}\), where \(\lambda\) is ncp_adjusted, \(\mathrm{df}_D\) is the number of constraints (df_restricted - df_full, the difference test's degrees of freedom), and N the prior sample size, the scaling this planner plans with; the usual sample formula divides by \(N - 1\) instead, which gives a value larger by the factor \(\sqrt{N/(N - 1)}\).

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.

Scope, and one thing this function is not for. The correction applies to a test whose statistic is large when the effect is present, so that publication selects large values. A chi square difference test between nested models is such a test: under the alternative it is approximately noncentral chi square with degrees of freedom equal to the number of constraints released and a noncentrality parameter proportional to sample size, and a paper reports the constrained path as supported when the difference test is significant. Steiger, Shapiro, and Browne (1985) give the asymptotic distribution of sequential chi square statistics for nested models and of the difference tests between them, and Satorra and Saris (1985) compute the power of the likelihood ratio test in covariance structure analysis from the noncentral chi square distribution, as is done here. The correction is therefore built for the difference test.

It is not appropriate for an omnibus model fit chi square, where a publishable result is a small statistic (a model that fits). The selection region is inverted there, so the truncated likelihood this function builds does not describe it.

The function refuses that case rather than only warning about it. This is why both models' degrees of freedom are required rather than only their difference. An omnibus fit chi square is the comparison of a model against the saturated model, and a saturated model has zero degrees of freedom, so df_full = 0 is its signature. Given only the difference, nothing distinguishes a fit test from a genuine comparison of two substantive models, and the mistake would pass silently. Supplying both is also a more faithful description of what was tested, and the two values are what a paper reports for each model. A just-identified full model (every parameter free, as in a path model compared with the same model with one path fixed at zero) also has zero degrees of freedom, and there the difference test is a genuine one. The plan depends only on the number of constraints, so for that case the call with df_full = 1 and df_restricted one more than the number of constraints gives the plan.

The correction uses a likelihood function of a truncated noncentral chi square distribution, the same construction the rest of the package uses for the noncentral F (see Taylor & Muller, 1996, Equation 2.1; Anderson, Kelley, & Maxwell, 2017, Equation 3; and Anderson & Maxwell, 2017): the truncated area between the critical value and the observed statistic, divided by the power of the test. Because the noncentrality parameter of a likelihood ratio test is proportional to sample size, the planned study's noncentrality parameter is the corrected one scaled by the ratio of the planned to the prior sample size.

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 chi square difference test is asymptotic, so both the correction and the planned-study power inherit that approximation. The likelihood ratio test is liberal in small samples, and more so the more parameters a model estimates. For a test that drops 2 of 6 parameters of a normal linear model, its actual Type I error is about .07 at a total sample size of 50, .055 at 200, and .052 at 400; dropping 5 of 21 parameters, it is about .08 at 200. At such sample sizes the real test's power can also sit above or below actual_power. Treat a planned sample size that is small relative to the number of parameters estimated with caution, and read a very small one as a signal that the prior effect was large rather than as a serious recommendation. The search for the planned sample size starts at the number of constraints plus 2 (df_restricted - df_full + 2), so no plan is smaller than that, even when a smaller study would already reach the desired power.

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

Satorra, A., & Saris, W. E. (1985). Power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50, 83–90. doi:10.1007/BF02294150

Savalei, V., Brace, J. C., & Fouladi, R. T. (2024). We need to change how we compute RMSEA for nested model comparisons in structural equation modeling. Psychological Methods, 29, 480–493. doi:10.1037/met0000537

Steiger, J. H., Shapiro, A., & Browne, M. W. (1985). On the multivariate asymptotic distribution of sequential chi-square statistics. Psychometrika, 50, 253–263. doi:10.1007/BF02294104

Author

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

Examples

# A path constrained to zero: the full model has 42 degrees of freedom and
# the restricted model 43, so the difference test has 1.
result <- 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)
result
#>  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

# A saturated full model means the comparison is an omnibus test of model
# fit, which this method does not describe, and the function says so:
try(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.

# Requesting more assurance than the prior result can support stops with an
# informative error naming the largest workable assurance (here near .95):
try(ss_buc_chisq_diff(chisq_observed = 9.5, N = 250, df_full = 42,
  df_restricted = 43, assurance = .99))
#> 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 .95; 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).