ReplicationSuccess (version 1.3.3)

powerSignificance: Computes the power for significance

Description

The power for significance is computed based on the result of the original study, the corresponding variance ratio, and the design prior.

Usage

powerSignificance(
  zo,
  c = 1,
  level = 0.025,
  designPrior = c("conditional", "predictive", "EB"),
  alternative = c("one.sided", "two.sided"),
  h = 0,
  shrinkage = 0,
  strict = FALSE
)

Value

The probability that a replication study yields a significant effect estimate in the specified direction.

Arguments

zo

Numeric vector of z-values from original studies.

c

Numeric vector of variance ratios of the original and replication effect estimates. This is usually the ratio of the sample size of the replication study to the sample size of the original study.

level

Significance level. Default is 0.025.

designPrior

Either "conditional" (default), "predictive", or "EB". If "EB", the power is computed under a predictive distribution, where the contribution of the original study is shrunken towards zero based on the evidence in the original study (with an empirical Bayes shrinkage estimator).

alternative

Either "one.sided" (default) or "two.sided". Specifies if the significance level is one-sided or two-sided. If the significance level is one-sided, then power calculations are based on a one-sided assessment of significance in the direction of the original effect estimates.

h

The relative between-study heterogeneity, i.e., the ratio of the heterogeneity variance to the variance of the original effect estimate. Default is 0 (no heterogeneity). Is only taken into account when designPrior = "predictive" or designPrior = "EB".

shrinkage

Numeric vector with values in [0,1). Defaults to 0. Specifies the shrinkage of the original effect estimate towards zero, e.g., the effect is shrunken by a factor of 25% for shrinkage = 0.25. Is only taken into account if the designPrior is "conditional" or "predictive".

strict

Logical vector indicating whether the probability for significance in the opposite direction of the original effect estimate should also be taken into account. Default is FALSE. Only taken into account when alternative = "two.sided".

Author

Leonhard Held, Samuel Pawel, Charlotte Micheloud, Florian Gerber

Details

powerSignificance is the vectorized version of the internal function .powerSignificance_. Vectorize is used to vectorize the function.

References

Goodman, S. N. (1992). A comment on replication, p-values and evidence, Statistics in Medicine, 11, 875--879. tools:::Rd_expr_doi("10.1002/sim.4780110705")

Senn, S. (2002). Letter to the Editor, Statistics in Medicine, 21, 2437--2444.

Held, L. (2020). A new standard for the analysis and design of replication studies (with discussion). Journal of the Royal Statistical Society: Series A (Statistics in Society), 183, 431-448. tools:::Rd_expr_doi("10.1111/rssa.12493")

Pawel, S., Held, L. (2020). Probabilistic forecasting of replication studies. PLOS ONE. 15, e0231416. tools:::Rd_expr_doi("10.1371/journal.pone.0231416")

Held, L., Micheloud, C., Pawel, S. (2022). The assessment of replication success based on relative effect size. The Annals of Applied Statistics. 16:706-720. tools:::Rd_expr_doi("10.1214/21-AOAS1502")

Micheloud, C., Held, L. (2022). Power Calculations for Replication Studies. Statistical Science. 37:369-379. tools:::Rd_expr_doi("10.1214/21-STS828")

See Also

sampleSizeSignificance, powerSignificanceInterim

Examples

powerSignificance(zo = p2z(0.005), c = 2)
powerSignificance(zo = p2z(0.005), c = 2, designPrior = "predictive")
powerSignificance(zo = p2z(0.005), c = 2, alternative = "two.sided")
powerSignificance(zo = -3, c = 2, designPrior = "predictive",
                  alternative = "one.sided")
powerSignificance(zo = p2z(0.005), c = 1/2)
powerSignificance(zo = p2z(0.005), c = 1/2, designPrior = "predictive")
powerSignificance(zo = p2z(0.005), c = 1/2, alternative = "two.sided")
powerSignificance(zo = p2z(0.005), c = 1/2, designPrior = "predictive",
                  alternative = "two.sided")
powerSignificance(zo = p2z(0.005), c = 1/2, designPrior = "predictive",
                  alternative = "one.sided", h = 0.5, shrinkage = 0.5)
powerSignificance(zo = p2z(0.005), c = 1/2, designPrior = "EB",
                  alternative = "two.sided", h = 0.5)

# power as function of original p-value
po <- seq(0.0001, 0.06, 0.0001)
plot(po, powerSignificance(zo = p2z(po), designPrior = "conditional"),
     type = "l", ylim = c(0, 1), lwd = 1.5, las = 1, ylab = "Power",
     xlab = expression(italic(p)[o]))
lines(po, powerSignificance(zo = p2z(po), designPrior = "predictive"),
      lwd = 2, lty = 2)
lines(po, powerSignificance(zo = p2z(po), designPrior = "EB"),
      lwd = 1.5, lty = 3)
legend("topright", legend = c("conditional", "predictive", "EB"),
       title = "Design prior", lty = c(1, 2, 3), lwd = 1.5, bty = "n")