Description Usage Arguments Details Value Author(s) References Examples

`ss.power.wa`

returns the necessary per-group sample size
to achieve a desired level of statistical power for a planned study testing
an omnibus effect using a one or two-way fully within-subjects ANOVA, 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, which can be transformed to
various effect size metrics.

1 2 3 4 | ```
ss.power.wa(F.observed, N, levels.A, levels.B = NULL,
effect = c("factor.A", "factor.B", "interaction"),
alpha.prior = 0.05, alpha.planned = 0.05, assurance = 0.8,
power = 0.8, step = 0.001)
``` |

`F.observed` |
Observed F-value from a previous study used to plan sample size for a planned study |

`N` |
Total sample size of the previous study |

`levels.A` |
Number of levels for factor A |

`levels.B` |
Number of levels for factor B, which is NULL if a single factor design |

`effect` |
Effect most of interest to the planned study: main effect of A
( |

`alpha.prior` |
Alpha-level |

`alpha.planned` |
Alpha-level ( |

`assurance` |
Desired level of assurance, or the long run proportion of times that the planned study power will reach or surpass desired level (assurance > .5 corrects for uncertainty; assurance < .5 not recommended) |

`power` |
Desired level of statistical power for the planned study |

`step` |
Value used in the iterative scheme to determine the noncentrality parameter necessary for sample size planning (0 < step < .01) (users should not generally need to change this value; smaller values lead to more accurate sample size planning results, but unnecessarily small values will add unnecessary computational time) |

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.power.wa`

uses the observed
*F*-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 non-central F distribution, where the truncation occurs due to small effect sizes being unobserved due to publication bias. The numerator of the likelihood function is simply 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. 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.

If the previous study of interest has not been subjected to publication
bias (e.g., a pilot study), `alpha.prior`

can be set to 1 to indicate
no publication bias. Alternative *α*-levels can also be
accommodated to represent differing amounts of publication bias. For
example, setting `alpha.prior`

=.20 would reflect less severe
publication bias than the default of .05. In essence, setting
`alpha.prior`

at .20 assumes that studies with *p*-values less
than .20 are published, whereas those with larger *p*-values are not.

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. Two alternatives are recommended. First, users can select a
lower value of assurance (e.g. .8 instead of .95). Second, users can reduce
the influence of publciation bias by setting `alpha.prior`

at a value
greater than .05. 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.

`ss.power.wa`

assumes sphericity for the within-subjects effects.

Suggested per-group sample size for planned study Publication bias and uncertainty- adjusted prior study noncentrality parameter

Samantha F. Anderson samantha.f.anderson@asu.edu, Ken Kelley kkelley@nd.edu

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,* 305-322.

Anderson, S. F., Kelley, K., & Maxwell, S. E. (2017). Sample size
planning for more accurate statistical power: A method correcting sample
effect sizes for uncertainty and publication bias. *Psychological
Science, 28,* 1547-1562.

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.

1 2 | ```
ss.power.wa(F.observed=5, N=60, levels.A=2, levels.B=3, effect="factor.B",
alpha.prior=.05, alpha.planned=.05, assurance=.80, power=.80, step=.001)
``` |

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