ss.power.it {BUCSS}

Necessary sample size to reach desired power for an independent t-test using an uncertainty and publication bias correction procedure

Description

ss.power.it 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, which can be transformed to various effect size metrics.

Usage

ss.power.it(t.observed, n, N, alpha.prior = 0.05, alpha.planned = 0.05,
  assurance = 0.8, power = 0.8, step = 0.001)

Arguments

t.observed

Observed t-value from a previous study used to plan sample size for a planned study

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

Alpha-level \alpha for the previous study or the assumed statistical significance necessary for publishing in the field; to assume no publication bias, a value of 1 can be entered

alpha.planned

Alpha-level (\alpha) assumed for the planned study

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)

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.power.it 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 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. In the two-group case, this formula reduces to the density of a truncated noncentral t-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 corrects 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 \alpha-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.it 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 calcualted 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. 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.

Value

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

Author(s)

Samantha F. Anderson [e-mail], Ken Kelley [e-mail]

References

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.

Examples

ss.power.it(t.observed=3, n=20, alpha.prior=.05, alpha.planned=.05,
assurance=.80, power=.80, step=.001)