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Workflow of 'x_from_power()' Using the 'power_curve' Algorithm

Source: vignettes/articles/x_from_power_workflow.Rmd
x_from_power_workflow.Rmd

Goal

This technical appendix describes the workflow of x_from_power() in power4mome when using the "power_curve" algorithm.

x_from_power()

The following is the workflow of x_from_power(), when the algorithm "power_curve" is used. Simulation is to be done for each value of x, and this can be slow when Monte Carlo or bootstrapping confidence intervals are involved in the test. It is not feasible, and also not necessary, to accurately estimate the levels of power along many values across a range of x, if the goal is to find the value of x with the target power. Therefore, steps are taken to balance speed and precision when finding the solution.

From Power to x (Sample Size or Effect Size [Parameter Value])

In x_from_power, x can be a sample size (n) or a population value (es, “effect size”) of the selected model parameter.

Annotation

  • by_x_1

    • The collection of all values tried and their results. It is updated whenever new value(s) is/are tried.

  • fit_1

    • The latest power curve estimated by power_curve, using the values tried, stored in by_x_1. It is updated whenever by_x_1 is updated.

  • x_j

    • The value(s) for which power levels will be estimated in a trial.

  • nrep_j

    • The number of replications to be used when estimate the power level for a value of x. In a trial, the numbers of replications can be different for different values, for efficiency.

  • by_x_j

  • x_out

    • The value of x which is a candidate solution (e.g., with estimated power closest to the target value).

  • power_out, nrep_out, ci_out, by_x_out

    • Results based on x_out.

  • ci_hit

    • Logical. Set to TRUE if there is at least one value of x with the confidence interval of the estimated power including the target power.

  • final_nrep

    • The desired number of replications for the solution. This value determines the desired level of precision (the width of the confidence interval) in the solution.

  • The sequences of values for nrep, R, and the number of x in a trial.

    • The initial number of replications (nrep) can be smaller than final_nrep, such that the initial trials, though with lower precision (wider confidence intervals), are faster to run. As the solution is likely to be be found (values of x with estimated power close to the target value found), nrep will be increased successively to final_nrep, such that a trial is slower to run but has a higher precision. Other values that affect the speed, such as the number of values of x (xs_per_trial) and the number of iterations (R) in Monte Carlo confidence intervals and bootstrapping, are also increased successively.

  • x_final

    • The value of x in the solution (e.g., with estimated power closest to the target value), if found.

  • power_final, nrep_final, ci_final, by_x_final

    • Results based on x_final.

  • Main functions used

    • power4test_by_n() and power4test_by_es(), for estimating the power levels for a set of values of x.

    • power_curve(), for estimating the relation between power and the value of x, based on the values of x having been examined.

    • The internal function estimate_x_range(), for determining the value(s) of x to be examined in a trial, given the value(s) examined so far and the tentative power curve.