R: Up-and-Down Target Calculation and Design Guidance

k2targ {upndown}

R Documentation

Up-and-Down Target Calculation and Design Guidance

Description

Up-and-down target calculation, as well as design options/guidance given a user-desired target.

Usage

k2targ(k, lowTarget = FALSE)

ktargOptions(target, tolerance = 0.1, maxk = 20)

g2targ(cohort, lower, upper)

gtargOptions(target, minsize = 2, maxsize = 6, tolerance = 0.1)

bcoin(target, fraction = FALSE, nameplate = FALSE, tolerance = 0.02)

Arguments

`k`	the number of consecutive identical responses required for dose transitions (k-in-a-row functions only).
`lowTarget`	logical, `k2targ()` only: is the design targeting below-median percentiles, with `k` repeated negative responses needed to level up and only one to level down - or vice versa? Default `FALSE`.
`target`	the desired target response rate (as a fraction in `(0,1)`), where relevant.
`tolerance`	For `⁠ktargOptions(), gtargOptions()⁠`: the half-width of the interval around `target` in which to search for design options. Default 0.1. For `bcoin()`: the half-width of the interval around 0.5 in which the function recommends to simply use classical UD without a coin, as well as the approximate amount of rounding to the returned coin probability (whether in decimal on rational terms). Default 0.02, and hard-coded to be no less than 0.0001.
`maxk`	`ktargOptions()` only: the maximum value of `k` to consider.
`cohort`, `lower`, `upper`	`g2targ()` only: the cohort (group) size, how many positive responses are allowed for a move upward, and how many are required for a move downward, respectively. For example `⁠cohort=3, lower=0, upper=2⁠` evaluates groups of 3 observations at a time, moves up if none are positive, down if `>=2` are positive, and repeats the same dose with 1 positive.
`minsize`, `maxsize`	`gtargOptions()` only: the minimum and maximum cohort size to consider. `minsize` has to be at least 2 (cohort size 1 is equivalent to classical UD).
`fraction`	`bcoin()` only: whether to report the coin probability as a rational rather than decimal fraction. Default `FALSE`.
`nameplate`	`bcoin()` only: in case `fraction = TRUE`, whether to return the "exact" rational probability, or allow some nudging of the resulting balance point towards the median. Default `FALSE`, and moot when `fraction = FALSE`.

Details

This suite of utilities helps users

Figure out the approximate target response-rate (a.k.a. the balance point), given design parameters;
Suggest potential design parameters, given user's desired target response-rate and other constraints.

Up-and-down designs (UDDs) generate random walks over dose space, with most dose-allocations usually taking place near the design's de-facto target percentile, called the "balance point" by some theorists to distinguish it from the user-designated target in case they differ (Oron and Hoff 2009, Oron et al. 2022).

Most k-in-a-row and group UDD parameter combinations yield balance points that are irrational percentiles of the dose-response function, and therefore are unappealing as official experimental targets.

However, since the UD dose distribution has some width, and since even the balance point itself is only a close approximation for the actual average of allocated doses, the user's target does not have to be identical to the balance point. It only needs to be "close enough".

The k2targ() and g2targ() utilities are intended for users who already have a specific k-in-a-row or group design in mind, and only want to verify its balance point. The complementary utilities ⁠ktargOptions(), gtargOptions()⁠ provide a broader survey of design-parameter options within user-specified constraints, given a desired target.

Lastly, bcoin() returns the biased-coin probabilities given the user's designated target. In contrast to the two other UDDs described above, the biased-coin design can target any percentile with a precisely matched balance point. That said, k-in-a-row and group UDDs offer some advantages over biased-coin in terms of properties and operational simplicity.

bcoin() can return the probability as a decimal (default) or approximate rational fraction. In the latter case, if nameplate is set to TRUE, you will get the exact "nameplate" coin probability \Gamma/(1 - \Gamma), with \Gamma being the target percentile between 0 and 1. However, the default nameplate = FALSE might nudge the coin to yield a balance point somewhat closer to the median. This choice is based upon the theoretical finding that the biased-coin design does tend to concentrate doses a bit further away from the median than the balance point would suggest (Oron and Hoff, 2009). See more information in bcoin()'s argument descriptions.

Value

⁠k2targ(), g2targ()⁠: the official balance point given the user-provided design parameters.
⁠ktargOptions(), gtargOptions()⁠: a data.frame with design parameters and official balance point, for all options that meet user-provided constraints. A printed string provides dose transition rule guidance.
⁠bcoin():⁠ a printed string that informs user of the biased-coin design rules, including the 'coin' probability in its user-chosen format (decimal or fraction). In case the user-desired target is 0.5 or very close to it, the string will inform user that they are better off just using classical UDD without a coin.

Author(s)

Assaf P. Oron <assaf.oron.at.gmail.com>

References

Durham SD, Flournoy N. Random walks for quantile estimation. In: Statistical Decision Theory and Related Topics V (West Lafayette, IN, 1992). Springer; 1994:467-476.
Gezmu M, Flournoy N. Group up-and-down designs for dose-finding. J Stat Plan Inference. 2006;136(6):1749-1764.
Oron AP, Hoff PD. The k-in-a-row up-and-down design, revisited. Stat Med. 2009;28:1805-1820.
Oron AP, Souter MJ, Flournoy N. Understanding Research Methods: Up-and-down Designs for Dose-finding. Anesthesiology 2022; 137:137–50.