bcdmat {upndown}

Transition Probability Matrices for Up-and-Down Designs

Description

Transition Probability Matrices for Common Up-and-Down Designs

Usage

bcdmat(cdf, target)

classicmat(cdf)

kmatMarg(cdf, k, lowTarget)

kmatFull(cdf, k, lowTarget, fluffup = FALSE)

gudmat(cdf, cohort, lower, upper)

Arguments

cdf

monotone increasing vector with positive-response probabilities. The number of dose levels M is deduced from vector's length.

target

the design's target response rate (bcdmat() only).

k

the number of consecutive identical responses required for dose transitions (k-in-a-row functions only).

lowTarget

logical k-in-a-row functions 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. See "Details" for more information.

fluffup

logical (kmatFull only): in the full k-in-a-row internal-state representation, should we "fluff" the matrix up so that it has Mk rows and columns (TRUE, default), or exclude k-1 "phantom" states near one of the boundaries?

cohort, lower, upper

gudmat 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.

Details

Up-and-Down designs (UDDs) generate random walk behavior, whose theoretical properties can be summarized via a transition probability matrix (TPM). Given the number of doses M, and the value of the cdf F at each dose (i.e., the positive-response probabilities), the specific UDD rules uniquely determine the TPM.

The utilities described here calculate the TPMs of the most common and simplest UDDs:

• The k-in-a-row or fixed staircase design common in sensory studies: ⁠kmatMarg(), kmatFull()⁠ (Gezmu, 1996; Oron and Hoff, 2009; see Note). Design parameters are k, a natural number, and whether k negative responses are required for dose transition, or k positive responses. The former is for targets below the median and vice versa.

• The Durham-Flournoy Biased Coin Design: bcdmat(). This design can target any percentile via the target argument (Durham and Flournoy, 1994).

• The original "classical" median-targeting UDD: classicmat() (Dixon and Mood, 1948). This is simply a wrapper for bcdmat() with target set to 0.5.

• Cohort or group UDD: gudmat(), with three design parameters for the group size and the up/down rule thresholds (Gezmu and Flournoy, 2006).

Value

An M\times M transition probability matrix, except for kmatFull() with k>1 which returns a larger square matrix.

Note

As Gezmu (1996) discovered and Oron and Hoff (2009) further extended, k-in-a-row UDDs with k>1 generate a random walk with internal states. Their full TPM is therefore larger than M\times M. However, in terms of random-walk behavior, most salient properties are better represented via an M\times M matrix analogous to those of the other designs, with transition probabilities marginalized over internal states using their asymptotic frequencies. This matrix is provided by kmatMarg(), while kmatFull() returns the full matrix including internal states.

Also, in kmatFull() there are two matrix-size options. Near one of the boundaries (upper boundary with lowTarget = TRUE, and vice versa), the most extreme k internal states are practically indistinguishable, so in some sense only one of them really exists. Using the fluffup argument, users can choose between having a more aesthetically symmetric (but a bit misleading) full Mk\times Mk matrix, or reducing it to its effectivelly true size by removing k-1 rows and columns.

Author(s)

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

References

• Dixon WJ, Mood AM. A method for obtaining and analyzing sensitivity data. J Am Stat Assoc. 1948;43:109-126.

• 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. The Geometric Up-and-Down Design for Allocating Dosage Levels. PhD Thesis. American University; 1996.

• 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.

See Also

• k2targ, ktargOptions to find the k-in-a-row target-response rate for specific k and vice versa.

• g2targ, , gtargOptions likewise for group up-and-down.

• pivec, currentvec, cumulvec, which provide probability vectors of dose-allocation distributions using Up-and-Down TPMs.

Examples

#  Let's use an 8-dose design, and  a somewhat asymmetric CDF

exampleF = pweibull(1:8, shape = 2, scale = 4)
# You can plot if you want: plot(exampleF)

# Here's how the transition matrix looks for the median-finding classic up-and-down

round(classicmat(exampleF), 2)
# Note how the only nonzero diagonals are at the opposite corners. That's how 
#   odd-n and even-n distributions communicate (see examples for vector functions).
# Also note how "up" probabilities (the 1st upper off-diagnoal) are decreasing, 
#   while "down" probabilities (1st lower off-diagonal) are increasing, and 
#   start exceeding "up" moves at row 4.

# Now, let's use the same F to target the 90th percentile, which is often
#    the goal of anesthesiology dose-finding studies.
#    We use the biased-coin design (BCD) presented by Durham and Flournoy (1994):

round(bcdmat(exampleF, target = 0.9), 2)

# Note that now there's plenty of probability mass on the diagonal (i.e., repeating same dose).

# Another option, actually with somewhat better operational characteristics, 
#   is "k-in-a-row". Let's see what k to use:

ktargOptions(.9, tolerance = 0.05)

# Even though nominally k=7's target is closest to 0.9, it's generally preferable
#    to choose a somewhat smaller k. So let's go with k=6.
# We must also specify whether this is a low (<0.5) or high target.

round(kmatMarg(exampleF, k = 6, lowTarget = FALSE), 2)

# Compare and contrast with the BCD matrix above! At what dose do the "up" and "down"
#   probabilities flip? 

# Lastly, if you want to see a 43 x 43 matrix - the full state matrix for k-in-a-row, 
#      run the following line:



  round(kmatFull(exampleF, k = 6, lowTarget = FALSE), 2)