Relevant, and possibly of interest- tl:dr- Percentile reinforcement schedules: rather than wait for most responses to meet criterion and then drastically reducing reinforcement frequency by shifting criteria infrequently, it is better to change criteria frequently to maintain both a relatively constant reinforcement density and an intermittent one.
SOURCES
Galbicka, Gregory. Shaping in the 21st Century: Moving Percentile Schedules into Applied Settings. n.d. (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1297861/pdf/jaba00010-0182.pdf) (p740-754)
...also, Fisher, Wayne W., Cathleen C. Piazza, and Henry S. Roane. Handbook of Applied Behavior Analysis. New York: Guilford Press, 2011. p240ish
OVERVIEW
(Shaping) used to be considered more an art than science. Now: “percentile schedules” involving rules for when to deliver reinforcers, and these momentary rules adjust based on recent (local) rates, duration, or types of responding. (ABA p240)
percentile schedules represent a formalization of the rules of shaping. (p755)
Percentile schedules, however, do more than automate shaping. In addition, they make explicit and objective the criteria that define responses as criterional or noncriterional throughout acquisition and maintenance, providing explicit prior control over reinforcement density as well as criterional response probability. Because of this, they provide almost complete independence from trainer- and subject-related variables. This allows all subjects to be trained in a specified manner despite changes in the trainer or the subject, or at different points in the differentiation. (p740)
(If you shaped by occasionally raising criterion for reinforcement) A plot of reinforcement density across time would reveal a pattern like a sawtooth; with each change in the criterion, reinforcement density drops abruptly, but as behavior gradually changes to include more and more criterional responses, reinforcement density gradually increases until the cydle repeats with the next criterion change. This cyclic change in reinforcement density is more pronounced following extended training (p243)
rather than wait for most responses to meet criterion and then drastically reducing reinforcement frequency by shifting criteria infrequently, it is better to change criteria frequently to maintain both a relatively constant reinforcement density and an intermittent one. Both characteristics decrease the likelihood of losing control
over responding prior to the acquisition of the terminal response. (p244)
The percentile solution, developed and expanded by Platt (1973) and colleagues, is momentarily to abandon
the exact physical characteristics of the response and treat it as an ordinal quantity. Ordinal quantities
are values that carry only an associated rank, as opposed to the more typical means of quantifying
observations by assigning a cardinal number and a standard unit. (p244)
NITTY GRITTY OF HOW TO DO IT
m previous observations create m + 1 intervals, one of which must contain the next observation. The counterintuitive notion that intervals of different sizes are equally likely to contain the next observation arises because the line represents a cardinal scale, but the question of which interval will contain the next observation relates to the ordinal properties of the observations. For the moment, ignore the fact that there are physical values attached to any of these observations, and treat them solely in terms of their ranks. In any distribution of values, there is one and only one value ranked 1st, 2nd, 3rd, and so forth. The question of interest is not "What is the expected value of the next observation (i.e., what distance will next be run)?" but rather is "Where will the next observation rank?" If the assumption of independence is met, it will be as likely to rank first or last or anywhere in between, depending on the number of prior observations. (p745)
Hence, the probability that the next observation will fall into any one of k intervals defined by m observations is k times the probability of falling into each interval, or k/(m+ 1). ...establishing a criterion at the kth rank. That is, rather than setting the criterion (for reinforcement) at a particular fixed, physical value, the criterion can specify that the next observation, to meet criterion, must rank higher than the value currently ranked k. When k = 1, responses will be considered criterional if they exceed the response currently ranked 1st (lowest).... The probability of a criterional response (denoted w) is ….w = 1 - [k/(m + 1)] . …. Thus, as the criterion is made more stringent (i.e., as k is increased), the probability of observing a criterional response decreases accordingly, as intuition would suggest. (p746)
(If you know you want w to be a set percentage of reinforcement, you can rewrite equation to find k.) (p746)
Instead of comparing current response to all previous responses (increasing m by one each iteration), use only the most recent responses to compare to. For example, only use the past 5 responses. (p747)
For ties- when current response is tied with response it must exceed: “The simplest solution is to select ties with a random probability equal to w and call them criterional.” (p749)
Percentile schedules appear to meet all the requirements for a viable procedure to formalize shaping except the last-they do not specify a terminal response. The criterion is never specified as an absolute; rather, it is described only in relative fashion (i.e., exceed the kth rank)...There is only one terminal response of all shaping-to do better on the next trial than on previous trials. This is what percentile schedules program, where "better" is defined as exceeding the kth rank and "previous trials" is given by the most recent m observations. Because criteria are evaluated relative to ongoing behavior, there is never a need to stop shaping (p750)
Although sequential dependencies (e.g. responses such as 1, 2, 3, 4, 1, 2, 3, 4, 1...etc) diminish the ability of percentile schedules to control criterional response probability, their effects can be minimized by increasing the comparison distribution size. (p753)
The other "limitation," that responding be ordinally rankable, could actually aid application of
percentile schedules...To illustrate, suppose we wish to train a developmentally disabled client to drink fluid though a straw. Prior observation of the behavior leads the shaper to suggest that the following five behaviors
might be involved: (1) holds glass, (2) directs glass toward mouth, (3) holds straw with other hand, (4) directs straw into mouth, and (5) sucks on straw. These five behaviors can easily be ranked 1 to 5, with 1 being furthest from the terminal response and 5 being dosest. A percentile schedule could be
imposed by recording the response value (i.e., 1 through 5) on each trial. Whether our conception of the response matches the subject's will be evident in the relative frequency of each of the different rankings. (So steps can be added or taken away depending on the subject’s responses). (p754)