Translating Quantitative Theories of Behavior into Improved Clinical Treatments for Problem Behavior

Abstract

The most important advancement in the treatment of destructive behavior has been the development of the functional analysis, which is used to prescribe effective treatments like functional communication training. Although this approach can be highly effective, extinction bursts and forms of treatment relapse commonly occur when function-based treatments are implemented by caregivers in natural community settings. In recent years, researchers have increasingly applied quantitative theories of behavior like behavioral momentum theory (BMT) and the temporally weighted matching law (TWML) to understand, prevent, or mitigate extinction bursts and treatment relapse. In this paper, we describe BMT and TWML and selectively review the basic, translational, and applied research supporting and opposing each theory. Then, we describe how function-based treatments may be refined based on these theories to improve the effectiveness, generality, and durability of function-based treatments for individuals with autism spectrum and related disorders who display problem behavior.

About 10% of individuals with developmental disabilities display severe problem behavior that poses a risk of harm to oneself (i.e., self-injurious behavior), others (i.e., aggression), or the environment (i.e., property destruction; Emerson et al., 2001; Holden & Gitlesen, 2006). The risk for problem behavior increases with communication deficits, intellectual disability severity, and co-occurring autism spectrum disorder (Holden & Gitlesen, 2006). The social impact of these behavior disorders on the quality of life for affected individuals and their families is difficult to overestimate. Problem behavior increases the risk of being subjected to restrictive procedures like physical restraint (Harris, 1993; Oliver et al., 1987) and placement in a residential facility (Black et al., 1990; McIntyre et al., 2002). Annual costs of problem behavior exceeded $3.5 billion in the U.S. in 1994 (Thompson & Gray, 1994), which is roughly equivalent to $6.6 billion in 2021. Aggressive behavior increases the risk for institutionalization, social isolation, physical restraint, medication overuse, service denial, and abuse (Antonacci et al., 2008). Self-injurious behavior causes health complications, such as blindness, self-amputation, fractures, brain trauma, and even death (Hyman et al., 1990).

The development and clinical deployment of functional-analysis procedures has markedly increased our understanding of and ability to treat severe problem behavior in individuals with autism and developmental disabilities (Campbell, 2003; Didden et al., 2006; Greer et al., 2016; Iwata et al., 1982/1994; Iwata et al., 1994). When conducting a functional analysis, the behavior analyst systematically manipulates the variables hypothesized to reinforce problem behavior using single-case designs (e.g., a multielement design in which the experimenter rapidly alternates experimental conditions, typically across sessions; DeRosa et al., 2021) and determines the effects of those manipulations with direct-observation measures of problem behavior. Common functions of problem behavior include social positive reinforcement in the form of access to social attention or preferred items (e.g., an iPad), social negative reinforcement in the form of escape from nonpreferred tasks, and automatic reinforcement in form of sensory stimulation (Iwata et al., 1994). In addition, researchers have also identified a variety of less common, idiosyncratic functions of problem behavior (Bowman et al., 1997; Fisher, Adelinis, et al., 1998; Fisher, Lindauer, et al., 1998; Owen et al., 2020; Tiger et al., 2009). For example, Bowman et al. and Owen et al. showed that, for some individuals, problem behavior was reinforced by increased caregiver compliance with the individual’s mands. A mand is a verbal operant that specifies its reinforcer (e.g., “iPad, please”). Randomized clinical trials, meta-analyses, and consecutive, case-controlled studies have demonstrated the effectiveness of function-based interventions for severe problem behavior like functional communication training (FCT) and noncontingent reinforcement (NCR; Didden et al., 2006; Greer, Fisher, Saini, et al., 2016; Hassiotis et al., 2009; Iwata et al., 1994; Lindgren et al., 2020; Phillips et al., 2017). FCT involves the delivery of the reinforcer identified by a function analysis (e.g., attention) contingent on an appropriate communication response whilst problem behavior is placed on extinction. NCR involves the delivery of the reinforcer identified by a functional analysis on a time-based schedule (e.g., continuous access to an iPad) whilst problem behavior is placed on extinction.

When results of an individual’s functional analysis identify a social function for their problem behavior, the most prescribed treatment is FCT. With FCT, once the function of problem behavior is identified (e.g., acquiring toys), the individual is taught to access that reinforcer through an appropriate mand called the functional communication response (FCR; e.g., “Toys, please” for problem behavior maintained by access to tangible items), and the contingency between problem behavior and the functional reinforcer is terminated (i.e., problem behavior is placed on extinction). Research has shown that FCT can effectively treat a variety of topographies of problem behavior (e.g., aggression, disruption, elopement, inappropriate vocalization, property destruction, SIB) emitted by individuals with a variety of developmental and related disorders (e.g., autism, attention-deficit/hyperactivity disorder, intellectual disability, language delay, traumatic brain injury; Fisher, Greer, & Fuhrman, 2015). In fact, researchers have found that when FCT is initially evaluated under optimal conditions, it typically reduces problem behavior by my more than 90% relative to baseline (Greer, Fisher, Saini, et al., 2016).

Despite the impressive results for function-based treatments like FCT, a variety of factors may disrupt the effectiveness of such treatments when they are implemented under less-than-optimal conditions. For example, when FCT is initiated, if the FCR does not produce reinforcement quickly, efficiently, and with minimal response effort, problem behavior may not decrease to clinically acceptable levels or may even result in an extinction burst. An extinction burst is a temporary increase in the rate or intensity of behavior that occurs when the response contacts extinction (Fisher et al., 2018; Horner & Day, 1991). Similarly, problem behavior may worsen after initial successful treatment when the treatment is challenged.

Treatment relapse is a general term used in medicine and the allied health fields to label the reappearance of a condition or symptoms previously in remission (Pritchard et al., 2014a). Behavior analysts have identified three specific forms of treatment relapse defined by the antecedent or consequent conditions under which problem behavior reemerges referred to as resurgence, renewal, and reinstatement. Resurgence is a form of treatment relapse in which the target response (e.g., problem behavior) increases after a period of successful treatment with differential reinforcement due to a worsening of current reinforcement conditions (e.g., the FCR goes unreinforced). Renewal is a form of treatment relapse in which the target response (e.g., problem behavior) increases when the individual is exposed to a change in context after a period of successful treatment in the original treatment context. Reinstatement is a form of treatment relapse in which the target response (e.g., problem behavior) increases after successful treatment with extinction when one or a few reinforcers are delivered either on a time-based schedule or contingent on the target response.

Nevin and Wacker (2013) indicated that behavior analysts typically test whether treatment effects maintain or problem behavior relapses after establishing steady-state responding with the treatment contingencies in effect (i.e., optimal conditions, with the FCR producing reinforcement on a fixed-ratio [FR] 1 schedule and problem behavior placed on extinction). They suggested that behavior analysts should also test whether treatment effects maintain under less-than-optimal conditions, when the treatment is challenged (e.g., during a resurgence challenge in which reinforcement is discontinued for the alternative response). During such challenges, problem behavior may increase, and Nevin and Wacker recommend that treatment should continue until treatment effects maintain during such challenges.

Quantitative Theories of Behavior

Quantitative theories of behavior, like behavioral momentum theory (BMT) and the temporally weighted matching law (TWML) provide quantitative accounts of why problem behavior often worsens when FCT and similar treatments undergo challenges, and these quantitative theories make specific predictions about procedures that behavior analysts may implement to mitigate or prevent treatment relapse. In addition, the TWML also provides a quantitative account of extinction bursts and suggests potential methods of mitigating or preventing such response bursts. In what follows, we will discuss: (a) the conceptual tenets and quantitative predictions BMT and the TWML; (b) the basic, translational, and applied research relevant to each theory; (c) the clinical implications of this research; and (d) potential directions for future quantitative models and research relevant to function-based treatments like FCT.

Behavioral Momentum Theory

Behavioral momentum theory (Nevin, 1992) is a quantitative theory of behavior in which the strength of a response is determined by its persistence when challenged by a disruptor or external force. This theory of behavior is metaphorically based on Newton’s first and second laws of motion. Newton’s first law of motion states that objects at rest remain at rest and objects in motion continue in motion at the same rate until acted upon by an external force. Similarly, according to BMT, a response maintains its ongoing or baseline rate of responding under static environmental conditions (e.g., in each stimulus context with a constant rate of reinforcement).

Newton’s second law of motion states that when an external force acts on a moving object, the change in the object’s velocity is directly proportional to the amount of force applied and inversely proportional to the mass of the object (i.e., objects with more mass require more force to speed up or slow down). Similarly, BMT posits that when environmental conditions change (e.g., the response contacts extinction), the change in the rate of the response is directly proportional to degree of environmental change and inversely proportional to the response’s resistance to change. Thus, whereas Newtonian physics mathematically defines the momentum of an object as its mass multiplied by its velocity, BMT correspondingly defines the momentum of a response as its resistance to change (the behavioral equivalent of an object’s mass) multiplied by its baseline rate of responding (the behavioral equivalent of an object’s velocity).

BMT Account of Resurgence

As previously mentioned, resurgence is an increase in problem behavior following successful treatment with differential reinforcement due to a worsening of reinforcement conditions (e.g., the FCR goes unreinforced). Resurgence is typically studied in a three-phase arrangement: (a) baseline, in which only the target response (e.g., problem behavior) produces reinforcement; followed by (b) extinction of the target response combined with reinforcement delivered for an appropriate alternative response (e.g., FCT) or on a time-based schedule (NCR) until the target response reaches low or zero levels; followed by (c) an extinction challenge in which all sources of programmed reinforcement are withdrawn. This type of arrangement simulates treatment-integrity errors that often occur in natural human environments when a caregiver is unable to deliver reinforcement for a child’s appropriate behavior due to competing responsibilities (e.g., when the caregiver is attending to other family responsibilities). In the applied literature on treatment-integrity errors, such occurrences are often referred to as errors of omission (St. Peter Pipkin et al., 2010).

General Principles of BMT

According to BMT, the baseline rate of a response is determined by response–reinforcer (or operant) contingencies, with the relative rate of a response matching its relative rate of reinforcement during baseline (e.g., Herrnstein, 1970). By contrast, BMT posits that stimulus–reinforcer (or Pavlovian) contingencies determine how resistant a response is to change, with increased stimulus–reinforcer pairings producing greater resistance to change. Under the assumptions of BMT, this distinction between the effects of operant contingencies on baseline response rates and the effects of Pavlovian contingencies on a response’s resistance to change has important implications for clinical treatments involving the delivery of differential and time-based reinforcement. That is, BMT predicts that delivering a high rate of alternative reinforcement during a differential-reinforcement treatment will suppress problem behavior relative to baseline if alternative reinforcement continues. However, BMT posits that if alternative reinforcement ceases (e.g., because a parent is busy tending to an infant sibling), problem behavior is likely to increase or resurge for two reasons. One reason is that the suppressive effects of alternative reinforcement on problem behavior are removed. The second reason, according to BMT, is that the delivery of alternative reinforcers produce additional Pavlovian pairing between the stimulus context and the delivery of the function reinforcer, and these pairings contribute to problem behavior’s resistance to change. That is, according to BMT, the delivery of differential or time-based reinforcement during function-based treatments contribute to reductions in problem behavior while the treatments are in place but also contribute to the persistence and worsening of problem behavior when alternative reinforcement is removed or reduced.

BMT Equation for Resurgence

Equation 1 shown below from Nevin and Shahan (Nevin & Shahan, 2011) provides a quantitative method for modeling the effects of various reinforcement-schedule parameters and disrupters to predict the rate of problem behavior (as a proportion of the baseline rate) when alternative reinforcement is withdrawn in the third phase of a resurgence test (see Greer, Fisher, Romani, et al., 2016, for a tutorial). On the left side of the equation, $B_t$ is the rate of the target response (e.g., problem behavior) at time $t$ of extinction, and $B_0$ is the baseline rate of responding. On the right side of the equation, the terms in the numerator represent variables that disrupt the target response, with larger values indicating greater disruption (hence the minus sign): $t$, which represents the disruptive effects of increasing time in extinction; $c$, which represents the disruptive effects of terminating the contingency between the target response and reinforcement; $d$, which scales the disruptive effects of removing $r$ reinforcers per unit of time for the target response; and $p$, which scales the disruptive effects of delivering alternative reinforcers $R_a$. It should be noted that BMT treats alternative reinforcement the same whether it is delivered contingent on an alternative response (e.g., FCT) or on a time-based schedule (e.g., NCR). The terms in the denominator of the right side of the equation represent variables that strengthen or increase the resistance to change of the target response, with $r$ representing the number of reinforcers delivered per unit of time for the target response and $R_a$ representing the number of alternative reinforcers delivered per unit of time.

$$\frac{B_t}{B_0}={10}^{\frac{-t(c+dr+p{R}_a)}{{(r+R_a)}^{0.5}}}$$ (1)

This BMT equation makes several specific predictions about how the variables depicted in the equation might be manipulated to mitigate or prevent resurgence of problem behavior. First, the equation predicts that higher reinforcement rates for problem behavior during baseline (represented by $r$) and higher alternative reinforcement rates during treatment (represented by $R_a$) should increase the momentum of problem behavior relative to lower rates, and thereby, increase the resurgence of problem behavior when alternative reinforcement is withdrawn or decreased. Conversely, the equation predicts that we should be able to decrease resurgence by lowering the rate of reinforcement for problem behavior during baseline, by lowering the rate of reinforcement for the alternative response during treatment, or by doing both.

Empirical Results Supporting BMT Predictions

Results from several basic, translational, and applied studies have supported these predictions of BMT (e.g., (Cançado et al., 2015; Fisher et al., 2019; Leitenberg et al., 1975; Podlesnik & Shahan, 2009; Pritchard et al., 2014a; Winterbauer & Bouton, 2012). For example, Fisher et al. tested the effects of higher and lower rates of reinforcement for problem behavior delivered during baseline with seven participants with autism or intellectual disability who displayed severe problem behavior. They correlated each baseline with a unique stimulus context, with the two contexts differing in terms of the color of each room and the matching-colored clothing worn by the experimenters. Next, they introduced identical schedules of alternative reinforcement for the FCR and placed problem behavior on extinction in each context. In the final phase, they conducted an extinction challenge in each context with both problem behavior and the FCR on extinction. Four of the seven participants showed significant levels of resurgence, and, consistent with BMT, higher levels of resurgence occurred in the context associated with higher rates of reinforcement for problem behavior.

Equation 1 also predicts that implementing Phase 2 for longer durations (i.e., increasing the time in treatment) should reduce resurgence relative to shorter durations. Early basic (Leitenberg et al., 1975) and applied (Wacker et al., 2011) studies provided some support for this prediction. However, more recent investigations have failed to replicate these earlier findings (Greer et al., 2020; Nall et al., 2018; Winterbauer et al., 2013). For example, Greer et al. evaluated whether conducting three times as many Phase-2 (treatment) sessions in one stimulus context would reduce resurgence relative to the control condition conducted in a separate stimulus context. They observed similar levels of resurgence during Phase 3 in the two contexts, indicating that treatment duration had little impact on resurgence.

Empirical Results Opposing BMT Predictions

Results of other basic studies on resurgence have contradicted the predictions of BMT (Craig & Shahan, 2016; Fujimaki et al., 2015; Winterbauer & Bouton, 2010). For example, Craig and Shahan exposed two groups of rats to either high (variable-interval [VI] 15-s) or low (VI 60-s) rates of reinforcement in Phase 1 (or baseline); they then subdivided each of these groups into three subgroups (creating six groups total) and provided either high (VI 15 s), low (VI 60 s), or no alternative reinforcement during Phase 2 (or treatment); and in Phase 3, they terminated all reinforcer deliveries (i.e., an extinction challenge). Contrary to results typically found in studies using multiple-schedule designs and contrary to the predictions of BMT, using these single-schedule arrangements across groups, they found that: (a) target-response persistence was negatively correlated with baseline-reinforcement rates for the two groups that received no alternative reinforcement during Phase 2 (i.e., the two groups that received treatment with extinction alone), (b) alternative reinforcement combined with extinction of the target response did not consistently reduce target responding more than extinction of target responding alone in Phase 2, and (c) target responding during Phase 3 was roughly equal in the groups that received higher and lower rates of alternative reinforcement in Phase 2.

Nevin et al. (2017) described another study that challenged the predictions of BMT in which the authors evaluated the effects of higher and lower rates of reinforcement using the Pavlovian-to-instrumental transfer paradigm (Estes, 1948). With this test, a specific stimulus is first repeatedly paired with reinforcer deliveries (i.e., Pavlovian conditioning). Next, a target response that was not available during the prior Pavlovian conditioning produces the reinforcer in the absence of the stimulus (operant conditioning). Finally, the stimulus that was paired with reinforcement during Pavlovian condition is repeatedly introduced and withdrawn during extinction of the target response. Estes showed that presentation of the stimulus during extinction of the target response increased the rate of that response relative to the rates observed in the absence of the stimulus. In the study described by Nevin et al., pigeons received 120 reinforcers per hour on a variable-time (VT) schedule in the presence of one stimulus (e.g., red house light) and 30 reinforcers per hour in the presence of another stimulus (e.g., white house light) according to a multiple-schedule design. Next, in Phase 2, the investigators trained the pigeons to peck a lighted key in the absence of the house-light stimuli. Finally, in Phase 3, they reintroduced the alternating red and white houselights while discontinuing reinforcement of key pecking (extinction). If Pavlovian processes determine the persistence of responding, as predicted by BMT, then greater resistance to extinction should have been observed in the presence of the stimulus correlated with a higher rate of reinforcement, which was not the case.

These and other inconsistencies between the predictions of BMT and the results of empirical investigations led Nevin et al. (2017) to conclude that BMT is flawed with regard to its assertion that Pavlovian processes determine behavioral persistence and its account of resurgence.

Enduring Clinical Relevance of BMT

Nevertheless, BMT may be useful in many applications. For example, Fisher, Greer, Craig et al. (2018) analyzed the data from four of their studies involving 12 applications of BMT with individuals with autism and related disabilities and found that the BMT-informed treatment reduced resurgence more than the alternative treatment in all but one case. That is, they observed that arranging relatively leaner schedules of reinforcement for the target response during baseline or relatively leaner schedules of alternative reinforcement during treatment typically reduced resurgence. Similarly, Fisher, Greer, Fuhrman, et al. (2018) produced favorable outcomes when they combined three predictions of BMT into a treatment package by reducing the rate of reinforcement for problem behavior in Phase 1 (baseline), reducing the rate of reinforcement for the alternative response in Phase 2 (treatment), and extending the duration of treatment in Phase 2. This combination of procedures reduced resurgence in Phase 3 by at least 50% in every participant (M = 65.1% reduction). These results suggest that although BMT may be a flawed theory of behavioral persistence and resurgence, its basic concepts may still be useful for conceptualizing and refining differential-reinforcement treatments for problem behavior like FCT.

BMT Account of Renewal

Renewal is an increase in problem behavior resulting from a context change. Renewal is typically studied in a three-phase arrangement: (a) baseline, in which only the target response (e.g., problem behavior) produces reinforcement in Context A (e.g., baseline contingencies in the home); followed by (b) successful treatment until the target response reaches low or zero levels in Context B (e.g., FCT implemented in the clinic); followed by (c) implementation of the treatment in Context A (e.g., transferring treatment to the home) or in a new context, Context C (e.g., transferring treatment to the classroom). If the target response increases when the treatment is implemented in the original or new context, the increase is called renewal. If the target response increases when the treatment is transferred back to the original baseline context, it is referred to as ABA renewal; if it increases when the treatment is transferred to a new context, it is referred to as ABC renewal (Thomas et al., 2003).

Unlike resurgence, which also can be conceptualized as a treatment-integrity error, renewal occurs when the treatment is implemented with high procedural integrity. Thus, renewal can be conceptualized in terms of generalization decrement, with more discriminable differences between the original treatment context and the treatment-transfer context producing less generalization of the treatment effects and more renewal when the treatment is transferred. For example, if baseline occurs in the home, and treatment is initiated in a clinic, and then the treatment is transferred to the classroom, we expect higher levels of renewal (i.e., poorer treatment transfer) if the classroom is more like the home than the clinic and lower levels of renewal (i.e., better treatment transfer) if the classroom is more like the clinic than the home.

Nevin et al. (2017) described how Equation 1 above can be extended to account for renewal. Equation 2 shows this extension:

$$\frac{B_t}{B_0}={10}^{\frac{-t(c+dr+p{R}_a+d_{sb}-d_{sc})}{{(r+R_a)}^{0.5}}}$$ (2)

Equation 2 is identical to Equation 1 except for addition of the terms $d_{SB}$ and $d_{SC}$ to the numerator, where $d_{SB}$ represents the effective stimulus difference between Context A and Context B, and $d_{SC}$ represents the effective stimulus difference between Context B and Context C. It is important to note that $d_{SC}$ can be set to zero or removed from the equation when modeling ABA renewal. This equation has not received much attention in the research literature. However, Nevin et al. applied the equation to data collected in a prior study by Berry et al. (2014) that evaluated ABA and ABC renewal with pigeons and found that the equation accounted for 95% of the variance for ABA renewal and 98% of the variance for ABC renewal.

Applied Studies Demonstrating Treatment Relapse due to Renewal

Saini et al. (2018) conducted interview-informed synthesized contingency analysis of problem behavior (e.g., aggression, disruption) in the home with four children, ages 7 to 8, diagnosed with autism spectrum disorder and referred for the assessment and treatment of problem behavior. The investigators used the test condition of contingency analysis conducted by caregivers as the baseline or A phase of an ABA renewal analysis. Next, therapists implemented FCT in the clinic as the B phase of the analysis. After the B Phase, the investigators trained caregivers to implement FCT in the home using modeling, roleplay (with an adult confederate playing the role of the child), and feedback until they implemented FCT with 100% accuracy. Finally, they had the caregivers implement FCT in the home with the child participants as the second A phase of the ABA renewal analysis. For three of the four children, problem behavior increased to levels equivalent to or higher than those observed during baseline, even though the caregivers accurately implemented FCT in the final phase of the ABA renewal analysis. Ibañez et al. (2019) conducted a similar analysis with three children referred for the assessment and treatment of food refusal and inappropriate mealtime behavior and observed clinically significant renewal with all three children.

Applied Studies on Mitigating Renewal

Fisher, Greer, Fuhrman, and Querim (2015) introduced multiple-schedule FCT for the treatment of problem behavior across contexts (i.e., across therapists or rooms) in accordance with a multiple-baseline design. With multiple-schedule FCT, discriminative stimuli are used to signal to the participant when reinforcement for the FCR is and is not available. For example, the therapist might wear a lanyard with a card that is green on one side and red on the other and use the green side to signal when reinforcement is available and the red side to signal when reinforcement is unavailable. Many studies have shown that multiple-schedule FCT can facilitate reinforcement schedule thinning (e.g., Betz et al., 2013; Greer, Fisher, Saini, et al., 2016; Roane et al., 2004). Fisher et al. evaluated whether multiple-schedule FCT could facilitate the transfer of treatment effects across contexts without engendering renewal of problem behavior. Results showed that they introduced multiple-schedule FCT, the FCR rapidly came under discriminative control and problem behavior remained at near-zero rates (i.e., no renewal). However, Fisher et al. did not include a control condition (e.g., FCT without the discriminative stimuli), so the results suggested, but did not demonstrate, that the discriminative stimuli of the multiple-schedule FCT mitigated renewal of problem behavior.

Kelley et al. (2018) conducted baseline and treatment sessions for escape-maintained problem behavior for two participants in accordance with an ABA renewal arrangement and observed clinically significant renew each time. They then combined stimulus features of the A and B contexts (e.g., had the caregiver in the room while a therapist conducted escape extinction) and subsequently observed less renewal. However, the effects of this renewal-mitigation technique were confounded with treatment-exposure time. Thus, this renewal-mitigation strategy also requires further study. Finally, Haney et al. (2021) extended the findings of Kelley et al. (2018) by introducing a similar, but more elaborate mitigation technique (i.e., gradually fading the caregiver into the B-phase treatment context) with one treatment target (e.g., refusing to consume liquids) but not with a control target (e.g., food refusal). This renewal-mitigation procedure was effective with all four participants.

BMT Account of Reinstatement

Reinstatement is an increase in problem behavior after one or a few reinforcers are delivered either on a time-based schedule or contingent on the target response. Reinstatement is typically evaluated in a three-phase arrangement: (a) baseline, in which only the target response (e.g., problem behavior) produces reinforcement; followed by (b) extinction of the target response (either alone or combined with alternative reinforcement) until the target response reaches low or zero levels; followed by (c) delivery of a few reinforcers either contingent on the target response or on a time-based schedule. When the reinforcers are delivered contingent on the target response in Phase 3, the arrangement simulates treatment-integrity errors that often occur in natural human environments when a caregiver inadvertently delivers reinforcement following problem behavior. In the applied literature on treatment-integrity errors, such occurrences are often referred to as errors of commission (St. Peter Pipkin et al., 2010). When the reinforcers are delivered on a time-based schedule during Phase 3, the arrangement simulates a situation in which the delivery of reinforcement functions as a discriminative stimulus that occasions problem behavior and may counteract the generalization-decrement effects of omitting reinforcers during Phase 2 (Nevin et al., 2017).

Nevin et al. (2017) described how Equation 1 above can be extended to account for reinstatement. Equation 3 shows this extension:

$$\frac{B_t}{B_0}={10}^{\frac{-t(c+dr+p{R}_a-d_x{r}_x)}{{(r+R_a)}^{0.5}}}$$ (3)

Equation 3 is identical to Equation 1 except for addition of the term $d_x{r}_x$, which represents the decrease in generalization decrement from baseline reinforcement of the target response due to the discriminative properties of reinforcement. Nevin et al. applied Equation 3 to data from an undergraduate lab class in which pigeons pecked a key on a VI 30-s schedule in one component of a multiple schedule and on a VI 120-s schedule in the other component during Phase 1 (baseline). After exposure to extinction in both components during Phase 2, the pigeons received a few time-based reinforcer deliveries in Phase 3. Consistent with the model, the pigeons showed reinstatement in both components because the delivery of time-based reinforcers in Phase 3 lessened the generalization decrement by making the stimulus conditions more like Phase 1 (baseline) and less like Phase 2 (extinction). When they applied Equation 3 to the data, it accounted for 95% of the observed variance.

A small number of investigations have evaluated reinstatement with clinical populations (DeLeon et al., 2005; Liggett et al., 2018; Pritchard et al., 2014b), and only one examined relative momentum effects on reinstatement. In a clinical translation of reinstatement, Pritchard et al. conducted a functional analysis of problem behavior (aggression and disruption) with a 16-year-old male with severe intellectual disability; the results showed that contingent attention reinforced his problem behavior. During the functional-analysis baseline, which served as Phase 1 of their reinstatement evaluation, they delivered attention for problem behavior on independent VI 60-s schedules in two components of a multiple schedule, with each component correlated with a different therapist. Problem behavior averaged 8.3 responses per min in each baseline component. In Phase 2 (treatment), the therapists prompted the participant to request attention via a communication device and delivered attention contingently on a VI schedule if the participant used the device and on a VT schedule if he didn’t use the communication device. The therapist in one component delivered a higher rate of VI VT reinforcement (120 per hour) than the therapist in the other component (30 per hour). Problem behavior decreased to 1.9 responses per min in the component with a higher rate or reinforcement and to 2.7 responses per min in the component with the lower rate of reinforcement. In Phase 3, the therapists in each component again delivered reinforcement for problem behavior on independent VI 60-s schedules. In the component associated with a higher rate of reinforcement in Phase 2 (treatment), problem behavior increased to rates considerably higher (20.7 per min) in Phase 3 than those observed in Phase 1, whereas in the component associated with a lower rate of reinforcement in Phase 2, problem behavior increased to rates similar (7.9 per min) in Phase 3 to those observed in Phase 1.

Although Prichard et al. did not apply a BMT equation to their data, the results are generally consistent with BMT in that the component with a higher rate of reinforcement during Phase 2 (treatment) produced a much larger reinstatement effect in Phase 3. That is, according to BMT, problem behavior increased more during Phase 3 in the component associated with a higher rate of reinforcement in Phase 2 because the additional reinforcer deliveries in Phase 2 increased the momentum of problem behavior through Pavlovian contingencies. However, because no momentum effect was observed during the Pavlovian-to-instrumental paradigm described above (Nevin et al., 2017), it is doubtful that Pavlovian contingencies explain the larger reinstatement effect observed by Pritchard et al.. Thus, additional research is needed to explain these and other clinical findings that are consistent with the predictions of BMT but that cannot be explained as Pavlovian processes.

Stimulus Control Refinements of BMT

According to BMT, the effects of prior reinforcement on the persistence of behavior during periods of extinction are specific to the stimulus context in which an organism received the reinforcers for that behavior. Thus, if reinforcement for the target response is delivered in one stimulus context, responding will persist in that context and in similar contexts during extinction, but responding is likely to be less persistent in dissimilar contexts. This prediction, as well as the predictions of other theories of resurgence (e.g., context theory [Bouton et al., 2012], resurgence as choice [RaC; Shahan & Craig, 2017]), is consistent with the generalization-decrement hypothesis, which states that when we reinforce a response in the presence of a specific stimulus (S^D), the effects tend to generalize more to similar stimuli and less to dissimilar ones.

Stimulus-control research has supported the generalization-decrement hypothesis by showing that differential reinforcement in the presence of an S^D during training produces increased responding for the S^D and for physically similar stimuli through generalization (e.g., Terrace, 1963a; Terrace, 1963b; Podlesnik & Fleet, 2014). Extinction in the presence of a separate stimulus correlated with extinction (S^Δ) during training produces decreased responding for the S^Δ as well as for physically similar stimuli (e.g., Bell et al., 2008). The similarity between the training and test stimuli determines the degree of generalization from the training to the test stimuli. Thus, we can view resurgence as generalization of the effects of reinforcement from the baseline and treatment phases to the treatment-challenge phase (Fisher et al., 2020). Therefore, when the treatment-challenge conditions are more like the baseline conditions than the treatment conditions, we should see greater resurgence of target responding. From this perspective, programming the treatment-challenge conditions to be more like the treatment conditions than the baseline conditions should decrease the resurgence of target responding. The generalization-decrement hypothesis makes similar predictions regarding renewal and reinstatement of problem behavior.

Based on this prediction, Mace et al. (2010) evaluated the effects of FCT in the presence of contextual stimuli. Therapists taught the FCR in one stimulus context and then conducted extinction in a separate stimulus context in which problem behavior did not have a history of reinforcement. The procedure was effective but has limited applicability given we do not have control over the reinforcement history of problem behavior (e.g., most individuals referred for the assessment and treatment of problem behavior likely have a history of reinforcement for problem behavior in the home and/or school setting). Betz et al. (2013) used a similar, but perhaps more practical procedure in which they conducted FCT and established discriminative control of the FCR using a multiple schedule (called multiple-schedule FCT) consisting of short periods in which the FCR either produced reinforcement or was on extinction (with problem behavior on extinction throughout). A salient discriminative stimulus (S^D; brightly colored wristband) signaled the reinforcement components of the multiple schedule, and the absence of this stimulus (S^Δ) signaled that reinforcement was not available. Thus, these procedures established the FCR in a stimulus context that was distinct from the stimulus context in which problem behavior previously produced reinforcement. Furthermore, Betz et al. taught the FCR in a stimulus context in which problem behavior was correlated with extinction, both when the FCR produced reinforcement (in the presence of the S^D) and when it did not (in the presence of the S^Δ). When Betz et al. quickly lengthened the extinction component of the multiple schedule, they did not observe an increase in problem behavior (i.e., no resurgence). These results suggest that the discriminative control provided by multiple schedules may mitigate resurgence of problem behavior when the FCR contacts a long period of extinction. However, Betz et al. alternated between the reinforcement and extinction components during multiple-schedule FCT, and it is unclear if they would have observed low levels of resurgence had they conducted a more traditional test for resurgence (e.g., consecutive sessions of extinction).

To extend the findings of Betz et al. (2013), Fuhrman et al. (2016) evaluated the resurgence-mitigating effects of multiple-schedule FCT with discriminative stimuli relative to traditional FCT (traditional FCT) without discriminative stimuli. Fuhrman et al. used a modified ABAB design in which each A and B condition included the standard three-phase resurgence design (i.e., baseline, treatment, extinction). In one condition (e.g., A) the researchers conducted a baseline followed by multiple-schedule FCT with schedule thinning, followed by an extinction phase. In the other condition (e.g., B) they conducted an identical baseline followed by traditional FCT (with no discriminative stimuli present), followed by an extinction phase. During the extinction phase that followed multiple-schedule FCT, the researchers signaled the absence of reinforcement for the FCR with the (S^Δ) they used in multiple-schedule FCT. By contrast, during the extinction phase that followed traditional FCT, the researchers did not include any stimuli that signaled the presence or absence of reinforcement (i.e., no S^Δ present). Findings revealed lower levels of problem behavior in the extinction phases that included the S^Δ and followed multiple-schedule FCT relative to the extinction phases that followed traditional FCT that did not include the S^Δ. Although the results were promising, procedural limitations (e.g., differing lengths of exposure to treatment and differing reinforcement rates during multiple-schedule FCT and traditional FCT) impact our ability to determine whether the S^Δ exerted stimulus control over problem behavior and resulted in the lower levels of resurgence.

Fisher et al. (2020) addressed the limitations of Fuhrman et al. (2016) to isolate the resurgence-mitigating effects of discriminative stimuli. They used a multielement design and three unique stimulus contexts to control for exposure to treatment and reinforcement rates. First, they conducted multiple-schedule FCT pretraining and thinned the schedule of reinforcement in one stimulus context. Next, they conducted a resurgence evaluation (baseline, FCT, extinction) in two novel contexts. In one of those contexts, they programmed the S^D to be present during FCT and the S^Δ to be present during extinction. In the other context, they excluded the discriminative stimuli in all three phases. Results showed 84% less resurgence of problem behavior in the context that included the S^Δ during the extinction phase relative to the extinction phase without the S^Δ. They hypothesized that the inclusion of the S^Δ during the extinction phase rendered that phase more like the extinction component of the mult-FCT pretraining phase, during which they programmed extinction for both problem behavior and the FCR and observed relatively low levels of both behaviors. These findings are consistent with the generalization-decrement hypothesis and the predictions of BMT and provide further empirical support for the use of multiple-schedule FCT in the treatment of problem behavior.

The results of these studies and other applied investigations have demonstrated the clinical utility of BMT in conceptualizing possible resurgence-mitigation techniques (e.g., Brown et al., 2020; Fisher, Greer, Craig et al., 2018; Wacker et al., 2011). However, as discussed above, many research findings that have directly tested the predictions of BMT have disconfirmed rather than supported the theory. In addition, most of the findings from applied studies that have supported the predictions of BMT have produced inconsistent findings across participants or small to moderate treatment effects (e.g., Fisher et al., 2019). The one clear exception to this conclusion involves the use of multiple-schedule FCT, which has produced clinically significant reductions in resurgence (Fuhrman et al., 2016; Fisher et al., 2020) and has facilitated the transfer of treatment effects across contexts with little or no renewal of problem behavior (Fisher, Greer, Fuhrman, & Querim, 2015). It is worth noting that the results of both basic and applied research support a key role for multiple schedules with respect to BMT findings (cf. Shahan & Craig, 2017).

In addition, limitations of the theory likely outweigh the benefit of continuing to investigate its use as a guide for refining current standard-of-care procedures. For example, the fact that BMT cannot account for some variables that are key components of standard-of-care treatments (e.g., effort of responses, quality of reinforcement) limits its applicability to clinical practice. Although BMT inspired a large amount of basic, translational, and applied investigations on resurgence and collaboration between basic and applied researchers, its shortcomings could lead to predictions that are inaccurate or countertherapeutic in applied settings.

Temporally Weighted Matching Law

The empirical and theoretical limitations of BMT as a model of resurgence led to an alternative conceptualization of resurgence, one based on the concatenated matching law (Baum & Rachlin, 1969), which has begun to receive considerable attention. The concatenated matching law predicts that behavior across two response alternatives (i.e., $B_1$ and $B_2$) will be allocated as a function of the relative values of reinforcement obtained by engaging in those response alternatives (i.e., $V_1$ and $V_2$) and that reinforcer value is determined by the product of individual reinforcement parameters such that

$$\frac{B_1}{B_1+B_2}=\frac{V_1}{V_1+V_2}=\frac{R_1{A}_1{I}_1}{R_1{A}_1{I}_1+R_2{A}_2{I}_2}$$ (4)

where $R_1$, $A_1$, and $I_1$ are the rates, amounts, and immediacies of reinforcement for $B_1$, and $R_2$, $A_2$, and $I_2$ are the same reinforcement parameters for $B_2$. Restating Equation 1 in terms of target and alternative behavior and rewriting the equation as the probability of the target response, given that either response occurs, produces

$$pT=\frac{V_T}{V_T+V_{Alt}}$$ (5)

When applying the concatenated matching law to extinction conditions, one quickly realizes that Equations 1 and 2 become unsolvable when reinforcers are unavailable (i.e., $V_T=0$ and $V_{alt}=0$) because dividing any numerator by 0 produces an undefined value. This inability to apply the concatenated matching law to periods of extinction is particularly problematic when modeling resurgence (i.e., when the functional reinforcer is withheld for problem and alternative behavior) and when simulating conditions in which extinction is the sole treatment component. The TWML¹ (Shahan & Craig, 2017; Shahan, under review) addresses this problem by infusing the concatenated matching law with a set of equations that effectively carry forward in time a history of reinforcement provided by each response alternative using an approach called the temporal weighting rule.

The temporal weighting rule (Devenport & Devenport, 1994) is the second major component of the TWML, and as suggested above, it carries forward in time a historical record of reinforcement for the response under consideration such that the effects of that history can continue to exert control over current responding. When incorporated within a choice framework, the temporal weighting rule predicts the relative probability of one response occurring over another under periods of transition (i.e., before responding has stabilized to the contingencies currently in effect), which is necessary when modeling resurgence and extinction bursts.

The temporal weighting rule assigns a weighting ($w_x$) to each experience in the organism’s history (e.g., one $w_x$ for Session 1, another for Session 2, and so forth), which allows for individual experiences to be weighted differentially. The temporal weighting rule weights recent events more heavily than temporally distal ones according to a hyperbolic-decay function, meaning that as the historical record of reinforcer deliveries recedes into the past, weights assigned to those associated experiences decline rather quickly at first but more slowly as time recedes further into the past. Also, although weights may approach 0, they never equal 0, either. Thus, their effects continue to affect behavior forever but do so with ever smaller degrees as the historical record stretches further and further back in time. Finally, the sum of all the weightings under consideration always equals 1, no matter the length of that history. Therefore, the shape of the weighting function changes with different lengths of historical records under consideration (see Figure 1 in Greer & Shahan, 2019, for example weighting functions). Quantitatively, the temporal weighting rule can be written as

$$w_x=\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$t_x$}\right.}{{\sum }_{i=1}^n\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$t_i$}\right.}$$ (6)

where $t_x$ is the time since a given experience and the present time, and the numerator is the recency from a past event or session.² That 1 is divided by $t_x$ in the numerator implies that shorter recencies will be weighted more heavily (i.e., increase $w_x$) and larger recencies will be weighted less heavily (i.e., decrease $w_x$), as discussed above. The denominator of Equation 3 sums these recencies, $n$ of which are currently under consideration.³

Transforming these weightings into values involves multiplying the weighting assigned to each session by the rate of reinforcement obtained for responding during that session, such that

$$V_T=\sum _x{w}_x{R}_{xT}{V}_{Alt}=\sum _x{w}_x{R}_{xAlt}$$ (7)

where all terms are as above for the session in question, $R_{xT}$ is the corresponding rate of reinforcement obtained for the target response, and $R_{xAlt}$ is the corresponding rate of reinforcement obtained for the alternative response.

A TWML Account of Resurgence

Applying the TWML to resurgence pulls together $V_T$ and $V_{Alt}$ and introduces parameters to scale target response rates in baseline ($k$), account for systematic bias for the target or the alternative response ($b$), and factor in the arousing effects of reinforcement ($A$). Formally, the TWML as applied to resurgence can be written

$$B_T=\frac{k{V}_T}{V_T+\raisebox{1ex}{$V_{Alt}$}\!\left/ \!\raisebox{-1ex}{$b$}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$A$}\right.}{B}_{Alt}=\frac{k\raisebox{1ex}{$V_{Alt}$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}{V_T+\raisebox{1ex}{$V_{Alt}$}\!\left/ \!\raisebox{-1ex}{$b$}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$A$}\right.}A=a(V_T+V_{Alt})$$ (8)

where $B_T$ is the absolute rate of the target response, $B_{Alt}$ is the absolute rate of the alternative response, and $a$ scales motivation for the functional reinforcer maintaining problem behavior. Elaboration on each of these parameters and their interaction is provided by Shahan and Craig (2017) and Greer and Shahan (2019). Like BMT, RaC and the TWML (described subsequently) treat alternative reinforcement the same whether it is delivered contingent on an alternative response (e.g., FCT) or on a time-based schedule (e.g., NCR).

Extension of the TWML to Account for Discrimination Processes

Shahan, Browning, and Nall (2020) recently extended the TWML to account for discrimination processes that have been shown to affect resurgence. In that study, they exposed groups of rats to differing durations of time in Phase 2 during an otherwise typical resurgence progression. Starting with the introduction of Phase 2, Phase 3 began on Day 4, 8, 16, 24, or 32 across groups. However, a sixth group of rats experienced alternating sessions in which alternative reinforcement was or was not available throughout Phase 2 (i.e., On/Off group). For this group, the first session of Phase 2 arranged reinforcement for the alternative response, whereas the second session did not. This alternation of alternative reinforcement being available and then not continued for this group until Phase 3 began on Day 32. Thus, every other session in Phase 2 for this group probed the extinction condition that later would be in effect continuously during Phase 3, and it allowed the researchers to compare levels of resurgence for this group at multiple timepoints relative to the groups that had constant alternative reinforcement in Phase 2 (i.e., on Days 4, 8, 16, 24, or 32). What the researchers found was that this unique history of alternative reinforcement availability in Phase 2 for the On/Off group mitigated resurgence at each comparable timepoint relative to the groups that had constant alternative reinforcement in Phase 2. In addition, as predicted by the TWML, more time in Phase 2 also helped mitigate resurgence for those groups that had alternative reinforcement constantly available in Phase 2.

Although the TWML fit the obtained resurgence data quite well for the groups that had constant alternative reinforcement in Phase 2 of Shahan, Browning, and Nall (2020), the model was unable to account for the resurgence data from the On/Off group. That is, the TWML had no way of addressing why such an experience in Phase 2 ought to mitigate resurgence as compared to the groups that had constant alternative reinforcement in Phase 2. To rectify this issue, the authors expanded the TWML to account for discrimination processes believed to affect the On/Off group, specifically. In essence, the authors proposed that the rats in the On/Off group learned that the unavailability of alternative reinforcement during “off” days did not mean that reinforcement was available for the target response, a learning history unlikely to have been acquired by any of the groups with constant alternative reinforcement in Phase 2. Therefore, the authors expanded the TWML to account for such a learning history (i.e., discrimination training) and did so by conceptualizing it as a growing bias away from the target response. The expanded TWML provided an excellent description of the resurgence data from all groups, including the On/Off group.

Extension of the TWML to Account for Extinction Bursts

More recently, Shahan (under review) reconsidered the extinction burst from the perspective of the TWML and, in the process, expanded the TWML in another important way. Like the missing ability to account for discrimination processes in the original incarnation of the TWML, the TWML also did not account for why target responding sometimes increases above baseline rates upon terminating dense schedules of reinforcement maintaining that response (i.e., an extinction burst). In looking at this issue closer, Shahan reanalyzed the extinction-burst data from Nist and Shahan (2021) and reasoned that consumption of or engagement with the reinforcer itself during baseline (i.e., reinforcement-related activity; defined here as approaching, collecting, and consuming the reinforcer) constitutes an alternative response, distinct from engaging in the target response to produce that reinforcer. In fact, with dense schedules of baseline reinforcement (e.g., FR 1), individuals often spend most of the overall session time engaged with the reinforcer, especially when the reinforcer is available for 20 s to 30 s or more during each delivery (cf. Fisher, Greer, Fuhrman, et al., 2018).

With a 30-s reinforcement interval (e.g., 30-s access to an iPad), an efficient clinical-research participant typically might display problem behavior in the first 3 to 4 seconds of the session and then spend the next 30 s in reinforcement-related activity. When the first reinforcement interval ends, the participant would again quickly display problem behavior and then spend the next 30 s in reinforcement-related activity, and this pattern would continue throughout the session (cf. Figure 1 in Fisher et al., in revision). Thus, over the course of a baseline session, an efficient participant would spend about 90% of baseline session time in reinforcement-related activity and 10% emitting problem behavior. When extinction is introduced (without alternative reinforcement), reinforcement-related activity is no longer available to compete with problem behavior and the individual reallocates their responding and displays more problem behavior, thereby producing the extinction burst.

Fisher et al. (in revision) observed that researchers are more likely to detect extinction bursts if reinforcement intervals are included when calculating baseline response rates. For example, in the example above, if the participant emitted 9 problem behaviors in a 5-min session and spent 270 s in reinforcement-related activity, the baseline response rate would be 1.8 responses per min (9 / 5 min) when including reinforcement intervals in the calculation and 18 response per min (9 / .5) min when excluding the 270 s of reinforcement intervals. If the participant then displayed 18 responses or 3.6 responses per min during the first session of extinction, this would be considered an extinction burst if one included reinforcement-interval time when calculating the baseline response rate (i.e., an increase from 1.8 to 3.6 responses per min) but not if one excluded reinforcement-interval time (i.e., a decrease from 18 to 3.6 responses per min).

Following this logic, Shahan (under review) noted that baseline response rates are suppressed because the organism must discontinue engagement with the target response to approach the reinforcer, collect it, consume it, and then return to the target response. Denser schedules of reinforcement, therefore, imply more disruption of target responding because of such reinforcement-related activity, which helps to account at least conceptually for why extinction bursts are more common following dense schedules of reinforcement for the target response in baseline (e.g., Nist & Shahan, 2021).

Shahan (under review) argued that reinforcement-related activity should be treated as a formal source of disruption for target responding within the TWML. Introducing this concept quantitatively allowed the TWML to predict increases in target responding above baseline rates because the disruption imposed by reinforcement-related activity in baseline is no longer present during extinction. Therefore, target responding, now unencumbered by such reinforcement-related activity, increases above baseline rates during extinction. The TWML accounts for reinforcement-related activity by treating it as an alternative response. That is, in baseline the individual has two response options: emitting the target response, which is available throughout a baseline session, and engaging in reinforcement-related activity, which is only available during reinforcement intervals. By treating reinforcement-related activity as an alternative response option in the TWML and applying the equation to the data from Nist and Shahan (2021), Shahan (under review) showed that the revised model can account for the extinction burst. This promising extension of the TWML offers new insights into potential mitigation techniques for yet another phenomenon that may compromise the success of function-based treatments in the clinic.

Basic Research Supporting/Refuting the TWML

The TWML has evolved to account for important conceptual and applied scenarios, meaning that support for TWML can be derived from several studies examining its preceding models (i.e., RaC; Shahan & Craig, 2017; RaC in Context; Shahan, Browning, & Nall, 2020). First, the previously described study by Shahan, Browning, and Nall (2020) on treatment duration and on/off cycling of reinforcement posed interesting challenges that a revised TWML could accommodate. Although BMT failed to predict the study’s findings (R² = .48), modifying the initial TWML equations to address discrimination processes facilitated a more robust account of the obtained data (increasing TWML’s R² from .56 to .92).

Second, this revised version of TWML that accounted for discrimination processes has been able to predict target behavior during procedures that extend beyond a typical resurgence arrangement. In a traditional resurgence test, alternative reinforcement is often suspended entirely (i.e., extinction), like the transitions from “On” periods to “Off” periods in Shahan, Browning, and Nall (2020). However, resurgence can occur during more subtle downshifts in reinforcement that do not involve a complete suspension of alternative reinforcement (Lattal et al., 2017).

In Shahan, Browning, Nist, and Sutton (2020), the authors examined resurgence following decrements in alternative reinforcement, including periods of extinction or simply leaner reinforcement schedules. In this study, the experimenters reinforced rats’ target lever presses according to a VI 30-s schedule before placing these lever presses on extinction and reinforcing alternative lever presses on a VI 10-s schedule. Then the experimenters examined target responding under the following reinforcement schedules for the alternative response: (a) VI 10-s schedule, (b) VI 20-s schedule, (c) VI 40-s schedule, (d) VI 80-s schedule, or (e) extinction. The researchers found that the magnitude of resurgence increased exponentially as alternative reinforcement decreased, with the largest increase in target responding corresponding to the leanest reinforcement schedules. Interestingly, the authors also found that downshifts in reinforcement conditions that involved some reinforcement of alternative behavior (i.e., VI 40 s, VI 80 s) resulted in persistent target behavior throughout the resurgence test above and beyond the patterns exhibited by rats who experienced a transition to extinction.

As the researchers noted, this paradoxical finding of low alternative-reinforcement conditions engendering more target responding than extinction is one that the TWML can predict whereas BMT does not. That is, the TWML predicts the current rate of target responding based on its current relative value, which is determined by the histories of reinforcement for the target and alternative response. This feature of TWML allows the model to predict that with different histories, alternative reinforcement can sometimes produce suppression and at other time produce persistence of target responding (see Figure 5 in Shahan, Browning, Nist, & Sutton, 2020). By contrast, BMT predicts that alternative reinforcement always suppresses target responding when it is present, regardless of the histories of reinforcement for the target and alternative response. The version of the TWML that incorporated discrimination processes related to reinforcer deliveries was an excellent predictor of responding (R² = .90 and .91 for male and female rats, respectively), including predictions of alternative behavior across experimental phases.

Third, the TWML has been used to successfully predict the effects of differential exposure to baseline and treatment conditions on resurgence. Specifically, the TWML predicts that resurgence will increase with longer baseline durations (i.e., lengthy reinforcement histories for target behavior) and resurgence will decrease with longer treatment durations (i.e., lengthy reinforcement histories for alternative behavior). Smith and Greer (2022) evaluated these predictions using a human-operant arrangement with Amazon Mechanical Turk workers, which arranged different contingencies across experimental phases when clicking response buttons and solving math problems. Responding in this way to a target button (e.g., the blue button) resulted in point delivery during baseline, with points dictating the amount of money earned (up to $6) at the conclusion of the experiment. Then, during the treatment phase, the experimenters extinguished responses to the target button and delivered points for responses to the alternative button (e.g., the yellow button). Finally, they placed both responses on extinction during the resurgence test. The experimenters assigned participants to one of four conditions that varied the duration of baseline (5 min or 20 min) and treatment (5 min or 20 min): (a) short baseline and short treatment, (b) short baseline and long treatment, (c) long baseline and short treatment, and (d) long baseline and long treatment. Smith and Greer observed less resurgence following longer treatment phases and more resurgence following longer baseline phases, though only the first finding reached statistical significance. When fitting the TWML to the obtained data, the authors found that the TWML provided an excellent account of the findings (R² ≥ .90 across groups).

Fourth, and as described earlier, Shahan (under review) recently examined whether the TWML could account for the conditions under which extinction bursts occur. Although extinction bursts can be challenging, particularly in applied situations when the target behavior poses risks to the individual or others, prevalence estimates of extinction bursts vary widely in clinical studies, ranging from about 15% of applications when extinction is implemented with other procedures (e.g., differential reinforcement) to about 62% of applications when extinction is implemented alone (e.g., Lerman & Iwata, 1995; Lerman et al., 1999; Woods & Borrero, 2019). Further, these bursts tend to resolve quickly after continued exposure to extinction (e.g., Nist & Shahan, 2021; Woods & Borrero, 2019). Thus, a quantitative theory that accounts for the extinction burst could improve both basic and clinical understanding of this complex phenomenon and lead to innovative clinical procedures for preventing or mitigating such bursts.

Clinical Translations of the TWML

The basic research noted above has provided a better understanding of phenomena that has immediate relevance to applied researchers and clinicians, particularly those providing assessment and treatment services for severe problem behavior. For example, recall the study by Shahan, Brown, Nist, and Sutton (2020) in which they observed that resurgence occurred during a worsening in reinforcement conditions even when alternative reinforcement was still available (e.g., transitioning reinforcement from a VI 10-s schedule to a VI 80-s schedule). Although this paper is important for conceptual reasons, it approximated the type of clinical transitions that occur routinely when thinning reinforcement during FCT, such as increasing periods of nonreinforcement for alternative behavior from 2 s to 5 s or 10 s (e.g., Greer, Fisher, Saini, et al., 2016). Several clinics that specialize in the assessment and treatment of severe problem behavior have used record-review procedures to detect the susceptibility of problem behavior to relapse during these increasing periods of nonreinforcement for alternative behavior (Briggs et al., 2018; Mitteer et al., in press; Muething et al., 2021). In these studies, the researchers have reviewed cases meeting certain inclusion criteria (e.g., high interobserver agreement, use of a single-case experimental design) and examined the rates of problem behavior before and after thinning reinforcement in the absence of other important relapse variables (e.g., contextual changes, treatment-integrity errors). Both Briggs et al. and Mitteer et al. detected resurgence in over three-fourths of FCT administrations, and Muething et al. found an even higher prevalence in their sample (i.e., 90% of FCT cases with resurgence). Thus, the TWML’s ability to predict target responding of non-human animals during similar downshifts in reinforcement might prove useful in understanding variables contributing to resurgence of clinically important target behavior during gradual schedule thinning.

Accordingly, Shahan and Greer (2021) used the TWML to analyze the conditions under which resurgence occurred during the two most recent retrospective studies of relapse during routine FCT schedule thinning (Mitteer et al., in press; Muething et al., 2021). Recall that the study by Shahan, Browning, Nist, and Sutton (2020) found that the magnitude of resurgence increased exponentially as a function of the decrement in reinforcement for the alternative behavior, with initially higher rates of target responding during the resurgence test in leaner schedules (e.g., VI 40 s, VI 80 s, extinction) than in denser schedules (e.g., VI 10-s maintenance sessions, VI 20 s). The TWML provided an excellent account of these findings in a basic preparation with non-human animals but also for the applied studies of resurgence with children experiencing treatment of their severe problem behavior. Specifically, Shahan and Greer found that resurgence of problem behavior during routine FCT schedule thinning increases in magnitude as a function of the decrement in alternative reinforcement. As the authors noted, “The similarity of findings across these studies [Shahan & Greer, 2021; Shahan, Browning, Nist, & Sutton, 2020] is noteworthy given the extensive differences between them, including the species, procedures employed, and nature of the measures used” (p. 246). Taken together, the findings of the basic and applied studies of resurgence suggest that the TWML provides a remarkable account of resurgence and may be helpful in informing future relapse-mitigation strategies.

However, other applied studies based on the predictions of the TWML are currently limited. Though described earlier in our discussion of BMT, the study on time in extinction by Greer et al. (2020) also failed to support the predictions of the TWML that extending the treatment phase (i.e., increasing the exposure of target behavior to extinction) would minimize resurgence. This is at odds with the translational study by Smith and Greer (2022) that found significant effects with this independent variable. Nevertheless, the basic research on TWML suggests a few areas that might be explored in the clinical setting.

The finding that cycling alternative reinforcement on and off can decrease resurgence (Shahan, Browning, & Nall, 2020) is in line with some applied work that incorporates intermittent periods of extinction to bolster treatment maintenance (e.g., Wacker et al., 2011). We now call this procedure contingency discrimination training (CDT) because it teaches the individual to discriminate when reinforcement is and is not available for the alternative response based solely on whether reinforcement predictably follows that response. Further research on CDT may be worth examining to better prepare individuals for periods of extinction that resemble traditional resurgence tests (e.g., when a reinforcer like the iPad is inaccessible while charging). In particular, the version of the TWML that incorporates discrimination processes and includes alternative behavior could be a valuable tool for researchers and practitioners using FCT with discriminative stimuli (e.g., Greer, Fisher, Saini, et al., 2016), where discrimination training is used to teach individuals when alternative reinforcement is or is not available.

It is worth noting that CDT teaches a different type of discrimination than that taught using multiple and chained schedules (Greer, Fisher, Saini, et al., 2016; Saini et al., 2016). Alternating periods of reinforcer availability and unavailability for the alternative response teaches the child about the underlying contingencies in place, not just for the alternative response, but also for problem behavior. Each transition from reinforcement availability to unavailability for the alternative response provides an opportunity for the individual to learn that the absence of reinforcement for the alternative response does not mean that problem behavior will again produce reinforcement (i.e., learning that the alternative response alone produces reinforcement but not always). This is a critical discrimination that patients may fail to learn under dense multiple and chained schedules of reinforcement, and this lack of learning becomes evident when caregivers fail to implement the treatment procedures precisely (e.g., not reinforcing the individual’s FCR when preoccupied with an infant child). Failing to learn this critical discrimination manifests as resurgence of problem behavior, estimated to have a prevalence as high as in 90% of patients and in 76% of FCT treatments with discriminative stimuli (Brigg et al., 2018; Kranak & Falligant, 2021; Mitteer et al., in press; Muething et al., 2021).

CDT avoids these pitfalls by teaching this critical discrimination explicitly and quickly. After having learned this important discrimination, other individually tailored procedures may be used to increase the probability of success post-discharge (e.g., further reinforcement schedule thinning, additional discrimination and generalization training). In sum, the TWML provides several fruitful avenues for future applied research even though the model is relatively new. We refer interested readers to Greer and Shahan (2019), which provides an extensive overview of some of the practical implications of the TWML to clinical research and practice. However, it is worth considering the wealth of data on other quantitative theories like BMT and how these predictions overlap with the TWML. Being mindful of reinforcement rates for target behavior during baseline, considering reinforcement rates and transitions during treatment, and acknowledging the importance of stimulus control in enhancing the durability of treatment are common threads among BMT and TWML. Although the models are by no means perfect, researchers and practitioners can lean on these quantitative theories to help inform treatment progressions and explore new methods of assisting individuals with problem behavior and their families, with the goal of minimizing the occurrence of such behavior during and following treatment.

Summary Comments and Directions for Future Research

According to Box (1976), all quantitative models are wrong (or imperfect representations of the phenomena they model), but some are useful (in that the illuminate important aspects of the phenomena). As we have described in this paper, basic research on BMT has shown that Pavlovian processes cannot account for behavioral persistence and that several of the key predictions of BMT about resurgence have been disconfirmed. By contrast, clinical research by our lab and others has shown that BMT may be useful for developing techniques for relapse prevention and mitigation. Although these two statements seem incompatible, they may not be. Basic research is often interested in testing and comparing quantitative theories of behavior, and as such, basic researchers look to set up experiments in which the alternative theories produce divergent predictions (to see which theory is more accurate). By contrast, clinical researchers look to test whether the predictions of quantitative theories can help us improve our clinical assessments and interventions, and we may have more success when we set up experiments in which alternative theories produce convergent predictions. For example, in our lab we have had the most success with mitigating resurgence using the discriminative stimuli of multiple-schedule FCT to signal periods during which neither problem behavior nor the FCR produces reinforcement (Fisher et al., 2020; Fuhrman et al., 2016). Our results in this area are consistent with the generalization-decrement hypothesis, which has been incorporated into both BMT and the TWML, and also context theory (Bouton et al., 2012)). Similarly, one of the important refinements to the TWML discussed above that may have important clinical implications (i.e., addressing discriminative processes) was based on context theory (Shahan et al., 2020). Thus, it may be useful for clinical researchers to look for and to study areas where different quantitative theories of behavior suggest common clinical refinements and to test the effects of those clinical enhancements before examining ones that are suggested by just one theory of behavior.