Why FCT Actually Works: The Contingency Strength Math

A +1 mand stacked against a -1 problem behavior gives you a 2.0 contingency strength gap, and that gap is the whole reason Functional Communication Training (FCT) works. Most Board Certified Behavior Analysts (BCBAs) learn FCT as a procedure: teach a Functional Communication Response (FCR), put the problem behavior on extinction, prompt the mand, fade prompts. That story is fine. It is also missing the math. If you can see FCT as two contingency strengths sitting side by side, you can predict whether your plan will work before you ever run it.

This page walks through that math in plain language. It is built from a CEU called "The Math Behind Behavior Reduction" by Matt Harrington. Watch the full session for the worked examples on the screen.

The two contingency strengths inside every FCT plan#

Every FCT plan has two behaviors competing for the same reinforcer. The mand. The problem behavior. Each one has its own contingency strength.

Contingency strength is a number between +1 and -1. It tells you the chance that a behavior produces the reinforcer, minus the chance the reinforcer shows up when the behavior does not happen. High strength means the behavior reliably gets paid. Low or negative strength means the behavior rarely or never gets paid.

When you write an FCT plan, you are setting up two of these numbers at once. The mand should be high. The problem behavior should be low. The bigger the gap, the more the learner picks the mand. That is the whole engine.

If you only think about FCT as "teach a mand and prompt it," you can miss the point. The prompts and the errorless teaching matter for skill acquisition. The reduction in problem behavior is driven by the gap.

The mand side: FR1 and a strength of +1.0#

In a clean FCT plan, the mand is on a Fixed Ratio 1 schedule. One mand, one reinforcer. Every time. That puts the mand all the way at the top of the scale.

The mand is on an FR1 schedule. Right? So, every time a mand appears, reinforcement closely follows. Every time a mand, they get reinforced. Therefore, the contingency strength is all the way up at one. From the talk — Matt Harrington

A strength of +1.0 means the learner can count on the response. If they ask, they get. There is no guessing. There is no waiting. That is what makes FR1 the right starting place for FCT. You want the mand to feel like a sure thing.

If your team can only run FR1 part of the time, the mand strength drops below 1.0. That gap shrinks. We will get to that.

The problem behavior side: extinction and a strength of -1.0#

The other half of FCT puts the problem behavior on extinction. Hits, drops, screams, head bangs. Whatever the topography. The learner does the behavior. Nothing changes in the environment. The demand stays. The toy does not return. The adult does not react.

When a behavior never produces the reinforcer, its contingency strength drops to -1.0. Some people call that a null, because dividing by zero is messy. The point is the same. From the learner's view, this behavior does not work anymore.

Now the learner is sitting in a room with two options.

if the learner has two options, they can either mand and have a, basically, 100% chance to get what they want, or they can have a severe behavior, and they have a 0% chance, or maybe even a negative percent chance to get what they want. From the talk — Matt Harrington

A +1.0 next to a -1.0. The math picks the winner. The learner picks the mand.

Why a 2.0 gap predicts behavior change#

The distance between +1.0 and -1.0 is 2.0. That is the biggest gap you can build with one mand and one problem behavior. It is also the reason FCT shows up in almost every behavior plan you read.

FCT works not because of, I would argue, not because of the prompts, not because of the stimulus, not because of the errorless learning, or the least to most. At its core, FCT works because of the mand, and a super low contingency strength for the severe behavior. From the talk — Matt Harrington

The prompts are scaffolding for the new skill. The errorless teaching keeps frustration low. But the reason the problem behavior drops is the gap. You moved the only working response from the problem behavior to the mand. The learner notices fast.

This is also why you should not over-credit any one component when FCT works. A team that uses sloppy prompts but holds extinction tight will still see reductions. A team that prompts beautifully but lets the problem behavior get paid once in a while will not. The gap is doing the work.

That's why it is the most common intervention, I would argue, in behavior analysis, because of that massive difference. From the talk — Matt Harrington

What happens when fidelity slips and the gap closes#

Here is the part most BCBAs skip when they write a plan. The +1.0 vs. -1.0 picture is true at 100% fidelity. Real sessions are not that clean.

Say the Registered Behavior Technician (RBT) misses a mand because they were charting. The mand was good. It did not get paid. The mand strength drops below 1.0.

Say the parent gives in once during a 20 minute tantrum on day three. The problem behavior gets paid one time. Its strength jumps off the floor.

Now the picture is something like a 0.7 mand against a 0.3 problem behavior. Still a positive gap. Much smaller. The learner is doing math too. A 0.7 sure thing vs. a 0.3 long shot still favors the mand, but the problem behavior is back on the menu. You will see it test the line.

A real example from the talk shows what this looks like in a contingency space spreadsheet. With a few problem behaviors paid and a few mands missed, the mand and the problem behavior can both end up at 0.5. A 0.5 vs. 0.5 plan does not reduce anything. The learner flips a coin. That is what fidelity drift feels like in numbers.

When a case stalls, the right question is not "are we doing FCT correctly?" The right question is "what does the gap look like right now?" If the gap closed, find the leak. Usually it is one of three things: a missed mand, a paid problem behavior, or a slow prompt that lets the problem behavior win the race.

Modifying FCT when extinction isn't safe#

Some problem behaviors cannot go on full extinction. Head banging on concrete. Aggression toward a younger sibling in a small house. Elopement near a road. The risk of an extinction burst is too high, or the family cannot hold the line.

You can still build the gap. You just need to move the other variables.

One option is to shrink the problem behavior's reinforcement schedule without going to zero. If the problem behavior used to get paid every time, drop it to once every few events. That moves its strength from +1.0 down toward 0.0. Not as good as -1.0, but the gap can still favor the mand.

Another option is to add a precursor as a second reinforced response. Now the learner has three options. Mand at +1.0. Precursor at something high, like +0.8. Problem behavior at +0.1. The learner has two ways to get paid before they ever reach the problem behavior. That is the math behind the enhanced choice model and a stripped down Skill Based Treatment (SBT) plan.

You can also shorten the response class. If the precursor is "say no," you can teach the team to honor that "no" instantly. The precursor gets a +1.0. The problem behavior never has time to start.

The point is not to memorize a new procedure. The point is that the gap is the tool. When extinction is off the table, you build the gap with whatever variables you have left.

Frequently asked questions#

What contingency strength gap is enough for FCT to work?

A clean FCT plan at 100% fidelity gives you about 2.0. In practice, a gap of around 0.5 or higher tends to drive change, as long as the mand is the stronger side. Below that, the learner is choosing between two roughly equal options, and the problem behavior keeps showing up. If your gap is under 0.5, find the fidelity leak before adding new components.

Does FCT still work if you can't put the problem behavior on full extinction?

Yes, if you can still build a gap. Drop the problem behavior's pay rate as much as is safe. Stack the mand and a precursor on the reinforced side. The math does not care whether the problem behavior is at -1.0 or +0.1, as long as the mand is meaningfully higher. The plan is harder to keep clean and the gains come slower, but the mechanism is the same.

Why do some kids return to problem behavior after FCT looked successful?

The gap closed. Something in the environment paid the problem behavior again. A new caregiver, a new setting, or a fidelity slip during a hard week. The mand has a long reinforcement history, so it usually stays in the response class. But the moment the problem behavior gets paid once, its strength jumps. Treat regression like a fidelity audit, not a new function or a new behavior.

Keep going#

The math behind FCT is the same math behind almost every behavior reduction plan you will ever write. Once you can see the gap, you can predict whether a plan will work, find leaks fast, and modify on the fly without leaving the framework. The full CEU walks through the contingency space equation, the percentile schedule for shaping, and a worked spreadsheet you can copy. Watch it when you have an hour.