Compare Conditional and Unconditional ABC Probability

To compare conditional and unconditional ABC probabilities, examine how often an event occurs in a defined sequence and across the relevant background. Report both fractions, then calculate an absolute difference or ratio only when units are compatible. Keep small denominators and zero base rates visible. The comparison describes association in sampled observations and does not establish a functional relation.

Define two compatible probabilities

State the conditional result as a complete question, such as P(break within ten seconds after response). State the unconditional result as P(break in a valid background unit). Use the same break definition, observation period, settings, and source records. Document response anchor, sequence window, background unit, overlap rule, and missing-state treatment.

Compatibility does not require identical denominators, but it does require a defensible opportunity comparison. Twelve ten-second background intervals and twelve response windows can still sample different contexts. Explain what each unit represents and avoid a ratio when event definitions, time spans, or observation coverage differ materially.

Preserve raw cells and coverage

For the conditional fraction, retain responses with break, responses without break, truncated windows, and missing sequences. For the background fraction, retain valid units with and without break plus missed, excluded, and unresolved units. Show planned and valid observation coverage by session or setting.

Do not let a sequence be selected because it contains the event. Build the response exposure list first. Do not treat an unobserved background interval as “no break.” When categories overlap, preserve break with other consequences instead of forcing the event into a single exclusive cell unless that rule was declared.

Calculate difference and ratio

The absolute probability difference is:

conditional probability − unconditional probability

It is reported in probability points or percentage points. A change from 20% to 60% is a 40-percentage-point difference. The conditional-to-background ratio is:

conditional probability / unconditional probability

The ratio is defined only when the background probability is above zero and the measures are comparable. A ratio can look very large when the background cell is rare, so always report the two fractions and absolute difference.

Verify Rosa’s arithmetic

Rosa has 15 response events with valid follow-up windows. Break follows 9: 9 / 15 = 0.60, or 60%. Six complete windows have no coded break. In 60 valid background units, break appears in 12: 12 / 60 = 0.20, or 20%. Forty-eight background units have no coded break.

The absolute difference is 0.60 − 0.20 = 0.40, or 40 percentage points. The ratio is 0.60 / 0.20 = 3.0. In this sample, break is three times as likely in the declared response window as in a background unit. That sentence describes the recorded probability comparison and does not state that the response caused the break.

Small-cell sensitivity remains visible. If one fewer response window contained break, 8 / 15 = 53.3%; the difference would be 33.3 points and the ratio about 2.67. If one more contained break, 10 / 15 = 66.7%; the difference would be 46.7 points and the ratio about 3.33.

Handle zero and near-zero background values

If break occurs in 0 of 60 background units and 9 of 15 response windows, the absolute difference is 60 points. The ratio is undefined because division by zero has no result. Do not add a small constant or replace zero with an arbitrary value unless a separate statistical method and rationale are declared.

A near-zero background estimate can also make the ratio unstable. One event in 60 units is 1.7%; a 60% conditional value would yield a ratio of 36. Report counts, uncertainty, and session distribution. The absolute difference is usually easier to interpret alongside sparse cells.

No valid response exposures produce an undefined conditional probability. No valid background units produce an undefined unconditional probability. These zero-denominator states signal insufficient exposure, not absence of association.

Examine window and session effects

The conditional result depends on the chosen follow-up span. A longer window creates more opportunity for break to occur after a response. Contingency space analysis illustrates why event probability after a response should be considered alongside background probability and explicit time windows.

Inspect results by session, routine, setting, and relevant context. Nine sequences concentrated in one session may reflect a different sampled pattern from nine distributed across weeks. Keep health, pain, task, staffing, communication access, and schedule conditions source-labeled. Avoid post hoc subdivision solely to find a larger contrast.

Graph and report the comparison

Use paired dots or bars labeled “9/15 response windows, 60%” and “12/60 background units, 20%.” Add the 40-point difference and 3.0 ratio as annotations. A session-level panel can display conditional and background cells over time. Do not graph the ratio alone.

Suggested wording: “Break occurred within the defined window after 9 of 15 responses (60%) and in 12 of 60 valid background units (20%). The observed difference was 40 percentage points and the conditional-to-background ratio was 3.0. Results depend on the response anchor, window, background unit, coverage, and sampled settings.”

Keep the inference descriptive

The comparison identifies a recorded association. It cannot establish that break reinforced the response, that the response produced break, or that the same relation would occur elsewhere. Both events may follow another condition, and descriptive observation can miss communication, pain, health, access, task, and social variables.

Antecedent-versus-consequent research supports comparing sequence and background probabilities. Comparative descriptive-method research reports limited correspondence with experimental outcomes, and research on reducing assessment ambiguity distinguishes descriptive association from experimental demonstration.

Review access and meaning with Rosa

Use Rosa’s preferred communication to ask how break access, transitions, discomfort, and observed responses are experienced. Keep AAC continuously available under ASHA guidance. Record assent, dissent, priorities, and context corrections.

Observation and calculation cannot justify withholding food, water, bathroom use, mobility, prescribed health care, rest, relationships, or emergency help. The BACB ethics hub and CASP public summary offer professional context. Medical, safety, privacy, payer, and legal decisions remain with qualified owners.

Clinician checklist and limitations

Before using the comparison, confirm:

conditional direction, response anchor, window, break definition, and background unit are explicit;
9 with break plus 6 without break equals 15 valid response windows;
12 with break plus 48 without break equals 60 valid background units;
missing, truncated, excluded, and unresolved observations remain visible;
60%, 20%, 40 percentage points, and ratio 3.0 reproduce;
zero and near-zero background cases are reported without invented ratios;
graphs show raw cells, coverage, sessions, contexts, and client input; and
conclusions stay descriptive with a qualified next-step owner and review date.

Rosa’s response denominator is small, and one cell change moves the comparison materially. Window length, background unit, setting mix, overlap, and missingness can change both estimates. The result does not confirm function, treatment need, or benefit. Reopen the worksheet when definitions, data, sampling, access, health, observer, or software changes, and preserve prior calculations.

How to Compare Conditional and Unconditional ABC Probabilities