PLOTTING AND ANALYZING CUMULATIVE RESPONSE CURVES IN OPERANT CONDITIONING STUDIES.
Set your cumulative-record axes with the arcsin formula so rate changes form clear angle bends you can spot in a second.
01Research in Context
What this study did
HERRICK (1965) wrote a how-to guide for drawing cumulative records.
He gave a math rule: pick axes so the first slope equals arcsin(sqrt(R1/R2)).
The goal is to make later rate changes jump out at your eyes.
What they found
The paper gives exact formulas, not new data.
Follow the rule and small rate shifts look like big angle bends on the page.
How this fits with other research
Richman et al. (2001) extends the 1965 idea. They plot minute-by-minute cumulatives inside one session to see if efficiency, not just reinforcement loss, causes downtrends.
Schaal (1996) also extends the rule. He scales the record as a percent of total responses against session time. This removes the need to pick time bins while still spotting local changes.
Chandler et al. (1992) and Rutland et al. (1996) show within-session patterns that are easier to read when you first set the axes the 1965 way.
Why it matters
Next time you graph a cumulative record, use the angle formula before you print. A quick calculator step turns flat lines into clear stories about when behavior speeds up or slows down. Your supervisor and future you will thank you for the cleaner visuals.
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Join Free →Before the next session, measure baseline rate R1 and expected rate R2, then rescale the y-axis so the initial slope equals arcsin(sqrt(R1/R2)).
02At a glance
03Original abstract
Because the response and time scales used in plotting cumulative response curves are often poorly selected, ineffective displays often result. The visual cue of a response rate change is the difference, [Formula: see text] between the angles, theta(1) and theta(2), representing the two rates, R(1) and R(2). These variables are related by: [Formula: see text] For a given rate change, the value of theta(1), namely, (M)theta(1), that yields the maximum value of [Formula: see text] namely, [Formula: see text] is given by (M)theta(1)=arc sin [Formula: see text] Ideally, the initial response rate should be represented by the (M)theta(1) appropriate for a given rate change. Because of practical considerations, however, some compromises with the ideal are allowable. Included in the discussion are (a) steps required to select appropriate response and time scales, with examples, and (b) guideposts for evaluating rate changes by means of angular changes.
Journal of the experimental analysis of behavior, 1965 · doi:10.1901/jeab.1965.8-59