Academic journal article Attention, Perception and Psychophysics

The Race Model Inequality for Censored Reaction Time Distributions

Academic journal article Attention, Perception and Psychophysics

The Race Model Inequality for Censored Reaction Time Distributions

Article excerpt

The race model inequality (RMI) introduced in Miller (1982) puts an upper limit on the amount of reaction time facilitation within the redundant-signals paradigm that is consistent with a race model. Here, it is shown through theoretical analysis and numerical simulation that inferences from the RMI test may become invalid when the experimenter misses a proportion of the responses by limiting the recording interval (right censoring) or excluding outliers from analysis (left and/or right censoring). Moreover, a correction of the inequality test for right-censored reaction time distributions is proposed.

(ProQuest: ... denotes formulae omitted.)

In the redundant-signals paradigm for simple reaction time (RT), the observer must initiate a response as quickly as possible following the detection of any stimulus onset. A typical finding is a redundancy gain: Responses are faster, on average, when two or more signals are presented simultaneously than when a single signal appears. This redundant-signals effect (RSE) has often, although not always, been replicated under different experimental settings-for example, comparing uni- versus multimodal stimulation (Diederich, 1995; Diederich & Colonius, 1987; Gielen, Schmidt, & Van den Heuvel, 1983; Miller, 1982, 1986; Molholm, Ritter, Javitt, & Foxe, 2004), single versus multiple stimuli within the same modality (e.g., Schwarz & Ischebeck, 1994), or monocular versus binocular stimulation (Hughes & Townsend, 1998; Westendorf & Blake, 1988)-and for specific populations (see, e.g., Corballis, 1998; Miller, 2004; Reuter-Lorenz, Nozawa, Gazzaniga, & Hughes, 1995; and Savazzi & Marzi, 2004, all for split-brain individuals; and Marzi et al., 1996, for hemianopics).

Raab (1962) proposed a race model for simple RT, postulating that (1) each individual stimulus elicits a (normally distributed) detection process performed in parallel to the others and (2) the winner's time determines the observable RT. The race model opens up the possibility that the RSE is generated by statistical facilitation: If detection latencies are interpreted as (nonnegative) random variables, the time to detect the first of several redundant signals is faster, on average, than the detection time for any single signal. Testing the race model amounts to probing whether an observed RT speedup is too large to be attributable to statistical facilitation (viz., probability summation), no matter which distributional assumptions have been made.

A test of general race models was developed by Miller (1978, 1982), showing that

Pr(RT^sub XY^ ≤ t) ≤ Pr(RT^sub X^ ≤ t) + Pr(RT^sub Y^ ≤ t) (1)

must hold for all t ≥ 0. This race model inequality (RMI) follows from

Pr^sub XY^ [min(X, Y ) ≤ t] ≤ Pr^sub X^ (X ≤ t) + Pr^sub Y^ (Y ≤ t) (2)

for any pair of random variables (X, Y ) with a joint probability distribution based on PrXY and with its marginal distributions identical to PrX and PrY. Thus, as was observed in Luce (1986, p. 130), the RMI test requires that the RT distributions in the single-signal conditions are identical to the corresponding (marginal) RT distributions in the redundant-signals condition (cf. Colonius, 1990). Note that, for fixed t, Inequality 2 corresponds to the well-known Boole's inequality (e.g., Billingsley, 1979). Neglecting possible additional components (such as motor time), the inequality stipulates that the RT distribution function for redundant stimuli is never larger than the sum of the RT distributions for the single stimuli. A violation of this inequality is interpreted as an indication of an underlying coactivation mechanism or some other strong form of nonindependence.

Miller's (1978, 1982) test has become a standard tool in numerous empirical RT studies (see the references above). Moreover, it has been the subject of various theoretical and methodological studies as well (Ashby & Townsend, 1986; Colonius, 1990, 1999; Colonius & Ellermeier, 1997; Colonius & Townsend, 1997; Colonius & Vorberg, 1994; Diederich, 1992; Miller, 1986, 1991, 2004; Miller & Ulrich, 2003; Mordkoff & Yantis, 1991; Townsend & Nozawa, 1995, 1997; Townsend & Wenger, 2004; Ulrich & Giray, 1986; Ulrich & Miller, 1997; Ulrich, Miller, & Schröter, 2007). …

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