Response Time Distribution

The analysis of RT distributions, in addition to mean RTs, can permit further insights into different cognitive processes, a point originally raised as early as 1950s in works such as those of Christie and Luce (1956), Hohle (1967), and Sternberg (1969). In essence, one early RT distribution account postulated that the RT is a functional output summing across a decision component that distributes exponentially and a residual component that distributes symmetrically (Hohle, 1965). This postulation later developed into the well-known ex-Gaussian distribution that convolves mathematically the exponential and Gaussian components. The ex-Gaussian function becomes popular mainly because of its capacity of accommodating positively skewed distributions, an observation commonly found in RT data before a representative values, such as mean, are averaged across several observers. A usual practice for analysing RT data is to average multiple observations for different experimental conditions in an observer, and the averaged values in each condition are averaged again across several observers. This practice presumes that the first-level averaged values catch the general shapes of RT distributions, thereby representing well the majority of RT data in a condition. This assumption conflicts with the observation of the skewed RT distributions, which cast doubt on the data generated from the (individual-level) mean RTs, because they may not represent some observations when distributions are skewed.

Further doubt on mean RTs comes from their ambiguity when answering the architecture question. Specifically, the finding of a display size effect and that of the slope ratio of target trial to blank trials calculated from mean RT in fact cannot determine how the cognitive architecture operates. This has been repeatedly demonstrated by Townsend and colleagues, showing that both serial and parallel models are able to predict the data from mean RTs (Townsend, 1971, 1990; see Townsend & Ashby, 1983 for a review; Townsend & Wenger, 2004).

1.4.1 Descriptive Models

The drawback of analysing only mean RTs appears to an alternative approach for analysing RT data, such as the distributional analyses (Lin, Heinke,

& Humphreys, 2015; Loft, Bowden, Ball, & Brewer, 2014; Payne & Stine-Morrow, 2014; Toeroek, Kolozsvari, Viragh, Honbolygo, & Csepe, 2014). The ex-Gaussian function, for example, breaks down a distribution into two mathematically and psychologically seemingly separable components: the Gaussian and exponential parts. The latter accounts mathematically for that why an RT distribution skews towards the short latency side. The former keeps the original symmetrical Gaussian part of a distribution. Although the data do not necessarily collaborate the dichotomy of decision and residual components into the Gaussian and the exponential parts of an RT distribution (Gholson & Hohle, 1968a, 1968b; see a recent review, Matzke & Wagenmakers, 2009), recent works show that the value of adopting an ex-Gaussian is that it provides a plausible model to describe positively skewed RT distributions (e.g., Matzke, Dolan, Logan, Brown, & Wagenmakers, 2013). This is in contrast to assuming the Gaussian distribution as the underlying function that generates RTs. The advance is crucial

because the Gaussian function may be only appropriate to account for the mean RTs across multiple observers, rather than the mean RTs across multiple observations within a condition in an observer (see Chapter 4 for the data supporting this point).

The initial attempt to conceptualise RTs, using the ex-Gaussian framework, distinguished perception and decision components and the components involved in the organization and execution of the motor responses (Hohle, 1965). For instance, Hohle’s original interpretation of the exponential component was that it reflects perception and decision processes – opposite to McGill and Gibbon’s interpretation of a residual motor latency (1965). The early conflicting interpretations and numerous succeeding works (Balota & Spieler, 1999; see a recent review in Matzke & Wagenmakers, 2009; Rohrer & Wixted, 1994) indicates that the mapping of ex-Gaussian components onto cognitive processes only results in paradigm-dependent interpretations. That is, the separation of a Gaussian component and an exponential component is meaningful only at the mathematical, but not cognitive, level; the resultant interpretations of exponential and Gaussian components varied with tested factors and experimental designs. As a consequence of the null finding, it is suggested that the ex-Gaussian model is most useful when been treated ‘as a descriptive first-order account of response latency’ (Heathcote, Popiel, &

Mewhort, D. J., 1991). This view is echoed later in Schwarz’s work (2001), where he used an ex-Wald function as a quantitative model, taking advantage of its Wald component to approximate a Wiener diffusion process, a link to the underlying cognitive processes. This was demonstrated in Schwarz’s go/nogo

digit comparison experiment, in which observers pressed a button upon detecting a go digit (6 or 9) and withheld button-press upon detecting a nogo digit (5). The results showed that (1) the a priori probabilities of the appearance of go digit (50%

vs. 75%) selectively affected only the evidence criterion (a statistical Wald parameter), (2) the numerical distance (6 vs. 9, comparing to 5) selectively affected the drift rate (a second statistical Wald parameter), and (3) the exponential parameter, γ, appeared insensitive to the two aforementioned experimental factors (Schwarz, 2001).

1.4.2 Non-Gaussian Distribution

The modest success by using the ex-Wald function to describe RT distributions lends support to researchers exploring other insights from analysing RT distributions, by comparing different experimental manipulations. Converging evidence supporting the descriptive approach of distributional analyses comes from Ashby and colleagues’ work (Ashby, Tein, & Balakrishnan, 1993). They showed, in a Sternberg memory-scanning task, a number of distribution-level predictions, such as the variance RTs and the shape of RT distributions, are inconsistent with the serial and the unlimited capacity parallel models. The descriptive approach to examining RT distributions by comparing different experimental conditions appears promising for understanding other cognitive processes, although not quite as much as the ambitious researchers originally envisioned (e.g., Hohle, 1965).

The success of descriptive distributional analyses using different probability functions and higher distributional moments suggests that the advantage of adopting the ex-Gaussian (or ex-Wald) function does not lie in the

probability function per se, but in its capacity to accommodate empirical RT distributions. The fact is that numerous other probability functions are capable of doing this (Dolan, van der Maas, & Molenaar, 2002; Feige et al., 2013; Heathcote, Brown, & Cousineau, 2004; E. M. Palmer, Horowitz, Torralba, & Wolfe, 2011;

Rouder, Lu, Speckman, Sun, & Jiang, 2005). In addition to the probability functions convolving with an exponential function, there are other generic functions, such as the 3-parameter Weibull function, gamma, log-normal, and Wald functions that are flexible enough to accommodate skewed distributions without convolving with an exponential function. These other plausible probability functions allow researchers to describe and compare the shape of RT distributions across different experimental conditions, with a strategy differing from the convolving functions. One of the strategies is to describe RT distributions using the location-scale form of the probability density¹⁰.

1.4.3 The Application of Descriptive Distribution Models to Visual Search The descriptive approach using plausible probability functions to model RT distributions seems promising to tackle also the problems in visual search paradigms. Specifically, the parallel, limited capacity model can produce linear function for the RT × display size relation, by dividing attention into multiple processing channels when search items increase. The increment of each search item dilutes the limited resources, rendering multiple processing channels sharing increasingly few resources, so lengthening response latencies.

10 The location-scale form refers to, in the case of the Gaussian function, mean and variance.

Further details specific to the probability functions adopted in this thesis will be explained in the later chapters.

One simple application of descriptive distributional analyses is to examine the spreads of RT distributions. RT standard deviations in fact correlate with RT means linearly, as has been showed recently by a meta-analysis for 3 visual search experiments. The finding was dubbed as ‘the linear law’ of RT (Wagenmakers & Brown, 2007). A more complicated application involves contrasting an entire distribution (i.e., the probability density function) and further its derivatives across different experimental conditions and/or participants. The distributional derivatives include, but are not limited to, the cumulative density function, the survivor function, the hazard function and different applications of these distributional functions based on specific experimental designs. The survivor function, for example, has been used to examine the effect of target-distractor similarity (VS theory originally called it interalternative similarities in Duncan & Humphreys, 1989) in a simple visual search paradigm (Fific et al., 2008). Fific and colleagues used the linguistic (Cyrillic) and nonlinguistic meaningless symbols in a simple two-item search task and found evidence favouring a parallel cognitive architecture with positive interacting channels against a serial architecture. This was supported by the data showing different patterns of survivor interaction contrast function, which calculated the difference between differences of the survivor functions of the four factorial conditions in a 2 by 2 factorial design experiment (Townsend & Nozawa, 1995) .

In summary, the descriptive distributional analyses suggest a new approach to examining visual search, and the thesis will explore one possible distributional analysis method, the 3-parameter Weibull function, to analyse visual search data.