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Describing An RT Distribution Shape Scale and Shift

Describing An RT Distribution Shape Scale and Shift

Weibull 

Green et al.s (1994, Biometrics)
$f\left(x;\mathrm{?}, \mathrm{?}, \mathrm{?}\right)=\left(\frac{\mathrm{?}}{\mathrm{?}}\right){\left(\frac{x?\mathrm{?}}{\mathrm{?}}\right)}^{\mathrm{?}?1}\exp \left\{?{\left(\frac{x?\mathrm{?}}{\mathrm{?}}\right)}^{\mathrm{?}}\right\}, x?\mathrm{?}, \mathrm{?}?0, \mathrm{?}?0.$

Rouder et al. (2008, PBR)
$f\left(t; \mathrm{?}, \mathrm{?}, \mathrm{?}\right)=\mathrm{??}{\left(t?\mathrm{?}\right)}^{\mathrm{?}?1}\exp \left[?\mathrm{?}{\left(t?\mathrm{?}\right)}^{\mathrm{?}}\right], t?\mathrm{?}$.

Rouder et al. (2005, PBR; also in 2003, Psychometrika)
$f\left(y_{\mathrm{ij}}; {\mathrm{?}}_i, {\mathrm{?}}_i, {\mathrm{?}}_i\right)=\frac{{\mathrm{?}}_i{\left(y_{\mathrm{ij}}?{\mathrm{?}}_i\right)}^{{\mathrm{?}}_i?1}}{{\mathrm{?}}_i^{{\mathrm{?}}_i}}\exp \left[?\frac{{\left(y_{\mathrm{ij}}?{\mathrm{?}}_i\right)}^{{\mathrm{?}}_i}}{{\mathrm{?}}_i^{{\mathrm{?}}_i}}\right], for y_{\mathrm{ij}}>{\mathrm{?}}_i$
Cousineau (2008, IEEE)
$f\left(x; \mathrm{?}, \mathrm{?}, \mathrm{?}\right)={\mathrm{??}}^{?\mathrm{?}}{\left(x?\mathrm{?}\right)}^{\mathrm{?}?1}\exp \left(?{\left(\frac{\left(x?\mathrm{?}\right)}{\mathrm{?}}\right)}^{\mathrm{?}}\right)$
? > 0 is shape responsible for the skew of the distribution; ? > 0 the scale; ? the shift, a lower bound (quoted from the Cousineaus article). 

	Shape	Scale	Shift
Green 1994	?	?	?
Rouder 2008	?	1/?	?
Rouder 2005	?i	?i	?i
Cousineau 2008	?	?	?

In Rouder et al. 2008 ? was called rate, which is an inverse of scale.  Equal variance assumption which frequently states in aggregated mean RT analysis equals to saying that only shift changes, but the shape and scale (or rate?) keep constant.  Thus, in this case, the change in shift equals to the change in mean.  In this article, the authors also discussed the functional role of shift, which serves as the minimum RT.  Contrast to the added constant term, T_ER in Ratcliffs (1978) diffusion model, hierarchical regression model using Bayesian approach allows shift to be modelled, whereas T_ER is a remedy for the diffusion model that predicts a 0 minimum RT when the sample size is increased.  The difference between the diffusion model and hierarchical regression model is that in the former the shift is a constant, whereas the latter frees the shift to vary participant-by-participant and item-by-item.  Few example, Rouder et al. mentioned were the task of detecting auditory tones has 130 and180 ms for minimum and mean RT, respectively.  Another example is 300 and 700 ms in lexical access tasks.

In their three-parameter Weibull model, the shift was designated as a possible index for minimum RT in large-sample limit (how large? 55 participants?). The shift and shape vary participant by participant, but the rate varies both with participants and items.  Rate (or log rate) is a good index to examine the effect resulting from experimental manipulations (covariates), because it affects not only mean but also standard deviation.  Also using rate as the "locus" assumes the minimum RT (shift?) is constant across covariates (different level of manipulation factors).

Cross-random effects model? (Snijders & Bosker, 1999; van den Noortgae, De Boeck, & Meulders, 2003).  Noninformative Jeffreys priors (Jeffreys, 1961)?

The Weibull model here according to Rouder et al. is best applied to cases when no participant-item replicates.


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