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. 2007 Aug;51(4):242-265.
doi: 10.1016/j.jmp.2007.01.001.

Consequences of Base Time for Redundant Signals Experiments

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Free PMC article

Consequences of Base Time for Redundant Signals Experiments

James T Townsend et al. J Math Psychol. .
Free PMC article

Abstract

We report analytical and computational investigations into the effects of base time on the diagnosticity of two popular theoretical tools in the redundant signals literature: (1) the race model inequality and (2) the capacity coefficient. We show analytically and without distributional assumptions that the presence of base time decreases the sensitivity of both of these measures to model violations. We further use simulations to investigate the statistical power model selection tools based on the race model inequality, both with and without base time. Base time decreases statistical power, and biases the race model test toward conservatism. The magnitude of this biasing effect increases as we increase the proportion of total reaction time variance contributed by base time. We marshal empirical evidence to suggest that the proportion of reaction time variance contributed by base time is relatively small, and that the effects of base time on the diagnosticity of our model-selection tools are therefore likely to be minor. However, uncertainty remains concerning the magnitude and even the definition of base time. Experimentalists should continue to be alert to situations in which base time may contribute a large proportion of the total reaction time variance.

Figures

Fig. 1
Fig. 1
Schematic of a standard redundant signals model.
Fig. 2
Fig. 2
Schematic of a redundant signals model including base time as in Eq. (1*).
Fig. 3
Fig. 3
The output function, RB=RfB shown as a function of the common standard deviation of the base time components BA, BV and BAV. When σ = 0 the function f * g exhibits a region of positivity near t = 500ms. For larger σ the function f * g is negative definite; the region of positivity has filtered away. Notice how the smoothing effect of the convolution operation increases as we increase σ.
Fig. 4
Fig. 4
The kernel g is a Normal density function with mean μ = 150ms and standard deviation σ = 15ms; we notice that convolution with kernel g does not mask the positivity in the original function f.
Fig. 5
Fig. 5
The kernel g is a Normal density function with mean μ = 150ms and standard deviation σ = 50ms; we notice that convolution with kernel g does indeed mask the positivity in the original function f.
Fig. 6
Fig. 6
The output function, RB=RfB shown as a function of the standard deviation of the base time variable. The base time is Weibull distributed with scale parameter 200 and a varying shape parameter. The smoothing effect of the convolution operation increases as we increase σ in a very similar manner to the effect produced by the Gaussian kernel.
Fig. 7
Fig. 7
The probability of making the correct model selection decision as a function of redundant signals facilitation in the generating model, and also as a function of the absence or presence of base time. Data shown here is for low variance base time, i.e. sB = 100 msec. When RSF < 0 the generating model produces less facilitation than expected from a UCIP race model: the correct decision is to retain the null (race) model; a rejection of the race model constitutes a false alarm. When RSF > 0 the generating model produces more facilitation than expected from a UCIP race model, and the correct decision is to reject; the probability of these correct rejections constitute the statistical power of the our model selection tool.
Fig. 8
Fig. 8
The probability of making the correct model selection decision as a function of redundant signals facilitation in the generating model, and also as a function of the absence or presence of base time. Data shown here is for high variance base time, i.e. sB = 200 msec. When RSF < 0 the generating model produces less facilitation than expected from a UCIP race model: the correct decision is to retain the null (race) model; a rejection of the race model constitutes a false alarm. When RSF > 0 the generating model produces more facilitation than expected from a UCIP race model, and the correct decision is to reject; the probability of these correct rejections constitute the statistical power of the our model selection tool.
Fig. 9
Fig. 9
The distribution of the maximum value of the sampled R*(t) (and RB(t)) functions with and without base time. When base time is present, its range is set at sB = 100msec. The maximum values of the true (infinite sample) R*(t) and RB(t) functions are indicated by a square and an asterisk respectively on the x-axes of each plot.
Fig. 10
Fig. 10
The distribution of the area under the positive component of the sampled R*(t) (and RB(t)) functions with and without base time. When base time is present, its range is set at sB = 100msec. The areas under the positive component of the true (infinite sample) R*(t) and RB(t) functions are indicated by a square and an asterisk respectively on the x-axes of each plot.
Fig. 11
Fig. 11
Plot of Capacity C(t) measured from a UCIP model with base time. A, V and BAV =df BA =df BV are all exponentially distributed. The common mean and standard deviation of the base time variables are varied by varying the exponential parameter β. The process variables had fixed exponential parameters α=ν=1450msec1.
Fig. 12
Fig. 12
Plot of Capacity C(t) measured from a UCIP model with base time. A and V are exponentially distributed; the base time variables are normally distributed. The common mean of the base time variables is μ = 100; the standard deviation σB of the base time varies between 0 and 60 msec. The process variables had fixed exponential parameters α=ν=1500msec1.
Fig. 13
Fig. 13
Here we attempt to reconstruct a model that produces the result that RTAV=min(A+MA,V+MV). In this conception of base time (Maris and Maris, 2003) both channels can initiate a motor process and the “minimum” operation is implicitly generated on RS trials because the response time measuring apparatus will only record the response from the channel which first completes both detection and motor processes.
Fig. 14
Fig. 14
Schematic of a general redundant signals model incorporating both ‘local’ and ‘global’ base times.

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