**Abstract**:

**Abstract**
**In multisensory settings such as the focused attention paradigm (FAP), subjects are instructed to respond to stimuli of the target modality only, yet reaction times tend to be shorter if an unattended stimulus is presented within a certain spatiotemporal vicinity of the target. The time window of integration (TWIN) model predicts successfully these observed cross-modal reaction time effects. It proposes that all the initially unimodal information must arrive at a point of integration within a certain time window in order to be integrated and thus to initiate response enhancements like the observed reaction time reductions. Here we conducted a parameter recovery study of the TWIN model for focused attention tasks, with five parameters (the durations of the visual and auditory unimodal and the integrated second stage, the width of the time window, and the effect size). Results show that parameter estimates were highly accurate (unbiased, constant error less than 5 ms) and precise (variable error less than 8 ms) throughout, speaking to a high reliability and criterion validity of the process. Further analyses ensured that the estimation procedure is consistent and sufficiently robust against contamination (faulty integration). It can thus be used to estimate reliably the point of integration and the width of the time window.**

*μ*representing the average second stage (central) processing time; (d) parameter

*ω*giving the width of the time window of integration; and (e) parameter Δ denoting the size of the cross-modal interaction effect in milliseconds.

^{1}If a stimulus from the target modality is the winner of the race in the peripheral channels, second-stage processing is initiated without any multisensory integration mechanism being involved.

^{2}These distributions are specified by numerical parameters that have to be estimated from the observed RT data. The model has been probed in a series of experiments under a variety of empirical conditions (Colonius et al., 2009; Diederich & Colonius, 2007b; Diederich & Colonius, 2008a). While these tests have generally supported the essential features of the model, the quality of a model test critically depends on one's ability to obtain good estimates of the model parameters. The purpose of the current simulation study therefore is to probe whether, and how well, our parameter estimation method is able to recover the correct parameters. To this end, RT data sets are randomly generated from the model's probability distributions with fixed and a-priori known parameter values. Next, estimates of these parameters are computed on these simulated data sets in order to check how well the original parameter values can be recovered. Obviously, the quality of the recovery results will depend both on properties of the model and on the efficiency of our estimation procedure. The next section provides details of the time window model and its simulation.

*V*and

*A*denoting the peripheral processing times for the visual and the acoustic stimulus with density with

*λ*> 0 and specific values

*λ*≡

*λ*and

_{A}*λ*≡

*λ*for the auditory and visual modality, respectively.

_{V}*τ*as SOA value and

*ω*as integration window width parameter, the above requirement for multisensory integration to take place is the realization of the event i.e., the nontarget stimulus wins the race in the first stage “opening the time window of integration” such that the termination of the target peripheral process

*V*falls into the window. Here, a positive

*τ*value indicates that the visual stimulus is presented before the acoustic, and a negative

*τ*value indicates the reverse presentation order. Observable total reaction time is the sum of first stage processing time of the target modality, here

*V*, and second stage processing time,

*M*, assumed to follow a normal distribution and comprising all subsequent processes including motor preparation and execution and, in the bimodal condition, multisensory integration. The mean of the distribution of

*M*differs depending on whether a unimodal or bimodal condition is considered. For unimodal trials,

*M*has a mean of

*μ*and variance

*σ*

^{2}(Figure 2A). For bimodal trials, second stage processing time depends on whether or not the condition for multisensory integration,

*I*,

_{τ}*, is met in a given trial (Figure 2B, C). Therefore, reaction time in the bimodal condition is mixture of two distributions with mean*

_{ω}*μ*and

*μ −*Δ, respectively, and mixing parameter

*P*(

*I*,

_{τ}*). Thus, mean reaction times in the unimodal and bimodal conditions are, respectively, with*

_{ω}*λ*denoting the exponential parameter for

_{V}*V*introduced above and

*E*the expectation operator (mean) of random variables. From Equation 2 it is obvious that when the cross-modal effect Δ equals zero or when the probability of integration

*P*(

*I*

_{τ}_{,ω}) is zero, expected reaction time in the unimodal and bimodal condition will be identical.

^{3}Probability of integration,

*P*(

*I*

_{τ}_{,ω}), is a function of the exponential parameters

*λ*and

_{V}*λ*for

_{A}*V*and

*A*, the window width parameter

*ω*, and the SOA value

*τ*that is determined by the experimental setup. Explicit expressions for

*P*(

*I*

_{τ}_{,ω}), depending on the sign of

*τ*and

*τ*+

*ω*can be found in the Appendix, see also Colonius and Diederich (2011).

^{4}

*λ*,

_{A}*λ*,

_{V}*μ*,

*ω*, and Δ, but only two “observables” are available, i.e., the unimodal and bimodal sample means for

*RT*and

_{V}*RT*. Thus, the model and its parameters are clearly not identifiable. However, the situation can easily be remedied by increasing the number of SOAs. With four SOA values there will be model equations for four bimodal means and one unimodal mean, i.e., an equal number of “observables” and parameters (i.e., a “saturated” model). Importantly, however, note that here we are not investigating formal model identifiability of TWIN using theoretical approaches as developed, for example, in Bamber and van Santen (2000). Rather, we want to ascertain in a pragmatic way whether our parameter estimation method is able to recover the model parameter values from data sets that have been generated under the model with those same parameters. Before we present the details of the simulation procedure in the next section, two more issues have to be addressed.

_{VA}*θ̂*for a parameter

*θ*is the

*mean-squared error*(MSE) defined by

*E*[(

*θ̂*−

*θ*)

^{2}]. It is well known (Casella & Berger, 2002, p. 330) that then where (

*E*[

*θ̂*]−

*θ*)

^{2}is (Bias

*θ̂*)

^{2}. Thus, MSE incorporates two components,

*Var*[

*θ̂*] measuring the variability of the estimator (

*precision*) and the other measuring its bias (

*accuracy*) (Casella & Berger, 2002, p. 330). In analogy to this definition, we evaluated the precision and accuracy of the parameter recovery resulting from the simulations by considering the variability and central tendency of the parameter estimates across different simulated data sets. Moreover, we tested for the robustness of the parameter estimation method, that is, its ability to recover the original parameter values when participants committed certain errors like integrating although event

*I*

_{τ}_{,ω}was not realized (or vice versa).

*λ*and

_{A}*λ*that will be estimated by model fitting at the level of means. However, the variance of the normally distributed processing duration in the second stage,

_{V}*σ*

^{2}, is not included in this model fitting procedure and, therefore, cannot be “recovered.” On the other hand, in order to generate random data sets from the model, this parameter needs to be specified as well. We chose to fix the value at

*σ*= 25 ms, a value that seems plausible given some experimental results obtained earlier (Diederich, Colonius, & Schomburg, 2008). Moreover, some pilot simulations suggested that the outcome of our main simulation study did not depend on the exact value of

*σ*.

Parameter | Value | % lower B | % valid | % upper B |

1/λ_{A} | Bounds: | [ 5, | … | , 250 ] |

20 | 20.3 | 79.7 | 0.0 | |

50 | 10.1 | 89.8 | 0.1 | |

100 | 4.3 | 94.7 | 1.0 | |

150 | 1.1 | 92.5 | 6.4 | |

1/λ_{V} | Bounds: | [ 5, | … | , 250 ] |

20 | 14.0 | 86.0 | 0.0 | |

50 | 8.4 | 91.4 | 0.2 | |

100 | 4.7 | 92.8 | 2.5 | |

150 | 2.4 | 87.4 | 10.2 | |

μ | Bounds: | [ 0, | … | … |

50 | 16.1 | 83.9 | 0.0 | |

100 | 6.8 | 93.2 | 0.0 | |

150 | 1.2 | 98.8 | 0.0 | |

200 | 0.0 | 100.0 | 0.0 | |

ω | Bounds: | [ 5, | … | , 1000 ] |

100 | 1.0 | 96.2 | 2.9 | |

200 | 0.3 | 93.6 | 6.1 | |

300 | 0.1 | 80.1 | 19.8 | |

Δ | Bounds: | [ 0, | … | , 175 ] |

20 | 7.0 | 85.4 | 7.6 | |

50 | 1.5 | 90.8 | 7.7 | |

100 | 0.2 | 88.7 | 11.1 |

*μ*, the duration of the secondary (i.e., postintegration or common) stage, were simply obtained by subtracting the 1/

*λ*values mentioned above from a typical simple-reaction time of 200 to 250 ms: 50, 100, 150, and 200 ms. The parameter Δ stands for the temporal gain, i.e., the reaction time reduction, resulting from the integration of the multimodal information. Here, we took 20 and 50 ms to be reasonable values and added 100 ms as a safe maximum gain. Note that, for simplicity, we do not consider the case of inhibition, i.e., negative values of Δ.

*ω*represents the width of the time window, i.e., the time interval during which integration may occur. In neurophysiological terms, this may be the duration for which the membrane potential of the integration cell remains excited. Using modern whole-cell in-vivo recording methods, duration estimates vary greatly between very short, e.g., 2 ms for cells in the avian cochlea (Reyes, Rubel, & Spain, 1994), and long durations, e.g., 50–140 ms in hippocampus (Andersen, Raastad, & Storm, 1990). In contrast, area MT (V5) cells integrate motion information across more than 200 ms (Bair & Movshon, 2004). To cover the whole range, here we utilized values of 100, 200, and 300 ms.

*unrealistic*vectors such as (20, 20, 50, 200, 100), which would on average result in negative RTs if integration occurred (mean

*RT*=

*μ*− Δ = 20 + 50 − 100 = −30). All of these vectors were tested and taken into the evaluation, irrespective of their plausibility.

^{5}

*t*= 0 and the acoustic stimulus would either precede, be in synchrony with, or lag behind the visual target stimulus by an SOA of −200, −150, −100, −50, 0, +50, +100, +150, and +200 ms, respectively. In all these conditions, a real subject would be instructed to react to the onset of the visual stimulus only (focused attention paradigm). As an additional ninth condition, we implemented a unimodal auditory condition (uniA), in which the subject would be presented only with an acoustic stimulus and would have to react to it.

*A*and

*V*are random variables with exponential distributions (cf. Equation 1), while

*M*is a random variable with a normal distribution

*σ*was not recovered during the process and is therefore not part of this evaluation study.

*z*-standardized differences between generated and predicted reaction times across all uni- and bimodal conditions,

*μ*between 5 and 300 ms).

*n*= 1 subject), the experiment level (

*n*= 40 subjects forming a group), and the study level (across all 576 experiments and thus

*n*= 23,040 subjects). While the main questions of this evaluation concern the study level, good and reliable results on subject and experiment levels are the necessary precondition for the analysis of the main results presented further below.

*n*= 200 trials per condition) along with the RTs predicted by the model based on the estimated parameters. The differences between these reaction times were fairly small, with approximately 1.9 ms per condition for the plausible parameter vector. As depicted in Panel B, across all

*n*= 40 subjects in all

*n*= 576 experiments, this prediction error had a mean of 5.95 ms (

*SD*= 3.64) per condition.

*μ*, and Δ), which are by themselves very close to the boundary and thus endangered from the outset, or very high values (for

*ω*).

*study*level, individual parameter estimates were averaged using the median across the 40 subjects of an experiment into group estimates. All of these group estimates stood clear off the boundaries and thus all further assessments (and the graphs shown in Figure 5) rely on the full amount of 576 estimates. Results for the evaluation of the recovery are shown in Figure 5 and Table 2. Graphs in Panels A to E show histograms for the five parameters, and while the leftmost graphs show histograms aggregated across all parameter values, the others (to the right) show them for each parameter value individually. To ease comparison, histograms are plotted against the distance of the recovered parameter to the original value.

Parameter | Value | Mean Diffs | SD | Median Diffs | MAD |

1/λ_{A} | All | −0.25 | 4.50 | −1.94 | 12.10 |

20 | 0.75 | 2.85 | 1.85 | 6.87 | |

50 | −0.52 | 3.40 | −1.19 | 10.5 | |

100 | −2.25 | 5.65 | −4.81 | 13.35 | |

150 | −0.75 | 5.85 | −3.59 | 15.07 | |

1/λ_{V} | All | −0.73 | 4.03 | −2.77 | 12.61 |

20 | −1.03 | 2.03 | −1.34 | 4.33 | |

50 | −1.95 | 4.20 | −4.21 | 10.27 | |

100 | 1.00 | 5.43 | −1.02 | 15.04 | |

150 | 0.00 | 6.45 | −4.55 | 16.70 | |

μ | All | 1.35 | 3.58 | 3.20 | 11.97 |

50 | 1.20 | 3.68 | 3.49 | 14.26 | |

100 | 1.48 | 3.30 | 3.18 | 12.13 | |

150 | 1.05 | 3.78 | 2.79 | 9.69 | |

200 | 1.70 | 3.93 | 3.33 | 11.49 | |

ω | All | −2.82 | 8.05 | 2.26 | 51.54 |

100 | −1.50 | 2.95 | −4.32 | 9.08 | |

200 | −4.05 | 5.65 | −9.71 | 23.59 | |

300 | −6.75 | 34.68 | 20.73 | 82.48 | |

Δ | All | 3.65 | 3.60 | 6.71 | 9.25 |

20 | 4.70 | 3.20 | 7.62 | 8.07 | |

50 | 2.78 | 3.18 | 6.41 | 10.05 | |

100 | 2.75 | 3.65 | 6.09 | 9.48 |

*μ*, the histograms ubiquitously are rather narrowly distributed around the expected value (zero on the

*x*axis). Medians (constant errors) of the group estimates for the first three parameters (

*μ*) range between −2.5 ms and +2.5 ms (means are between −5 and 5 ms) for individual parameter values, and level out across all parameter values to about half that size, thereby exhibiting a very high accuracy of the estimates. Median absolute deviations (MAD) are lower than 7 ms (

*SD*≤ 16 ms), speaking to a high precision. Given the low absolute distances and the three times higher deviation measures, the estimates have to be classified as

*unbiased*.

*ω*, the same assessment (narrow distribution, unbiasedness) holds true for the first two parameter values of 100 and 200 ms. Although the medians are negative, they are close to zero (medians > −5 ms; means > −10 ms), while the median absolute deviation is again lower than 7 ms (

*SD*< 24 ms). In contrast, for

*ω*= 300 ms, the constant error is higher (median = −13 ms, mean = −18 ms) and the distribution is much wider (MAD = 35 ms,

*SD*= 82 ms) and shallower than in all other cases.

*consistent*if it converges to the true (population) value when sample size becomes infinitely large. Note that a consistent estimator may be biased and an unbiased estimator may not be consistent. For many estimators, like maximum likelihood estimators, consistency holds under rather general conditions. However, the parameter estimation method utilized here does not guarantee consistency of the estimators. On the other hand, consistency follows if both the bias of the estimator and its variance converge to zero for sample size approaching infinity (Lehmann, 1983). Therefore, we tested the accuracy and precision behavior of our estimates with increasing sample size.

*μ*,

*ω*, Δ) = (20, 50, 100, 200, 50), with the standard number of 40 subjects and with a number of trials per condition (sample size) increasing from 40 to 200 in steps of 20 and from 200 to 1000 in steps of 100 trials. Results in Figure 6 show that the bias for the median decreases with the square root of the sample size (black symbols) and that same holds true for the median absolute deviation (gray symbols) for all five parameters. This strongly suggest that the parameter estimation procedure produces consistent estimators.

*λ*parameters of the first stage and the mean

*μ*of the second stage processing. This issue becomes especially critical when one tries to interpret differences among parameter values across experimental conditions. For example, increasing stimulus intensity should be reflected in differences in the

*λ*parameters but not in Δ because intensity affects first stage processing and integration probability, but not the amount of facilitation or inhibition in the second stage, according to TWIN assumptions.

*r*> 0.98 between

*μ*and between

*μ*; further a strong, positive correlation between the estimates for the first two parameters and Δ with

*r*> 0.88, and a comparably high but negative correlation between the estimates for

*μ*and Δ, with

*r*= −0.87; finally a moderate (

*r*= 0.38) but significant negative correlation between

*ω*and Δ. Correlations between

*ω*and the first three parameters were not significant. While some of these correlations reflect expected trade-off effects of the estimation procedure, they also point to some caveats for interpreting parameter differences between experimental conditions.

*p*)% of the due cases, whereas RT was not reduced in the other

*p*%. At the same time, reaction times were generated incorporating the reduction in

*p*% of the cases for which integration should not have taken place. In six experiments, the percentage

*p*was systematically increased from 0% to 25% in steps of 5%, using the same plausible parameter vector (20, 50, 150, 200, 50) as before. Generated RTs were averaged and presented to the parameter estimation procedure.

*ω*)? And (d) can the robustness of the parameter estimation be increased by using the median over the mean in averaging the RT data?

*μ*and

*ω*become underestimated. However, while accuracy worsens with contamination, the variable error (gray lines), and thus the precision, remains stable across a wide range of contamination, meaning that the error in parameter estimation is quite systematic.

*ω*value had any effect on the robustness, since we had observed some gross under- and overestimates for

*ω*in the main analyses. We therefore assessed robustness for three values of

*ω*. While we fixed

*μ*, and Δ to the values used before (standard vector), we increased

*ω*from 100 to 200 (standard value) to 300 ms. Results (Figure 8B) for the first three parameters seem to be rather constant across the three panels and thresholds do not differ. The only effect the increase seems to have, is a systematic decrease of the threshold for

*ω*associated with an earlier underestimation of

*ω*. The effect resembles the pattern observed for the

*ω*estimates in Figure 5D. However, while this is true for the deviations in absolute measures, the relative deviation from the original parameter value becomes lower and lower with the increasing

*ω*. These effects are paralleled by a general overestimation of Δ, which is additionally reduced with increasing

*ω*.

*μ*,

*ω*, Δ) were systematically combined into a total of 576 different parameter sets (vectors of length five) for each of which a virtual experiment was conducted involving 40 subjects producing 200 responses per experimental condition. Data were generated by sampling from the probability distributions postulated in the model, best-fitting parameter values were obtained by minimizing an objective function defined for the model, and the resulting distributions of parameter estimates were evaluated with respect to accuracy (constant error) and precision (variable error).

*ω*. Further investigation revealed that the distribution of estimates for

*ω*= 300 ms can be divided into two subpopulations (Figure 9A). For

*ω*estimates is rather narrow (MAD = 11 ms). In contrast, for the complementary case,

*P*(

*I*) on the intensity parameters

*λ*and

_{A}*λ*across the SOA range (cf. the equations in the Appendix). The corresponding graphs in Figure 9B depict the two opposite cases relating to the two subpopulations defined by

_{V}*ω*, from

*ω*= 100 ms (lightest gray) to

*ω*= 700 ms (darkest gray). Obviously, for

*ω*values of 200 ms and above the curves on the right (where

*ω*.

*P*(

*I*) × Δ, the relative insensitivity of

*P*(

*I*) with respect to the exact

*ω*value for higher window widths makes determination of optimal

*ω*values less precise for the subpopulation defined by

*λ*and

_{V}*λ*and second stage parameter

_{A}*μ*is not surprising given the additivity of the two stages at the mean RT level. A similar argument holds for the high negative correlation between

*ω*and the (positive) Δ values and the moderate negative correlation between

*ω*and Δ. These relationships must be kept in mind when interpreting parameter values across different experimental conditions. On the other hand, the absence of correlation between window width

*ω*and parameters

*λ*,

_{V}*λ*, and

_{A}*μ*is encouraging for testing certain hypotheses. For example, one may probe the inverse-effectiveness hypothesis by comparing experimental conditions with high and low intensity values. If a larger window width is associated with low intensity

*λ*parameters, and vice versa, a conclusion in favor of that hypothesis may be less prone to an artifact of correlations among the parameter estimates.

*SD*) by the median absolute deviation (MAD) in the objective function, contamination does not contribute to such a great extent (Figure 8C). Thus for future research, we recommend to use the objective function (Equation 6) in this modified version.

*ω*, the parameter estimates can be used to calculate the point of integration in the (cortical) processing stream and the gain with a confidence interval of ±33 ms; and if only one experiment is run, the width of the time window of integration (TWIN) can be estimated with a C.I. of ±70 ms . Still, as the time from first activation of a cell in V1 (latency of 45 ms after stimulus onset) to that of the last cell in FEF (latency of 100 ms) is only 55 ms (Schmolesky et al., 1998), our parameter estimates cannot be used to decide upon the localization within the visuo-saccadic processing stream.

*μ*) is systematically underestimated by the same amount. As a consequence, one might falsely localize the point of integration away from early processing areas towards motor areas in the brain. While we did not set a fixed threshold, given a total reaction time in experiments with real subjects of 200 to 250 ms, we considered errors of up to 10 ms as acceptable. Precision of the parameter estimates may even be of more importance. For example, assume one investigates whether the time window is larger (wider) in one condition compared to another condition or if the window width measured for a subject differs from the optimal window, with optimality derived from a Bayesian argument (Colonius & Diederich, 2010; Diederich et al., 2008). If the window estimates (

*ω*) for the two conditions are 100 and 150 ms, say, one needs to know about the precision (the reciprocal of the variance) to infer whether this difference is real or just a product of chance.

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^{2}Some distribution-free model tests have been performed as well; see Diederich and Colonius (2008b).

^{3}Note that the parameter of the auditory peripheral processing time only enters this equation via

*P*(

*I*

_{τ}_{,ω}). One might argue that termination of first stage processing should be determined by the minimum of

*V*and

*A*. This is a plausible model version but experimental tests did not find strong support for it (Colonius & Arndt, 2001).

^{5}Including implausible and unrealistic parameter values presents an additional stress test of the model. While a high degree of flexibility of a model to accommodate even implausible parameter combinations may be considered a drawback, it should be realized that the model does not know about their implausibility and, as long as plausible parameters are recoverable, there is no reason to be worried.

*V*for the visual and

*A*for the acoustic stimulus have an exponential distribution with parameters

*λ*and

_{V}*λ*, respectively. Thus, for

_{A}*t*≥ 0, and

*f*(

_{V}*t*) =

*f*(

_{A}*t*) ≡ 0 for

*t*< 0. The corresponding distribution functions are referred to as

*F*(

_{V}*t*) and

*F*(

_{A}*t*), respectively. The visual stimulus is the target and the acoustic stimulus is the nontarget. By the model where

*τ*denotes the SOA value and

*ω*is the width of the integration window. Computing the integral expression requires that we distinguish between three cases for the sign of

*τ*+

*ω*: