Integration of anatomical and external response mappings explains crossing effects in tactile localization: A probabilistic modeling approach

Badde, Stephanie; Heed, Tobias; Röder, Brigitte

doi:10.3758/s13423-015-0918-0

Integration of anatomical and external response mappings explains crossing effects in tactile localization: A probabilistic modeling approach

Theoretical Review
Published: 08 September 2015

Volume 23, pages 387–404, (2016)
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Integration of anatomical and external response mappings explains crossing effects in tactile localization: A probabilistic modeling approach

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Stephanie Badde^1,2,
Tobias Heed¹ &
Brigitte Röder¹

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Abstract

To act upon a tactile stimulus its original skin-based, anatomical spatial code has to be transformed into an external, posture-dependent reference frame, a process known as tactile remapping. When the limbs are crossed, anatomical and external location codes are in conflict, leading to a decline in tactile localization accuracy. It is unknown whether this impairment originates from the integration of the resulting external localization response with the original, anatomical one or from a failure of tactile remapping in crossed postures. We fitted probabilistic models based on these diverging accounts to the data from three tactile localization experiments. Hand crossing disturbed tactile left–right location choices in all experiments. Furthermore, the size of these crossing effects was modulated by stimulus configuration and task instructions. The best model accounted for these results by integration of the external response mapping with the original, anatomical one, while applying identical integration weights for uncrossed and crossed postures. Thus, the model explained the data without assuming failures of remapping. Moreover, performance differences across tasks were accounted for by non-individual parameter adjustments, indicating that individual participants’ task adaptation results from one common functional mechanism. These results suggest that remapping is an automatic and accurate process, and that the observed localization impairments in touch result from a cognitively controlled integration process that combines anatomically and externally coded responses.

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Introduction

In touch, an object’s location is initially encoded in a skin-based, anatomical reference frame: The spatial arrangement of neurons in the primary and secondary somatosensory cortex reflects the position of their receptive fields on the body surface (Penfield and Boldrey 1937). Yet, to plan actions toward the object, its location must be transformed into another, posture-dependent reference frame (Pouget et al. 2002; Sober and Sabes 2005). Any perceived tactile stimulus seems to be transformed into an external-spatial reference frame (Driver and Spence 1998; Yamamoto and Kitazawa 2001a; Shore et al. 2002; Spence et al. 2004; Soto-Faraco et al. 2004; Röder et al. 2004; Schicke and Röder 2006; Heed and Azañón 2014; Heed et al. 2015), even when such recoding is currently not required (Kitazawa 2002; Azañón et al. 2010a). This process of coordinate transformation is addressed as tactile remapping (Driver and Spence 1998) and has been associated with regions of the intraparietal sulcus in posterior parietal cortex (Azañón et al. 2010b; Bolognini and Maravita 2007; Renzi et al. 2013). Crucially, both the original, anatomical representation as well as the remapped, external representation are maintained by the brain (Heed and Röder 2010; Buchholz et al. 2011; 2013), that is, after transformation both reference frames are available to estimate the location of the touch. Consequently, the brain might base tactile localization by default on information coded in both reference frames, rather than relying responses on the external reference frame alone.

To differentiate between different tactile reference frames, researchers have manipulated body posture to create situations in which anatomical and external reference frames point to conflicting left–right responses. For example, when the hands are crossed, the right hand (anatomical reference frame) is located in the left hemispace (external reference frame). A well-known consequence of hand crossing is that participants’ ability to judge which of two successive tactile stimuli, one applied to each hand, occurred first is markedly impaired compared to an uncrossed hands condition (Yamamoto and Kitazawa 2001a; Shore et al. 2002). These so-called crossing effects in temporal order judgments (TOJ) are thought to arise from conflicting left–right mappings in the different spatial reference frames (Yamamoto and Kitazawa 2001a, b; Shore et al. 2002; Röder et al. 2004; Schicke and Röder 2006; Heed et al. 2012; Heed and Azañón 2014; Azañón et al. 2015). However, it is unclear whether this conflict bears on the remapping process, impairing the establishment of the external response mapping (Yamamoto and Kitazawa 2001a; Röder et al. 2004; Kóbor et al. 2006; Azañón and Soto-Faraco 2007; Kitazawa et al. 2008), or whether it unfolds its effect at the stage of information integration once remapping is complete (Shore et al. 2002; Badde et al. 2013; 2014; Badde et al. 2015). Additionally, it has been suggested that crossing effects might simply be caused by the additional demands of the unusual, presumably uncomfortable crossed posture (Azañón et al. 2010b; Longo et al. 2010; Haggard et al. 2003) or by additional hemispheric transmission costs, which are specific to the crossed posture (Buchholz et al. 2012; Canzoneri et al. 2014). Furthermore, it has been debated whether crossing effects critically depend on the application of more than one stimulus in the TOJ task (Yamamoto and Kitazawa 2001a; Kitazawa 2002; Shore et al. 2002). To address these questions, we used a two-step strategy. First, we tested whether tactile TOJ crossing effects are task- and stimulation-specific. To this end, we modified the original TOJ paradigm, by varying, within participants, the task instructions and the number of stimuli. Second, we compared the ability of different models to account for participants’ behavior in these different tasks. These models were designed on the two theoretical accounts for crossing effects, that is, non-reliable externally coded response mapping in the crossed posture versus integration of conflicting left–right mappings in the crossed posture.

Materials and methods

Behavioral experiments

Participants

Nineteen right-handed participants (six male, aged 20-37 years, mean 25 years) from the community of Hamburg volunteered for the experiment. All had normal or corrected-to-normal vision and did not report any impairments in tactile sensitivity. Participants received course credit or were compensated with 7 Euro/hour. The experiment was conducted in accordance with the general guidelines of the Declaration of Helsinki (World Medical Association, 2008).

Apparatus

Participants were seated at a table, resting their hands and elbows on the table surface. The index fingers were placed on response buttons. The arms were positioned either crossed or uncrossed. A foam cushion was placed underneath one arm to avoid skin contact between the hands and arms in the crossed posture. The distance between the response buttons for the two hands was 25 cm. Tactile stimulators (Oticon bone conductors, type BC 461-012, sized about 1.6 x 1 x 0.8 cm) were taped to the middle fingers, covering the whole fingernail and some proximate skin. Stimuli consisted of 15-ms-long vibrations at a frequency of 200 Hz. The experiment was controlled by the software Presentation, version 14.5 (Neurobehavioral Systems, Albany, CA, USA), which interfaced with custom-built hardware to drive the stimulators and to record responses. To shield off any auditory cues produced by the tactile stimulators, participants wore ear plugs as well as headphones that played white noise. Instructions and a fixation cross were displayed on a monitor placed 80 cm in front of the participant.

Tasks

Temporal order judgment (TOJ)

Two tactile stimuli were successively presented, one stimulus to each hand. Participants were instructed to report the temporal order of the two stimuli and to respond by a button press with the hand that was stimulated first (Fig. 1).

First touch localization (FTL)

Two tactile stimuli were successively presented, one stimulus to each hand. Participants were instructed to localize the first stimulus as fast as possible and to ignore the second stimulus. Responses were given by button press with the hand at which participants had perceived the relevant, first touch. Thus, the FTL task differed from the TOJ task only with respect to the task instructions.

Single touch localization (STL)

A single tactile stimulus was applied, either to the right or to the left hand. Participants had to localize the stimulus and respond by a button press with the respective hand. Thus, the STL task differed from the TOJ and FTL tasks in the number of presented stimuli. Instructions for all three localization tasks, TOJ, FTL, and STL, stressed response accuracy but not response speed.

Detection task

One tactile stimulus was applied, either to the right or to the left hand. The task was to detect the stimulus and to respond as fast as possible with the designated response hand. The response hand was fixed throughout experimental blocks and was not related to the stimulated hand.

Design

Two within-participant factors were manipulated in all experiments: hand posture (factor: posture, levels: uncrossed and crossed) and the hand which received the relevant (first) stimulus (factor: hand, levels: left and right hand). In the TOJ and FTL tasks, the time interval between the two stimuli was varied in addition (factor: stimulus onset asynchrony (SOA), levels: 50, 80, 110, 150, 200, 250, and 300 ms). In the detection task, the stimulus either occurred at the designated response hand or at the other hand (factor: congruency, levels: congruent and incongruent). The response hand was either the left or the right hand (factor: response hand).

Procedure

Every trial lasted 3000 ms. Each experiment comprised 200 trials, divided into four blocks of 50 trials each. Participants were encouraged to rest between blocks. The factors hand and SOA varied within blocks, whereas posture varied every other block. Condition order and the order of experiments were counterbalanced across participants.

Data preparation

Trials with reaction times below 150 ms and above 1500 ms were excluded from the analysis (3.5 % of all trials).

TOJ and FTL performance were quantified in two ways: first, by mean accuracy averaged across all SOAs (Cadieux et al. 2010), and second, by slope values of a probit regression over SOAs up to 250 ms (with negative SOA indicating “left hand first” stimuli; Fig. 2) (Shore et al. 2002). For the probit analysis (Finney 1947), response accuracy was transformed into “right hand first” responses, indicating whether participants judged the anatomically right hand to be stimulated first (Yamamoto and Kitazawa 2001a; Shore et al. 2002). Then, probit values of the ratio of “right hand first” responses per SOA were calculated by applying the inverse normal distribution (probit(p _{right hand first})=z⇔Φ_0,1(z)=p _{right hand first}). The resulting probit values were linearly regressed onto their corresponding SOA. The regression slope was used as a measure of TOJ performance with steeper probit slopes indicating better performance (Shore et al. 2002; Heed and Azañón 2014).

STL performance was quantified by mean accuracy only, as this task involved just one touch.

In the detection task, participants rarely missed a stimulus. Therefore, reaction time (RT) rather than accuracy was analyzed.

Data analysis

A repeated measures ANOVA with factors posture and experiment was conducted on accuracy data from the three localization experiments (TOJ, FTL, and STL). Paired t tests on mean accuracy and, where applicable, probit slopes were used to assess crossing effects separately for each experiment. Difference scores between uncrossed and crossed postures were calculated for accuracy and, where applicable, probit slopes. First, these difference scores were compared across experiments using pairwise paired t tests. Second, for each pair of experiments Pearson’s r of the difference scores was calculated and tested for significant deviation from zero.

For the detection task, a repeated measures ANOVA (type III sum of squares) with factors posture, response hand, and congruency was conducted on RT.

p-values were adjusted for multiple comparisons according to Holm (1979). All reported results were significant at a type I error level of 5 % unless noted otherwise. η ² and d were used as standardized effect size measures. Note that d was calculated from the mean difference and the standard deviation of the differences and, thus, is not directly comparable to d values obtained in between subjects designs.

Model structure

We compared several variants of two main models. We termed the first main model integration model: this model explained crossing effects as resulting from integration of potentially conflicting anatomical and external left–right response mappings. We termed the second main model non-integration model: this model explained crossing effects as resulting from reduced reliability of the external response mapping in the crossed posture.

Integration models

Our integration model incorporates the idea that the probability of a right-hand response (l o g i t ⁻¹(𝜃))^{Footnote 1} is based on the weighted combination of the response codes (ρ) derived from anatomical and external reference frames.

$$ \theta = \omega_{anatomical} \rho_{anatomical} + \omega_{external} \rho_{external}^{1} $$

(1)

In the experiments that we report in the present study, tactile stimuli were always applied to one of two locations, the left hand or the right hand, with each hand positioned in the left or right hemispace. Thus, for both reference frames, the task-relevant spatial information could be reduced to left–right categories. We arbitrarily used -1 to code for left and 1 to code for right. The left–right mapping of the tactile stimuli depended not only on the stimulated hand, but in addition on hand posture as well. The anatomical mapping was constant across hand postures, whereas the external left–right mapping switched in the crossed posture: For example, the left crossed hand was located in right space, and, therefore, its external response code (ρ _{e
x
t
e
r
n
a
l}) was 1 rather than -1. As a consequence, anatomical and external response mappings were congruent in the uncrossed posture, so that the combination of the anatomical and external response codes (𝜃) was determined by the sum of their weights. In contrast, anatomical and external left–right mappings were opposed in the crossed posture, so that 𝜃 depended on the difference of their weights.

Responses were always coded relative to the right hand, as is custom in the analysis of TOJ crossing effects (Yamamoto and Kitazawa 2001a; Shore et al. 2002). To allow comparisons across all three tasks, we initially averaged across SOAs in the TOJ and FTL tasks. Thus, the data consisted of the number of “right-hand” responses per participant, stimulated hand, and posture for each experiment. For each condition, the number of “right-hand” responses was described as a binomially distributed random variable. The inverse logit of 𝜃, that is, the inverse logit of the combined anatomical and external left–right response codes corresponded to the Bernoulli parameter of the Binomial, describing the probability to give a “right-hand” response in a single trial. The graphical model of one integration model variant is depicted in Fig. 3.

In a separate analysis, we focused on the tasks that comprised two tactile stimuli per trial, that is, FTL and TOJ tasks. This analysis included effects of the time interval between the stimuli, that is, the factor SOA, in the task comparison. The proportion of “right-hand” responses per SOA (with negative SOA indicating “left-hand” stimuli) was fitted with a logistic function. The SOA-dependent probability of a right-hand response, l o g i t ⁻¹(𝜃 ^′), depended on 𝜃 as the slope parameter of the logistic function and on an independent estimate of the inflection point of the sigmoid, the point of subjective simultaneity (PSS) (𝜃 ^′(s)=𝜃⋅SOA(s)/c o n s t+PSS).

We tested several variants of this integration model to explore the consistency of the incorporated reference frames weights (ω _anatomical and ω _external) across experiments, postures, and participants. In model terms, we gradually reduced the number of free parameters based on theoretical assumptions. Model variants are listed in Table 1.

Table 1 Goodness of fit of all tested models

Full size table

First, we tested a model that only had the constraint that stimulus localization to the left and to the right hand (h)^{Footnote 2} was described with the same set of weight parameters, but allowed for separate anatomical and external weight parameter estimates for each individual (i), posture (p), and experiment (e) (Table 1, Model 1):

$$\begin{array}{@{}rcl@{}} \mathrm{Model 1:} \quad\theta(i,h,p,e) &= \omega_{\text{anatomical}}(i,p,e) \rho_{\text{anatomical}}(h,p) \\ & \quad + \omega_{\text{external}} (i,p,e) \rho_{\text{external}}(h,p)^{1,2} \end{array} $$

Yet, our experiments were based on the hypothesis that crossing effects are caused by conflicting anatomical and external left–right mappings in the crossed posture, implying that crossing effects are related in all three localization experiments. Consequently, we devised a model that described the data from all three localization experiments simultaneously by introducing one task context parameter for each reference frame (δ _{a
n
a
t
o
m
i
c
a
l} and δ _{e
x
t
e
r
n
a
l}). These task context parameters adjusted the individual anatomical and external weights across all participants to capture general differences between responses in the three experiments. The anatomical and external task contexts (δ _{a
n
a
t
o
m
i
c
a
l} and δ _{e
x
t
e
r
n
a
l}) of the TOJ task served as reference for the other two tasks and were, accordingly, set to 1. Task context parameters for the STL and FTL tasks were allowed to vary across postures. Thus, in this model variant the individual reference frame weights (ω _{a
n
a
t
o
m
i
c
a
l} and ω _{e
x
t
e
r
n
a
l}) still varied across postures but were now constant across experiments (Table 1, Model 2):

$$\begin{array}{@{}rcl@{}} \mathrm{Model~2:} \quad\theta(i,h,p,e) &= \omega_{\text{anatomical}}(i,p) \rho_{\text{anatomical}}(h,p) \delta_{\text{anatomical}}(p,e) \\ & \quad + \omega_{\text{external}} (i,p) \rho_{\text{external}}(h,p)\delta_{\text{external}}(p,e)^{1,2} \end{array} $$

In the next model variant, we tested whether participants changed their reference frame weighing with posture, by restricting the participant-specific weights (ω _{a
n
a
t
o
m
i
c
a
l} and ω _{e
x
t
e
r
n
a
l}) to be constant across postures. We term this model “pure conflict model”, because it attempts to explain crossing effects only by individually weighted anatomical and external left–right response codes. Accordingly, the task context parameters (δ _{a
n
a
t
o
m
i
c
a
l} and δ _{e
x
t
e
r
n
a
l}) were also held constant across postures (Table 1, Fig. 3, Model 3):

$$\begin{array}{@{}rcl@{}} \mathrm{Model~3:\!\!\!} \quad\theta(i,h,p,e) &\!= \!\omega_{\text{anatomical}}(i) \rho_{\text{anatomical}}(h,p) \delta_{\text{anatomical}}(e) \\ &\quad + \omega_{\text{external}} (i) \rho_{\text{external}}(h,p)\delta_{\text{external}}(e)^{1,2} \end{array} $$

To further reduce the number of free parameters, we next tested whether it is necessary to assume that both anatomical and external weights vary across participants. To address this question, either the anatomical or the external reference frame weights (ω _{a
n
a
t
o
m
i
c
a
l} and ω _{e
x
t
e
r
n
a
l}) were held constant across participants (Table 1, Models 4 and 5):

$$\begin{array}{@{}rcl@{}} \mathrm{Model~4:} \quad\theta(i,h,p,e) & =& \omega_{\text{anatomical}} \rho_{\text{anatomical}}(h,p) \delta_{\text{anatomical}}(e) \\ &+& \omega_{\text{external}} (i) \rho_{\text{external}}(h,p)\delta_{\text{external}}(e)\\ \mathrm{\mathrm{Model~5:}} \quad\theta(i,h,p,e) &=& \omega_{\text{anatomical}}(i) \rho_{\text{anatomical}}(h,p) \delta_{\text{anatomical}}(e) \\ &+& \omega_{\text{external}} \rho_{\text{external}}(h,p)\delta_{\text{external}}(e)^{1,2} \end{array} $$

Next, we tested an integration model variant with anatomical and external reference frame weights held constant across participants. This model explained inter-individual differences only by a general performance level parameter (π)(Table 1, Model 6):

$$\begin{array}{@{}rcl@{}} \mathrm{Model~6:}\!\!\! \quad\theta(i,h,p,e) &\,=\, \pi(i) \cdot \left( \omega_{\text{anatomical}} \rho_{\text{anatomical}}(h,p) \delta_{\text{anatomical}}(e) \right.\\ & \qquad \qquad\!\!+ \left. \omega_{\text{external}} \rho_{\text{external}}(h,p)\delta_{\text{external}}(e)\right)^{1,2} \end{array} $$

We designed two logistic models to assess the effects of SOA and fitted them to the data from TOJ and FTL tasks. In addition, these models tested whether participants had an individual bias towards one of the hands by including a free PSS parameter (Table 1, Models 7 vs. 8):^{Footnote 3}

$$\begin{array}{@{}rcl@{}} \mathrm{Model\; 7:} \quad\theta^{\prime}(i,h,p,e,s) &\! = \text{SOA}(s)/const \cdot (\omega_{\text{anatomical}}(i) \rho_{\text{anatomical}}(p) \delta_{\text{anatomical}}(e) \\ &\qquad\qquad\qquad\qquad+ \omega_{\text{external}} (i) \rho_{\text{external}}(p)\delta_{\text{external}}(e))\\ \mathrm{Model\; 8:}\quad\theta^{\prime}(i,h,p,e,s) & = \text{SOA}(s)/const \cdot (\omega_{\text{anatomical}}(i) \rho_{\text{anatomical}}(p) \delta_{\text{anatomical}}(e) \\ &\qquad\qquad\qquad\qquad+ \omega_{\text{external}} (i) \rho_{\text{external}}(p)\delta_{\text{external}}(e))\\ &\quad + \text{PSS}(i)^{1,2,3} \end{array} $$

Non-integration models

The second set of models were based on the hypothesis that crossing effects in tactile localization tasks result from unreliable external left–right mappings in the crossed but not in the uncrossed posture. This assumption was modeled by varying the weight given to the correctly mapped external response code between crossed and uncrossed postures. In contrast to the integration models, the non-integration models comprised the external, but not the anatomical left–right response code. Consequently, only one weight rather than two weights were estimated to calculate the logit of the probability of a right-hand response (𝜃). This design reflects that integration of the transformed external with the original, skin-based response codes is not assumed within this account. Again, the number of free parameters was reduced over a series of model variants.

We started with a binomial model without any constraints. That is, separate response probabilities, l o g i t ⁻¹(𝜃), (Model 9) were calculated for each participant, stimulated hand, posture, and experiment. As in the integration model, l o g i t ⁻¹(𝜃) was implemented as a Bernoulli parameter. This model had no explanatory value, as one parameter per data point was fitted, but rather provided a measure of the goodness of fit that could be maximally achieved by fitting a binomial model to our data (Table 1, Model 9):

$$\mathrm{Model~9:} \quad\theta(i,h,p,e) = \omega_{\text{external}}(i,h,p,e) \rho_{\text{external}}(h,p)^{1,2} $$

First, we constrained the non-integration model to describe the localization of stimuli on both hands with the same set of weights, but allowing the external reference frame weight to differ across participants, postures, and experiments (Table 1, Model 10):

$$\mathrm{Model~10:} \quad\theta(i,h,p,e) = \omega_{\text{external}}(i,p,e) \rho_{\text{external}}(h,p)^{1,2} $$

Second, we constrained the non-integration model to explain data from all three experiments simultaneously (Table 1, Model 11), allowing individual weights for each posture, but using these parameters for all three experiments. Differences between experiments were again captured by a task context parameter (δ _{e
x
t
e
r
n
a
l}) that varied across experiments and postures. The number of parameters in this non-integration model variant was equal to the number of parameters of the pure conflict model (Model 3, Fig. 3).

$$\mathrm{Model\ 11:} \quad\theta(i,h,p,e) = \omega_{\text{external}}(i,p) \rho_{\text{external}}(h,p)\delta_{\text{external}}(p,e)^{1,2} $$

In analogy to the integration model, we extended the non-integration model by implementing the factor SOA for the TOJ and FTL tasks. In these logistic models (Models 12 and 13), the weighted external left–right response code 𝜃 was used as the slope of a logistic function and, additionally, a separate PSS parameter was estimated for each sigmoid. Again, we started with an unconstrained model, estimating separate parameters for each participant, stimulated hand, posture, and experiment, to show the maximal fit achievable with a logistic model (Table 1, Model 12).

$$\begin{array}{@{}rcl@{}} \mathrm{Model\ 12:} \quad\theta^{\prime}(i,p,e,s)= \omega(i,p,e) \rho(p) \cdot \text{SOA}(s)/const + \text{PSS}(i,p,e)^{1,2,3} \end{array} $$

Finally, we constructed a logistic variant of the non-integration model that allowed the external weight to vary across participants and postures, but not across experiments (Table 1, Model 12). This model had the same number of free parameters as the corresponding integration model, Model 8.

$$\begin{array}{@{}rcl@{}} \mathrm{Model~13:} \quad\theta^{\prime}(i,p,e,s) &&\!\!=\omega_{\text{external}}(i,p) \rho_{\text{external}}(h,p) \delta_{\text{external}}(p,e) \cdot \text{SOA}(s) \\ &&+\text{PSS}(i)^{1,2,3}\end{array} $$

Model implementation

The posterior distributions of each model parameter were approximated with Markov Chain Monte Carlo (MCMC) sampling using WinBUGS (Lunn et al. 2000). Three Markov chains of 1000 samples were initialized with random values. The first 100 samples were disregarded to allow the chains to reach their equilibrium distribution (Rouder and Lu 2005).

We employed a hierarchical model design that allowed predicting response probabilities of new participants. Thus, free parameters that varied across participants were drawn from an overarching population distribution, rather than being estimated independently. The only free parameters that varied across participants were the anatomical and external weight parameters (ω _{a
n
a
t
o
m
i
c
a
l} and ω _{e
x
t
e
r
n
a
l}). In model variants in which the weights varied not only across participants but also across experiments or postures, independent population distributions were estimated for each of these variables. The population distributions were Gaussians, for which mean and standard deviation were estimated during model fitting. In the binomial models, these Gaussians had $\mathcal {N}(1, 2)$ priors for the mean values and Uniform(0,2) priors for the standard deviations. In the logistic models, priors of the mean values were $\mathcal {N}(2, 2)$ distributions. When the weights were held constant across participants we used an $\mathcal {N}(1, 2)$ prior. Some model variants additionally estimated the task context parameters δ _{a
n
a
t
o
m
i
c
a
l} and δ _{e
x
t
e
r
n
a
l}. The free task context parameters (STL compared to TOJ and FTL compared to TOJ) had an $\mathcal {N}(1, 2)$ prior. The logistic model variants additionally contained a free PSS parameter which always had an $\mathcal {N}(0, 2)$ prior.

Goodness of fit and model comparison

For model evaluation, the posterior predictive distribution was sampled: In each iteration, a data point (k, number of right- hand responses) was generated based on the current parameter estimates.

To visualize the goodness of fit, the posterior predictive distribution was plotted together with the observed values. A formalized test of the goodness of fit was performed using posterior predictive checks (Gelman et al. 1996), which compare the lack of fit of the model for the observed data with the lack of fit of the model when fitted to random data generated by the same model. The χ ²-statistic was used to quantify the respective lack of fit. The comparison of the two resulting χ ² values determined the Bayesian p value of the model. As these two discrepancies are in the ideal case identical, a suitable model has a Bayesian p value near 0.5, whereas values close to 0 or close to 1 indicate that the chosen model is not appropriate. Additionally, we calculated the determination coefficient R ² and the root mean squared deviance (RMSD) of all samples from the posterior predictive distribution compared to the experimentally observed data. Note that RMSD is estimated larger and R ² smaller, when using the full posterior predictive distribution rather than the maximum a posteriori (MAP) estimate as is often done. For model comparison, the Bayesian predictive information criterion (BPIC) (Ando 2007) was assessed with model complexity p V measured as half the posterior variance of the model-level deviance (Gelman et al. 2004).

Model validation

We used a leave-one-out approach to identify the predictive capabilities of the best binomial and logistic models. Using MCMC-sampling, we acquired posterior predictive distributions rather than point estimates for the prediction of missing values. First, participants performance in uncrossed postures was predicted based on their performance in crossed posture conditions, and vice versa. Second, participants’ performance in one task was used to predict performance in the other task(s). Third, individual participants’ performance was predicted based on the data from all other participants.

The quality of these predictions was compared to the fit of posterior predictive distributions generated by the respective unconstrained models (Models 9 and 11 for the binomial and logistic models, respectively), and to the predictions of a Bayesian full linear regression model with random intercepts (for the binomial model) and a Bayesian full probit regression model with random intercepts (for the logistic model), respectively. For each comparison, we calculated paired t tests on participant level RMSD values.

Parameter analysis

During model evaluation, the posterior distribution of each free parameter was approximated. We characterized these distributions by calculating their MAP estimates and the respective 95 % credible intervals (CI). Weight parameters ω _{a
n
a
t
o
m
i
c
a
l} and ω _{e
x
t
e
r
n
a
l} were interpreted as reliably influential if their CI did not contain zero. This is because a weight of 0 would indicate that the respective response code was not considered for the response at all. Task context parameters δ _{a
n
a
t
o
m
i
c
a
l} and δ _{e
x
t
e
r
n
a
l} were deemed reliable if their CI did not include 1. This is because, a task context parameter of 1 would denote identity of the weights with the reference weights, that is, with the weights of the TOJ task, indicating that the parameter did not express a difference between tasks.

To further analyze the estimated weight parameters, we correlated the MAPs of the individual anatomical weights (ω _{a
n
a
t
o
m
i
c
a
l}) and the individual external weights (ω _{e
x
t
e
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l}), as well as and each of these weight parameters with the participants’ average crossing effects (that is, accuracy in uncrossed minus accuracy in crossed conditions).