The size congruity effect: Is bigger always more?
Introduction
In order to act adaptively in a constantly changing environment, animals need to keep track of different quantities, including time, length, size, area, and volume. In recent evolutionary history, also abstract quantities (numbers) have been added to this list. Research in different disciplines has recently focused on understanding the cognitive foundations of these different quantities. A still unresolved issue, however, concerns the relation between them (for reviews, see Cantlon et al., 2009, Cohen Kadosh et al., 2008).
An effect that plays a pivotal role in this discussion is the size congruity effect (Besner and Coltheart, 1979, Henik and Tzelgov, 1982). In the original size congruity paradigm (Besner & Coltheart, 1979), two numbers are presented in different physical sizes, and the participant’s task is to choose the numerically larger (or smaller) stimulus. The size congruity effect entails that it is easier to pick the numerically larger (or smaller) of two digits when this is also the physically larger (smaller) stimulus. Similarly, when the task is to pick the physically larger (or smaller) of the two digits, the task is easier if the physically larger stimulus is also numerically larger (Henik & Tzelgov, 1982). Size congruity effects are also found when only one stimulus is presented per trial. In this paradigm, the task is to judge whether the presented digit is larger or smaller than a fixed standard (e.g., standard 5) (Schwarz and Heinze, 1998, Schwarz and Ischebeck, 2003). The current paper addresses the origins of this effect using both computational and behavioral studies.
The size congruity effect shows that numerical and physical (size) dimensions are not processed in complete independence. An interaction between different quantities must be assumed somewhere along the processing stream. This interaction can take place at the input, representation, decision, or output level (Verguts & Fias, 2008). It is still debated at which of these levels the size congruity effect originates. In the literature, two opposing accounts have been put forward (e.g., Cohen Kadosh et al., 2007, Schwarz and Heinze, 1998). Schwarz and Heinze (1998) have clearly identified both positions, so we take their definitions as examples. In the first account, the size congruity effect is taken as evidence for representational overlap between the dimensions of number and physical size. According to this account, it is the case that “[…] both, the digit’s physical size and its numerical value are first mapped onto an integrated internal analog representation, which is then processed further to activate the appropriate response” (Schwarz & Heinze, 1998, p. 1168). We will call this the shared representation account (Fig. 1A). In this account, size congruity originates at the representational level, i.e. a level where numerical and physical size would be jointly represented in an analog format. A stimulus configuration in which numerical size correlates positively with physical size is congruent (e.g., in the stimulus configuration 2 8, the numerical and physical size are both small for the left stimulus and both large for the right stimulus). A stimulus configuration in which the two dimensions correlate negatively, is incongruent (e.g., in the stimulus configuration 8 2). An instance of this shared representation theory is Walsh’s influential theory of magnitude (ATOM), according to which there are common metrics for different quantitative dimensions, including space, time, and number (Bueti and Walsh, 2009, Walsh, 2003). Many recent studies on congruity effects between numerical size and other quantities have been interpreted in terms of ATOM (e.g., Chiou et al., 2009, Cohen Kadosh et al., 2008, Cohen Kadosh and Henik, 2006, Cohen Kadosh et al., 2007, Kaufmann et al., 2005, Pinel et al., 2004, Xuan et al., 2007). A theoretical problem with this proposal, and with the shared representation account more generally, is that the common metric for the shared dimensions has remained unspecified. Put simply, it is unclear whether 1 s (quantity in the time dimension) corresponds to 1 cm or 1 m (quantity in the length dimension); or to 1 g or 1 kg (quantity in the weight dimension); and so on. One exception is the proposal by Meck and Church (1983) for counting and time representations in rats. In their mode-control model, both counting and timing rely on a pacemaker that sends pulses to an accumulator. Depending on the task, the accumulator fills up to represent either time or the number of discrete elements (i.e., counting). They even proposed that a count corresponds to a passage of time of 200 ms (Meck & Church, Experiment 4). However, it is unclear how general this count-to-time mapping is. Furthermore, the model cannot easily be extended to other quantitative dimensions.
The second account for explaining the size congruity effect suggests that it results from the fact that, although numerical and physical size are initially processed separately, the two separate processing pathways interact at the decision level. According to this second account it is the case that “[…] size and numerical information are first processed in parallel, functionally independent channels and that both of them can activate a specific “subresponse” ” (Schwarz & Heinze, 1998, p. 1168). We can think of these subresponses as the decision alternatives that are imposed by the task and we therefore call this account the shared decisions account (Fig. 1B). In a standard size congruity paradigm, the decision alternatives (e.g. left larger or right larger) have a one-to-one mapping to the responses (e.g. left hand or right hand). Therefore, dissociating between the decision level and the output level is neither possible nor relevant here. Importantly, the shared decisions account assumes that the size congruity effect is due to the fact that the two dimensions activate the same (in the congruent case) or different (in the incongruent case) task-relevant, discrete codes. Hence, according to this account, congruity can be defined only relative to the task at hand. For example, in a magnitude comparison task (but not necessarily in other tasks), the stimulus configuration 2 8 is congruent because the numerical and physical size both activate the right larger code. In contrast, the stimulus configuration 8 2 is incongruent because the numerical dimension activates the left larger code, whereas the physical dimension activates the right larger code. In a sense, the activation of the decision units also establishes a representation, namely of the decision that is made. This could cause confusion about what is exactly meant by a shared representation account. However, the definitions of Schwarz and Heinze (1998) are clear: In the shared representation account, the size congruity effect originates at the level where numerical and physical magnitude are jointly coded in an analog format. This information about numerical and physical size is lost at the decision level. We reserve the term “shared representation” for a representation that codes numerical and physical size in a shared analog format, before task-specific operations are applied to it.
Although many studies have discussed the interaction between numerical and physical size (e.g. Algom et al., 1996, Ansari et al., 2006, Henik and Tzelgov, 1982, Kaufmann et al., 2005, Pinel et al., 2004, Schwarz and Ischebeck, 2003), few have been designed to directly oppose the shared representation and the shared decisions account. Schwarz and Heinze (1998) recorded event-related potentials (ERPs) from six electrodes to determine the locus of the size congruity effect. In their results, the first reliable effects of congruity at the frontal electrodes occur at 280 ms after stimulus presentation for the physical comparison task and at 368 ms for the numerical comparison task. The authors interpreted this finding as evidence for early interactions and hence for a shared representation. They also examined the lateralized readiness potential (LRP), an ERP component that is thought to reflect motor preparation and execution (Coles, 1989, Hackley and Miller, 1995, Masaki et al., 2004). An initial deflection of the LRP towards the incorrect response in incongruent trials compared to congruent trials is considered as evidence for response competition between the correct response activated by the relevant dimension and the incorrect response automatically activated by the irrelevant dimension (e.g., Gevers, Ratinckx, De Baene, & Fias, 2006). Schwarz and Heinze (1998) interpreted the absence of such a deflection in their data as additional support for a shared representation. However, as also argued by Szücs and Soltész (2008), the absence of an initial LRP deflection cannot be taken as evidence in favor of a shared representation. If the irrelevant dimension is processed more slowly than the relevant dimension (e.g. because the relevant dimension is emphasized by task instructions), then it is possible that the irrelevant dimension does not cause an initial LRP deflection and yet slows down selecting the correct response. Szücs and Soltész (2008) also recorded ERPs in a size congruity paradigm. They found both early (between 150 and 250 ms after stimulus presentation) and late (between 300 and 430 ms) facilitation and interference effects of numerical and physical dimensions. They suggested that the size congruity effect appears at multiple levels of stimulus processing and response preparation. Cohen Kadosh et al. (2007) combined ERP measures with functional magnetic resonance imaging (fMRI) in both a numerical and a physical comparison task. A region of interest analysis of the fMRI data from the primary motor cortex showed an interference effect in the hemisphere ipsilateral to the hand for the correct response. It was concluded that congruity is not completely resolved until response initiation. In their ERP analyses, the authors found an initial deflection of the LRP when numerical distance was large, but not when numerical distance was small. It was argued that response competition plays a role only when cognitive load is low. Under a high cognitive load, a shared representation would cause the congruity effect. However, as discussed above, the absence of an LRP effect does not justify this conclusion. A more general problem with the available ERP evidence is that there is no consensus on what constitutes an “early” or “late” ERP component. For example, both Schwarz and Heinze (1998) and Cohen Kadosh et al. (2007) categorize the congruity effect around 300 ms after stimulus onset as early, whereas Szücs and Soltész (2008) categorize this as a late component.
In sum, the available behavioral and neuroimaging results are not conclusive in dissociating between the shared representation and shared decisions accounts. A crucial point that blurs the discussion is the fact that both accounts so far have remained underspecified. Only when both are specified in sufficient detail will it be possible to resolve this debate. As mentioned above, specifying the shared representation account has an inherent indeterminacy problem (what is the common metric for quantities?). On the other hand, the shared decisions account can be straightforwardly modeled with a dual route architecture. This approach has proven successful in congruity paradigms similar to numerical and physical size congruity. Cohen, Dunbar, and McClelland (1990) for example simulated the well-known Stroop effect (Stroop, 1935) with a computational model implementing a dual route architecture. In their model, an “ink color” and a “word” route both activate one of the two possible responses (“red” or “green”). An attentional bias strengthens the task-relevant route (i.e. “ink color” in a classic Stroop task) to ensure that the correct response is selected (e.g. response “red” when the word the ink color is “red”). The automatically activated task-irrelevant route (“word”) causes facilitation in congruent trials (e.g. the word “red” printed in red) by activating the correct response unit and it causes interference in incongruent trials (e.g. the word “green” printed in red) by activating the incorrect response unit. Later, also other congruity effects were successfully modeled with a dual route architecture, including the Eriksen flanker effect (Cohen et al., 1992, Yeung et al., 2004), the Simon effect (Zorzi & Umiltà, 1995) and the SNARC effect (Gevers, Verguts, Reynvoet, Caessens, & Fias, 2006).
In the current study, we took a similar approach to model the size congruity effect. For this purpose, we started from our earlier computational model of number processing (Verguts, Fias, & Stevens, 2005). This model simulates behavioral performance in a number of tasks, but here we focus on number comparison. The model uses place coding (i.e. number-selective coding) to represent numerical magnitude on two input layers, one for each of the to be compared numbers. These input layers then activate the output layer, which codes for the decision that has to be made. If the instruction is to choose the larger number, the units of the output layer code for “left digit is larger” and “right digit is larger” (if the responses are arranged left and right). After training the model on the comparison task, the pattern of weights between the input and output layers produces a comparison distance effect (for details, see Verguts et al., 2005). Here, we implement only the comparison mechanism of the Verguts, Fias, and Stevens model in a simplified way. To simulate the size congruity paradigm, we add to this a similar mechanism for physical size comparison. Like the model of Cohen et al. (1990), the task-relevant route (e.g. numerical size comparison) and the task-irrelevant route (e.g. physical size comparison) interact only at the decision level. At this level, the units that code for the decision alternatives (e.g. “left digit is larger” and “right digit is larger”) are shared between the numerical and the physical dimension. Hence, the resulting model is an instance of the shared decisions account. Notably, the model is always trained on numerical and physical size comparison separately. Only in the test phase, the model is confronted with the size congruity paradigm. The distinction between the relevant dimension and the irrelevant dimension is made by simply attenuating the activation from the irrelevant dimension (for details, see Appendix B). With four behavioral experiments and four simulations, we show that the principle of physical and numerical routes operating independently but converging at the decision level can account for many aspects of the size congruity effect. In Experiment 1, we first replicate the typically observed interactions of numerical and physical distance with size congruity and test whether the shared decisions model can simulate these effects. In Experiments 2–4, we directly oppose the two accounts. In Experiment 2, it is tested whether a size congruity effect is present in both a magnitude judgment task and a parity judgment task because only the shared representation account predicts that it will be observed in both tasks. We further investigate the influence of the task on the size congruity effect in Experiment 3. Here, we use a close/far task (Santens & Gevers, 2008) to test whether modulations of the size congruity effect follow the predictions of the shared representation account or of the shared decisions account. In the first three experiments, numerical size is the relevant dimension and physical size the irrelevant dimension. In Experiment 4, we investigate whether the conclusions based on Experiment 3 still hold when changing the relevant and the irrelevant dimension. We use the same close/far task as in Experiment 3, but now with physical size as the relevant dimension and numerical size as the irrelevant dimension.
Section snippets
Experiment 1
The size congruity effect can be affected by manipulating the discriminability between the values on the relevant and irrelevant dimension (Pansky & Algom, 1999). Therefore, we calibrated the physical sizes of the stimuli used in this study in order to guarantee equal discriminability on the two dimensions. In Experiment 1, this was done separately for each participant on the first day of the experiment (see Appendix A). On the second day, we used the calibrated stimuli in a standard size
Simulation 1
In Simulation 1, we show that the shared decisions account can be implemented in a computational model with a dual route architecture. We evaluate whether the model can simulate the effects of Experiment 1 and explain how these effects originate from the model.
Experiment 2
According to the shared decisions account, the size congruity effect originates from the fact that the numerical size route and the physical size route activate shared decision alternatives. These alternatives are set by the task instructions. In Experiment 1 and Simulation 1, the decision alternatives were “left larger” and “right larger” or “left smaller” and “right smaller”. In the current experiment, we investigate what happens when the physical and numerical magnitude of all stimuli stays
Method
The models used in Simulation 2 are presented in Fig. 5A and B. They are similar to the model of Simulation 1, but now only 1 stimulus is present for both dimensions. The models were trained on both magnitude judgment and parity judgment. After training, the performance of the models was tested using the 16 stimuli from Experiment 2 that varied in numerical and physical size.
Results and discussion
After training, the model responded correctly to all stimuli. Simulated RT is plotted in Fig. 4. The model showed a
Experiment 3
In Experiment 2, we showed that the size congruity effect is absent when the decision alternatives required for the task are not automatically activated by the irrelevant dimension. However, caution should be taken in making strong conclusion based on this null effect. It could be argued that in the parity task of Experiment 2, participants were able to completely ignore the physical dimension, because parity does not apply to this dimension. Therefore, in Experiment 3, we further explore the
Method
The model used for the magnitude judgment task in Simulation 2 was adapted to give the decision close (to the mean size) or far (from the mean size) for both numerical size and physical size. After training, simulated RT was obtained for the same 16 stimuli as used in Experiment 3.
Results and discussion
The model gave the correct answer on all stimuli. Simulated RT is plotted in Fig. 7. The results from the model are in accordance with the behavioral results. When congruity is defined according to the shared
Experiment 4
So far, in our behavioral experiments and simulations, we investigated the interaction between numerical size as the relevant dimension and physical size as the irrelevant dimension. In the shared decisions model, the numerical and the physical size route are formally equivalent and therefore interchangeable. The model thus predicts a similar congruity effect when the relevant dimension is physical size and the irrelevant dimension is numerical size. In experiment 4, we investigate whether the
Method
The model simulation was exactly the same as in Simulation 3, except that now the activity from the numerical dimension was attenuated relative to the activity from the physical dimension.
Results and discussion
The model gave the correct response to all stimuli. Simulated RT is plotted in Fig. 8. The results from the model are in accordance with the behavioral results and very similar to the results of Simulation 3. In particular, simulated RT was sensitive to congruity as defined by the shared decision account
General discussion
In this study, we compared the shared representation and the shared decisions accounts of the size congruity effect. Both accounts were hitherto underspecified. In addition, we have argued that the shared representation account suffers from a conceptual indeterminacy problem. The shared decisions account, on the other hand, can be considered as belonging to a broader theoretical framework, explaining congruity effects in terms of dual route processing. We specified this account in a dual route
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