Most theories of numerical cognition assume that the perception of a quantity is independent of that which the quantity describes (termed an abstract quantity representation). Beck’s cognitive theory of depression, in contrast, assumes that depressed individuals maintain negative perceptual biases and that depressed individuals’ perception of quantity will be dependent on that which the quantity describes. Here, we explore the nature of quantity representations by assessing whether level of depression and valence of events influences individuals’ perceptions of numerical quantities. In a number bisection task, we presented participants with three quantities: one associated with the time until a positive event, one associated with the time until a negative event, and a target number. The participant was asked to judge whether the quantity denoted by the target number was closer to the time until the positive or negative event. Results indicated that event valence influenced the perception of quantity and this perceptual bias interacted with the level of depression. Thus, these findings indicate that quantity representations are malleable and are represented non-abstractly in the brain.
There is an ongoing debate in the field of numerical cognition concerning whether there exists a single, abstract psychological representation of quantity that links to all numerical symbols (e.g.,
Numbers are symbols that denote quantity. A number’s quantity representation refers to the constellation of features that are encoded, stored, or derived by the brain with respect to the magnitude denoted by a numerical symbol. Accepting the law of perceptual variability (
There is a continuing debate concerning whether there exists a single, abstract quantity representation that links to all expressions of quantity (e.g.,
McCloskey’s single representation theory states that the cognitive number system is comprised of three separate modules – number comprehension, production, and calculation – that all communicate through a single abstract quantity representation (
Support has emerged for the various mechanisms involved in McCloskey’s number processing system, primarily through the study of subjects with brain damage (
Recently, the view that numbers are represented by an abstract quantity representation has been challenged by theories proposing that quantity is not independent of the symbolic system or the items it represents (
Cohen’s Multiple Quantity Representation model suggests that each different numerical format activates a separate, non-abstract quantity representation (
Whether quantities are represented abstractly or non-abstractly has implications beyond the field of numerical cognition. For example, there is abundant evidence suggesting that depressed and non-depressed individuals perceive the world differently (
Beck’s cognitive theory of depression posits that depressed individuals have specific negative cognitions that revolve around pessimistic views concerning the self, the world, and the future (known as the
Numerous studies have demonstrated support for Beck’s cognitive theory of depression (e.g.,
Negativity biases have also been demonstrated to influence a depressed individual’s interpretation and expectation of what the future holds (
Beck’s cognitive theory of depression, therefore, predicts that level of depression should influence how individuals process and understand numerical quantities when the numerical quantities are associated with affective content. For example, Beck’s theory of depression predicts that depressed individuals should perceive the magnitude associated with a particular number (e.g., 5) as shorter if it describes the time (5 months) until a negative event (lose your job) relative to when it describes a time until a positive event (win the lottery). This prediction, however, can only manifest if quantities are represented non-abstractly. Otherwise, perceived quantity will be unaffected by that which the quantity describes.
Here, we assess whether quantity representations are abstract or non-abstract by testing the relation between participant’s scores on a depression scale (the BDI-IIi) and their perceptions of quantity. We assessed participants’ perception of quantity with a modified number bisection taskii. Past research has found the numerical bisection task to be an adequate assessment of underlying quantity representations (e.g.,
Our number bisection task consisted of two numbers – one on the right, and one on the left – each associated with a valenced event statement. One of the statements (e.g., the left) was negative (e.g., “You will lose your job in 3 months”), and one of the statements (e.g., the right) was positive (e.g., “You will win the lottery in 48 months”). A third number was positioned between the two event statements (termed the
Consistent with non-abstract quantity representation theories, we predict as BDI scores increase, participants will perceive quantities describing the time until the negative event as smaller than those same quantities describing the time until a positive event. This perception will push their perceived midpoint towards the positive event. As BDI scores decrease, this midpoint bias towards the positive event will decrease. These predicted results, where midpoint biases are influenced by the affect of the event as depression scores increase, would support non-abstract quantity representation theories.
Two hundred and thirty-three undergraduate students were recruited to participate in exchange for class credit. Sample size was determined by (a) setting a minimum number of participants, (b) estimating the time necessary to collect that number of participants, (c) posting all available experimental slots for the time estimated in “b,” and (d) running all participants who signed up. We set the minimum number of participants at 200 to ensure enough power to accurately estimate the relation between the numerical bias functions and depression. This procedure resulted in the collection of more than the minimum number of participants because of a higher than expected sign-up rate.
All stimuli (with the exception of the BDI) were presented on a 24-in. LED color screen controlled by a Mac mini. Participants completed a paper version of the Beck’s Depression Inventory II (BDI-II;
The life events used in the current study were compiled during an earlier study that created a comprehensive list of validated and standardized affect-related life events (
We used Beck’s Depression Inventory II (BDI-II) to assess participants’ depressive symptoms.
Each trial of the number bisection task consisted of two affective event statements presented on the right and left side of the screen. One event was always positive and one was always negative. The side of the negative event was randomized. The affective event statements each contained a number that indicated the time (in months) until that event would occur. In between the two event statements (in the center of the computer screen) was a probe number. The participant’s task was to judge whether the probe number was quantitatively closer to the number on the right or the number on the left (see
At the beginning of each trial, participants were presented the events with the numbers masked. This was done so that the participants could read and understand the events without being able to gather any information about the numerical symbols. The structure of this part of the trial went as follows. First, the event statement with a mask covering its associated number on the left side of the screen was presented for 300 ms per word. The numbers were masked with three pound signs (i.e., “###”). A timer bar was also presented to the left of the event. The timer bar was gradually filled in with white from top to bottom as the time ran out to read the event. The 300 milliseconds per word time limit gave participants enough time to read the event without making them wait too long for the timer to complete. Once the allotted time to read the left event elapsed, the second event was presented on the right side of the screen with its associated number also masked for 300 ms per word. At this point, both events were visible. Once the allotted time to read the right event elapsed, the timer bar completely filled and turned blue.
At this point, participants were asked to judge the valence of the events. This was done to ensure that participants read and paid attention to the events presented. Half of the participants were instructed to identify the event that was more positive, whereas the other half of participants were instructed to identify the event that was more negative. Half of the participants were assigned to use the mouse to make responses and half of the participants used buttons on the keyboard. We implemented both input methods to ensure that response input method did not influence the results (the data confirmed they did not).
Once participants made a valence response, the two numbers of the interval within the event statements were unmasked and a probe number appeared centered between the events. All three numbers were presented on the same horizontal line on the computer screen so that participants could quickly see the numbers once they were unmasked. As quickly as possible, participants were to identify the event statements that contained the number that was numerically closest to the probe number.
The numbers paired with the affective events represented the to-be-bisected interval. Similar to
Each participant completed a total of 56 trials during the number bisection task. For 28 of the trials, a positive event was linked to the smaller number and a negative event to the larger number. For the remaining 28 trials, a positive event was linked to the larger number and a negative event to the smaller number. The order of the trials was randomized.
To make a bisection response, participants determined which number (within either the positive or negative event statement) was quantitatively closest to the probe number. The probe number ranged from -4 to 4 units from the true midpoint of the presented interval. This range included 0 so that the true midpoint was presented. For every trial, the probe number was chosen at random from within the -4 to 4 unit range. Between subjects, this resulted in varying numbers of trials encountered for each of the nine possible distances from the true midpoint.
Participants made their responses in one of two ways, either by using the mouse or by pressing designated keys on the keyboard. Half of the participants made their responses by using the mouse and clicking on either the left or right event. For these participants, the cursor was not visible on the computer screen. Instead, a yellow outlined box appeared to indicate which side of the interval the cursor was hovering over. The yellow box automatically adjusted in size depending on the length of the event statement. The other half of the participants made their responses by using the keyboard, where the “d” key selected the event on the left and the “k” key selected the event on the right.
To ensure that participants did not have enough time to perform any calculations, a reaction time pressure was implemented. This was done by having a beep sound that informed participants if their bisection responses were too slow. For the first trial, the RT deadline was set at 3 seconds (
Participants were run individually in a small, dark room on a Mac Mini running OSX with a 24-inch LED screen. The ceiling light in each participant’s testing room was turned off to eliminate distraction, make it easier for participants to view the experiment on the computer screen, and to ensure that all participants completed the experiment under the same environmental conditions. At the start of the experiment, the computer presented instructions that explained the number bisection task and instructed participants to put on a pair of headphones for use during the task. The instructions provided information about the task format, how to make event ratings (e.g., “you are to choose which event you feel is more negative”), and how to make quantitative judgments (e.g., “identify which of the life event numbers is quantitatively closest to the center number.” Here, the probe number is more simply referred to as the center number in the instructions). In addition, the instructions included examples that illustrated how to complete the task and provided information about the reaction time pressure (e.g., “you will hear a beep if your response was too slow”). Participants were required to wear headphones in order to hear the beeping sound presented throughout the experiment (the time pressure manipulation used is described in detail above). Each participant completed four practice trials that were identical to the experimental trials. Participants completed 56 experimental trials as described above. Participant’s accuracy levels and reaction times were recorded for all trials.
Upon finishing the number bisection task, participants completed Beck’s Depression Inventory (BDI-II), which was used to measure the presence of depression symptoms for each participant. Once they completed the BDI-II, participants were thanked for their participation and were given course credit, marking the end of the experiment.
Six participants did not complete the experiment due to technical issues and were removed from data analysis.
Participants’ depression levels were estimated by their scores on the BDI-II, where higher scores indicate higher levels of depression. Scores on the BDI-II ranged from 0-42, with an average depression score of 10.34 (
To ensure that our analyses reflected participants who relied on estimates rather than calculations, we removed any participants who performed 40% or more of their bisection judgments in longer than 3 secondsiii. This removed a total of 27 participants. Nonetheless, the patterns of observed results are virtually identical regardless of whether all participants are included in the analysis. In addition, we removed all trials where the bisection judgment exceeded 5 seconds. This constraint eliminated 1.2% of the data.
To determine whether the valence of presented events influenced the perceived midpoint, we separated the data into two conditions: negative event on the left side of the interval (termed
For the Negative Right condition, there was a significant positive relationship found between distance of the probe number from the midpoint and proportion of “above” responses (intercept = 0.47, slope = 0.03), with distance from the midpoint accounting for a significant proportion of the variation in “above” responses,
To test the hypothesis that participants’ BDI-II scores would be related to their midpoint bias, we first calculated each participant’s proportion of “above” responses in the Negative Right and Negative Left conditions. When the participant’s midpoint bias is closer to positively valenced events in the Negative Right condition, the proportion above will be relatively large. In contrast, when the participant’s midpoint bias is closer to positively valenced events in the Negative Left condition, the proportion above will be relatively small. To detect a consistent bias, we subtracted each participant’s proportion of “above” responses in the Negative Right condition from their proportion of “above” responses in the Negative Left condition. From these calculations, a negative number indicates a stronger midpoint bias towards the positive event (i.e., negativity bias - compressed quantities associated with a negative event), whereas a positive number indicates a stronger midpoint bias towards the negative event (i.e., positivity bias - compressed quantities associated with a positive event). To assess the influence of depression scores on midpoint bias, we (i) averaged the bias scores for each BDI II score, and (ii) calculated a linear regression with depression scores squared as the predictor variable and the difference in “above” responses as the criterion variable. The results indicated that depression scores accounted for a significant proportion of the variation in the difference of “above” responses,
We assessed whether state of mind and the information associated with a quantity influenced the perception of the quantity. The predominant theories of numerical cognition assume that the psychological representation of quantity is abstract. That is, there exists a single quantity representation that is separate and independent from the information associated with it (e.g.,
Independence between the perception of a quantity and that which the quantity describes is a fundamental property of an abstract quantity representation. Therefore, if quantity representation is abstract, people’s perception of quantity should not be influenced by the valence associated with an event or an object – regardless of the participant’s state of mind. Beck’s cognitive theory of depression, in contrast, is founded on the assertion that depressed individuals maintain negative perceptual biases (
In the current study, we tested these predictions by examining how varying levels of depression influence participants’ perceptions of quantities when quantities are paired with positive and negative events in a number bisection task. The number bisection task revealed a significant effect of event valence on perceived midpoint. Specifically, when a quantity is associated with a time until a positive event, participants in general (collapsed across BDI-II scores) perceive that quantity as smaller than when that same quantity is associated with a time until a negative event. We also found that as depression scores increased, participants had stronger biases to perceive quantities associated with a time until a negative event as smaller than quantities associated with a time until a positive event. These findings (a) contradict the predictions stemming from an assumption of abstract quantity representations and (b) are predicted by Beck’s cognitive theory of depression.
Because the present data contradict predictions stemming from an assumption of an abstract representation of quantity, theories that rest on this assumption must be viewed more critically. Cohen and colleagues (e.g.,
Our data not only support the predictions of Beck’s cognitive theory of depression, but also the theory of depressive realism. According to the depressive realism theory, non-depressed individuals systematically demonstrate unrealistic cognitions that are influenced and sustained by optimistic biases (e.g.,
In addition, the influence of depression score on quantity perception sheds light on a largely unexplored aspect of the literature regarding the negativity biases of depressed individuals. Particularly, current literature does not explain whether negativity biases become stronger as depression levels increase. The current study begins to explore whether negativity biases are more robust in severely depressed individuals (those with a BDI score of 29 or higher) in comparison to mildly depressed individuals (those with a BDI score of 14-19). By assessing perceptions of quantity across the continuum of BDI scores, our results indicate that individuals’ estimation biases do become more drastic as depression levels increase. Thus, our data suggest that the negativity biases of depressed individuals become stronger as depression levels increase. Given the non-clinical sample used in the current study and the limited amount of information that can be drawn from BDI-II scores, future research examining clinically depressed populations is recommended to provide more insight into this perceptual bias.
In sum, the results of the current study indicate that the valence of an event, as well as an individual’s level of depression (as measured by the BDI-II), influence an individual’s perception of quantity. Specifically, low BDI scorers overestimate the quantities associated with the time until a future negative event and underestimate the quantities associated with the time until a future positive event. As BDI scores increase, this trend reverses. These results (1) challenge models of numerical cognition based on abstract quantity representations, and (2) support both Beck’s Cognitive Theory of Depression and the theory of depressive realism.
For this study, the data and
The data and the
Here, we will refer to high BDI scorers to represent individuals who experience depressive symptoms and low BDI scorers to represent individuals who experience mild or no depressive symptoms. Due to the nonclinical sample recruited for the current study, we apply these terms in order to outline the current hypotheses. It is important to note that all analyses assessed the relation between scores on the BDI-II and number perception. We did not dichotomize the scale into high and low scorers. We simply use these terms here to simplify predictions.
Although stimulus valence has not been shown to influence number perception, there is some evidence that stimulus valence can influence line bisection tasks. Unfortunately, this evidence is a bit contradictory. When the to-be-bisected line is composed of affect words, the emotional content magnified the typical leftward bias (
There was a trend towards a negative between RT and BDI scores,
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The authors have declared that no competing interests exist.
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