The role of probability and intentionality in preschoolers causal generalizations. David M. Sobel 1. Jessica A. Sommerville 2. Lea V.

Transcription

1 Running Head: PROBABALISTIC CAUSAL GENERALIZATION The role of probability and intentionality in preschoolers causal generalizations David M. Sobel 1 Jessica A. Sommerville 2 Lea V. Travers 3 Emily J. Blumenthal 2 Emily Stoddard 1 1 Department of Cognitive and Linguistic Sciences, Brown University 2 Department of Psychology, University of Washington 3 Department of Psychology, Loyola University Chicago in press, Journal of Cognition and Development

2 Probabilistic Causal Generalization 2 Abstract Three experiments examined whether preschoolers recognize that the causal properties of objects generalize to new members of the same set given either deterministic or probabilistic data. Experiment 1 found that 3- and 4-year-olds were able to make such a generalization given deterministic data, but were at chance when they observed probabilistic information. Five-year-olds reliably generalized in both situations. Experiment 2 found that 4-year-olds could make some probabilistic inferences, particularly when comparing sets that had no efficacy with sets with in which some members had efficacy. Children had some difficulty discriminating between completely effective sets and stochastic ones. Experiment 3 examined whether 3- and 4-year-olds could reason about probabilistic data when provided with information about the experimenter s beliefs about causal outcomes. Children who were more successful on standard false belief measures were more likely to respond as if the data were deterministic. These data suggest that children s probabilistic inferences develop into early elementary school, but preschoolers might have some understanding of probability when reasoning about causal generalization. We discuss the nature of that probabilistic understanding.

3 Probabilistic Causal Generalization 3 The role of probability and intentionality in preschoolers causal generalizations Before children reach elementary school they have sophisticated causal reasoning abilities across a variety of domains (Bullock, Gelman, & Baillargeon, 1982; Schulz & Gopnik, 2004; Shultz, 1982). Not only can children interpret events they observe in terms of their causal structure, they can explain and reason counterfactually across domains of knowledge (Harris, German, & Mills, 1996; Schult & Wellman, 1997; Sobel, 2004). Such investigations have led researchers to consider the nature of children s causal inference and learning abilities, which have often been described as quite sophisticated (e.g., Gopnik, Sobel, Schulz, & Glymour, 2001; Schulz & Sommerville, 2006; Sobel, Tenenbaum, & Gopnik, 2004, see also Gopnik et al., 2004). The present investigation has two goals concerning children s causal inference. First, we examine children s ability to generalize objects causal properties from category information. Category-based induction has been examined at great length in young children (e.g., Gelman, 2003; Sloutsky & Fisher, 2004). Preschoolers recognize that objects in a common category tend to share common properties, including their causal efficacy on other objects (e.g., Gelman & Markman, 1986; Gopnik & Sobel, 2000; Welder & Graham, 2001). But these investigations have only presented children with deterministic data; the present studies examine whether children s generalization abilities persist when they observe probabilistic data. Many investigations of preschoolers reasoning suggest that they have difficulty with probabilistic concepts (e.g., Davies, 1965; Piaget & Inhelder, 1975; Schlottman, 2000). Piaget and Inhelder (1975), for example, showed that 3-7-year-olds choices on

4 Probabilistic Causal Generalization 4 various probabilistic tasks were arbitrary, and that it was not until the concrete operational stage that children could differentiate between deterministic and probabilistic relations. They suggested that preoperational children only focused on surface associations, like magnitude estimations. However, other investigations have suggested that 4-6-year-olds have some understanding of probability (e.g., Beck et al., 2006; Fischbein, 1975; Kushnir & Gopnik, 2005; Kuzmak & Gelman, 1986, Perner, 1979). For example, Perner (1979) found that 4-5-year-olds could reason appropriately about probabilistic contrasts when the contrast was quite large (7-1 vs. 1-7). Further, he found that 4- and 5-year-olds did show a learning effect: correct performance on probability judgments was more likely at the end of the experimental session after children saw the results of 24 trials. Although he also found evidence that 4- and 5-year-olds could be tricked when magnitude estimations conflicted with representations of probability, these data suggest that 4- and 5-year-olds might be an important age group to examine with regards to their inferential abilities about probabilistic data. Similar evidence comes from work by Kushnir and Gopnik (2005), who presented 4-year-olds with a machine that activated when certain objects were placed on it. They varied the frequency with which individual objects activated the machine, so that some of the objects that children observed activated the machine every time they were placed on it, while others did so only some of the time. Specifically, they showed children an object that made the machine go three out of three times that it was placed on the machine and an object that made the machine go only one out of the three times that it was placed on it (specifically, the first time). Children were more likely to choose the object that always

5 Probabilistic Causal Generalization 5 activated the machine when asked to pick the best one to make the machine go. They also found that when children observed one object activate the machine two out of three times, while another activated it one out of three, their responses depended on the efficacy observed the last time children saw the object on the machine. Two open questions are raised by this study. The first question is whether preschoolers make similar inferences about various probabilistic contrasts, or if there was something particular about these frequencies that allowed children to reason correctly. The second question is whether children can generalize the efficacy they observe to novel exemplars. Kushnir and Gopnik (2005) examined children s ability to reason about those particular objects we consider whether children can extend this information to make inferences about other members of the same object set. In the present experiments, we examined inferences over a variety of probabilistic contrasts (within the limits of children s memory) and used a generalization method in which children observe two sets of objects that each individually have or do not have efficacy on a machine similar to the one used by Kushnir and Gopnik (2005). Children were then asked to infer that efficacy to novel members of the two sets. In Experiment 1, three-, 4-, and 5-year-olds were asked to generalize a novel causal property of an object from observing whether other members of the same set had this property. The data children were shown were either fully deterministic (all members of one category and no member of the other category possess the property) or fully probabilistic (some members of each set possessed the property, but unequally). We expected all children to extend the causal property appropriately when shown deterministic data. At issue was whether they would make the same inference when

6 Probabilistic Causal Generalization 6 shown probabilistic data. Experiment 2 then considered 4-year-olds ability to make these inferences under a variety of probabilistic contrasts in which one set had deterministic efficacy (either all or none of that set had efficacy) while the other efficacy in the other set was probabilistic. Kushnir and Gopnik (2005) also found that contextual factors affected children s inferences. They found that the efficacy of the children s own actions affected their judgments. In their conflicting intention condition, children saw the experimenter place each block on the machine twice, one activating it both times, the other activating it neither time. Then, the children were given a chance to place each block on the machine once, and the efficacies of the objects were reversed. Children chose the object they had placed on the machine as the one that would make the machine go, and did so more often than when they just observed all three of these interventions. Kushnir, Wellman, and Gelman (2009) argued that these data suggested that children are not generally biased by their own actions; rather, they proposed that children use their own actions as a source of information when presented with ambiguity. These investigations have all focused on how children learn and make inferences from information they directly observe. However, much causal knowledge is not learned by direct observation and interaction with the world, but rather through communication with other people. A growing body of literature in cognitive development has examined whether children learn differently from reliable and unreliable sources of knowledge. Preschoolers appear to take the reliability of an individual as a source of information into account when making inferences about the data s/he generates (e.g., Birch et al., 2008; Harris & Koenig, 2006; Jaswal & Malone, 2007; Koenig, Clement, & Harris, 2004;

7 Probabilistic Causal Generalization 7 Koenig & Harris, 2005). Children also appear capable of such inferences even when given probabilistic information. Pasquini et al. (2007) found that 4-year-olds were more likely to trust an informant who was 75% accurate than one who was 25% accurate (3- year-olds struggled with this inference, and responded at chance given this information). These data suggest that 4-year-olds might have some probabilistic reasoning abilities with regards to their generalization, as children extend an individual s reliability to novel information. But these investigations focused on whether individuals are reliable. In Experiment 3, we examined whether children recognize that particular data points are reliable or unreliable. Several researchers have argued that children believe causal relations are deterministic (Bullock et al., 1982; Schulz & Sommerville, 2006). Schulz and Sommerville (2006) in particular, found that when children observe probabilistic data, they attempt to seek out hidden causes of that stochastic information. If children believe a hidden cause is present that could influence a particular object s efficacy, they might treat the probabilistic aspects of the data they observe as unreliable, and reason as if the causal properties of the objects are deterministic. To do this, Experiment 3 replicated the probabilistic trials used in Experiment 1, but added the experimenter s reaction to each object s efficacy. For the objects in the set that made the machine go two out of three times, the experimenter reacted as if the efficacy of the object was appropriate and the one ineffective object was an abnormality; for the objects in the other set, s/he acted conversely treating the one object that made the machine go as abnormal and the two that failed to activate the machine as appropriate. Children might believe that the objects in the set have deterministic efficacy, and that the probabilistic data they observe results from a hidden cause. If children think

8 Probabilistic Causal Generalization 8 that the data they observe represents deterministic information, they should be more likely to make inferences like the deterministic trials in Experiment 1 as opposed to the probabilistic ones. While it is possible that all children would be sensitive to this contextual information, to understand that another s surprising reaction indicates the possibility of a hidden cause being present, children must know something about the belief state of the experimenter. Thus, we predict that not all preschoolers will benefit from this information; rather, children who succeed on standard measures of false belief will be more likely to show this benefit. We presented a false belief measure across all the experiments to ensure that we only observed such a relation in Experiment 3. Experiment 1 Three-, 4-, and 5-year-olds were given a category generalization task under two different conditions. These ages were examined because some of the previous investigations suggested that children s probabilistic understanding begins to emerge at this age (Kushnir & Gopnik, 2005; Kuzmak & Gelman, 1986; Perner, 1979). On some trials, children were shown deterministic data, in which all members of one set of objects possessed a novel causal property and no member of another set of objects possessed that property. On other trials, children were shown probabilistic data in which some objects in each set possessed this property although one set was more likely to possess the property than the other. At issue was whether children could use this frequency information to generalize the causal property to novel objects: in both cases, we showed children a new member of each set and asked which one of these novel objects has the causal property.

9 Probabilistic Causal Generalization 9 Methods Participants. The final sample consisted of sixteen 3-year-olds (8 girls, M = months, Range: months), sixteen 4-year-olds (8 girls, M = months, Range: months), and sixteen 5-year-olds (8 girls, M = months, Range: months) who were recruited from a list of hospital births and several preschools in a suburban area. Six additional children were tested, but not included in the final sample: four because of experimenter error, one because of parent interference, and one had a diagnosis of autism. Forty-three children were Caucasian, 1 was Hispanic, 2 were Asian, 1 was from middle-eastern descent, and 1 was from mixed descent. All children came from middle- to upper-class backgrounds, but no formal measure of SES was administered. Materials. The machine was a 16.3 cm x 13.5 cm x 5.5 cm wooden box (painted grey) with a red Lucite top. When an object was placed on the machine, the top depressed, and if a hidden switch (controlled by the experimenter) was in the on position, the machine would light up (red) and play music as long as the object was on it (if the switch was in the off position, the machine would never activate). In the main task, we used four pairs of two sets of four identical blocks (i.e., 32 blocks). Each set was a different color and distinguishable shape from the other sets: yellow triangles paired with small orange cubes, blue spheres paired with green rectangles, large black cubes paired with brown cans, small white cylinders (thinner and shorter than the brown cans) paired with beige square rods. Two others blocks (a purple rectangle, shaped differently from the green blocks, and a pale yellow cylinder shaped different from the white cylinders) were used in the warm-up. A crayon box containing candles and a Band-aids

10 Probabilistic Causal Generalization 10 box containing pencils was also used. Figure 1 shows the machine, as well as an example set of demonstration and test objects. [Insert Figure 1 Approximately Here] Procedure. Each participant was allowed time to become familiar with the experimenter and was then brought into a game room, where they were told they would play a game. The child sat opposite the experimenter at a table. The experimenter placed the machine on the table and said, This is my machine. It s a very special machine. Some things make it go, and some things don t. Let s take a look. Pretest. Children were shown two blocks, different in appearance. The experimenter placed each block on the machine; one activated it and the other did not. Children were then asked, Which block will make the machine go? All children chose the object they had just observed activate the machine. Test Trials. At the start of the trial, children were shown two sets of three identical objects, placed on either side of the machine (see Figure 1). Each block was placed on the machine individually. On deterministic trials, all three blocks in one set activated the machine and no block in the other set did so. On probabilistic trials, two out of three blocks in one set activated the machine while one out of three blocks in the other set did so. After children observed this demonstration, they were shown the two test blocks, and were asked which one would activate the machine. If children responded that both objects would make the machine go (which they did on 3% of the deterministic trials and 11% of the probabilistic trials), they were asked to choose the best one that would make the machine go.

11 Probabilistic Causal Generalization 11 On the probabilistic trials, the order of the activation was always that the first objects activated the machine the last object that was placed on the machine in both sets failed to activate it. Thus, if children were only paying attention to the causal efficacy of the last object in each set, they would respond at chance levels. Children received two deterministic and two probabilistic trials. Each trial used blocks with different appearances; care was taken such that no particular shape or category of shape (e.g., square vs. round objects) reliably activated the machine more often. The spatial location of the higher frequency object was counterbalanced across trials. These trials were presented in one of four orders in which the two types of trials were always blocked together, but which counterbalanced whether the child saw deterministic trials first. False Belief Task. Children were also given a standard unexpected contents false belief task. Children were shown either a Crayola crayon or Band-aids box and were asked what was inside. All children responded appropriately. Children were then shown the actual contents of the box, which were birthday candles or pencils, respectively, and children were told specifically what was inside of the box. The actual contents were then put back into the box and the box was closed. The experimenter then asked the falsebelief-other question: Let s say your <friend another child who was mentioned before the task began> comes in here. He/She has never seen this box before. What will he/she think is in the box? After children responded, the experimenter then asked the falsebelief-self question: Before I showed you what was in the box, what did you think was in the box? After children responded, the experimenter asked a control question: What is really in the box?

12 Probabilistic Causal Generalization 12 Results and Discussion We scored the number times the child chose the object whose corresponding set always activated the machine (in the deterministic condition) or whose corresponding set activated the machine more often (in the probabilistic condition). Preliminary analyses revealed that responses did not significantly differ between the two deterministic or two probabilistic trials in any condition, sign tests ns, so these data were combined into scores that ranged from 0-2. These scores are shown in Table 1. Preliminary analyses also revealed no difference in responses between genders on either the deterministic or probabilistic trials, both χ 2 (2, N = 48)-values < 1.00, both p-values ns., or among the four orders, both F(3, 44)-values < 1.11, both p-values ns. As a result, we did not consider these variables further. [Insert Table 1 Approximately Here] A 2 (Probability: Deterministic vs. Probabilistic trials) x 3 (Age Group) mixed Analysis of Variance was conducted with probability as a within-subject factor and age as a between-subject factor. This analysis revealed a significant main effect of probability: children had higher scores on the deterministic than probabilistic trials, F(1, 45) = 37.71, p <.001, Partial η 2 =.46. The main effect of age group was also significant: the three age groups differed in their overall scores, F(2, 450) = 6.25, p <.005, Partial η 2 =.22. A significant interaction between these variables was not found. All three age groups were more likely to use the frequency of the demonstration objects to generalize causal efficacy on the deterministic than probabilistic trials, t(30) = 2.76, 4.33, and 3.87 for the 3-, 4-, and 5-year-olds respectively, all p-values <.05, Cohen s d-values = 0.80, 1.51, and 1.37 respectively. Combining across the two types of

13 Probabilistic Causal Generalization 13 trials, the 5-year-olds scored significantly higher than the 3-year-olds overall, t(30) = 3.36, p <.05 (Scheffé Correction), Cohen s d = 1.23, but the difference between the 3- and 4-year-olds overall was not significant, nor was the difference between the 4- and 5- year-olds. Performance by each age group was then compared to what would be expected by chance (50%). The distribution of each of the three age groups responses to the deterministic trial were significantly different from what would be expected by chance, χ 2 (2, N = 16) = 17.00, 43.35, and for the 3-, 4-, and 5-year-olds respectively, all p- values <.01. Only the 5-year-olds distribution of responses was significantly different from chance on the probabilistic trials, χ 2 (2, N = 16) = 8.00, p <.05. Finally, we considered performance on the false belief measure. One child (a 3- year-old) was not administered the false belief test because of experimental error. Children received a score of 1 for each of the false belief other and false belief self question they answered correctly for an overall false belief score between 0-2. Children received an overall score of 0 if they answered the control question incorrectly. Threeyear-olds had an overall score of 0.40 (SD = 0.63), 4-year-olds had an overall score of 0.94 (SD = 0.93), and 5-year-olds had an average score of 1.69 (SD = 0.60). Unsurprisingly, there was a significant correlation between children s age and their false belief score, r(47) = 0.60, p <.001. There was also a significant correlation between performance on the probabilistic (but not the deterministic) trials and children s score on the false belief measure, r(47) = 0.29, p <.05. However, when age was factored out of that correlation, the relation was insignificant, r(44) = 0.09, ns. This suggests that the

14 Probabilistic Causal Generalization 14 correlation between false belief performance and responses on the probabilistic trials was due to the child s age. These data suggest that 3- to 5-year-olds can extend the causal properties of a set of objects onto a novel member of ostensibly the same set when all members of one set and none of the other demonstrate the causal property. Three- and 4-year-olds inferences about probabilistic data were no different from chance, but 5-year-olds did appear to recognize the difference in probabilities reliably. The distinction in performance between these two types of trials might result from preschoolers inability to reason about probabilistic data, as suggested by several classic investigations (e.g., Piaget & Inhelder, 1975). The ability to reason about probabilistic data might develop after the fifth birthday. A question that emerges from this experiment is whether preschoolers are incapable of making any kind of probabilistic inference and can only respond to deterministic data or if there are probabilistic contrasts that preschoolers do recognize. Experiment 2 investigated this issue by asking 4-year-olds to make the same inference as in Experiment 1, but contrasting other probabilities between 0 out of 3 and 3 out of 3. In doing so, we can consider several hypotheses concerning preschoolers inferential ability. One is that preschoolers had difficulty with the probabilistic contrast in Experiment 1 because neither set was deterministic. This would posit that preschoolers cannot recognize all probabilistic contrasts per se, rather they recognize that one set acts deterministically and the other does not, and respond based on that determinism (e.g., if one set never make the machine go, then the item from the other set must be the right choice). In Experiment 2, all of the trials given to children involved one set with

15 Probabilistic Causal Generalization 15 deterministic efficacy (either all of the objects made the machine go or none of them did) and one set whose members had probabilistic efficacy. If preschoolers can recognize this determinism, then they should be able to make all of these inferences. A second hypothesis is that children s understanding of determinism might also be unidirectional. Preschoolers might only make appropriate inferences when all of the objects in one set activate the machine, regardless of the efficacy demonstrated in the other set. Alternatively, they might also only make appropriate inferences when all of the objects in one set fail to activate the machine, regardless of the efficacy of the other objects. The two types of trials in Experiment 2 were designed to test this possibility. A third hypothesis is that children are not necessarily paying attention to these specific pieces of deterministic data, and instead are focused on the contrast in probabilities. When that contrast is narrow (as in the case of the 2/3 vs. 1/3 trials in Experiment 1, where one object separates the demonstration sets), preschoolers might have struggled to keep track of the information. On this view, widening the gap in probability (separating the difference in the demonstration sets by two objects) might improve performance overall. This might explain why 4-year-olds in Experiment 1 struggled making the 2/3 vs. 1/3 probabilistic inference, but Kushnir and Gopnik s (2005) 4-year-olds were able to distinguish a probabilistic contrast of 3/3 vs. 1/3 this is one of the probabilistic contrasts that we attempt to replicate in Experiment 2, and on this view, this inference should be easier for 4-year-olds than a contrast of 3/3 vs. 2/3. Of course, these hypotheses might not be mutually exclusive, so children might only be sensitive to the probabilistic distance when all of the objects have efficacy as opposed to when one set of objects does not have efficacy.

16 Probabilistic Causal Generalization 16 Experiment 2 Experiment 1 suggested that 3- and 4-year-olds could not generalize from probabilistic data. However, we only considered one probabilistic contrast (2/3 vs. 1/3). Here, we consider all the other possible contrasts using three objects in each set. We examined children s inferences when one set acted deterministically, but there was noise in the other set. Four-year-olds were asked to make the same kinds of inferences as in Experiment 1, based on data in which one set always activated the machine while the other set did so probabilistically, or in which one set never activated the machine, while the other set did so probabilistically. We chose to investigate only 4-year-olds because the results of Experiment 1 suggested a general transition between 3- and 5-year-olds ability to reason about the probabilistic data. We wanted to see if 4-year-olds could only reason about data when both sets had deterministic efficacy, or if 4-year-olds could generalize if only one set acted consistently. Method Participants. The final sample consisted of thirty-four 4-year-olds (19 girls, M = months, Range: months), who were recruited from a list of hospital births. Six additional children were tested, but not included in the final sample: five because of experimenter error, and one failed the pretest (see below). Thirty-three children were Caucasian, 1 was from mixed descent. All children came from middle- to upper-class backgrounds, but no formal measure of SES was administered. Materials. The same materials from Experiment 1 were used here. Procedure. The basic procedure here paralleled that of Experiment 1. Children were introduced to the machine in the same manner, and given the same pretest. One

17 Probabilistic Causal Generalization 17 child failed the pretest, and his data were replaced as it was uncertain whether he understood the basic nature of the task. The test trials were also similar to Experiment 1. On each, two sets of three identical objects were brought out, and each object s efficacy was demonstrated on the machine. After this demonstration, two new objects one identical to the members of the first set and one identical to the members of the second set were brought out, and children were asked which of these two objects would make the machine go. Note that children never observed these new objects placed on the machine. Children were randomly assigned to one of two conditions. Approximately half (n = 18) of the children were in the one-apart condition. They were shown four trials in which the difference between the two sets was the efficacy of one object. Specifically, children observed two trials in which all members of one set activated the machine and two out of the three members of the other set did so (3/3 vs. 2/3, called all trials). On one of these trials, the two objects that activated the machine were the first two placed on it; on the other trial, these were the last two objects placed on it. As such, children could not reliably respond correctly on both trials just by focusing on the efficacy of the first or last object placed on the machine in each set. Similarly, children observed two trials in which no member of one set activated the machine and one out of the three objects in the other set did so (1/3 vs. 0/3, called none trials). Again, for the set that had only one efficacious object, it was either the first one demonstrated (on one trial) or the last one demonstrated (on the other trial), so that the order in which objects were efficacious did not influence responding.

18 Probabilistic Causal Generalization 18 After being shown this demonstration, children were asked which of the two novel objects made the machine go. If they responded both (which they did on 9.5% of the trials overall), they were asked to just pick one. The overall order of trials was presented in one of four quasi-random orders, such that the two all trials and the two none trials were blocked together. The side that contained the higher frequency set was counterbalanced across these trials. The other children (n = 16) were in the two-apart condition. They were shown similar all and none trials, but the contrasts were 3/3 vs. 1/3 for the all trials and 2/3 vs. 0/3) for the none trials. As in Experiment 1, children were also given a false belief task (using the same method as Experiment 1) after these test trials as a control, to compare to performance in Experiment 3. Results and Discussion Children received a score of 1 if they chose the object from the higher frequency demonstration set on each trial. Preliminary analyses revealed that responses did not significantly differ between the two all or the two none trials across either condition, sign tests ns, so these data were combined into scores that ranged from 0-2. The means and distributions of these scores are shown in Table 2. Preliminary analyses also revealed no difference in responses between genders on either the all or none trials, both χ 2 (2, N = 34)-values < 2.05, both p-values ns., or among the four orders, both F(3, 33)-values < 0.80, both p-values ns. As a result, we did not consider these variables further. [Insert Table 2 Approximately Here] A 2 (Trial: All vs. None) x 2 (Condition: One-Apart vs. Two-Apart) mixed Analysis of Variance was conducted with trial as a within-subject factor and condition as

19 Probabilistic Causal Generalization 19 a between-subject factor. This analysis revealed a significant main effect of condition: children in the two-apart condition were more likely to choose the higher frequency object than children in the one-apart condition, F(1, 32) = 4.05, p =.05, Partial η 2 =.11. No main effect of trial was found. A significant interaction between condition and trial was also found F(1, 32) = 4.92, p <.05, Partial η 2 =.13. The interaction resulted from children in the one-apart condition scoring higher on the none trials than the all trials, t(17) = 2.36, p <.05, Cohen s d = This was not the case in the two-apart condition, where no difference between the trials was found, t(15) = 0.57, ns. Further, on the all trials, children scored significantly higher in the two-apart condition than the one-apart condition, t(32) = 3.05, p <.01, Cohen s d = There was no difference between the one-apart and two-apart condition on the none trials, t(32) = 0.40, ns. We also compared the distributions of responses with that of chance. Chi-Squared goodness of fit tests revealed that children in the two-apart condition responded differently from chance expectations on both the all and none trials (contrasting 3/3 vs. 1/3 and 2/3 vs. 0/3), χ 2 (2, N = 16) = and respectively, both p-values <.001. Children in the one-apart condition also responded differently from chance on the none trials (contrasting 1/3 vs. 0/3), χ 2 (2, N = 18) = 14.33, p =.001. However, children in the one-apart condition did not respond differently from chance on the all trials (specifically contrasting 3/3 vs. 2/3), χ 2 (2, N = 18) = 3.78, ns. Finally, we examined performance on the false belief task. Four children (two in each condition) were not administered false belief tasks because of experimental error. This test was scored in the same manner as Experiment 1. Children had an overall score of 0.97 (SD = 0.81). However, there was not a significant correlation between

20 Probabilistic Causal Generalization 20 performance on either the all or none trials and children s score on the false belief measure, both r(30)-values <.05, ns. These data suggest that 4-year-olds can reason about certain kinds of probabilistic data when making a generalization inference. For example, we replicated the probabilistic inferences that Kushnir and Gopnik (2005) showed 4-year-olds could make (3/3 vs. 1/3, our all trials in the two-apart condition). In general, 4-year-olds reliably inferred the contrast when one set of objects never activated the machine compared with another set that does, regardless of whether the efficacy of objects in the other set was deterministic or probabilistic. When one set of objects appeared to always make the machine go, 4-year-olds struggled to generalize to that set when most of the objects in the other set also made the machine go. This difficulty in the one-apart condition is consistent with their difficulty in the probabilistic trials in Experiment 1, where the contrast was only one object. Taken together with that experiment, 4-year-olds appear able to infer that if no objects in a set have efficacy, the reliability of the other set doesn t matter as long as at least one activates the machine. However, if both sets have members with efficacy, then the relative distance between the frequencies of that efficacy might affect inferences. An open question is whether this is the extent of preschoolers probabilistic reasoning abilities; we wanted to consider whether preschoolers could generalize based on these probabilistic data. In Experiment 3, we added a piece of contextual information to the procedure that we believed would improve the younger children s performance. We presented 3- and 4-year-olds with the same kind of probabilistic inference as in Experiment 1 (2/3 vs. 1/3), but added another piece of information on each trial: the

21 Probabilistic Causal Generalization 21 experimenter s belief about the efficacy of each object. This was done by having the experimenter react after each object demonstrated its efficacy on the machine. The reaction indicated that the experimenter expected the data to be deterministic that is, he/she reacted surprised when the one object whose efficacy was different from the other two objects in each set produced or failed to manifest that causal property. For the other two objects, the experimenter acted knowledgeable, as if the observed causal efficacy was expected. At issue is how children will treat the experimenter s reactions. We hypothesize that children are integrating the causal information they observe with contextual information inherent in these reactions. Specifically, if children look for hidden causes when shown probabilistic evidence (Schulz & Sommerville, 2006), then they might be more inclined to believe that the causal efficacy of the objects is deterministic here than in Experiment 1 as the experimenter s reaction would indicate that s/he also expected a deterministic outcome. This inference is potentially based on treating adults as reliable sources of information; research on word learning suggests that preschoolers treat adults as reliable sources until shown information to the contrary (e.g., Corriveau, Meints, & Harris, in press). Based on this work, in this procedure, children might believe that the experimenter was a reliable source of information and treat his/her reactions as relevant to the data they observe. Importantly, we are agnostic as to whether the means by which children might monitor probabilistic data is the same as how they monitor an informant s relative accuracy (e.g., Pasquini et al., 2007). But in order to appreciate that a particular piece of data is unreliable from another s reaction to it, children must represent the experimenter s knowledge state about

22 Probabilistic Causal Generalization 22 the object and its causal efficacy previous to that efficacy being demonstrated. Because we present this epistemic knowledge in the form of another s behavior, we expect children to be sensitive to this information only if they understand the relation between one s behavior and beliefs. As such, unlike Experiments 1-2, we expected performance on these trials to be related to their performance on a standard false belief task, independent of age. Because 5-year-olds in Experiment 1 responded to the probabilistic trials at better than chance levels, we investigated only 3- and 4-year-olds here. Although it would be unsurprising to find that 5-year-olds have superior understanding of false belief than 3- and 4-year-olds, these older children might potentially make probabilistic generalizations because of more than just their superior abilities on false-belief measures. Experiment 3 Three- and 4-year-olds were given the same introduction to the machine and objects as in Experiment 1. They observed only probabilistic trials, but on each trial, children were given another piece of information a reaction to each object s causal efficacy that implied the Experimenter believed the objects were deterministically related to the machine s activation. What might this reaction mean? We hypothesized that the experimenter has a belief about the nature of the causal efficacy of each object: that all members of one set and no member of the other set will activate the machine. At issue is whether children recognize this information and are influenced by it. Do they recognize that they should discount the one object that failed to activate the machine in the 2/3 set and the one object that did activate the machine in the 1/3 set, making these trials more like the deterministic trials from Experiment 1?

23 Probabilistic Causal Generalization 23 Methods Participants. The final sample consisted of sixteen 3-year-olds (9 girls, M = months, Range: months) and sixteen 4-year-olds (8 girls, M = months, Range: months), who were recruited from a list of hospital births and several preschools in a suburban area. Three additional children were tested, but not included in the final sample all because of experimental error. Twenty-nine children were Caucasian, 2 were Hispanic, and 1 was Asian. All children came from middle- to upper-class backgrounds, but no formal measure of SES was administered. Materials. The same materials from Experiment 1 were used. Procedure. Children were introduced to the machine in the same manner as Experiment 1, and given the same familiarization trial with one exception. Immediately after each object was placed on the machine, which either activated or not, the experimenter reacted in a positive manner, indicating that he expected that object to have that efficacy (e.g., Yeah, see, that s right, look at that. ). Test Trials. The structure of the test trials was the same as in Experiments 1-2. Children observed two sets of three identical objects, each placed on the machine individually. All four test trials were akin to the probabilistic trials in Experiment 1. Two out of three objects in one set and one out of three objects in the other demonstration set activated the machine. Immediately after each object was placed on the machine, the experimenter reacted to the machine s response. That reaction was consistent with the experimenter expecting the data to be deterministic. For the set that activated the machine 2 out of 3 times, the experimenter reacted positively for the two objects that activated the machine (e.g., See, that s right. Look at that. ), and for the one object that did not, the

24 Probabilistic Causal Generalization 24 experimenter reacted in a surprised manner (e.g., Huh? Oh, that s weird. That s funny. ). Similarly, for the set that activated the machine 1 out of 3 times, the experimenter reacted positively for the two objects that did not activate the machine and surprised for the one object that did activate the machine. The side of the higher frequency object was counterbalanced, as was whether the surprising event for each set occurred first or last. After the demonstration, the same test question as in Experiments 1-2 was asked: Children were shown two new objects, one of which was identical to each of the two sets, and children were asked which one of these new objects made the machine go. If children responded that both objects made the machine activate (which they did on 5% of the trials), they were encouraged that only one made the machine go, and to chose the one that did so. Children were administered a false-belief task (using the same method as in Experiment 1-2), either before or after they participated in the main task with the machine (order of tasks randomly determined). Results and Discussion Preliminary analysis demonstrated there were no differences in responses among the four trials, so these data were combined. Children received a score from 0-4 dependent on how many of the objects from the higher frequency set they chose. Preliminary analyses revealed that there was no difference in scores between children who received a trial in which the surprising event was the first thing children observed or the last thing they observed. There was also no difference in scores between children who observed a trial in which the higher frequency was on the right first or the left first. Table 3 shows the distribution of responses from both age groups.

25 Probabilistic Causal Generalization 25 [Insert Table 3 Approximately Here] Three-year-olds had an average score of 2.00 (SD = 0.73); 4-year-olds had an average score of 2.50 (SD = 1.16). These scores did not significantly differ, t(31) = -1.31, ns. As in Experiment 1, three-year-olds distribution of responding was no different from chance responding, χ 2 (4, N = 16) = 2.67, ns. Their overall accuracy level was also not different from chance, t(15) = 0.00, ns. Four-year-olds distribution was different from chance responding, χ 2 (4, N = 16) = 10.67, p <.05. However, their overall accuracy was only marginally different from chance, t(15) = 1.73, p =.10. This suggests that age might not have been a factor in accurate responses. We next considered performance on the false-belief task, which was scored in the same manner as in Experiments 1-2. One child (a 3-year-old) was not administered the false-belief test because of experimental error. Three-year-olds had an average score of 0.60 (out of 2, SD = 0.63), while 4-year-olds had an average score of 0.88 (SD = 0.96). These scores were not significantly different from one another, t(29) = -0.94, ns. However, we did observe a significant correlation between false belief scores and the number of choices on the test measure based on the greater frequency with which the objects activated the machine, r(31) = 0.49, p <.01. To investigate this correlation further, we performed a hierarchical linear regression on the total number of trials children made higher frequency choices. We first examined the effect of the child s age (scored in months), which was a non-significant factor, ΔR 2 = 0.05, F(1, 29) = 1.53, ns. We next considered how adding children s false-belief scores affected the model. This addition explained a significant amount of additional variance, ΔR 2 = 0.21, F(1, 28) = 7.93, p <.05.

26 Probabilistic Causal Generalization 26 To demonstrate that children who understood the experimenter s false belief were most likely to respond based on the higher frequency object, we considered the distribution of responses of children with a score of 2 (i.e., success on all false belief questions) against the distribution of children who did not achieve this score. These distributions (also shown in Table 3) were significantly different, χ 2 (3, N = 31) = 15.85, p =.001, φ = Because the sample size was relatively small, a linear-by-linear analysis was also performed, which also suggested a difference between the distributors, χ 2 (1, N = 31) = 7.76 p =.005. Children who scored highest on the false belief measure also outperformed children who did not when we compared their overall number of higherfrequency choices on the test trials, Mann-Whitney U = 35.00, z = -2.42, p <.05, r = As in Experiment 1, preschoolers overall judgments about probabilistic data were at chance levels. However, when the experimenter s reaction to each object s efficacy was provided, which suggested the experimenter believed the data should be deterministic, a subset of the preschoolers inferences appeared similar to the older children from Experiment 1. Specifically, children who succeeded on a standard false belief measure, regardless of age, appeared to register that the experimenter s reactions were relevant towards making an inference. An open question is exactly how these children are reasoning about this information. We will discuss several possibilities in the general discussion. General Discussion Three-, 4-, and 5-year-olds had little difficulty generalizing the causal properties of an object from a demonstrated set when shown deterministic data that all members

27 Probabilistic Causal Generalization 27 of one set had the causal property and no member of the other set did so. Preschoolers were generally unable to make the same kind of inference when shown particular probabilistic information that two out of three members of one set and one out of three members of the other set had the property. By age five, children appeared able to recognize this difference. But 4-year-olds ability to make probabilistic inferences was not completely bereft. Experiment 2 found that they could make probabilistic inferences under some circumstances: when there was large probabilistic contrast between the two sets or when one set never made the machine go. Experiment 3 found that a particular piece of contextual information helped some preschoolers inferences even when both sets contained effective objects and the difference in efficacy between the sets was small. When the experimenter expressed surprise at the different efficacy of the objects, children who were successful on a standard measure of false belief seemed able to integrate that reaction and discounted the data from the object that had the anomalous efficacy. These preschoolers appeared to reason more like the 5-year-olds given probabilistic data in Experiment 1. Experiments 1 and 2 suggest that children have no trouble with causal generalization from deterministic data, and have some probabilistic inference abilities as well. We speculate that preschoolers can recognize certain aspects of probabilistic inference. First, they can recognize that some efficacy is better than no efficacy. Across Experiments 1 and 2, children always chose the object from the higher frequency set when one set did not contain an efficacious object. Second, when both sets contain efficacious objects (i.e., both probabilities are greater than zero), children might struggle with small probabilistic contrast. In Experiment 2, preschoolers had no problem

28 Probabilistic Causal Generalization 28 reasoning about the 3/3 vs. 1/3 trials, reliably choosing the object from the set that always activated the machine, whereas they responded at chance levels when the contrast was smaller across the experiments (3/3 vs. 2/3 or 2/3 vs. 1/3). Thus, these data suggest that preschoolers understand some aspects of probabilistic reasoning at earlier ages than previously thought (e.g., Piaget & Inhelder, 1975; Schlottman, 2000). These data are also consistent with a growing body of literature that suggests preschoolers have some probabilistic reasoning abilities, particularly about causal relations (e.g., Schulz, Bonowitz, & Griffiths, 2007; Sobel et al., 2004; Sobel & Munro, 2009). However, responses to probabilistic data appear to change; there does appear to be a difference in how preschoolers and 5-year-olds (and presumably adults, e.g., Einhorn & Hogarth, 1985) reliably generalize from probabilistic data. The older children might be more likely to pick up on the magnitude of the probabilistic contrast. The results of Experiment 3, however, suggest that there are cases where preschoolers inferences can look like older children s if they can discount the anomalous data they observe. In this experiment, preschoolers who recognized that the experimenter believed that the data should have been deterministic responded as if the data were deterministic (similar to the level of performance generated by the 5-year-olds in Experiment 1 who were asked to make the same inference without the experimenter s reaction). In both cases, children chose the higher frequency object more often than chance levels. Preschoolers in Experiment 3 who did not possess the capacity to understand another s false belief states responded at a similar level of performance (i.e., no different from chance) to the preschoolers in Experiment 1.

29 Probabilistic Causal Generalization 29 This interpretation generally parallels work by Kushnir and colleagues (Kushnir & Gopnik, 2005; Kushnir, Wellman, & Gelman, 2009), who suggest various ways in which children s causal inferences are influenced by factors beyond observed data. For instance, Kushnir et al. (2009) demonstrated that preschoolers would use another s knowledge when making a causal inference only when that knowledge was applicable to the inference in question (see also Sobel & Corriveau, in press). These investigations suggest that preschoolers are not just reasoning about the data they observe, but rather thinking through what produced those data whether those data were representative of that underlying mechanism. Further, across the experiments, external perceptual similarity denoted category membership. Perceptual similarity is certainly a good cue for category membership, and potentially a default when not given information to the contrary (e.g., Landau, Smith, & Jones, 1988). However, there are other cues to category membership, which might aid children s inferences. Children use causal properties as a guide to category membership, and infer category membership from objects labels (Gopnik & Sobel, 2000). If all the objects in each set were also labeled (e.g., all the orange cubes were called blickets, and all the yellow triangles called tibs ), preschoolers might have been more inclined to generalize from probabilistic data. On the other hand, if each object was given a distinct label (either a common or a proper noun), preschoolers might be less inclined to generalize from these data. While the child s understanding of the experimenter s belief states informed their inferences, more generally, contextual factors affect children s causal inferences (e.g., Sobel & Sommerville, 2009). Demonstrating that the way in which children construe the objects category membership affects their generalization

30 Probabilistic Causal Generalization 30 ability would be consistent with the hypothesis that children s causal reasoning is affected by this kind of contextual information. This is a topic for further investigation. To conclude, the present data suggest that children have little difficulty making causal generalization inferences given deterministic data, but before the age of 5 appear to struggle when shown some kinds of probabilistic data. Four-year-olds can engage in some kinds of reasoning about probabilistic data, particularly when the contrast is between a probabilistic outcome and one that never occurs, or when the contrast is between an outcome that always happens and one that only sometimes happens if the latter is infrequent. Preschoolers also seem to be influenced by a particular contextual factor concerning the data. As such, we conclude that preschoolers do have some understanding of probabilistic inference, but exploring what other factors influence preschoolers understanding of probability is an open question for future investigation.

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36 Probabilistic Causal Generalization 36 Author Note David M. Sobel, Department of Cognitive and Linguistic Sciences, Box 1978, Brown University, Providence, RI, 02912, USA. This work was supported by NSF (DLS ) to DMS. We would like to thank all of the parents and children who participated in this research. We would also like to thank Jon Cohen for assistance in data collection and Esra Aksu, Sheridan Brett, Karis Casagrande, Claire Cook, Rebecca Crossin, Emily Hopkins, Karina Ikesoe, Caroline Kleeman, Rachel Shelley-Abrahamson, Cesalie Stepney for assistance in participant recruitment and data analysis. Tamar Kushnir provided helpful discussion about this manuscript. Address comments to D. Sobel, Department of Cognitive and Linguistic Sciences, Box 1978, Brown University, Providence, RI Phone: (401) Fax: (401)

37 Probabilistic Causal Generalization 37 Figure Captions Figure 1. The machine and sets of demonstration and test blocks used across the Experiments. One of the demonstration blocks is shown activating the machine.

38 Figure 1. Probabilistic Causal Generalization 38