Introduction

The question that has dominated the literature on children’s causal reasoning for the last decade is how new causal relations are learned and represented [1, 2]. A line of studies has investigated children’s ability to make causal inferences from a very young age on [3, 4, 5, 6, 7, 8, 9]. The majority of these studies have used the blicket detector paradigm to create an artificial, novel setting for testing this ability [4, 6].The paradigm consists of a machine that activates (lights up and plays music) when some objects (blickets), but not others, are placed on it. Each blicket detector trial measures a learner’s ability to make a specific type of causal inference, such as screening-off, indirect screening-off, or backwards blocking.

Even though this line of work did not focus on investigating the development of causal inference, it has uncovered some developmental differences. It has been shown that the majority of 19-, 24-, 30-month-olds [5, 8] and 3- to 5-year-olds [5, 7] are capable of making screening-off inferences. This implies not considering a target object to be a blicket in the context where the target object activates the machine in the presence of an established blicket, but not by itself [10]. It has been shown that in contrast to the majority of 19-month-olds [8], the majority of 24-month-olds [8], and 3- and 4-year-olds [9] are capable of making indirect screening-off inferences. This implies considering a target object to be a blicket in the context where the target object activates the machine in the presence of a second object, and the second object does not to activate the machine by itself. Additionally, it has been shown that in contrast to the majority of 3-year-olds, the majority of 4-year-olds are capable of making backwards blocking inferences [9]. This implies not considering a target object to be a blicket in the context where the target object activates the machine in the presence of a second object, and the second object activates the machine by itself. These findings have led researchers to stress the need for future research to further investigate the developmental trajectory of causal inference, e.g. [9].

In previous work on preschoolers’ causal inference, conclusions on the existence of developmental differences have been based on the average behavior of age groups. This approach stands in contrast with the sizable variability in behavior within age groups that has been observed in these studies. Within groups that were considered to be capable of making a specific type of inference, percentages of succeeding children ranged from 72 to 100 for screening-off inferences [5, 7, 8] and from 76 to 100 for indirect screening-off and backwards blocking inferences [8, 9] As mapping the variability in children’s behavior is considered of great importance in studying development [1], the goal of the present study is to further investigate the development of causal inference by studying variability in behavior.

Variability in children’s behavior could originate from different sources. Possibly subgroups of children apply different strategies for solving causal inference problems (we will refer to this source of variability as strategy use). By “strategy” we mean a process, either conscious or unconscious, resulting in a certain approach to solving causal inference trials, cf. [11]. Different strategies either imply the use of different learning mechanisms, such as Bayesian methods or causal learning on the basis of associative models (for an overview see Gopnik et al., 2004) [3], or the use of different task-related strategies, such as imitation of the experimenter or guessing, e.g. [5, 8]. Alternatively, variability in behavior is explained by children making errors in applying strategies (we will refer to this source of variability as accuracy in strategy execution). Differences in accuracy can, for example, be related to differences in information processing abilities [9]. To render a complete picture of the development of children’s causal inference, it is important to discriminate between these sources of variability and relate them to age. From this analysis roughly two developmental models could be proposed: either children of different ages apply different strategies (strategy use), or children of different ages apply the same strategy, but become better at executing this strategy with age (accuracy in strategy execution). Investigating which of these models best describes development is an important first step in further investigating the development of children’s causal inference. Ultimately, one would want to know what strategies children apply, and how strategy use is related to age and the circumstances a child faces. However, before being able to study these interesting, but highly complex issues, a necessary first step is to determine whether variability in behavior can be explained by strategy use at all.

The approach that has been used in previous work on children’s causal inference, does not allow one to discriminate between variability in responses stemming from strategy use and variability stemming from accuracy in strategy execution, because average behavior was analyzed over no more than one or two causal inference trials [3, 4, 5, 6, 7, 8, 9]. Therefore, in the present study, we look at individuals’ response patterns over a larger number of trials. To this end, children of a relatively broad age range, 2- to 5-year-olds, are administered a relatively wide range of causal inference trials: a screening-off trial [5], two versions of an indirect-screening-off trial [9, 7], a backwards blocking trial [9], and a non-causal association trial [4, 12] (see Method). Mainly, these trials are selected for their frequent use and tendency to bring about variable responses in previous work (see Method). Individuals’ response patterns over trials are analyzed with a latent variable technique [13, 14] (see Method). The methodological approach allows us to map the variability in children’s causal inference behavior by comparing the relative fit of models that explain variability in responses in different ways, and this way to contribute uniquely to the research on the development of this ability. The present study’s first research question is: Can variability in 2- to 5-year-olds’ behavior be explained by strategy use? To answer this question, the fit to the data of a 1-class-model (not indicative of variability being explained by strategy use) will be compared to the fit of multiple-class-models (indicative of variability being (partially) explained by strategy use). The detection of different subgroups producing comparable responses for the four (different) trials is a first indication of strategy use. Stability of responses over multiple, equivalent items (e.g., indirect-screening-off trials) would provide a stronger indication. However, this young age group does not allow for administering more trials. The second research question is: Is children’s strategy use related to age? To answer this question, the fit to the data of models with and without age as a covariate on the class membership probabilities will be compared. The third question is: Does age-related strategy use better explain the variability in children’s behavior than an age-related increase in accuracy of executing a single strategy? To answer this question, the fit to the data of a 1-class-model (indicative of variability being explained by accuracy of execution) and a multiple-class-model (indicative of variability being explained by age-related strategy use) with age as a covariate on the response probabilities will be compared. Please see the Methods and Materials and Results sections for further explanation of the latent variable techniques, including the model comparisons. In the Discussion we will elaborate on the selected model and provide directions for future research by speculating on possible interpretations of the strategies of the model.

Material and methods

Results

Comparison results to previous studies

Several test trials (trial 1: screening-off, trial 3: indirect screening-off B, and trial 4: backwards blocking) had been administered with the exact same procedures as in previous studies. Hence, before performing the individual difference analyses, we compared the proportions of children succeeding on these trials in the present study to those of children of comparable ages in previous studies (see Table 1). The proportion of 2-year-olds making a screening-off inference did not differ between this study and Sobel and Kirkham’s (2006) study [8]. The proportions of 3-year-olds making indirect screening-off and backwards blocking inferences did not differ between this study and Sobel et al.’s (2004) study [9]. The proportion 4-year-olds making an indirect screening-off inference did not differ between this study and Sobel et al.’s study. The proportion 4-year-olds making a backwards blocking inference did not differ between this study and Sobel et al.’s first experiment, but did differ between this study and Sobel et al.’s second experiment. So overall, our data were highly comparable to previous data.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Children’s responses on causal inference trials in the present study and comparison of these responses with previous studies; numbers (and percentages) of participants succeeding (S) and not succeeding (NS).

https://doi.org/10.1371/journal.pone.0195019.t001

Studying variability 1: Can variability in children’s behavior be explained by strategy use?

First, we investigated whether the existence of different strategies could explain variability in children’s behavior (strategy use). To this end, latent class models (LCM; see Method) with 1, 2, 3, and 4 classes were fit to the data: children’s binary response patterns on four test trials. A 1-class-model showing the best fit, would indicate that variability could not explained by strategy use. A multiple-class-model showing the best fit, would indicate that variability could (at least partially) explained by strategy use. Fig 1A shows the goodness-of-fit measures of the 1-, 2-, 3-, and 4-class-models (see arrow 1). The 3-class-model had the lowest BIC value, implying the best and most parsimonious fit to the data. This demonstrates that variability in behavior was (at least partially) explained by strategy use.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Goodness-of-fit measures for latent class models.

Note: L = Log Likelihood, Number of parameters = number of feely estimated parameters minus the number of parameters estimated at the boundary, BIC = Bayesian Information Criterion. The arrows correspond to the first three steps in studying variability (see Results section).

https://doi.org/10.1371/journal.pone.0195019.g001

Studying variability 4: The selected model

To conclude, on the basis of the above analyses a LCM with 3 classes and age as a covariate on class membership was selected. The selected model captures the data well as indicated by the likelihood ratio Chi-Square (G2 = 4.26 with df = 1). As there were a number of empty and small valued cells in the data table, the p-value for this G2 was obtained by using a parametric bootstrap procedure [32] and had value p = 0.35.

Table 2 shows the response probabilities for the selected model. These are the probabilities of succeeding on a specific trial given membership of a specific class. As mentioned above, succeeding was defined as making the intended causal inference, such as screening-off or backwards blocking (see Method). Children applying the first strategy had high probabilities of succeeding on all test trials. This was a relatively large group (N = 51) mostly consisting of children in the older age-range. Compared to children in the first group, children applying the second strategy had lower probabilities of succeeding on the first two test trials (screening-off and indirect screening-off A). On the last two trials they showed a qualitatively different pattern of responses compared to first group: they succeeded on trial 3 (indirect screening-off B), but not on trial 4 (backwards blocking). This was a smaller group (N = 27) consisting of children in the younger age-range. Last, children applying the third strategy did not succeed on the first two trials (screening-off and indirect screening-off A), but did succeed on the last two trials (indirect screening-off B and backwards blocking). This was a relatively small group (N = 12) consisting of children of all ages.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Selected latent class model: Response probabilities.

https://doi.org/10.1371/journal.pone.0195019.t002

Fig 2 shows how strategy use was related to age in the selected model (see Method). The use of the first strategy increased with age, and the use of the second strategy decreased with age. The majority of 2-year-olds applied the second strategy, while the majority of 3-, 4- and 5-year-olds applied the first strategy. In all age groups, a small proportion of children applied the third strategy.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Selected latent class model: Percentages of children per strategy by age group.

https://doi.org/10.1371/journal.pone.0195019.g002

Discussion

The aim of the present study was to further investigate the development of causal inference by mapping the variability in young children’s behavior. Mainly, we set out to discriminate between two possible sources of variability, strategy use and accuracy in strategy execution, and relate these to age. In other words, we asked whether children of different ages apply different strategies when confronted with causal inference problems (strategy use), or whether they apply the same strategy, but become better at executing this strategy with age (accuracy in strategy execution). To the end of discriminating between these sources of variance, a relatively wide range of causal inference trials was administered to children of a relatively broad age range: 2- to 5-year-olds. Subsequently, individuals’ response patterns over trials were analyzed with a latent variable technique [13, 14]. It was shown that variability in behavior could largely be explained by age-related strategy use: children applied three different strategies when confronted with causal inference problems. And importantly, this age-related strategy use better explained the variability in children’s behavior than age-related increase in accuracy of executing a single strategy.

Hypotheses on development relating to both strategy use and accuracy in strategy execution have been proposed in the literature. Concerning strategy use, two types of strategies have been distinguished. First, children of different ages could apply different learning mechanisms. Both computational approaches, such as Bayesian methods [33, 34], and psychological approaches, such as causal learning on the basis of associative models [35, 36, 37] have been proposed (for an overview see Gopnik et al., 2004) [3]. Second, children of different ages could use different task-related strategies. For example, they could guess or imitate the experimenter, e.g. [5, 8]. Concerning accuracy in strategy execution, it has been suggested that with age children become better at executing a single strategy due to developing motor, statistical, or information processing abilities, or developing domain-specific knowledge [8, 9, 38]. For example, Sobel and Kirkham (2006) [8] demonstrated that in contrast to the majority of 24-month-olds, the majority of 19-month-olds failed to make indirect screening-off inferences in a blicket detector paradigm. The authors hypothesized this failure did not reflect the use of a different learning mechanism, but instead reflected the developing association between knowledge and action in the younger age group. By using an eye-gaze paradigm they found support for their hypothesis: 8-month-olds’ inferences were similar to those of the 24-month-olds. The present study’s results suggest a way of combining these existing hypotheses into a tentative model of the development of causal inference. It was shown that continuous development of motor, statistical or information processing abilities does not lead to a continuous increase in accuracy of executing a single strategy for causal inference. Instead, the development of causal inference is better described in terms of the application of multiple age-related strategies. This finding suggests that the development of certain abilities leads children to select and apply different strategies, either sequentially or according to an overlapping waves model [39, 40]. In other words, strategy use mediates the relation between children’s developing motor, statistical or information processing abilities and their causal inference. Previous work has found support for a mediatory role of strategy use in the development of transitive reasoning, free classification, and discrimination learning [22, 24, 25]. For example, Schmittmann et al. (2012) [24] found a relation between the use of discrimination learning strategies and children’s working memory and attentional control after controlling for age.

This study takes a first step in introducing a fruitful approach for investigating variability in behavior to the research on children’s causal inference. By using the approach we focused on the behavior that children of different ages demonstrate when confronted with causal inference problems, and the results can be considered indicative for children’s behavior when facing these types of problems in daily life. The findings show that 2- to 5-year-olds apply three different age-related strategies, that is, consistent patterns of behavior. Ultimately, one would want to know what learning mechanisms or task-induced strategies are reflected by these strategies, and how strategy use is related to age and the circumstances a child faces. Note that on the basis of this study’s results no definite conclusions can be drawn on the interpretations of the detected strategies. Future research, however, might address questions on the nature of strategies children of different ages apply by performing more confirmatory modeling approaches. Confirmative approaches require the formulation of specific hypotheses about the nature of the expected strategies. Therefore, in the following paragraph we elaborate on more and less plausible interpretations of the in the present study detected strategies based on the existing literature on children’s learning mechanisms and task-induced strategies. We do not suggest that, based on the data of the present study, definite conclusions can be drawn on interpretation of strategies, but with the following paragraph we intend to do suggestions for future work by elaborating on, in our opinion, more and less plausible interpretations.

Children applying the first strategy had high probabilities of succeeding on all test trials. The group’s responses are consistent with rational models, such as so called Bayesian inference over Causal Graphical Models, as proposed by Gopnik et al. (2004) [3] and Sobel et al. (2004) [9], but some authors claim that associative models could explain these data too [41]. One could possibly refer to this first strategy as the “rational strategy”. A clear difference between the first two groups was that children applying the second strategy showed a qualitatively different pattern of responses on the last two trials (indirect screening-off B and backwards blocking) compared to children applying the first strategy. As the associative strength of the target object is the same in these trials (that is, in both trials the target object has a 100% positive association with the outcome), the symmetry in children’s classification of the target object as a blicket on both these trials suggests the possible use of a learning mechanism on the basis of an associative model, such as the Rescorla-Wagner model [9, 37]. This interpretation is consistent with the group’s lower age and the fact that associative reasoning is a powerful learning mechanism early in life [42, 43]. Note that the pattern of responses does not reflect a yes bias, as the large majority of this group (89%) did not label the non-target object as a blicket on trial 3 (indirect screening-off B). However, this hypothesis is inconsistent with the results if the rational model would suggest continuous development. One could possibly refer to this second strategy as the “associative strategy”. Children applying the third strategy did not succeed on the first two trials (screening-off and indirect screening-off A), but did succeed on the last two trials (indirect screening-off B and backwards blocking). A closer look at this group’s responses on the first two trials shows that the majority (70% on trial 1 and 61% on trial 2) responded by putting the causally effective block plus one or two ineffective blocks on the machine. These responses suggest the possible use of a task-related strategy directed at avoiding uncertainty: children in this group might have chosen to respond in a manner that they had observed to work most frequently or simply by putting all available objects on the machine. One could possibly refer to this third strategy as the “uncertainty avoidance strategy”. Looking at the developmental pattern, it was shown that the use of the “rational strategy” increased with age, the use of the “associative strategy” decreased with age, and the “uncertainty avoidance strategy” was applied by a small proportion of children in all age groups. Hence, there are appealing developmental interpretations of the model outcomes, but more research is needed to be able to draw more definite conclusions about the reasoning processes leading to the observed, different types of response patterns.

Several recommendations could be given for future work. To be able to interpret detected strategies in more detail, the mostly exploratory latent variable approach of the present study could be replaced by a more confirmatory approach. Using Rule Assessment Methodology [26, 44], trials could be selected that optimally distinguish between children applying different learning mechanisms [26, 44]. For example, base rate trials could be included to distinguish response patterns consistent with Bayesian inference over Causal Graphical Models from response patterns consistent with certain associative models [9, 45, 46]. Or, alternatively children's inference models could be estimated directly from data by adopting one causal reasoning framework, cf. [47]. In addition, the variability approach could be used to investigate whether strategy use differs over domains, such as the physical and psychological domains [7, 46, 48], or over task characteristics, such as deterministic or probabilistic tasks [49, 50, 51], and how strategy use is related to age under these different circumstances. Moreover, it is of high interest to relate children’s cognitive abilities, such as their statistical or information processing abilities, to their causal inference strategy, cf. [24]. To conclude, the approach for investigating variability in behavior has shed a new light on the development causal inference, and holds the promise of initiating a new line of research.

Acknowledgments

We would like to thank the children, parents, daycare centers and primary school for their participation in the study. We would like to thank Heleen Drapers and Miriam Maan for collecting data, Coos Hakvoort for providing technical support, and Dorothy Mandell for commenting on earlier drafts on the paper.