"1. INTRODUCTION. Making causal inferences based on non experimental statistical data has been a controversial topic [Freedman, 1987, Rogosa, 1987, Denis, 2006]. While randomized controlled trials are the standard approach to take care of intervening third variables, causal modeling is an alternative method of making causal inferences [Pearl, 2009, Sprites et al., 2001] based on observational data making certain causal assumptions. We are using the causal modeling approach to analyze and explore the data from a game-like math tutor, Monkey’s Revenge. We created four different versions of Monkey’s Revenge with varying degree of game-like properties. A total of 297 middle school (12-14 year olds) students from four Northeastern schools in the United States participated in the study. We logged their tutor activities and asked 16 survey questions using 5 point Likert scale from “strongly disagree” to “strongly agree.” We then used factor analysis to reduce the variables into six categories: likeMath, mathSelfConcept, pedagogical preference (prefer Computers to books; find real world examples helpful for learning Math.), tutorHelpful, tutorConfusing, likeTutor. From students’ log data, we calculated variables %correct (ratio of correct problems to total problems); avgAttemptTime (average time spent on each attempt) and avgHints (average number of hints asked on each question). We also collected preTestScore and prePostGain and used a variable game-like as an ordinal parameter (taking on values 1 through 4). Along with using causal modeling to explore and analyze our data, we analyze the causal modeling approach itself. We compare it against the standard statistical approaches of correlation and multiple regression. 2. CAUSAL MODELING, CORRELATION MATRIX. Based on the data we collected, we used TETRAD, a free causal modeling software package [Glymour et al., 2004] with the PC search algorithm to generate a causal graph (fig 1). We also generated a graph based on the correlation matrix, computed by finding the correlations between the variables (fig 2). Figure 1 Causal model from PC algorithm without domain knowledge. True negatives (indirect and spurious associations): Correlation is not causation as there might be possible confounders causing the spurious association, and causal modeling controls for all third variables regarding them as possible confounders. From the correlation matrix, we see that likeTutor and %correct are correlated which would suggest that students who like the tutor performed better. This result could be interpreted as evidence for student engagement, since students who liked the tutor are presumably more engaged while using it. But the causal model (Fig 1) infers that this is a spurious association confounded by likeMath. Students who like math tend to like tutor more and to have better performance. Once we control for likeMath, there is no relation between likeTutor and %correct. Thus, the scientific interpretation of the data changes. Figure 2 Graph based on correlation matrix. Still, the causal model is limited to assertions about the observed variables as there might be other confounders which we have not observed. Causal modeling makes also distinction between direct and indirect association. For example, likeMath and avgHints are negatively correlated, which suggests that the students who like math ask for fewer hints. But once we control for %correct, that correlation is gone (see Fig 1), suggesting that the students who like math ask for fewer hints only because they already know the correct responses and so do not need as much help—there is no direct effect. False negatives (reduced statistical power and multicollinearity): Controlling on third variables reduces statistical power and we might get false negatives if we have few data. Multicollinearity is an extreme case when the independent variables are correlated among themselves. For example: avgAttemptTime is correlated with both %correct (0.3**) and preTestScore(0.3**). But since, %correct and preTestScore are highly correlated among themselves (0.6**), avgAttemptTime is conditionally independent to both of them. We can see that avgAttemptTime is an isolated node in figure 1; in contrast, the correlation graph (Figure 2) indicates avgAttemptTime is related to both preTestScore and %correct. 3. CAUSAL STRUCTURE, PATH ORIENTATION AND DOMAIN KNOWLEDGE. In the causal model, some edges have plausible orientations ( e.g: likeMath ÆlikeTutorÅ game-like). Using the information that likeTutor is correlated with both likeMath and game-like, but likeMath and game-like are independent between themselves, search algorithm correctly identifies that it is not likeTutor influencing likeMath and game-like but the other way round. However, we see that there are other edges which are incorrectly oriented such as %correctÆpreTestScore; student performance on the tutor cannot have influenced a pretest that occurred before students began using the tutor. Correlation underdetermines causality as covariance in statistical data is rarely sufficient to disambiguate causality. Therefore, even after we use search algorithms to find some structure, there are a number of “Markov equivalent” structures. In TETRAD, we can add domain knowledge in the form of knowledge tiers which represent the casual hierarchy. Causal links are only permitted to later tiers, and cannot go back to previous tiers. We used the following knowledge tier based on our knowledge of assumed causal hierarchy and temporal precedence. i. Gender ii. Game-like, mathSelfConcept iii. likeMath, Pedagogical preference iv. preTestScore v. %correct, avgAttemptTime, avgHints, tutorConfusing, tutorHelpful vi. likeTutor vii. prePostGain We see from Figure 1 and Figure 3 that adding domain knowledge not only fixes the path orientations (preTestScoreÆ%correct), but have changed the whole causal structure adding some new causal links (genderÆmathSelfConcept, pedagogicalPreferenceÆtutorHelpful, correctÆavgAttemptTime). At first, it may appear that knowledge of causal hierarchy only helps to orient the edges specifying which one is cause and which one is effect. However, besides distinguishing variables as potential causes and effects, the domain knowledge also restricts the set of variables to be considered as confounders and mediators. Note that many data analyses have such causal hierarchies implicit in the analysis and conclusions. Figure 3 Causal model with domain knowledge. Sometimes, we do not know about the causal hierarchy of the variables we are trying to analyze and may not know which is the cause and which is the effect, but having information of the causal hierarchy of third variables, such as whether they are a potential confounder or a potential mediator, can help infer if there is any causal path between the variables of interest. We can illustrate this with a concrete example in education. Suppose we have observed that engagement and learning are correlated, but want to understand the causal relation between them. Imagine there are two other variables, prior knowledge, a potential confounder, and performance, a potential mediator. Consider two scenarios: if partialling out prior knowledge removes the correlation, then we know there is no causal relationship between engagement and learning, and the causal structure is engagementÅprior knowledgeÆlearning. On the other hand, if partialing out performance removes the correlation between engagement and learning, then there is still an indirect causal effect between the two, either engagement->performance->learning, or learning->performance->engagement. Interestingly, adding domain knowledge can also address the problem of multicollinearity. Since we have set preTestScore on higher causal tier than %correct, %correct cannot be a possible confounder or mediator and therefore, the partial correlation (preTestScore, avgAttemptTime | %correct) is not calculated and the correlation between preTestScore to avgAttemptTime is maintained. 4. CAUSAL MODELING AND MULTIPLE REGRESSION. Causal modeling is a sophisticated extension to multiple regression which employs a series of multiple regression. Multiple regression only looks at direct effect but fails at identifying indirect effects. While multiple regression can be equally robust when it comes to predictive accuracy, causal modeling provides a better representation and framework to understand interrelationships of variables. Since causal modeling allows multiple layers of associations of variables, it adds affordance to insert domain knowledge in the form of a causal hierarchy. On top of the statistical assumptions used by statistical methods such as regression, causal modeling adds causal assumptions such as faithfulness and causal sufficiency [Sprites et al., 2001]. Stronger assumptions add more analytical power but also higher chances of inaccuracy. It is up to researcher to select these assumptions based on their data and domain. We have accepted the causal assumptions made by TETRAD since they seem reasonable for our data and purpose. 5. CONFIRMATORY, EXPLORATORY AND GRAPHICAL TOOL. Causal modeling can be used to test goodness of fit of a model constructed a priori from theory. We did not start with such a model, but the causal model has supported some of our prior hypotheses (pedagogicalPreferenceÆtutorHelpfulÆlikeTutor) and (mathSelfConceptÆtutorConfusingÆlikeTutor). Using causal modeling as an exploratory tool to generate new theories is controversial as the possibility of unobserved confounders and under determination of causality from correlation pose serious limitation to generate new valid conclusions. But, conditional independencies in data and domain knowledge can offer some new inferences which can be helpful in guiding us towards further analyses and examination. For example, in our causal model, we found that likeMath has both direct (likeMathÆ%correct) and indirect (likeMathÆpreTestScoreÆ%correct) effect on %correct. Based on this, we are considering two possible causal models as shown in figure 4. Model I suggests that pretestScore does not capture all of the variance in prior knowledge of the student, as represented by the latent node “Prior knowledge.” So, students who like math and have high prior knowledge may have a low pre test score but they have high performance nonetheless. Model II on the other hand suggests that students who like math both have higher prior knowledge and are more engaged, and have therefore higher performance. In other words, likeMath affects both prior knowledge and engagement. Figure 4 Two possible causal models linking likeMath and %correct. We were not able to make any conclusive findings with the causal model, but it has at least made interesting inferences and raised questions that are very important for us. It has directed towards the possibilities that we would like to make further examination and possibly run some controlled randomized trials. Even if researchers are skeptical of the domain knowledge we have brought to bear and are dubious of the causal modeling assumptions, it is still possible to consider Figure 1 without the assumption that the edges represent causality. This graph would be a compact representation of the partial correlation relationships among the variables 6. CONCLUSIONS. We have used a causal modeling approach to explore the data from a game-like math tutor. Based on statistical independence within data and our domain knowledge, TETRAD’s causal inference algorithm generated a causal model which not only confirmed some of our prior hypotheses about data but also made some interesting new findings. Causal modeling does a good job of identifying not only cause and effect but also confounders so that we can find spurious associations and mediators so that we know both direct and indirect effects. But those inferences cannot be claimed as accurate since causal modeling cannot identify spurious association caused by unobserved confounders and there are multiple Markov equivalent causal models that can be generated from the same data. Still, causal modeling is the best approach we have found, particularly when compared with common statistical techniques such as correlation and multiple regression to generate most plausible inferences from observational educational data sets."