Background: Observation Studies, Unmeasured Confounding, and Sensitivity Analysis: An important part of educational research is identifying important, potentially causal, factors that influence children's learning from observational studies. However, it is well-known that discovering such factors from observational studies can be biased due to unmeasured confounders. For example, several works have explored whether attending charter schools perform better on standardized tests than those enrolled in regular public schools (Abdulkadiroglu, Angrist, Dynarski et al., 2011; Tuttle, Gill, Gleason et al., 2013). Figures 1(a) and (b) depict two potential causal graphs for charter school attendance (i.e., C), and test scores (i.e. Y). Both figures include parental background (i.e., P), such as parental education levels and family income, and directly influence students' academic achievement (i.e., the path from P to Y). Concurrently, parental background often reflects the socioeconomic status and resources provided by families and may also predict charter school attendance (i.e. the path from P to C). In short, parental background is a confounder. If parental background are measurable and it is the only variable that influences charter schooling and student achievement (i.e. Figure 1(a)), there are methods such as matching or weighting to adjust for confounding and assess the desired causal effect. However, as shown in Figure 1(b), there may be unobservable factors (i.e. U) that influence charter schooling and test scores. One potential unmeasured confounder (U) could be the neighborhood where charter schools may be more prevalent in areas with disadvantaged neighborhoods and the characteristics of neighborhoods may negatively impact students' academic achievement. Most importantly, If the true causal structure is Figure 1(b), but the investigator conducted an analysis assuming the causal structure in Figure 1(a), the estimated causal effect of attending charter schools on test scores may be biased. Another example is from the Early Childhood Longitudinal, Kindergarten Class of 1998-1999, a longitudinal, observational study of students in the United States. Researchers found a statistically significant and positive effect of attending school-based pre-k programs on students' learning (Magnuson, Meyers, Ruhm, & Waldfogel, 2004). After adjusting for family and demographic variables, participation in center-based care was associated with a four-percentile increase in reading scores relative to children who experienced only parental care. However, as the authors caution in their work, the observed association may exhibit bias because their models "only control for few child characteristics, raising the possibility that [they] may not be correcting adequately for potential selection effects", such as children's developmental status or developmental disabilities. Sensitivity analysis is a framework to assess the robustness of potentially causal associations discovered from observational studies. The most famous example of a sensitivity analysis is by Cornfield and co-authors (Cornfield, et al.,1959), who used sensitivity analysis to conclude that it would take an extraordinary amount of evidence to nullify the positive association between smoking and lung cancer from observational studies. Specifically, Cornfield et al showed that to nullify the positive association, one would need another variable (U) besides smoking that is (i) a near-perfect predictor of lung cancer and (ii) is nine times more likely to be present among smokers than non-smokers, (i.e. ). Given that no such variable has been hypothesized in the literature, Cornfield's sensitivity analysis provided a precise and quantifiable framework to strengthen evidence from observational studies and paved the way for several important public health policies on smoking in the United States, most notably the 1964 Surgeon General's Report which concluded that smoking causes lung cancer. Given its history in health sciences, the vast majority of sensitivity analyses focus on studying associations between a binary, independent variable (e.g. smoking versus no smoking; treatment versus control/placebo) and a dependent variable with statistical tests that measure the difference between two groups (e.g. two-sample t-test; Wilcoxon sign-rank test). However, in educational observational studies, researchers often measure associations between a non-binary independent variable(s), such as classroom size, types of education programs (Early et al., 2007), duration of physical activity (Carlson, et al, 2008), and a dependent variable, such as achievement scores. Also, when measuring the association between two non-binary, categorical variables, the chi-square test of independence is arguably one of the most popular tests of the association. Yet, to the best of our knowledge, there is no sensitivity analysis for the chi-square test. Overall, existing methods for sensitivity analysis are inadequate to deal with data structures common in educational settings and new methods are needed to strengthen associations discovered from observational studies in education. Method: Sensitivity analysis for arbitrary test statistics from a contingency table: The current paper proposes a universal framework to conduct sensitivity methods for arbitrary I x J contingency tables. Briefly, contingency tables are widely used in education and psychometrics to describe associations between two categorical variables (Everitt, 1992). Unfortunately, as mentioned earlier, there does not exist a sensitivity analysis for contingency tables greater than a 2x2 table. Our work extends the Rosenbaum sensitivity model (Rosenbaum & Krieger, 1990) to assess the sensitivity of a large class of test statistics that quantify associations between the rows and the columns of an I x J contingency table (See Figure 2). The class of test statistics includes the chi-square test for independence, the likelihood ratio test, the global odds ratio test, and the linear by linear association test. We apply our method to assess the association between three types of pre-kindergarten (pre-k) care and students' overall math achievement, measured on a discrete scale, from the Early Childhood Longitudinal Study--Kindergarten cohort. After controlling for socioeconomic and demographic factors, we find that in the absence of any unmeasured confounding, the association between pre-k programs and math performance is strong, especially for black and Hispanic female students (two-sided p-values: 0.0034 and 0.0086, respectively). Also, the observed associations are no longer statistically significant at the significance level of 0.05 if there is an unmeasured confounder (U) that simultaneously changes the odds of enrolling into a pre-k program and success in math achievement by a factor of 2.4.