What's the most appropriate statistical analysis to run for a correlational study?

I am designing a correlational study experiment to test children on their performance on a task at $t_1$ and then at $t_2$ a year later, specifically using the balance-scale paradigm (Siegler, 1981). I’m not sure which statistical tests will be appropriate to run.

I want to find out if the early use of a sophisticated decision rule at $t_1$ is predictive of children using even more sophisticated rules in $t_2$. Namely, is a child who can deploy a complex decision rule quicker than most of her peers at $t_1$ more likely to use even more complex decision rules at $t_2$? I will compare these results with the $t_1$ vs $t_2$ results of children who failed to deploy that more complicated decision rule at $t_1$. These two decision rules will be operationalised as non-continuous binary variables.

What kind of statistical testing would be most appropriate here?

---

References

Siegler, R. S. (1981). Developmental sequences within and between concepts. Monographs of the Society for Research in Child Development, 46(2), 84