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