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
Although you have two times of testing, you really just have two binary variables and you'd like to know if they are associated, there are no repeated measures here. You'd like to know if the value of one binary variable depends on the value of the other binary variable. I'll call them A,B and X,Y.
If there is no relationship between A,B and X,Y, then you expect the ratio of X:Y to be the same for A as it is for B (this is your null hypothesis). You can examine your data by building a 2x2 contingency table with cells AX, AY, BX, and BY.
You can test the null hypothesis described with a chi-square test for independence, or Fisher's exact test if the sample size is fairly small.