We showcase applications of the functinal chi-square (FunChisq) test on several types of discrete patterns. Here we use the row to represent independent variable \(X\) and column for the dependent variable \(Y\). The FunChisq test statistically determines whether \(Y\) is a function of \(X\).
A pattern represents a perfect function if and only if function index is 1; otherwise, the pattern represents an imperfect function. A function pattern is statistically significant if the p-value from the FunChisq test is less than or equal to 0.05.
A significant perfect functional pattern:
An significant perfect many-to-one functional pattern:
An insignificant perfect functional pattern:
A perfect constant functional pattern:
We contrast four imperfect patterns to illustrate the differences in FunChisq test results.
p4 represent the same non-monotonic function pattern in different sample sizes;
p2 is the transpose of
p1, no longer functional; and
p3 is another non-functional pattern. Among the first three examples,
p3 is the most statistically significant, but
p1 has the highest function index \(\xi_f\). This can be explained by a larger sample size but a smaller effect in
p1. However, when
p1 is linearly scaled to
p4 to have exactly the same sample size with
p3, both the \(p\)-value and the function index \(\xi_f\) favor
p3 for representing a stronger function.