Detecting functional direction
If a function is a many-to-one mapping, then its inverse is not a function. In this context, the three methods can behave sharply differently depending on table shape: square (same numbers of rows and columns), landscape (more columns than rows), or portrait (more rows than columns). Here the row variable is the independent variable \(X\) and the column variable is the dependent variable \(Y\). Each method tests whether \(Y\) is a function of \(X\) on the original table; then it tests whether \(X\) is a function of \(Y\) on the transposed table. We use \(p\)-values for the two FunChisq methods and the conditional entropy \(H(Y|X)\). In all three methods, a smaller value indicates a stronger functional direction.
We will show by three examples that only AdpFunChisq performs
consistently well on all three table types. Below is a generic function
compare.methods to visualize the input contingency table,
perform the tests, and report the correctness of the function
direction.
require(FunChisq)
require(infotheo)
require(DescTools)
compare.methods <- function(func)
{
plot_table(func, xlab = 'Y', ylab = 'X', value.cex = 0.9, col="seagreen",
main = expression("Function:"~italic(Y%~~%f(X))))
plot_table(t(func), xlab = 'X', ylab = 'Y', value.cex = 0.9, col="firebrick1",
main = expression("Inverse non-function:"~ italic(X != f^-1*(Y))))
sample.matrix <- Untable(func)
x <- as.numeric(sample.matrix[, 1])
y <- as.numeric(sample.matrix[, 2])
H.y.given.x <- condentropy( y, x )
H.x.given.y <- condentropy( x, y )
if(H.y.given.x < H.x.given.y) {
cat("Conditional entropy picked CORRECT direction X to Y: H(Y|X) =",
format(H.y.given.x, digits = 2), "< H(X|Y) =",
format(H.x.given.y, digits = 2), "\n")
} else {
cat("Conditional entropy picked *WRONG* direction Y to X: H(Y|X) =",
format(H.y.given.x, digits = 2), ">= H(X|Y) =",
format(H.x.given.y, digits = 2), "\n")
}
func.fc.pval <- fun.chisq.test(func)$p.value
ifunc.fc.pval <- fun.chisq.test(t(func))$p.value
if(func.fc.pval < ifunc.fc.pval) {
cat("Original FunChisq picked CORRECT direction X to Y: p(X->Y) =",
format(func.fc.pval, digits = 2), "< p(Y->X) =",
format(ifunc.fc.pval, digits = 2), "\n")
} else {
cat("Original FunChisq picked *WRONG* direction Y to X: p(X->Y) =",
format(func.fc.pval, digits = 2), ">= p(Y->X) =",
format(ifunc.fc.pval, digits = 2), "\n")
}
func.afc.pval <- fun.chisq.test(func, method="adapted")$p.value
ifunc.afc.pval <- fun.chisq.test(t(func), method="adapted")$p.value
if(func.afc.pval < ifunc.afc.pval) {
cat("Adapted FunChisq picked CORRECT direction X to Y: p(X->Y) =",
format(func.afc.pval, digits = 2), "< p(Y->X) =",
format(ifunc.afc.pval, digits = 2), "\n")
} else {
cat("Adapted FunChisq picked *WRONG* direction Y to X: p(X->Y) =",
format(func.afc.pval, digits = 2), ">= p(Y->X) =",
format(ifunc.afc.pval, digits = 2), "\n")
}
}Example 1. A function in a square table versus its inverse
In this example, all three methods performed correctly.
#> Conditional entropy picked CORRECT direction X to Y: H(Y|X) = 0.77 < H(X|Y) = 0.88
#> Original FunChisq picked CORRECT direction X to Y: p(X->Y) = 0.027 < p(Y->X) = 0.059
#> Adapted FunChisq picked CORRECT direction X to Y: p(X->Y) = 0.027 < p(Y->X) = 0.059Example 2. A function in a landscape table versus its inverse
In this example, conditional entropy failed to detect the correct functional direction.
#> Conditional entropy picked *WRONG* direction Y to X: H(Y|X) = 1 >= H(X|Y) = 0.9
#> Original FunChisq picked CORRECT direction X to Y: p(X->Y) = 0.022 < p(Y->X) = 0.12
#> Adapted FunChisq picked CORRECT direction X to Y: p(X->Y) = 0.017 < p(Y->X) = 0.12Example 3. A function in a portrait table versus its inverse
In this example, the original FunChisq could not detect the correct functional direction.
#> Conditional entropy picked CORRECT direction X to Y: H(Y|X) = 0.45 < H(X|Y) = 0.9
#> Original FunChisq picked *WRONG* direction Y to X: p(X->Y) = 0.049 >= p(Y->X) = 0.046
#> Adapted FunChisq picked CORRECT direction X to Y: p(X->Y) = 0.049 < p(Y->X) = 0.34
