### Overview

The package provides statistical hypothesis testing methods for
inferring model-free functional dependency. Functional test statistics
are asymmetric and functionally optimal, unique from other related
statistics. The test significance is based on either asymptotic
chi-squared or exact distributions.

The tests include an asymptotic *functional chi-squared test*
[@zhang2013deciphering], *an
adapted functional chi-squared test* [@Kumar2022AFT], and an *exact
functional test* [@zhong2019eft;@Nguyen2020EFT].
The *normalized* functional chi-squared test was used by Best
Performer NMSUSongLab in HPN-DREAM (DREAM8) Breast Cancer Network
Inference Challenges (Hill et al., 2016) <10.1038/nmeth.3773>.

To measure the effect size, one can use the asymmetric *function
index* [@Zhong2019FANTOM5;@KumarZSLS18].
Its value is minimized to 0 by perfectly independent patterns and
maximized to 1 by perfect non-constant functions.

A simulator [@sharma2017simulating] can
generate functional, non-functional, and independent patterns as
contingency tables. The simulator provides options to control row and
column marginal distributions and the noise level.

### When to use the package

Tests in this package can be used to reveal evidence for causality
based on the causality-by-functionality principle. They target
model-free inference without assuming a parametric model. For continuous
data, these tests offer an advantage over regression analysis when a
parametric functional form cannot be assumed. Data can be first
discretized, e.g., by R packages ‘Ckmeans.1d.dp’
or ‘GridOnClusters’.
For categorical data, they provide a novel means to assess directional
dependency not possible with symmetrical Pearson’s chi-squared or
Fisher’s exact tests. They are a better alternative to conditional
entropy in many aspects.

### To download and install the
package

`install.packages("FunChisq")`

### Citing the package