# Power Divergence Test¶

PowerDivergenceTest(x [,y] [, lambda] [,theta0] )

If x is a matrix with one row or column, or if x is a vector and y is not given, then a goodness-of-fit test is performed (x is treated as a one- dimensional contingency table. The entries of x must be non-negative integers. In this case, the hypothesis tested is whether the population probabilities equal those in theta0, or are all equal if theta0 is not given.

If x is a matrix with at least two rows and columns, it is taken as a two-dimensional contigency table: the entries of x must be non-negative integers. Otherwise, x and y must be vectors of the same length. The contigency table is calculated using counts from Statsbase. Then the power divergence test is performed of the null hypothesis that the joint distribution of the cell counts in a 2-dimensional contingency table is the product of the row and column marginals.

The power divergence test is given by

$\dfrac{2}{\lambda(\lambda+1)}\sum_{i=1}^I \sum_{j=1}^J n_{ij}\left[(n_{ij}/\hat{n}_{ij})^\lambda -1\right]$

where $$n_{ij}$$ is the cell count in the $${i}$$ th row and $${j}$$ th column and $$\lambda$$ is a real number. Note that when $$\lambda = 1$$, this is equal to Pearson’s chi-squared statistic, as :mathlambda to 0, it converges to the likelihood ratio test statistic, as $$\lambda \to -1$$ it converges to the minimum discrimination information statistic (Gokhale and Kullback 1978), for $$\lambda=-2$$ it equals Neyman modified chi-squared (Neyman 1949), and for $$\lambda=-1/2$$ it equals the Freeman-Tukey statistic (Freeman and Tukey 1950). Under regulairty conditions, their asymptotic distributions are identical (see Drost et. al. 1989). The chis-squared null approximation works best for $$\lambda$$ near $${2/3}$$.

Implements: pvalue, confint

References:

• Agresti, Alan. Categorical Data Analysis, 3rd Edition. Wiley, 2013.
ChisqTest(x [,y] [,theta0])

Convenience function for power divergence test with $$\lambda=1$$.

MultinomialLRT(x [,y] [,theta0])

Convenience function for power divergence test with $$\lambda=0$$.