P-value from Chi sq test statistic in Python

Question

I have computed a test statistic that is distributed as a chi square with 1 degree of freedom, and want to find out what P-value this corresponds to using python.

I

Answer 1

If you want to understand the math, the p-value of a sample, x (fixed), is

P[P(X) <= P(x)] = P[m(X) >= m(x)] = 1 - G(m(x)^2)

where,

• P is the probability of a (say k-variate) normal distribution w/ known covariance (cov) and mean,
• X is a random variable from that normal distribution,
• m(x) is the mahalanobis distance = sqrt( < cov^{-1} (x-mean), x-mean >. Note that in 1-d this is just the absolute value of the z-score.
• G is the CDF of the chi^2 distribution w/ k degrees of freedom.

So if you're computing the p-value of a fixed observation, x, then you compute m(x) (generalized z-score), and 1-G(m(x)^2).

for example, it's well known that if x is sampled from a univariate (k = 1) normal distribution and has z-score = 2 (it's 2 standard deviations from the mean), then the p-value is about .046 (see a z-score table)

In [7]: from scipy.stats import chi2

In [8]: k = 1

In [9]: z = 2

In [10]: 1-chi2.cdf(z**2, k)
Out[10]: 0.045500263896358528