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Everywhere online there is how to calculate the Chi-Square density value given a confidence level: $\alpha$/p value; but I can not find how one calculates the inverse? How to calculate the $\alpha$/p-value knowing only the density value?

For example, if df = 4, and $\alpha$=0.95, then how does one calculate the $\chi^2_4$?

In R this would be qchisq(0.95, 4)= 9.4877. What is the math behind this?

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Given the degrees of freedom and an alpha value, qchisq essetnialy finds the point $x$ such that $F(x)=\alpha$, where $F$ is the CDF.

This is known as the $\alpha^{th}$ quantile. The q in the q* functions stands for quantile, and so this class of R functions are the quantile functions for teir respective distributions. You feed them a quantile, $q$, then return the $x$ such that $F(x)=q$.

How this is done in practice depends on what distribution you're working with. Inverting the CDF may be a fun exercise, but not one I'm interested in doing right now. You could use any number of root finding methods to solve

$$F(x)-q=0$$

for $x$ which should yield the correct quantile with sufficiently many iterations.

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  • $\begingroup$ Ok so but how does it find that point? Does it loop over the alpha until a value is found that matches the desired density? $\endgroup$
    – notMyName
    Commented May 22, 2020 at 21:36
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    $\begingroup$ One way would be to write down the CDF and then invert it algebraically. That isn't likely possible with the chi-square density, so the other option is to use root finding methods to solve $F(x)-q=$ for $x$. Finally, you can actually see how qchisq is implemented by looking up the source code (qchisq calls the method for the quantile function for the gamma distirbution. Find the code here) $\endgroup$
    – Demetri Pananos
    Commented May 22, 2020 at 21:44

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