Chi Suqare

${\chi }^2$ Goodness-of-fit test

1. State

• $H_0$: The distribution of [In context] is the same as the claimed distribution
• $H_a$: The distribution of [In context] is not the same as the claimed distribution
  $H_0$ : $P_1=?/P_2=?/.../P_n=?$
  $H_a$ : At least 2 of the $P_i$ is incorrect

2. Check Condition

• Random: The data come from a random sample or a randomized experiment
• Large Sample Size: All expected counts are at least 5
• Independent: Individual observations are independent. When sampling without replacement, check that the population is at least 10 times as large as the sample (the 10% condition).

3. Calculation

$${\chi }^2=\sum _{i=1}^n\frac{({\text{Observed}}_i-{\text{Expected}}_i{)}^2}{{\text{Expected}}_i}$$

• $df=\text{Number of categories}-1$
• ${\text{Expected}}_i=n_i\times p_i$

Given that ${\chi }^2$ and $df$，$p_{val}$ is calculated by:

$$p_{val}=P({\chi }^2\ge X)=1-\frac{\gamma \left(\frac{df}{2},\frac{{\chi }^2}{2}\right)}{\Gamma \left(\frac{df}{2}\right)}$$

• $\gamma (s,x)$ is *Lower incomplete gamma function:

$$\gamma (s,x)={\int }_0^x{t}^{s-1}{e}^{-t}dt$$

• $\Gamma (s)$ is Gamma function:

$$\Gamma (s)={\int }_0^{\infty }{t}^{s-1}{e}^{-t}dt$$

• Regularized Incomplete Gamma Function:

$$\frac{\gamma (s,x)}{\Gamma (s)}$$

Also known as $P(s, x)$, `gammainc(s, x)`。

Using scipy.special.gammainc：

import scipy.special as sp
 
def chi2_p_value(chi2_val, df):
    p_val = 1 - sp.gammainc(df / 2, chi2_val / 2)
    return p_val
 
# example
chi2_val = 10.83  # Chi Squanre
df = 4            # Degree of freedom
p_value = chi2_p_value(chi2_val, df)
print(f"P-value: {p_value:.5f}")

• $p_{val}\ge \alpha$ : Fail to reject $H_0$
• $p_{val}\le \alpha$ : Reject $H_0$
  If reject, which category contribute the most to the rejection?
  Check the CompList

4. Conclusion

We have sufficient statistical evidence that [in context]