Pearsons Chi-Squared Test

Pearson’s ${\chi }^2$ Test for Multinomial Data

Pearson’s chi-squared test is used for multinomial data. If $X=(X_1,\dots ,X_n)$ has a multinomial $(n,p)$ distribution, then the MLE of $p$ is $\hat{p}=({\hat{p}}_1,\dots ,{\hat{p}}_k)=(X_1/n,\dots ,X_k/n)$.

Let $p_0=(p_{01},\dots ,p_{0k})$ be some fixed vector and suppose we want to test

$$H_0:p=p_0\mathrm{versus}{H}_1:p\ne p_0.$$

Definition

Pearson’s ${\chi }^2$ statistic is

$$T=\sum _{j=1}^k\frac{(X_j-n{p}_{0j}{)}^2}{n{p}_{0j}}=\sum _{j=1}^k\frac{(X_j-E_j{)}^2}{E_j}$$

where $E_j=\mathbb{E}(X_j)=n{p}_{0j}$ is the expected value of $X_j$ under $H_0$.

Theorem

Under $H_0$, $T⇝{\chi }_{k-1}^2$. Hence the test: reject $H_0$ if $T>{\chi }_{k-1,\alpha }^2$ has asymptotic level $\alpha$. The p-value is $\mathbb{P}({\chi }_{k-1}^2>t)$ where $t$ is the observed value of the test statistic.