Hypothesis Test for a 2 by 2 Contingency Table

Definition

The hypothesis test for a $2\times 2$ contingency table is used to determine if there’s a significant association between two categorical variables

Consider a $2\times 2$ Contingency Table

	Dead	Alive
Population 1	$a$	$b$	$n_1$
Population 2	$c$	$d$	$n_2$
	$m_1$	$m_2$	$n$
i.e. $a\sim B(n_1,p_1)$ and $c\sim B(n_2,p_2)$

We want to test $H_0:p_1=p_2$. Let ${\hat{p}}_1=\frac{a}{n_1}$, ${\hat{p}}_2=\frac{c}{n_2}$, $\hat{p}=\frac{m_1}{n}$, and $B(r,n,p)=\left(\genfrac{}{}{0ex}{}{n}{r}\right)p^r(1-p{)}^{n-r}$

Uncorrelated Chi-squared Test

The test statistic is defined as $T=\frac{n(ad-bc{)}^2}{n_1{n}_2{m}_1{m}_2}\sim {\chi }^2(1)$

We reject $H_0$ if $T\ge {\chi }_{\alpha }^2(1)$

Yates’ Corrected Chi-squared Test

The test statistic is defined as $T=\frac{n(|ad-bc|-\frac{n}{2}{)}^2}{n_1{n}_2{m}_1{m}_2}\sim {\chi }^2(1)$

We reject $H_0$ if $T\ge {\chi }_{\alpha }^2(1)$

Fisher’s Exact Test

The test statistic is defined as $T=\sum _{j=x_1}^{x_2}\frac{\left(\genfrac{}{}{0ex}{}{n_1}{j}\right)\left(\genfrac{}{}{0ex}{}{n_2}{m_1-j}\right)}{\left(\genfrac{}{}{0ex}{}{n}{m_1}\right)}$ where $x_1=\max (0,m_1-n_2)$, $x_2=\min (n_1,m_1)$

We reject $H_0$ if $T\le \alpha$

Corrected Chi-squared Test

The test statistic is defined as $T=\frac{n(|ad-bc|-\frac{n}{4}{)}^2}{n_1{n}_2{m}_1{m}_2}\sim {\chi }^2(1)$

We reject $H_0$ if $T\ge {\chi }_{\alpha }^2(1)$

Liddell’s Exact Test

The test statistic is defined as $T=\sum _{x=0}^{n_1}\sum _{y=0}^{n_2}B(x,n_1,\hat{p})B(y,n_2,\hat{p})I(\left|\frac{x}{n_1}-\frac{y}{n_2}\right|\ge |a-c|)$

We reject $H_0$ if $T\le \alpha$

Exact Unconditional Test

The test statistic is defined as $T=\underset{0\le p\le 1}{\sup }\sum _{x=0}^{n_1}\sum _{y=0}^{n_2}B(x,n_1,p)B(y,n_2,p)I(|z(x,y)|\ge |z(a,c)|)$ where $z(x,y)=\frac{(\frac{x}{n_1}-\frac{y}{n_2})}{\sqrt{\frac{1}{n_1}(\frac{x}{n_1})(1-\frac{x}{n_1})+\frac{1}{n_2}(\frac{y}{n_2})(1-\frac{y}{n_2})}}$

We reject $H_0$ if $T\le \alpha$

Approximate Unconditional Test

The test statistic is defined as $T=\sum _{x=0}^{n_1}\sum _{y=0}^{n_2}B(x,n_1,\hat{p})B(y,n_2,\hat{p})I(|z^{\ast }(x,y)|\ge |z^{\ast }(a,c)|)$ where $z^{\ast }(x,y)=\frac{(\frac{x}{n_1}-\frac{y}{n_2})}{\sqrt{(\frac{1}{n_1}+\frac{1}{n_2})\hat{p}(1-\hat{p})}}$

We reject $H_0$ if $T\le \alpha$